Subgroups — C₂×C₂₃×#3544 #19210375

Order 5888.

7086 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3544	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₂₃×#563	C₂	1	4	yes	maximal, complemented
3	C₂×Q₈²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#555	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#554	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#552	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#547	C₂	1	4	yes	maximal, complemented
8	C₂×C₂₃×#544	C₂	1	4	yes	maximal, complemented
9	C₂×C₂₃×#542	C₂	1	4	yes	maximal, complemented
10	C₂×C₂₃×#536	C₂	1	4	yes	maximal, complemented
11	C₂²×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#529	C₂	1	2	yes	maximal, complemented
13	C₂²×#41×C₂₃	C₂²	1	4	yes	—
14	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
15	C₂₃×#563	C₂²	1	64	yes	complemented
16	Q₈²×C₂₃	C₂²	1	16	yes	complemented
17	C₂₃×#555	C₂²	1	16	yes	complemented
18	C₂₃×#554	C₂²	1	32	yes	complemented
19	C₂₃×#552	C₂²	1	64	yes	complemented
20	C₂₃×#547	C₂²	1	64	yes	complemented
21	C₂₃×#544	C₂²	1	64	yes	complemented
22	C₂₃×#542	C₂²	1	64	yes	complemented
23	C₂₃×#536	C₂²	1	64	yes	complemented
24	C₂×C₂₃×C₄⋊Q₈	C₂²	1	4	yes	—
25	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
28	C₂×C₂₃×C₄⋊Q₈	C₂²	1	4	yes	—
29	C₂×C₂₃×C₄⋊Q₈	C₂²	1	4	yes	—
30	C₂×C₂₃×C₄⋊Q₈	C₂²	1	4	yes	—
31	C₂×C₂₃×C₄⋊Q₈	C₂²	1	16	yes	complemented
32	C₂₃×#529	C₂²	1	32	yes	complemented
33	C₂×C₂₃×#126	C₂²	1	4	yes	—
34	C₂×C₂₃×#126	C₂²	1	8	yes	—
35	C₂×C₂₃×#125	C₂²	1	2	yes	—
36	C₂×C₂₃×#125	C₂²	1	4	yes	—
37	C₂×C₂₃×#125	C₂²	1	8	yes	—
38	C₂×C₂₃×#124	C₂²	1	2	yes	—
39	C₂×C₂₃×#124	C₂²	1	4	yes	—
40	C₂×C₂₃×#124	C₂²	1	4	yes	—
41	C₂×C₂₃×#123	C₂²	1	4	yes	—
42	C₂×C₂₃×#123	C₂²	1	8	yes	—
43	C₂×C₂₃×#123	C₂²	1	8	yes	—
44	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
45	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	8	yes	—
46	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	8	yes	—
47	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
48	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	8	yes	—
49	C₂×C₂₃×#121	C₂²	1	2	yes	—
50	C₂×C₂₃×#121	C₂²	1	4	yes	—
51	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
52	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	—
53	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
54	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
55	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
56	C₂×C₂₃×#117	C₂²	1	4	yes	—
57	C₂×C₂₃×#117	C₂²	1	4	yes	—
58	C₂×C₂₃×#117	C₂²	1	4	yes	—
59	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
60	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
61	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
62	C₂×#41×C₂₃	—	4	16	no	—
63	C₂×#41×C₂₃	C₂³	1	32	yes	—
64	C₂²×Q₈×C₂₃	—	4	8	no	—
65	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
66	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
67	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
68	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
69	C₂²×Q₈×C₂₃	C₂³	1	16	yes	—
70	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
71	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
72	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
73	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
74	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
76	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
77	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
78	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
79	C₂₃×C₄⋊Q₈	C₂³	1	32	yes	—
80	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
81	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
82	C₂₃×C₄⋊Q₈	C₂³	1	32	yes	—
83	C₂₃×C₄⋊Q₈	C₂³	1	16	yes	—
84	C₂₃×C₄⋊Q₈	C₂³	1	32	yes	—
85	C₂₃×C₄⋊Q₈	C₂³	1	32	yes	—
86	C₂₃×#126	C₂³	1	64	yes	—
87	C₂₃×#126	C₂³	1	32	yes	—
88	C₂₃×#125	C₂³	1	32	yes	—
89	C₂₃×#125	C₂³	1	64	yes	—
90	C₂₃×#125	C₂³	1	16	yes	—
91	C₂₃×#124	C₂³	1	32	yes	—
92	C₂₃×#124	C₂³	1	16	yes	—
93	C₂₃×#124	C₂³	1	32	yes	—
94	C₂₃×#123	C₂³	1	64	yes	—
95	C₂₃×#123	C₂³	1	64	yes	—
96	C₂₃×#123	C₂³	1	32	yes	—
97	C₂₃×C₂²⋊Q₈	C₂³	1	64	yes	—
98	C₂₃×C₂²⋊Q₈	C₂³	1	16	yes	—
99	C₂₃×C₂²⋊Q₈	C₂³	1	32	yes	—
100	C₂₃×C₂²⋊Q₈	C₂³	1	64	yes	—
101	C₂₃×C₂²⋊Q₈	C₂³	1	64	yes	—
102	C₂₃×#121	C₂³	1	16	yes	—
103	C₂₃×#121	C₂³	1	32	yes	—
104	C₄×Q₈×C₂₃	C₂³	1	32	yes	—
105	C₄×Q₈×C₂₃	C₂³	1	16	yes	—
106	C₄×D₄×C₂₃	C₂³	1	32	yes	—
107	C₄×D₄×C₂₃	C₂³	1	32	yes	—
108	C₄×D₄×C₂₃	C₂³	1	16	yes	—
109	C₂₃×#117	C₂³	1	32	yes	—
110	C₂₃×#117	C₂³	1	32	yes	—
111	C₂₃×#117	C₂³	1	32	yes	—
112	C₂×C₄⋊C₄×C₂₃	C₂³	1	32	yes	—
113	C₂×C₄⋊C₄×C₂₃	C₂³	1	32	yes	—
114	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
115	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	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	8	yes	—
119	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
120	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
121	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
122	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
123	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
124	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
125	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
126	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
127	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
128	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
129	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
130	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
131	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
132	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
133	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
134	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
136	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
137	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
138	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
139	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
140	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
141	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
142	#41×C₂₃	—	4	64	no	—
143	C₂×Q₈×C₂₃	—	2	8	no	—
144	C₂×Q₈×C₂₃	—	4	4	no	—
145	C₂×Q₈×C₂₃	—	4	8	no	—
146	C₂×Q₈×C₂₃	—	4	4	no	—
147	C₂×Q₈×C₂₃	—	4	32	no	—
148	C₂×Q₈×C₂₃	—	4	8	no	—
149	C₂×Q₈×C₂₃	C₂⁴	1	8	yes	—
150	C₂×Q₈×C₂₃	C₂⁴	1	8	yes	—
151	C₂×Q₈×C₂₃	C₂⁴	1	16	yes	—
152	C₂×Q₈×C₂₃	C₂⁴	1	16	yes	—
153	C₂×Q₈×C₂₃	C₂⁴	1	8	yes	—
154	C₂×D₄×C₂₃	—	2	16	no	—
155	C₂×D₄×C₂₃	—	2	8	no	—
156	C₂×D₄×C₂₃	—	4	8	no	—
157	C₂×D₄×C₂₃	C₂⁴	1	8	yes	—
158	C₂×D₄×C₂₃	C₂⁴	1	16	yes	—
159	C₂×D₄×C₂₃	C₂⁴	1	16	yes	—
160	C₂²×C₄×C₂₃	—	2	4	no	—
161	C₂²×C₄×C₂₃	—	2	16	no	—
162	C₂²×C₄×C₂₃	—	2	8	no	—
163	C₂²×C₄×C₂₃	—	2	16	no	—
164	C₂²×C₄×C₂₃	—	4	16	no	—
165	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
166	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₂₃	C₂⁴	1	16	yes	—
168	C₂²×C₄×C₂₃	C₂⁴	1	16	yes	—
169	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
172	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₂₃	C₂⁴	1	32	yes	—
178	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
179	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
180	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
181	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
182	C₄⋊C₄×C₂₃	C₂⁴	1	32	yes	—
183	C₄⋊C₄×C₂₃	C₂⁴	1	32	yes	—
184	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
185	C₄⋊C₄×C₂₃	C₂⁴	1	32	yes	—
186	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
187	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
188	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
189	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
190	C₄⋊C₄×C₂₃	C₂⁴	1	32	yes	—
191	C₂²⋊C₄×C₂₃	C₂⁴	1	16	yes	—
192	C₂²⋊C₄×C₂₃	C₂⁴	1	16	yes	—
193	C₂²⋊C₄×C₂₃	C₂⁴	1	32	yes	—
194	C₂²⋊C₄×C₂₃	C₂⁴	1	16	yes	—
195	C₂²⋊C₄×C₂₃	C₂⁴	1	32	yes	—
196	C₂²⋊C₄×C₂₃	C₂⁴	1	16	yes	—
197	C₂²⋊C₄×C₂₃	C₂⁴	1	16	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	16	yes	—
203	C₄²×C₂₃	C₂⁴	1	16	yes	—
204	C₂×#3544	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂³×C₂₃	—	2	4	no	—
206	C₂³×C₂₃	—	2	8	no	—
207	C₂³×C₂₃	—	2	8	no	—
208	C₂³×C₂₃	—	2	8	no	—
209	C₂³×C₂₃	H(5,𝔽₂)	1	4	yes	—
210	C₂³×C₂₃	C₂⁵	1	8	yes	—
211	C₂³×C₂₃	C₂⁵	1	2	yes	—
212	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
213	Q₈×C₂₃	—	2	16	no	—
214	Q₈×C₂₃	—	4	8	no	—
215	Q₈×C₂₃	—	4	16	no	—
216	Q₈×C₂₃	—	4	8	no	—
217	Q₈×C₂₃	—	4	16	no	—
218	D₄×C₂₃	—	2	32	no	—
219	D₄×C₂₃	—	2	16	no	—
220	D₄×C₂₃	—	4	16	no	—
221	C₂×C₄×C₂₃	—	2	8	no	—
222	C₂×C₄×C₂₃	—	2	8	no	—
223	C₂×C₄×C₂₃	—	2	8	no	—
224	C₂×C₄×C₂₃	—	2	2	no	—
225	C₂×C₄×C₂₃	—	2	2	no	—
226	C₂×C₄×C₂₃	—	2	32	no	—
227	C₂×C₄×C₂₃	—	2	8	no	—
228	C₂×C₄×C₂₃	—	2	4	no	—
229	C₂×C₄×C₂₃	—	2	8	no	—
230	C₂×C₄×C₂₃	—	2	8	no	—
231	C₂×C₄×C₂₃	—	2	16	no	—
232	C₂×C₄×C₂₃	—	2	32	no	—
233	C₂×C₄×C₂₃	—	4	32	no	—
234	C₂×C₄×C₂₃	2₋¹⁺⁴	1	8	yes	—
235	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
236	C₂×C₄×C₂₃	C₂⁵	1	4	yes	—
237	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
238	C₂×C₄×C₂₃	C₂⁵	1	2	yes	—
239	C₂×C₄×C₂₃	C₂⁵	1	2	yes	—
240	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
241	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
242	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
243	C₂×C₄×C₂₃	C₂⁵	1	4	yes	—
244	#3544	C₂×C₂₃	1	32	yes	direct product factor
245	C₂×#563	C₂×C₂₃	1	4	yes	complemented
246	C₂×Q₈²	C₂×C₂₃	1	1	yes	characteristic, complemented
247	C₂×#555	C₂×C₂₃	1	1	yes	characteristic, complemented
248	C₂×#554	C₂×C₂₃	1	2	yes	complemented
249	C₂×#552	C₂×C₂₃	1	4	yes	complemented
250	C₂×#547	C₂×C₂₃	1	4	yes	complemented
251	C₂×#544	C₂×C₂₃	1	4	yes	complemented
252	C₂×#542	C₂×C₂₃	1	4	yes	complemented
253	C₂×#536	C₂×C₂₃	1	4	yes	complemented
254	C₂²×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
255	C₂×#529	C₂×C₂₃	1	2	yes	complemented
256	C₂²×C₂₃	—	2	4	no	—
257	C₂²×C₂₃	—	2	8	no	—
258	C₂²×C₂₃	—	2	8	no	—
259	C₂²×C₂₃	—	2	8	no	—
260	C₂²×C₂₃	—	2	8	no	—
261	C₂²×C₂₃	—	4	16	no	—
262	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	4	yes	—
263	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
264	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	4	yes	—
265	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
266	C₄×C₂₃	—	2	8	no	—
267	C₄×C₂₃	—	2	8	no	—
268	C₄×C₂₃	—	2	2	no	—
269	C₄×C₂₃	—	2	2	no	—
270	C₄×C₂₃	—	2	8	no	—
271	C₄×C₂₃	—	2	4	no	—
272	C₄×C₂₃	—	2	8	no	—
273	C₄×C₂₃	—	2	8	no	—
274	C₄×C₂₃	C₂×2₋¹⁺⁴	1	8	yes	—
275	C₂²×#41	C₂²×C₂₃	1	4	yes	—
276	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
277	#563	C₂²×C₂₃	1	64	yes	complemented
278	Q₈²	C₂²×C₂₃	1	16	yes	complemented
279	#555	C₂²×C₂₃	1	16	yes	complemented
280	#554	C₂²×C₂₃	1	32	yes	complemented
281	#552	C₂²×C₂₃	1	64	yes	complemented
282	#547	C₂²×C₂₃	1	64	yes	complemented
283	#544	C₂²×C₂₃	1	64	yes	complemented
284	#542	C₂²×C₂₃	1	64	yes	complemented
285	#536	C₂²×C₂₃	1	64	yes	complemented
286	C₂×C₄⋊Q₈	C₂²×C₂₃	1	4	yes	—
287	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
288	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
289	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
290	C₂×C₄⋊Q₈	C₂²×C₂₃	1	4	yes	—
291	C₂×C₄⋊Q₈	C₂²×C₂₃	1	4	yes	—
292	C₂×C₄⋊Q₈	C₂²×C₂₃	1	4	yes	—
293	C₂×C₄⋊Q₈	C₂²×C₂₃	1	16	yes	complemented
294	#529	C₂²×C₂₃	1	32	yes	complemented
295	C₂×#126	C₂²×C₂₃	1	4	yes	—
296	C₂×#126	C₂²×C₂₃	1	8	yes	—
297	C₂×#125	C₂²×C₂₃	1	2	yes	—
298	C₂×#125	C₂²×C₂₃	1	4	yes	—
299	C₂×#125	C₂²×C₂₃	1	8	yes	—
300	C₂×#124	C₂²×C₂₃	1	2	yes	—
301	C₂×#124	C₂²×C₂₃	1	4	yes	—
302	C₂×#124	C₂²×C₂₃	1	4	yes	—
303	C₂×#123	C₂²×C₂₃	1	4	yes	—
304	C₂×#123	C₂²×C₂₃	1	8	yes	—
305	C₂×#123	C₂²×C₂₃	1	8	yes	—
306	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
307	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	8	yes	—
308	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	8	yes	—
309	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	—
310	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	8	yes	—
311	C₂×#121	C₂²×C₂₃	1	2	yes	—
312	C₂×#121	C₂²×C₂₃	1	4	yes	—
313	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
314	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	—
315	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
316	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
317	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
318	C₂×#117	C₂²×C₂₃	1	4	yes	—
319	C₂×#117	C₂²×C₂₃	1	4	yes	—
320	C₂×#117	C₂²×C₂₃	1	4	yes	—
321	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
322	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
323	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
324	C₂×C₂₃	—	2	4	no	—
325	C₂×C₂₃	—	4	8	no	—
326	C₂×C₂₃	#3544	1	4	yes	direct product factor
327	C₂×C₂₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
328	C₂×C₂₃	C₂²×2₋¹⁺⁴	1	2	yes	—
329	C₂×#41	—	4	16	no	—
330	C₂×#41	C₂³×C₂₃	1	32	yes	—
331	C₂²×Q₈	—	4	8	no	—
332	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
333	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
334	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
335	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
336	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
337	C₂²×Q₈	C₂³×C₂₃	1	16	yes	—
338	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
339	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
340	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
341	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
342	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
343	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
344	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
345	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
346	C₄⋊Q₈	C₂³×C₂₃	1	32	yes	—
347	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
348	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
349	C₄⋊Q₈	C₂³×C₂₃	1	16	yes	—
350	C₄⋊Q₈	C₂³×C₂₃	1	32	yes	—
351	C₄⋊Q₈	C₂³×C₂₃	1	32	yes	—
352	C₄⋊Q₈	C₂³×C₂₃	1	32	yes	—
353	#126	C₂³×C₂₃	1	32	yes	—
354	#126	C₂³×C₂₃	1	64	yes	—
355	#125	C₂³×C₂₃	1	16	yes	—
356	#125	C₂³×C₂₃	1	32	yes	—
357	#125	C₂³×C₂₃	1	64	yes	—
358	#124	C₂³×C₂₃	1	16	yes	—
359	#124	C₂³×C₂₃	1	32	yes	—
360	#124	C₂³×C₂₃	1	32	yes	—
361	#123	C₂³×C₂₃	1	32	yes	—
362	#123	C₂³×C₂₃	1	64	yes	—
363	#123	C₂³×C₂₃	1	64	yes	—
364	C₂²⋊Q₈	C₂³×C₂₃	1	16	yes	—
365	C₂²⋊Q₈	C₂³×C₂₃	1	64	yes	—
366	C₂²⋊Q₈	C₂³×C₂₃	1	64	yes	—
367	C₂²⋊Q₈	C₂³×C₂₃	1	32	yes	—
368	C₂²⋊Q₈	C₂³×C₂₃	1	64	yes	—
369	#121	C₂³×C₂₃	1	16	yes	—
370	#121	C₂³×C₂₃	1	32	yes	—
371	C₄×Q₈	C₂³×C₂₃	1	16	yes	—
372	C₄×Q₈	C₂³×C₂₃	1	32	yes	—
373	C₄×D₄	C₂³×C₂₃	1	32	yes	—
374	C₄×D₄	C₂³×C₂₃	1	16	yes	—
375	C₄×D₄	C₂³×C₂₃	1	32	yes	—
376	#117	C₂³×C₂₃	1	32	yes	—
377	#117	C₂³×C₂₃	1	32	yes	—
378	#117	C₂³×C₂₃	1	32	yes	—
379	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
380	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
381	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
382	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
383	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
384	C₂×C₄⋊C₄	C₂³×C₂₃	1	32	yes	—
385	C₂×C₄⋊C₄	C₂³×C₂₃	1	32	yes	—
386	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
387	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
388	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
389	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
390	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
391	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
392	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
393	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
394	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
395	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
396	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
400	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
401	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
402	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
403	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
404	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
405	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
406	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
407	C₂₃	C₂×#3544	1	1	yes	characteristic, direct product factor, Sylow
408	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
409	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
410	#41	—	4	64	no	—
411	C₂×Q₈	—	2	8	no	—
412	C₂×Q₈	—	4	4	no	—
413	C₂×Q₈	—	4	8	no	—
414	C₂×Q₈	—	4	4	no	—
415	C₂×Q₈	—	4	32	no	—
416	C₂×Q₈	—	4	8	no	—
417	C₂×Q₈	C₂⁴×C₂₃	1	8	yes	—
418	C₂×Q₈	C₂⁴×C₂₃	1	8	yes	—
419	C₂×Q₈	C₂⁴×C₂₃	1	16	yes	—
420	C₂×Q₈	C₂⁴×C₂₃	1	16	yes	—
421	C₂×Q₈	C₂⁴×C₂₃	1	8	yes	—
422	C₂×D₄	—	2	16	no	—
423	C₂×D₄	—	2	8	no	—
424	C₂×D₄	—	4	8	no	—
425	C₂×D₄	C₂⁴×C₂₃	1	8	yes	—
426	C₂×D₄	C₂⁴×C₂₃	1	16	yes	—
427	C₂×D₄	C₂⁴×C₂₃	1	16	yes	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	16	no	—
430	C₂²×C₄	—	2	8	no	—
431	C₂²×C₄	—	2	16	no	—
432	C₂²×C₄	—	4	16	no	—
433	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
434	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
435	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
436	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
437	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
438	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
439	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
440	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
441	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
442	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
443	C₂²×C₄	C₂⁴×C₂₃	1	16	yes	—
444	C₂²×C₄	C₂⁴×C₂₃	1	16	yes	—
445	C₂²×C₄	C₂⁴×C₂₃	1	32	yes	—
446	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
447	C₄⋊C₄	C₂⁴×C₂₃	1	16	yes	—
448	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
449	C₄⋊C₄	C₂⁴×C₂₃	1	16	yes	—
450	C₄⋊C₄	C₂⁴×C₂₃	1	32	yes	—
451	C₄⋊C₄	C₂⁴×C₂₃	1	32	yes	—
452	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
453	C₄⋊C₄	C₂⁴×C₂₃	1	32	yes	—
454	C₄⋊C₄	C₂⁴×C₂₃	1	16	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	C₂²⋊C₄	C₂⁴×C₂₃	1	16	yes	—
460	C₂²⋊C₄	C₂⁴×C₂₃	1	16	yes	—
461	C₂²⋊C₄	C₂⁴×C₂₃	1	32	yes	—
462	C₂²⋊C₄	C₂⁴×C₂₃	1	16	yes	—
463	C₂²⋊C₄	C₂⁴×C₂₃	1	32	yes	—
464	C₂²⋊C₄	C₂⁴×C₂₃	1	16	yes	—
465	C₂²⋊C₄	C₂⁴×C₂₃	1	16	yes	—
466	C₄²	C₂⁴×C₂₃	1	8	yes	—
467	C₄²	C₂⁴×C₂₃	1	4	yes	—
468	C₄²	C₂⁴×C₂₃	1	4	yes	—
469	C₄²	C₂⁴×C₂₃	1	16	yes	—
470	C₄²	C₂⁴×C₂₃	1	16	yes	—
471	C₄²	C₂⁴×C₂₃	1	16	yes	—
472	C₂³	—	2	4	no	—
473	C₂³	—	2	8	no	—
474	C₂³	—	2	8	no	—
475	C₂³	—	2	8	no	—
476	C₂³	C₂₃×H(5,𝔽₂)	1	4	yes	—
477	C₂³	C₂⁵×C₂₃	1	8	yes	—
478	C₂³	C₂⁵×C₂₃	1	2	yes	—
479	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
480	Q₈	—	2	16	no	—
481	Q₈	—	4	8	no	—
482	Q₈	—	4	16	no	—
483	Q₈	—	4	8	no	—
484	Q₈	—	4	16	no	—
485	D₄	—	2	32	no	—
486	D₄	—	2	16	no	—
487	D₄	—	4	16	no	—
488	C₂×C₄	—	2	8	no	—
489	C₂×C₄	—	2	8	no	—
490	C₂×C₄	—	2	8	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	32	no	—
494	C₂×C₄	—	2	8	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	8	no	—
497	C₂×C₄	—	2	8	no	—
498	C₂×C₄	—	2	16	no	—
499	C₂×C₄	—	2	32	no	—
500	C₂×C₄	—	4	32	no	—
501	C₂×C₄	C₂₃×2₋¹⁺⁴	1	8	yes	—
502	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
503	C₂×C₄	C₂⁵×C₂₃	1	4	yes	—
504	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
505	C₂×C₄	C₂⁵×C₂₃	1	2	yes	—
506	C₂×C₄	C₂⁵×C₂₃	1	2	yes	—
507	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
508	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
509	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
510	C₂×C₄	C₂⁵×C₂₃	1	4	yes	—
511	C₂²	—	2	8	no	—
512	C₂²	—	2	4	no	—
513	C₂²	—	2	8	no	—
514	C₂²	—	2	8	no	—
515	C₂²	—	2	8	no	—
516	C₂²	—	4	16	no	—
517	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	4	yes	—
518	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
519	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	4	yes	—
520	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
521	C₄	—	2	8	no	—
522	C₄	—	2	8	no	—
523	C₄	—	2	2	no	—
524	C₄	—	2	2	no	—
525	C₄	—	2	8	no	—
526	C₄	—	2	4	no	—
527	C₄	—	2	8	no	—
528	C₄	—	2	8	no	—
529	C₄	C₂×C₂₃×2₋¹⁺⁴	1	8	yes	—
530	C₂	—	2	4	no	—
531	C₂	—	4	8	no	—
532	C₂	C₂₃×#3544	1	4	yes	direct product factor
533	C₂	C₂²×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
534	C₂	C₂²×C₂₃×2₋¹⁺⁴	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3544	C₂	direct (N × H)
2. C₂×C₂₃×#563	C₂	semidirect
3. C₂×Q₈²×C₂₃	C₂	semidirect
4. C₂×C₂₃×#555	C₂	semidirect
5. C₂×C₂₃×#554	C₂	semidirect
6. C₂×C₂₃×#552	C₂	semidirect
7. C₂×C₂₃×#547	C₂	semidirect
8. C₂×C₂₃×#544	C₂	semidirect
9. C₂×C₂₃×#542	C₂	semidirect
10. C₂×C₂₃×#536	C₂	semidirect
11. C₂²×C₂₃×C₄⋊Q₈	C₂	semidirect
12. C₂×C₂₃×#529	C₂	semidirect
15. C₂₃×#563	C₂²	semidirect
16. Q₈²×C₂₃	C₂²	semidirect
17. C₂₃×#555	C₂²	semidirect
18. C₂₃×#554	C₂²	semidirect
19. C₂₃×#552	C₂²	semidirect
20. C₂₃×#547	C₂²	semidirect
21. C₂₃×#544	C₂²	semidirect
22. C₂₃×#542	C₂²	semidirect
23. C₂₃×#536	C₂²	semidirect
31. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
32. C₂₃×#529	C₂²	semidirect
204. C₂×#3544	C₂₃	direct (N × H)
244. #3544	C₂×C₂₃	direct (N × H)
245. C₂×#563	C₂×C₂₃	semidirect
246. C₂×Q₈²	C₂×C₂₃	semidirect
247. C₂×#555	C₂×C₂₃	semidirect
248. C₂×#554	C₂×C₂₃	semidirect
249. C₂×#552	C₂×C₂₃	semidirect
250. C₂×#547	C₂×C₂₃	semidirect
251. C₂×#544	C₂×C₂₃	semidirect
252. C₂×#542	C₂×C₂₃	semidirect
253. C₂×#536	C₂×C₂₃	semidirect
254. C₂²×C₄⋊Q₈	C₂×C₂₃	semidirect
255. C₂×#529	C₂×C₂₃	semidirect
277. #563	C₂²×C₂₃	semidirect
278. Q₈²	C₂²×C₂₃	semidirect
279. #555	C₂²×C₂₃	semidirect
280. #554	C₂²×C₂₃	semidirect
281. #552	C₂²×C₂₃	semidirect
282. #547	C₂²×C₂₃	semidirect
283. #544	C₂²×C₂₃	semidirect
284. #542	C₂²×C₂₃	semidirect
285. #536	C₂²×C₂₃	semidirect
293. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
294. #529	C₂²×C₂₃	semidirect
326. C₂×C₂₃	#3544	direct (N × H)
407. C₂₃	C₂×#3544	direct (N × H)
532. C₂	C₂₃×#3544	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×#3544	2
C₂×C₂₃×#563	2
C₂×Q₈²×C₂₃	2
C₂×C₂₃×#555	2
C₂×C₂₃×#554	2
C₂×C₂₃×#552	2
C₂×C₂₃×#547	2
C₂×C₂₃×#544	2
C₂×C₂₃×#542	2
C₂×C₂₃×#536	2
C₂²×C₂₃×C₄⋊Q₈	2
C₂×C₂₃×#529	2
C₂×#3544	23

Subgroups — C2×C23×#3544 #19210375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3544 #19210375