Subgroups — C₂₃×#3494 #15671113

Order 2944.

1142 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#555	C₂	1	2	yes	maximal, complemented
2	C₂₃×#553	C₂	1	2	yes	maximal, complemented
3	C₂₃×#553	C₂	1	2	yes	maximal, complemented
4	C₂₃×#551	C₂	1	4	yes	maximal, complemented
5	C₂₃×#547	C₂	1	2	yes	maximal, complemented
6	C₂₃×#546	C₂	1	4	yes	maximal, complemented
7	C₂₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#542	C₂	1	2	yes	maximal, complemented
9	C₂₃×#540	C₂	1	2	yes	maximal, complemented
10	C₂₃×#535	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#532	C₂	1	2	yes	maximal, complemented
12	C₂₃×#529	C₂	1	2	yes	maximal, complemented
13	C₂₃×#525	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₂₃×#121	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×#41×C₂₃	C₂²	1	4	yes	complemented
20	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
22	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
24	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
25	C₂₃×#127	C₂²	1	2	yes	complemented
26	C₂₃×#126	C₂²	1	4	yes	complemented
27	C₂₃×#126	C₂²	1	4	yes	complemented
28	C₂₃×#125	C₂²	1	2	yes	complemented
29	C₂₃×#125	C₂²	1	2	yes	complemented
30	C₂₃×#124	C₂²	1	4	yes	complemented
31	C₂₃×#124	C₂²	1	4	yes	complemented
32	C₂₃×#124	C₂²	1	2	yes	complemented
33	C₂₃×#123	C₂²	1	2	yes	—
34	C₂₃×#123	C₂²	1	4	yes	complemented
35	C₂₃×#123	C₂²	1	4	yes	complemented
36	C₂₃×#123	C₂²	1	4	yes	complemented
37	C₂₃×#123	C₂²	1	4	yes	complemented
38	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
39	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	complemented
40	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
41	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
42	C₂₃×#121	C₂²	1	2	yes	—
43	C₂₃×#121	C₂²	1	2	yes	—
44	C₂₃×#121	C₂²	1	2	yes	—
45	C₂₃×#121	C₂²	1	2	yes	—
46	C₂₃×#121	C₂²	1	2	yes	complemented
47	C₂₃×#121	C₂²	1	2	yes	complemented
48	C₂₃×#121	C₂²	1	4	yes	complemented
49	C₂₃×#121	C₂²	1	4	yes	complemented
50	C₂₃×#121	C₂²	1	4	yes	complemented
51	C₂₃×#121	C₂²	1	1	yes	characteristic, complemented
52	C₂₃×#121	C₂²	1	1	yes	characteristic, complemented
53	C₂₃×#120	C₂²	1	4	yes	—
54	C₂₃×#120	C₂²	1	2	yes	—
55	C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
56	C₄×D₄×C₂₃	C₂²	1	4	yes	—
57	C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
58	C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
59	C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
60	C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
61	C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
62	C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
63	C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
64	C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
65	C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
66	C₂₃×#117	C₂²	1	2	yes	—
67	C₂₃×#117	C₂²	1	1	yes	characteristic
68	C₂₃×#117	C₂²	1	1	yes	characteristic
69	C₂₃×#117	C₂²	1	1	yes	characteristic
70	C₂₃×#117	C₂²	1	4	yes	complemented
71	C₂₃×#117	C₂²	1	1	yes	characteristic, complemented
72	C₂₃×#117	C₂²	1	2	yes	complemented
73	C₂₃×#117	C₂²	1	1	yes	characteristic, complemented
74	C₂₃×#117	C₂²	1	2	yes	complemented
75	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
77	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
78	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
79	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
80	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
81	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
82	C₂×C₄²×C₂₃	C₂²	1	2	yes	complemented
83	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
84	C₂⁴×C₂₃	C₂³	1	2	yes	—
85	#41×C₂₃	—	2	4	no	—
86	#41×C₂₃	—	4	8	no	—
87	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
88	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
89	C₂×D₄×C₂₃	—	2	4	no	—
90	C₂×D₄×C₂₃	—	4	4	no	—
91	C₂×D₄×C₂₃	C₂³	1	2	yes	—
92	C₂×D₄×C₂₃	C₂³	1	2	yes	—
93	C₂×D₄×C₂₃	C₂³	1	2	yes	—
94	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₂₃	C₂³	1	2	yes	—
96	C₂×D₄×C₂₃	C₂³	1	4	yes	—
97	C₂×D₄×C₂₃	C₂³	1	2	yes	—
98	C₂×D₄×C₂₃	C₂³	1	4	yes	—
99	C₂×D₄×C₂₃	C₂³	1	4	yes	—
100	C₂×D₄×C₂₃	C₂³	1	4	yes	—
101	C₂²×C₄×C₂₃	—	2	4	no	—
102	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
104	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
105	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
106	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
107	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
108	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
109	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
110	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
111	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
112	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
113	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
114	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
115	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
116	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
117	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
118	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
119	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
120	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
121	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
122	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
123	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
124	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
125	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
126	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
127	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
128	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
129	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
130	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
131	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
132	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
133	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
134	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
135	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
136	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
137	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
138	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
139	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
140	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
141	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
142	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
143	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
144	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
145	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
146	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
147	C₄²×C₂₃	C₂³	1	2	yes	—
148	C₄²×C₂₃	C₂³	1	2	yes	—
149	C₄²×C₂₃	C₂³	1	2	yes	—
150	C₄²×C₂₃	C₂³	1	2	yes	—
151	C₄²×C₂₃	C₂³	1	2	yes	—
152	C₄²×C₂₃	C₂³	1	4	yes	—
153	C₂³×C₂₃	—	2	2	no	—
154	C₂³×C₂₃	—	2	2	no	—
155	C₂³×C₂₃	—	2	4	no	—
156	C₂³×C₂₃	—	4	1	no	—
157	C₂³×C₂₃	—	4	2	no	—
158	C₂³×C₂₃	—	4	2	no	—
159	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
160	C₂³×C₂₃	C₂⁴	1	4	yes	—
161	C₂³×C₂₃	C₂⁴	1	2	yes	—
162	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
163	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
164	Q₈×C₂₃	—	2	2	no	—
165	Q₈×C₂₃	—	4	4	no	—
166	D₄×C₂₃	—	2	4	no	—
167	D₄×C₂₃	—	2	8	no	—
168	D₄×C₂₃	—	2	4	no	—
169	D₄×C₂₃	—	2	4	no	—
170	D₄×C₂₃	—	2	4	no	—
171	D₄×C₂₃	—	2	2	no	—
172	D₄×C₂₃	—	4	4	no	—
173	D₄×C₂₃	—	4	2	no	—
174	D₄×C₂₃	—	4	4	no	—
175	D₄×C₂₃	—	4	4	no	—
176	C₂×C₄×C₂₃	—	2	2	no	—
177	C₂×C₄×C₂₃	—	2	4	no	—
178	C₂×C₄×C₂₃	—	2	2	no	—
179	C₂×C₄×C₂₃	—	2	2	no	—
180	C₂×C₄×C₂₃	—	2	2	no	—
181	C₂×C₄×C₂₃	—	2	2	no	—
182	C₂×C₄×C₂₃	—	2	4	no	—
183	C₂×C₄×C₂₃	—	4	4	no	—
184	C₂×C₄×C₂₃	—	4	2	no	—
185	C₂×C₄×C₂₃	—	4	2	no	—
186	C₂×C₄×C₂₃	—	4	4	no	—
187	C₂×C₄×C₂₃	—	4	4	no	—
188	C₂×C₄×C₂₃	#41	1	4	yes	complemented
189	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
190	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
191	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
192	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
193	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
194	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
195	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
196	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
197	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
198	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
199	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
200	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
201	#3494	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂²×C₂₃	—	2	1	no	—
203	C₂²×C₂₃	—	2	1	no	—
204	C₂²×C₂₃	—	2	1	no	—
205	C₂²×C₂₃	—	2	1	no	—
206	C₂²×C₂₃	—	2	1	no	—
207	C₂²×C₂₃	—	2	2	no	—
208	C₂²×C₂₃	—	2	4	no	—
209	C₂²×C₂₃	—	2	2	no	—
210	C₂²×C₂₃	—	2	4	no	—
211	C₂²×C₂₃	—	2	1	no	—
212	C₂²×C₂₃	—	2	4	no	—
213	C₂²×C₂₃	—	2	2	no	—
214	C₂²×C₂₃	—	2	1	no	—
215	C₂²×C₂₃	—	4	4	no	—
216	C₂²×C₂₃	—	8	4	no	—
217	C₂²×C₂₃	C₂×#41	1	2	yes	—
218	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
219	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
220	C₄×C₂₃	—	2	1	no	—
221	C₄×C₂₃	—	2	1	no	—
222	C₄×C₂₃	—	2	2	no	—
223	C₄×C₂₃	—	2	2	no	—
224	C₄×C₂₃	—	2	2	no	—
225	C₄×C₂₃	—	2	4	no	—
226	C₄×C₂₃	—	2	2	no	—
227	C₄×C₂₃	—	2	2	no	—
228	C₄×C₂₃	—	2	2	no	—
229	C₄×C₂₃	—	2	2	no	—
230	C₄×C₂₃	C₂×#41	1	2	yes	—
231	C₄×C₂₃	C₂×#41	1	2	yes	—
232	#555	C₂×C₂₃	1	2	yes	complemented
233	#553	C₂×C₂₃	1	2	yes	complemented
234	#553	C₂×C₂₃	1	2	yes	complemented
235	#551	C₂×C₂₃	1	4	yes	complemented
236	#547	C₂×C₂₃	1	2	yes	complemented
237	#546	C₂×C₂₃	1	4	yes	complemented
238	#544	C₂×C₂₃	1	1	yes	characteristic, complemented
239	#542	C₂×C₂₃	1	2	yes	complemented
240	#540	C₂×C₂₃	1	2	yes	complemented
241	#535	C₂×C₂₃	1	1	yes	characteristic, complemented
242	#532	C₂×C₂₃	1	2	yes	complemented
243	#529	C₂×C₂₃	1	2	yes	complemented
244	#525	C₂×C₂₃	1	1	yes	characteristic, complemented
245	C₂×#121	C₂×C₂₃	1	1	yes	characteristic, complemented
246	#520	C₂×C₂₃	1	1	yes	characteristic, complemented
247	#518	C₂×C₂₃	1	1	yes	characteristic, complemented
248	C₂×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
249	C₂×C₂₃	—	2	2	no	—
250	C₂×C₂₃	—	2	2	no	—
251	C₂×C₂₃	—	4	2	no	—
252	C₂×C₂₃	—	4	1	no	—
253	C₂×C₂₃	—	4	4	no	—
254	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
255	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
256	C₂×C₂₃	#585	1	1	yes	characteristic
257	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
258	C₂×#41	C₂²×C₂₃	1	4	yes	complemented
259	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
260	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
261	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
262	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
263	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
264	#127	C₂²×C₂₃	1	2	yes	complemented
265	#126	C₂²×C₂₃	1	4	yes	complemented
266	#126	C₂²×C₂₃	1	4	yes	complemented
267	#125	C₂²×C₂₃	1	2	yes	complemented
268	#125	C₂²×C₂₃	1	2	yes	complemented
269	#124	C₂²×C₂₃	1	2	yes	complemented
270	#124	C₂²×C₂₃	1	4	yes	complemented
271	#124	C₂²×C₂₃	1	4	yes	complemented
272	#123	C₂²×C₂₃	1	2	yes	—
273	#123	C₂²×C₂₃	1	4	yes	complemented
274	#123	C₂²×C₂₃	1	4	yes	complemented
275	#123	C₂²×C₂₃	1	4	yes	complemented
276	#123	C₂²×C₂₃	1	4	yes	complemented
277	C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	—
278	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
279	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
280	C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	complemented
281	#121	C₂²×C₂₃	1	2	yes	—
282	#121	C₂²×C₂₃	1	2	yes	—
283	#121	C₂²×C₂₃	1	2	yes	—
284	#121	C₂²×C₂₃	1	2	yes	—
285	#121	C₂²×C₂₃	1	1	yes	characteristic, complemented
286	#121	C₂²×C₂₃	1	1	yes	characteristic, complemented
287	#121	C₂²×C₂₃	1	4	yes	complemented
288	#121	C₂²×C₂₃	1	2	yes	complemented
289	#121	C₂²×C₂₃	1	2	yes	complemented
290	#121	C₂²×C₂₃	1	4	yes	complemented
291	#121	C₂²×C₂₃	1	4	yes	complemented
292	#120	C₂²×C₂₃	1	2	yes	—
293	#120	C₂²×C₂₃	1	4	yes	—
294	C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
295	C₄×D₄	C₂²×C₂₃	1	4	yes	—
296	C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
297	C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
298	C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
299	C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
300	C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
301	C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
302	C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
303	C₄×D₄	C₂²×C₂₃	1	4	yes	complemented
304	C₄×D₄	C₂²×C₂₃	1	4	yes	complemented
305	#117	C₂²×C₂₃	1	1	yes	characteristic
306	#117	C₂²×C₂₃	1	1	yes	characteristic
307	#117	C₂²×C₂₃	1	2	yes	—
308	#117	C₂²×C₂₃	1	1	yes	characteristic
309	#117	C₂²×C₂₃	1	2	yes	complemented
310	#117	C₂²×C₂₃	1	1	yes	characteristic, complemented
311	#117	C₂²×C₂₃	1	1	yes	characteristic, complemented
312	#117	C₂²×C₂₃	1	2	yes	complemented
313	#117	C₂²×C₂₃	1	4	yes	complemented
314	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
315	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
316	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
317	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
318	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
319	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
320	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
321	C₂×C₄²	C₂²×C₂₃	1	2	yes	complemented
322	C₂₃	#3494	1	1	yes	characteristic, direct product factor, Sylow
323	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
324	C₂⁴	C₂³×C₂₃	1	2	yes	—
325	#41	—	2	4	no	—
326	#41	—	4	8	no	—
327	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
328	C₂×Q₈	C₂³×C₂₃	1	4	yes	—
329	C₂×D₄	—	2	4	no	—
330	C₂×D₄	—	4	4	no	—
331	C₂×D₄	C₂³×C₂₃	1	2	yes	—
332	C₂×D₄	C₂³×C₂₃	1	2	yes	—
333	C₂×D₄	C₂³×C₂₃	1	2	yes	—
334	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
335	C₂×D₄	C₂³×C₂₃	1	2	yes	—
336	C₂×D₄	C₂³×C₂₃	1	4	yes	—
337	C₂×D₄	C₂³×C₂₃	1	2	yes	—
338	C₂×D₄	C₂³×C₂₃	1	4	yes	—
339	C₂×D₄	C₂³×C₂₃	1	4	yes	—
340	C₂×D₄	C₂³×C₂₃	1	4	yes	—
341	C₂²×C₄	—	2	4	no	—
342	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
343	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
344	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
345	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
346	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
347	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
348	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
349	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
350	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
351	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
352	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
353	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
354	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
355	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
356	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
357	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
358	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
359	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
360	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
361	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
362	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
363	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
364	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
365	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
366	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
367	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
368	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
369	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
370	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
371	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
372	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
373	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
374	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
375	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
376	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
377	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
378	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
379	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
380	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
381	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
382	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
383	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
384	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
385	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
386	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
387	C₄²	C₂³×C₂₃	1	2	yes	—
388	C₄²	C₂³×C₂₃	1	2	yes	—
389	C₄²	C₂³×C₂₃	1	2	yes	—
390	C₄²	C₂³×C₂₃	1	2	yes	—
391	C₄²	C₂³×C₂₃	1	2	yes	—
392	C₄²	C₂³×C₂₃	1	4	yes	—
393	C₂³	—	2	2	no	—
394	C₂³	—	2	2	no	—
395	C₂³	—	2	4	no	—
396	C₂³	—	4	1	no	—
397	C₂³	—	4	2	no	—
398	C₂³	—	4	2	no	—
399	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
400	C₂³	C₂⁴×C₂₃	1	4	yes	—
401	C₂³	C₂⁴×C₂₃	1	2	yes	—
402	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
403	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
404	Q₈	—	2	2	no	—
405	Q₈	—	4	4	no	—
406	D₄	—	2	4	no	—
407	D₄	—	2	8	no	—
408	D₄	—	2	4	no	—
409	D₄	—	2	4	no	—
410	D₄	—	2	4	no	—
411	D₄	—	2	2	no	—
412	D₄	—	4	4	no	—
413	D₄	—	4	4	no	—
414	D₄	—	4	2	no	—
415	D₄	—	4	4	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	4	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	4	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	#41×C₂₃	1	4	yes	complemented
429	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
430	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
431	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
432	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
433	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
434	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
435	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
436	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
437	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
438	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
439	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
440	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
441	C₂²	—	2	1	no	—
442	C₂²	—	2	1	no	—
443	C₂²	—	2	1	no	—
444	C₂²	—	2	1	no	—
445	C₂²	—	2	1	no	—
446	C₂²	—	2	2	no	—
447	C₂²	—	2	4	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	2	4	no	—
450	C₂²	—	2	1	no	—
451	C₂²	—	2	4	no	—
452	C₂²	—	2	2	no	—
453	C₂²	—	2	1	no	—
454	C₂²	—	4	4	no	—
455	C₂²	—	8	4	no	—
456	C₂²	C₂×#41×C₂₃	1	2	yes	—
457	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
458	C₂²	C₂⁵×C₂₃	1	1	yes	characteristic
459	C₄	—	2	1	no	—
460	C₄	—	2	1	no	—
461	C₄	—	2	2	no	—
462	C₄	—	2	2	no	—
463	C₄	—	2	2	no	—
464	C₄	—	2	4	no	—
465	C₄	—	2	2	no	—
466	C₄	—	2	2	no	—
467	C₄	—	2	2	no	—
468	C₄	—	2	2	no	—
469	C₄	C₂×#41×C₂₃	1	2	yes	—
470	C₄	C₂×#41×C₂₃	1	2	yes	—
471	C₂	—	2	2	no	—
472	C₂	—	2	2	no	—
473	C₂	—	4	1	no	—
474	C₂	—	4	4	no	—
475	C₂	—	4	2	no	—
476	C₂	C₂²×#41×C₂₃	1	1	yes	characteristic
477	C₂	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
478	C₂	C₂₃×#585	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂₃×#555	C₂	semidirect
2. C₂₃×#553	C₂	semidirect
3. C₂₃×#553	C₂	semidirect
4. C₂₃×#551	C₂	semidirect
5. C₂₃×#547	C₂	semidirect
6. C₂₃×#546	C₂	semidirect
7. C₂₃×#544	C₂	semidirect
8. C₂₃×#542	C₂	semidirect
9. C₂₃×#540	C₂	semidirect
10. C₂₃×#535	C₂	semidirect
11. C₂₃×#532	C₂	semidirect
12. C₂₃×#529	C₂	semidirect
13. C₂₃×#525	C₂	semidirect
14. C₂×C₂₃×#121	C₂	semidirect
15. C₂₃×#520	C₂	semidirect
16. C₂₃×#518	C₂	semidirect
17. C₂×C₂₃×#117	C₂	semidirect
19. C₂×#41×C₂₃	C₂²	semidirect
23. C₂₃×C₄⋊Q₈	C₂²	semidirect
24. C₂₃×C₄⋊Q₈	C₂²	semidirect
25. C₂₃×#127	C₂²	semidirect
26. C₂₃×#126	C₂²	semidirect
27. C₂₃×#126	C₂²	semidirect
28. C₂₃×#125	C₂²	semidirect
29. C₂₃×#125	C₂²	semidirect
30. C₂₃×#124	C₂²	semidirect
31. C₂₃×#124	C₂²	semidirect
32. C₂₃×#124	C₂²	semidirect
34. C₂₃×#123	C₂²	semidirect
35. C₂₃×#123	C₂²	semidirect
36. C₂₃×#123	C₂²	semidirect
37. C₂₃×#123	C₂²	semidirect
39. C₂₃×C₂²⋊Q₈	C₂²	semidirect
40. C₂₃×C₂²⋊Q₈	C₂²	semidirect
41. C₂₃×C₂²⋊Q₈	C₂²	semidirect
46. C₂₃×#121	C₂²	semidirect
47. C₂₃×#121	C₂²	semidirect
48. C₂₃×#121	C₂²	semidirect
49. C₂₃×#121	C₂²	semidirect
50. C₂₃×#121	C₂²	semidirect
51. C₂₃×#121	C₂²	semidirect
52. C₂₃×#121	C₂²	semidirect
55. C₄×Q₈×C₂₃	C₂²	semidirect
57. C₄×D₄×C₂₃	C₂²	semidirect
58. C₄×D₄×C₂₃	C₂²	semidirect
59. C₄×D₄×C₂₃	C₂²	semidirect
60. C₄×D₄×C₂₃	C₂²	semidirect
61. C₄×D₄×C₂₃	C₂²	semidirect
62. C₄×D₄×C₂₃	C₂²	semidirect
63. C₄×D₄×C₂₃	C₂²	semidirect
64. C₄×D₄×C₂₃	C₂²	semidirect
65. C₄×D₄×C₂₃	C₂²	semidirect
70. C₂₃×#117	C₂²	semidirect
71. C₂₃×#117	C₂²	semidirect
72. C₂₃×#117	C₂²	semidirect
73. C₂₃×#117	C₂²	semidirect
74. C₂₃×#117	C₂²	semidirect
76. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
77. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
78. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
82. C₂×C₄²×C₂₃	C₂²	semidirect
188. C₂×C₄×C₂₃	#41	semidirect
201. #3494	C₂₃	direct (N × H)
232. #555	C₂×C₂₃	semidirect
233. #553	C₂×C₂₃	semidirect
234. #553	C₂×C₂₃	semidirect
235. #551	C₂×C₂₃	semidirect
236. #547	C₂×C₂₃	semidirect
237. #546	C₂×C₂₃	semidirect
238. #544	C₂×C₂₃	semidirect
239. #542	C₂×C₂₃	semidirect
240. #540	C₂×C₂₃	semidirect
241. #535	C₂×C₂₃	semidirect
242. #532	C₂×C₂₃	semidirect
243. #529	C₂×C₂₃	semidirect
244. #525	C₂×C₂₃	semidirect
245. C₂×#121	C₂×C₂₃	semidirect
246. #520	C₂×C₂₃	semidirect
247. #518	C₂×C₂₃	semidirect
248. C₂×#117	C₂×C₂₃	semidirect
258. C₂×#41	C₂²×C₂₃	semidirect
262. C₄⋊Q₈	C₂²×C₂₃	semidirect
263. C₄⋊Q₈	C₂²×C₂₃	semidirect
264. #127	C₂²×C₂₃	semidirect
265. #126	C₂²×C₂₃	semidirect
266. #126	C₂²×C₂₃	semidirect
267. #125	C₂²×C₂₃	semidirect
268. #125	C₂²×C₂₃	semidirect
269. #124	C₂²×C₂₃	semidirect
270. #124	C₂²×C₂₃	semidirect
271. #124	C₂²×C₂₃	semidirect
273. #123	C₂²×C₂₃	semidirect
274. #123	C₂²×C₂₃	semidirect
275. #123	C₂²×C₂₃	semidirect
276. #123	C₂²×C₂₃	semidirect
278. C₂²⋊Q₈	C₂²×C₂₃	semidirect
279. C₂²⋊Q₈	C₂²×C₂₃	semidirect
280. C₂²⋊Q₈	C₂²×C₂₃	semidirect
285. #121	C₂²×C₂₃	semidirect
286. #121	C₂²×C₂₃	semidirect
287. #121	C₂²×C₂₃	semidirect
288. #121	C₂²×C₂₃	semidirect
289. #121	C₂²×C₂₃	semidirect
290. #121	C₂²×C₂₃	semidirect
291. #121	C₂²×C₂₃	semidirect
294. C₄×Q₈	C₂²×C₂₃	semidirect
296. C₄×D₄	C₂²×C₂₃	semidirect
297. C₄×D₄	C₂²×C₂₃	semidirect
298. C₄×D₄	C₂²×C₂₃	semidirect
299. C₄×D₄	C₂²×C₂₃	semidirect
300. C₄×D₄	C₂²×C₂₃	semidirect
301. C₄×D₄	C₂²×C₂₃	semidirect
302. C₄×D₄	C₂²×C₂₃	semidirect
303. C₄×D₄	C₂²×C₂₃	semidirect
304. C₄×D₄	C₂²×C₂₃	semidirect
309. #117	C₂²×C₂₃	semidirect
310. #117	C₂²×C₂₃	semidirect
311. #117	C₂²×C₂₃	semidirect
312. #117	C₂²×C₂₃	semidirect
313. #117	C₂²×C₂₃	semidirect
315. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
316. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
317. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
321. C₂×C₄²	C₂²×C₂₃	semidirect
322. C₂₃	#3494	direct (N × H)
428. C₂×C₄	#41×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#555	2
C₂₃×#553	2
C₂₃×#551	2
C₂₃×#547	2
C₂₃×#546	2
C₂₃×#544	2
C₂₃×#542	2
C₂₃×#540	2
C₂₃×#535	2
C₂₃×#532	2
C₂₃×#529	2
C₂₃×#525	2
C₂×C₂₃×#121	2
C₂₃×#520	2
C₂₃×#518	2
C₂×C₂₃×#117	2
#3494	23

Subgroups — C23×#3494 #15671113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3494 #15671113