Subgroups — C₂×C₂₃×#3455 #19210311

Order 5888.

7302 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3455	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₂₃×#542	C₂	1	4	yes	maximal, complemented
3	C₂×C₂₃×#541	C₂	1	2	yes	maximal, complemented
4	C₂×C₂₃×#540	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#539	C₂	1	4	yes	maximal, complemented
6	C₂×C₂₃×#538	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#529	C₂	1	8	yes	maximal, complemented
10	C₂×C₂₃×#525	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×#41×C₂₃	C₂	1	2	yes	maximal, complemented
12	C₂²×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₂₃	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#542	C₂²	1	64	yes	complemented
16	C₂₃×#541	C₂²	1	32	yes	complemented
17	C₂₃×#540	C₂²	1	32	yes	complemented
18	C₂₃×#539	C₂²	1	64	yes	complemented
19	C₂₃×#538	C₂²	1	64	yes	complemented
20	C₂₃×#533	C₂²	1	16	yes	complemented
21	C₂₃×#532	C₂²	1	16	yes	complemented
22	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
24	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic, complemented
25	C₂₃×#529	C₂²	1	128	yes	complemented
26	C₂×C₂₃×#126	C₂²	1	8	yes	—
27	C₂×C₂₃×#126	C₂²	1	8	yes	complemented
28	C₂×C₂₃×#125	C₂²	1	8	yes	complemented
29	C₂×C₂₃×#125	C₂²	1	2	yes	complemented
30	C₂×C₂₃×#124	C₂²	1	8	yes	—
31	C₂×C₂₃×#124	C₂²	1	2	yes	complemented
32	C₂₃×#525	C₂²	1	32	yes	complemented
33	C₂×C₂₃×#123	C₂²	1	4	yes	—
34	C₂×C₂₃×#123	C₂²	1	8	yes	—
35	C₂×C₂₃×#123	C₂²	1	8	yes	complemented
36	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
37	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
38	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	8	yes	complemented
39	C₂×C₂₃×#121	C₂²	1	4	yes	—
40	C₂×C₂₃×#121	C₂²	1	8	yes	—
41	C₂×C₂₃×#121	C₂²	1	4	yes	—
42	C₂×C₂₃×#120	C₂²	1	4	yes	—
43	C₄×#41×C₂₃	C₂²	1	32	yes	complemented
44	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
45	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
46	C₂×C₄×D₄×C₂₃	C₂²	1	8	yes	—
47	C₂×C₄×D₄×C₂₃	C₂²	1	8	yes	—
48	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
49	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
50	C₂×C₂₃×#117	C₂²	1	4	yes	—
51	C₂×C₂₃×#117	C₂²	1	2	yes	—
52	C₂×C₂₃×#117	C₂²	1	2	yes	—
53	C₂×C₂₃×#117	C₂²	1	2	yes	—
54	C₂×C₂₃×#117	C₂²	1	16	yes	complemented
55	C₂×C₂₃×#117	C₂²	1	4	yes	complemented
56	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
57	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
58	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
59	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
60	C₂²×C₄²×C₂₃	C₂²	1	4	yes	complemented
61	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
62	C₂×#41×C₂₃	—	2	16	no	—
63	C₂×#41×C₂₃	C₂³	1	16	yes	—
64	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
65	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
66	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
67	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
68	C₂²×D₄×C₂₃	C₂³	1	8	yes	—
69	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
70	C₂³×C₄×C₂₃	—	2	8	no	—
71	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
73	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
74	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
75	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
76	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
77	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
78	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
79	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
80	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	complemented
81	C₂₃×C₄⋊Q₈	C₂³	1	16	yes	complemented
82	C₂₃×#127	C₂³	1	8	yes	complemented
83	C₂₃×#126	C₂³	1	64	yes	—
84	C₂₃×#126	C₂³	1	64	yes	complemented
85	C₂₃×#125	C₂³	1	64	yes	complemented
86	C₂₃×#125	C₂³	1	16	yes	complemented
87	C₂₃×#124	C₂³	1	64	yes	—
88	C₂₃×#124	C₂³	1	16	yes	complemented
89	C₂₃×#123	C₂³	1	32	yes	—
90	C₂₃×#123	C₂³	1	64	yes	—
91	C₂₃×#123	C₂³	1	64	yes	complemented
92	C₂₃×C₂²⋊Q₈	C₂³	1	32	yes	—
93	C₂₃×C₂²⋊Q₈	C₂³	1	32	yes	—
94	C₂₃×C₂²⋊Q₈	C₂³	1	64	yes	complemented
95	C₂₃×#121	C₂³	1	32	yes	—
96	C₂₃×#121	C₂³	1	64	yes	—
97	C₂₃×#121	C₂³	1	32	yes	—
98	C₂₃×#120	C₂³	1	32	yes	—
99	C₄×Q₈×C₂₃	C₂³	1	32	yes	complemented
100	C₄×Q₈×C₂₃	C₂³	1	32	yes	complemented
101	C₄×D₄×C₂₃	C₂³	1	64	yes	—
102	C₄×D₄×C₂₃	C₂³	1	64	yes	—
103	C₄×D₄×C₂₃	C₂³	1	32	yes	complemented
104	C₄×D₄×C₂₃	C₂³	1	32	yes	complemented
105	C₂₃×#117	C₂³	1	32	yes	—
106	C₂₃×#117	C₂³	1	16	yes	—
107	C₂₃×#117	C₂³	1	16	yes	—
108	C₂₃×#117	C₂³	1	16	yes	—
109	C₂₃×#117	C₂³	1	32	yes	complemented
110	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
112	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
113	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
114	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
115	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
116	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
117	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
118	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
119	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
120	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
121	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
122	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
123	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
124	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
125	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
126	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
127	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
128	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
131	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
132	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
133	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
134	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
135	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
136	C₂×C₄²×C₂₃	C₂³	1	32	yes	complemented
137	C₂⁴×C₂₃	—	2	4	no	—
138	C₂⁴×C₂₃	—	4	4	no	—
139	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
140	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
141	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
142	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
143	#41×C₂₃	—	2	64	no	—
144	C₂×Q₈×C₂₃	—	2	8	no	—
145	C₂×Q₈×C₂₃	—	4	8	no	—
146	C₂×Q₈×C₂₃	C₂⁴	1	8	yes	—
147	C₂×Q₈×C₂₃	C₂⁴	1	16	yes	—
148	C₂×D₄×C₂₃	—	2	16	no	—
149	C₂×D₄×C₂₃	—	2	8	no	—
150	C₂×D₄×C₂₃	—	4	8	no	—
151	C₂×D₄×C₂₃	—	4	16	no	—
152	C₂×D₄×C₂₃	C₂⁴	1	16	yes	—
153	C₂×D₄×C₂₃	C₂⁴	1	8	yes	—
154	C₂×D₄×C₂₃	C₂⁴	1	32	yes	—
155	C₂×D₄×C₂₃	C₂⁴	1	16	yes	—
156	C₂²×C₄×C₂₃	—	2	4	no	—
157	C₂²×C₄×C₂₃	—	2	8	no	—
158	C₂²×C₄×C₂₃	—	2	8	no	—
159	C₂²×C₄×C₂₃	—	2	32	no	—
160	C₂²×C₄×C₂₃	—	2	16	no	—
161	C₂²×C₄×C₂₃	—	4	4	no	—
162	C₂²×C₄×C₂₃	—	4	16	no	—
163	C₂²×C₄×C₂₃	—	4	4	no	—
164	C₂²×C₄×C₂₃	#41	1	8	yes	—
165	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
169	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₂₃	C₂⁴	1	16	yes	—
171	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
174	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
175	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
176	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
177	C₂²×C₄×C₂₃	C₂⁴	1	32	yes	—
178	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
181	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
182	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
183	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
184	C₄⋊C₄×C₂₃	C₂⁴	1	32	yes	—
185	C₄⋊C₄×C₂₃	C₂⁴	1	32	yes	—
186	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
187	C₄⋊C₄×C₂₃	C₂⁴	1	32	yes	—
188	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
189	C₂²⋊C₄×C₂₃	C₂⁴	1	16	yes	—
190	C₂²⋊C₄×C₂₃	C₂⁴	1	16	yes	—
191	C₂²⋊C₄×C₂₃	C₂⁴	1	16	yes	—
192	C₂²⋊C₄×C₂₃	C₂⁴	1	32	yes	—
193	C₂²⋊C₄×C₂₃	C₂⁴	1	32	yes	—
194	C₂²⋊C₄×C₂₃	C₂⁴	1	32	yes	—
195	C₂²⋊C₄×C₂₃	C₂⁴	1	32	yes	—
196	C₄²×C₂₃	C₂⁴	1	8	yes	—
197	C₄²×C₂₃	C₂⁴	1	8	yes	—
198	C₄²×C₂₃	C₂⁴	1	8	yes	—
199	C₄²×C₂₃	C₂⁴	1	4	yes	—
200	C₄²×C₂₃	C₂⁴	1	4	yes	—
201	C₄²×C₂₃	C₂⁴	1	16	yes	—
202	C₄²×C₂₃	C₂⁴	1	32	yes	—
203	C₂×#3455	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂³×C₂₃	—	2	4	no	—
205	C₂³×C₂₃	—	2	8	no	—
206	C₂³×C₂₃	—	2	4	no	—
207	C₂³×C₂₃	—	2	8	no	—
208	C₂³×C₂₃	—	2	16	no	—
209	C₂³×C₂₃	—	2	8	no	—
210	C₂³×C₂₃	—	4	8	no	—
211	C₂³×C₂₃	—	4	16	no	—
212	C₂³×C₂₃	C₂×#41	1	4	yes	—
213	C₂³×C₂₃	C₂⁵	1	4	yes	—
214	C₂³×C₂₃	C₂⁵	1	8	yes	—
215	C₂³×C₂₃	C₂⁵	1	2	yes	—
216	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
217	Q₈×C₂₃	—	2	16	no	—
218	Q₈×C₂₃	—	4	16	no	—
219	D₄×C₂₃	—	2	32	no	—
220	D₄×C₂₃	—	2	16	no	—
221	D₄×C₂₃	—	4	16	no	—
222	D₄×C₂₃	—	4	32	no	—
223	C₂×C₄×C₂₃	—	2	8	no	—
224	C₂×C₄×C₂₃	—	2	4	no	—
225	C₂×C₄×C₂₃	—	2	8	no	—
226	C₂×C₄×C₂₃	—	2	8	no	—
227	C₂×C₄×C₂₃	—	2	4	no	—
228	C₂×C₄×C₂₃	—	2	16	no	—
229	C₂×C₄×C₂₃	—	2	4	no	—
230	C₂×C₄×C₂₃	—	2	16	no	—
231	C₂×C₄×C₂₃	—	2	16	no	—
232	C₂×C₄×C₂₃	—	2	32	no	—
233	C₂×C₄×C₂₃	—	4	8	no	—
234	C₂×C₄×C₂₃	—	4	32	no	—
235	C₂×C₄×C₂₃	—	4	8	no	—
236	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
237	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
238	C₂×C₄×C₂₃	C₂×#41	1	16	yes	—
239	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
240	C₂×C₄×C₂₃	C₂⁵	1	4	yes	—
241	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
242	C₂×C₄×C₂₃	C₂⁵	1	16	yes	—
243	C₂×C₄×C₂₃	C₂⁵	1	4	yes	—
244	C₂×C₄×C₂₃	C₂⁵	1	4	yes	—
245	C₂×C₄×C₂₃	C₂⁵	1	2	yes	—
246	C₂×C₄×C₂₃	C₂⁵	1	2	yes	—
247	#3455	C₂×C₂₃	1	32	yes	direct product factor
248	C₂×#542	C₂×C₂₃	1	4	yes	complemented
249	C₂×#541	C₂×C₂₃	1	2	yes	complemented
250	C₂×#540	C₂×C₂₃	1	2	yes	complemented
251	C₂×#539	C₂×C₂₃	1	4	yes	complemented
252	C₂×#538	C₂×C₂₃	1	4	yes	complemented
253	C₂×#533	C₂×C₂₃	1	1	yes	characteristic, complemented
254	C₂×#532	C₂×C₂₃	1	1	yes	characteristic, complemented
255	C₂×#529	C₂×C₂₃	1	8	yes	complemented
256	C₂×#525	C₂×C₂₃	1	2	yes	complemented
257	C₂×C₄×#41	C₂×C₂₃	1	2	yes	complemented
258	C₂²×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
259	C₂²×C₂₃	—	2	4	no	—
260	C₂²×C₂₃	—	2	8	no	—
261	C₂²×C₂₃	—	2	4	no	—
262	C₂²×C₂₃	—	2	8	no	—
263	C₂²×C₂₃	—	2	16	no	—
264	C₂²×C₂₃	—	2	8	no	—
265	C₂²×C₂₃	—	4	16	no	—
266	C₂²×C₂₃	—	4	16	no	—
267	C₂²×C₂₃	—	4	8	no	—
268	C₂²×C₂₃	C₂²×#41	1	4	yes	—
269	C₂²×C₂₃	C₂²×#41	1	2	yes	—
270	C₂²×C₂₃	#585	1	4	yes	—
271	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
272	C₄×C₂₃	—	2	4	no	—
273	C₄×C₂₃	—	2	8	no	—
274	C₄×C₂₃	—	2	8	no	—
275	C₄×C₂₃	—	2	4	no	—
276	C₄×C₂₃	—	2	16	no	—
277	C₄×C₂₃	—	2	4	no	—
278	C₄×C₂₃	C₂²×#41	1	4	yes	—
279	C₄×C₂₃	C₂²×#41	1	4	yes	—
280	C₂²×#41	C₂²×C₂₃	1	2	yes	—
281	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
282	#542	C₂²×C₂₃	1	64	yes	complemented
283	#541	C₂²×C₂₃	1	32	yes	complemented
284	#540	C₂²×C₂₃	1	32	yes	complemented
285	#539	C₂²×C₂₃	1	64	yes	complemented
286	#538	C₂²×C₂₃	1	64	yes	complemented
287	#533	C₂²×C₂₃	1	16	yes	complemented
288	#532	C₂²×C₂₃	1	16	yes	complemented
289	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
290	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
291	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic, complemented
292	#529	C₂²×C₂₃	1	128	yes	complemented
293	C₂×#126	C₂²×C₂₃	1	8	yes	—
294	C₂×#126	C₂²×C₂₃	1	8	yes	complemented
295	C₂×#125	C₂²×C₂₃	1	8	yes	complemented
296	C₂×#125	C₂²×C₂₃	1	2	yes	complemented
297	C₂×#124	C₂²×C₂₃	1	8	yes	—
298	C₂×#124	C₂²×C₂₃	1	2	yes	complemented
299	#525	C₂²×C₂₃	1	32	yes	complemented
300	C₂×#123	C₂²×C₂₃	1	4	yes	—
301	C₂×#123	C₂²×C₂₃	1	8	yes	—
302	C₂×#123	C₂²×C₂₃	1	8	yes	complemented
303	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	—
304	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	—
305	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	8	yes	complemented
306	C₂×#121	C₂²×C₂₃	1	4	yes	—
307	C₂×#121	C₂²×C₂₃	1	8	yes	—
308	C₂×#121	C₂²×C₂₃	1	4	yes	—
309	C₂×#120	C₂²×C₂₃	1	4	yes	—
310	C₄×#41	C₂²×C₂₃	1	32	yes	complemented
311	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	complemented
312	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	complemented
313	C₂×C₄×D₄	C₂²×C₂₃	1	8	yes	—
314	C₂×C₄×D₄	C₂²×C₂₃	1	8	yes	—
315	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	complemented
316	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	complemented
317	C₂×#117	C₂²×C₂₃	1	4	yes	—
318	C₂×#117	C₂²×C₂₃	1	2	yes	—
319	C₂×#117	C₂²×C₂₃	1	2	yes	—
320	C₂×#117	C₂²×C₂₃	1	2	yes	—
321	C₂×#117	C₂²×C₂₃	1	16	yes	complemented
322	C₂×#117	C₂²×C₂₃	1	4	yes	complemented
323	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
324	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
325	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
326	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
327	C₂²×C₄²	C₂²×C₂₃	1	4	yes	complemented
328	C₂×C₂₃	—	2	4	no	—
329	C₂×C₂₃	—	4	8	no	—
330	C₂×C₂₃	—	4	4	no	—
331	C₂×C₂₃	#3455	1	4	yes	direct product factor
332	C₂×C₂₃	C₂³×#41	1	1	yes	characteristic
333	C₂×C₂₃	C₂×#585	1	2	yes	—
334	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
335	C₂×#41	—	2	16	no	—
336	C₂×#41	C₂³×C₂₃	1	16	yes	—
337	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
338	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
339	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
340	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
341	C₂²×D₄	C₂³×C₂₃	1	8	yes	—
342	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
343	C₂³×C₄	—	2	8	no	—
344	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
345	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
346	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
347	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
348	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
349	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
350	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
351	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
352	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
353	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	complemented
354	C₄⋊Q₈	C₂³×C₂₃	1	16	yes	complemented
355	#127	C₂³×C₂₃	1	8	yes	complemented
356	#126	C₂³×C₂₃	1	64	yes	—
357	#126	C₂³×C₂₃	1	64	yes	complemented
358	#125	C₂³×C₂₃	1	64	yes	complemented
359	#125	C₂³×C₂₃	1	16	yes	complemented
360	#124	C₂³×C₂₃	1	64	yes	—
361	#124	C₂³×C₂₃	1	16	yes	complemented
362	#123	C₂³×C₂₃	1	32	yes	—
363	#123	C₂³×C₂₃	1	64	yes	—
364	#123	C₂³×C₂₃	1	64	yes	complemented
365	C₂²⋊Q₈	C₂³×C₂₃	1	32	yes	—
366	C₂²⋊Q₈	C₂³×C₂₃	1	32	yes	—
367	C₂²⋊Q₈	C₂³×C₂₃	1	64	yes	complemented
368	#121	C₂³×C₂₃	1	32	yes	—
369	#121	C₂³×C₂₃	1	64	yes	—
370	#121	C₂³×C₂₃	1	32	yes	—
371	#120	C₂³×C₂₃	1	32	yes	—
372	C₄×Q₈	C₂³×C₂₃	1	32	yes	complemented
373	C₄×Q₈	C₂³×C₂₃	1	32	yes	complemented
374	C₄×D₄	C₂³×C₂₃	1	64	yes	—
375	C₄×D₄	C₂³×C₂₃	1	64	yes	—
376	C₄×D₄	C₂³×C₂₃	1	32	yes	complemented
377	C₄×D₄	C₂³×C₂₃	1	32	yes	complemented
378	#117	C₂³×C₂₃	1	32	yes	—
379	#117	C₂³×C₂₃	1	16	yes	—
380	#117	C₂³×C₂₃	1	16	yes	—
381	#117	C₂³×C₂₃	1	16	yes	—
382	#117	C₂³×C₂₃	1	32	yes	complemented
383	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
384	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
385	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
386	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
387	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
388	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
389	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
390	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
391	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
392	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
393	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
394	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
395	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
396	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
400	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
401	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
402	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
403	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
404	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
405	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
406	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
407	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
408	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
409	C₂×C₄²	C₂³×C₂₃	1	32	yes	complemented
410	C₂₃	C₂×#3455	1	1	yes	characteristic, direct product factor, Sylow
411	C₂⁴	—	2	4	no	—
412	C₂⁴	—	4	4	no	—
413	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
414	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
415	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
416	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
417	#41	—	2	64	no	—
418	C₂×Q₈	—	2	8	no	—
419	C₂×Q₈	—	4	8	no	—
420	C₂×Q₈	C₂⁴×C₂₃	1	8	yes	—
421	C₂×Q₈	C₂⁴×C₂₃	1	16	yes	—
422	C₂×D₄	—	2	16	no	—
423	C₂×D₄	—	2	8	no	—
424	C₂×D₄	—	4	8	no	—
425	C₂×D₄	—	4	16	no	—
426	C₂×D₄	C₂⁴×C₂₃	1	16	yes	—
427	C₂×D₄	C₂⁴×C₂₃	1	8	yes	—
428	C₂×D₄	C₂⁴×C₂₃	1	32	yes	—
429	C₂×D₄	C₂⁴×C₂₃	1	16	yes	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	8	no	—
432	C₂²×C₄	—	2	8	no	—
433	C₂²×C₄	—	2	32	no	—
434	C₂²×C₄	—	2	16	no	—
435	C₂²×C₄	—	4	4	no	—
436	C₂²×C₄	—	4	16	no	—
437	C₂²×C₄	—	4	4	no	—
438	C₂²×C₄	#41×C₂₃	1	8	yes	—
439	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
440	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
441	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
442	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
443	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
444	C₂²×C₄	C₂⁴×C₂₃	1	16	yes	—
445	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
446	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
447	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
448	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
449	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
450	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
451	C₂²×C₄	C₂⁴×C₂₃	1	32	yes	—
452	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
453	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
454	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
455	C₄⋊C₄	C₂⁴×C₂₃	1	16	yes	—
456	C₄⋊C₄	C₂⁴×C₂₃	1	16	yes	—
457	C₄⋊C₄	C₂⁴×C₂₃	1	16	yes	—
458	C₄⋊C₄	C₂⁴×C₂₃	1	32	yes	—
459	C₄⋊C₄	C₂⁴×C₂₃	1	32	yes	—
460	C₄⋊C₄	C₂⁴×C₂₃	1	16	yes	—
461	C₄⋊C₄	C₂⁴×C₂₃	1	32	yes	—
462	C₄⋊C₄	C₂⁴×C₂₃	1	16	yes	—
463	C₂²⋊C₄	C₂⁴×C₂₃	1	16	yes	—
464	C₂²⋊C₄	C₂⁴×C₂₃	1	16	yes	—
465	C₂²⋊C₄	C₂⁴×C₂₃	1	16	yes	—
466	C₂²⋊C₄	C₂⁴×C₂₃	1	32	yes	—
467	C₂²⋊C₄	C₂⁴×C₂₃	1	32	yes	—
468	C₂²⋊C₄	C₂⁴×C₂₃	1	32	yes	—
469	C₂²⋊C₄	C₂⁴×C₂₃	1	32	yes	—
470	C₄²	C₂⁴×C₂₃	1	8	yes	—
471	C₄²	C₂⁴×C₂₃	1	8	yes	—
472	C₄²	C₂⁴×C₂₃	1	8	yes	—
473	C₄²	C₂⁴×C₂₃	1	4	yes	—
474	C₄²	C₂⁴×C₂₃	1	4	yes	—
475	C₄²	C₂⁴×C₂₃	1	16	yes	—
476	C₄²	C₂⁴×C₂₃	1	32	yes	—
477	C₂³	—	2	4	no	—
478	C₂³	—	2	8	no	—
479	C₂³	—	2	4	no	—
480	C₂³	—	2	8	no	—
481	C₂³	—	2	16	no	—
482	C₂³	—	2	8	no	—
483	C₂³	—	4	8	no	—
484	C₂³	—	4	16	no	—
485	C₂³	C₂×#41×C₂₃	1	4	yes	—
486	C₂³	C₂⁵×C₂₃	1	4	yes	—
487	C₂³	C₂⁵×C₂₃	1	8	yes	—
488	C₂³	C₂⁵×C₂₃	1	2	yes	—
489	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
490	Q₈	—	2	16	no	—
491	Q₈	—	4	16	no	—
492	D₄	—	2	32	no	—
493	D₄	—	2	16	no	—
494	D₄	—	4	16	no	—
495	D₄	—	4	32	no	—
496	C₂×C₄	—	2	8	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	8	no	—
499	C₂×C₄	—	2	8	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	16	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	16	no	—
504	C₂×C₄	—	2	16	no	—
505	C₂×C₄	—	2	32	no	—
506	C₂×C₄	—	4	8	no	—
507	C₂×C₄	—	4	32	no	—
508	C₂×C₄	—	4	8	no	—
509	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
510	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
511	C₂×C₄	C₂×#41×C₂₃	1	16	yes	—
512	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
513	C₂×C₄	C₂⁵×C₂₃	1	4	yes	—
514	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
515	C₂×C₄	C₂⁵×C₂₃	1	16	yes	—
516	C₂×C₄	C₂⁵×C₂₃	1	4	yes	—
517	C₂×C₄	C₂⁵×C₂₃	1	4	yes	—
518	C₂×C₄	C₂⁵×C₂₃	1	2	yes	—
519	C₂×C₄	C₂⁵×C₂₃	1	2	yes	—
520	C₂²	—	2	8	no	—
521	C₂²	—	2	4	no	—
522	C₂²	—	2	4	no	—
523	C₂²	—	2	8	no	—
524	C₂²	—	2	16	no	—
525	C₂²	—	2	8	no	—
526	C₂²	—	4	16	no	—
527	C₂²	—	4	8	no	—
528	C₂²	—	4	16	no	—
529	C₂²	C₂²×#41×C₂₃	1	4	yes	—
530	C₂²	C₂²×#41×C₂₃	1	2	yes	—
531	C₂²	C₂₃×#585	1	4	yes	—
532	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
533	C₄	—	2	4	no	—
534	C₄	—	2	8	no	—
535	C₄	—	2	8	no	—
536	C₄	—	2	4	no	—
537	C₄	—	2	16	no	—
538	C₄	—	2	4	no	—
539	C₄	C₂²×#41×C₂₃	1	4	yes	—
540	C₄	C₂²×#41×C₂₃	1	4	yes	—
541	C₂	—	2	4	no	—
542	C₂	—	4	4	no	—
543	C₂	—	4	8	no	—
544	C₂	C₂₃×#3455	1	4	yes	direct product factor
545	C₂	C₂³×#41×C₂₃	1	1	yes	characteristic
546	C₂	C₂×C₂₃×#585	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₂₃
socle	C₂³×C₂₃	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂₃×#3455	C₂	direct (N × H)
2. C₂×C₂₃×#542	C₂	semidirect
3. C₂×C₂₃×#541	C₂	semidirect
4. C₂×C₂₃×#540	C₂	semidirect
5. C₂×C₂₃×#539	C₂	semidirect
6. C₂×C₂₃×#538	C₂	semidirect
7. C₂×C₂₃×#533	C₂	semidirect
8. C₂×C₂₃×#532	C₂	semidirect
9. C₂×C₂₃×#529	C₂	semidirect
10. C₂×C₂₃×#525	C₂	semidirect
11. C₂×C₄×#41×C₂₃	C₂	semidirect
12. C₂²×C₂₃×#117	C₂	semidirect
15. C₂₃×#542	C₂²	semidirect
16. C₂₃×#541	C₂²	semidirect
17. C₂₃×#540	C₂²	semidirect
18. C₂₃×#539	C₂²	semidirect
19. C₂₃×#538	C₂²	semidirect
20. C₂₃×#533	C₂²	semidirect
21. C₂₃×#532	C₂²	semidirect
22. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
23. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
24. C₂×C₂₃×#127	C₂²	semidirect
25. C₂₃×#529	C₂²	semidirect
27. C₂×C₂₃×#126	C₂²	semidirect
28. C₂×C₂₃×#125	C₂²	semidirect
29. C₂×C₂₃×#125	C₂²	semidirect
31. C₂×C₂₃×#124	C₂²	semidirect
32. C₂₃×#525	C₂²	semidirect
35. C₂×C₂₃×#123	C₂²	semidirect
38. C₂×C₂₃×C₂²⋊Q₈	C₂²	semidirect
43. C₄×#41×C₂₃	C₂²	semidirect
44. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
45. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
48. C₂×C₄×D₄×C₂₃	C₂²	semidirect
49. C₂×C₄×D₄×C₂₃	C₂²	semidirect
54. C₂×C₂₃×#117	C₂²	semidirect
55. C₂×C₂₃×#117	C₂²	semidirect
60. C₂²×C₄²×C₂₃	C₂²	semidirect
80. C₂₃×C₄⋊Q₈	C₂³	semidirect
81. C₂₃×C₄⋊Q₈	C₂³	semidirect
82. C₂₃×#127	C₂³	semidirect
84. C₂₃×#126	C₂³	semidirect
85. C₂₃×#125	C₂³	semidirect
86. C₂₃×#125	C₂³	semidirect
88. C₂₃×#124	C₂³	semidirect
91. C₂₃×#123	C₂³	semidirect
94. C₂₃×C₂²⋊Q₈	C₂³	semidirect
99. C₄×Q₈×C₂₃	C₂³	semidirect
100. C₄×Q₈×C₂₃	C₂³	semidirect
103. C₄×D₄×C₂₃	C₂³	semidirect
104. C₄×D₄×C₂₃	C₂³	semidirect
109. C₂₃×#117	C₂³	semidirect
136. C₂×C₄²×C₂₃	C₂³	semidirect
203. C₂×#3455	C₂₃	direct (N × H)
247. #3455	C₂×C₂₃	direct (N × H)
248. C₂×#542	C₂×C₂₃	semidirect
249. C₂×#541	C₂×C₂₃	semidirect
250. C₂×#540	C₂×C₂₃	semidirect
251. C₂×#539	C₂×C₂₃	semidirect
252. C₂×#538	C₂×C₂₃	semidirect
253. C₂×#533	C₂×C₂₃	semidirect
254. C₂×#532	C₂×C₂₃	semidirect
255. C₂×#529	C₂×C₂₃	semidirect
256. C₂×#525	C₂×C₂₃	semidirect
257. C₂×C₄×#41	C₂×C₂₃	semidirect
258. C₂²×#117	C₂×C₂₃	semidirect
282. #542	C₂²×C₂₃	semidirect
283. #541	C₂²×C₂₃	semidirect
284. #540	C₂²×C₂₃	semidirect
285. #539	C₂²×C₂₃	semidirect
286. #538	C₂²×C₂₃	semidirect
287. #533	C₂²×C₂₃	semidirect
288. #532	C₂²×C₂₃	semidirect
289. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
290. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
291. C₂×#127	C₂²×C₂₃	semidirect
292. #529	C₂²×C₂₃	semidirect
294. C₂×#126	C₂²×C₂₃	semidirect
295. C₂×#125	C₂²×C₂₃	semidirect
296. C₂×#125	C₂²×C₂₃	semidirect
298. C₂×#124	C₂²×C₂₃	semidirect
299. #525	C₂²×C₂₃	semidirect
302. C₂×#123	C₂²×C₂₃	semidirect
305. C₂×C₂²⋊Q₈	C₂²×C₂₃	semidirect
310. C₄×#41	C₂²×C₂₃	semidirect
311. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
312. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
315. C₂×C₄×D₄	C₂²×C₂₃	semidirect
316. C₂×C₄×D₄	C₂²×C₂₃	semidirect
321. C₂×#117	C₂²×C₂₃	semidirect
322. C₂×#117	C₂²×C₂₃	semidirect
327. C₂²×C₄²	C₂²×C₂₃	semidirect
331. C₂×C₂₃	#3455	direct (N × H)
353. C₄⋊Q₈	C₂³×C₂₃	semidirect
354. C₄⋊Q₈	C₂³×C₂₃	semidirect
355. #127	C₂³×C₂₃	semidirect
357. #126	C₂³×C₂₃	semidirect
358. #125	C₂³×C₂₃	semidirect
359. #125	C₂³×C₂₃	semidirect
361. #124	C₂³×C₂₃	semidirect
364. #123	C₂³×C₂₃	semidirect
367. C₂²⋊Q₈	C₂³×C₂₃	semidirect
372. C₄×Q₈	C₂³×C₂₃	semidirect
373. C₄×Q₈	C₂³×C₂₃	semidirect
376. C₄×D₄	C₂³×C₂₃	semidirect
377. C₄×D₄	C₂³×C₂₃	semidirect
382. #117	C₂³×C₂₃	semidirect
409. C₂×C₄²	C₂³×C₂₃	semidirect
410. C₂₃	C₂×#3455	direct (N × H)
544. C₂	C₂₃×#3455	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3455	2
C₂×C₂₃×#542	2
C₂×C₂₃×#541	2
C₂×C₂₃×#540	2
C₂×C₂₃×#539	2
C₂×C₂₃×#538	2
C₂×C₂₃×#533	2
C₂×C₂₃×#532	2
C₂×C₂₃×#529	2
C₂×C₂₃×#525	2
C₂×C₄×#41×C₂₃	2
C₂²×C₂₃×#117	2
C₂×#3455	23

Subgroups — C2×C23×#3455 #19210311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3455 #19210311