Subgroups — C₂₃×#36652 #19184278

Order 5888.

2570 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#525	C₂	1	2	yes	maximal, complemented
2	C₂₃×#2817	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2816	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2815	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2814	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2813	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2812	C₂	1	4	yes	maximal, complemented
8	C₂₃×#2395	C₂	1	4	yes	maximal, complemented
9	C₂₃×#2392	C₂	1	4	yes	maximal, complemented
10	C₂₃×#2388	C₂	1	4	yes	maximal, complemented
11	C₂₃×#2280	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₂₃	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#525	—	2	16	no	—
15	C₂×C₂₃×#123	C₂²	1	4	yes	—
16	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
17	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
18	C₂×C₂₃×#121	C₂²	1	2	yes	—
19	C₂×C₂₃×#121	C₂²	1	2	yes	—
20	C₂×C₂₃×#120	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
22	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
23	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
24	C₂×C₂₃×#117	C₂²	1	2	yes	—
25	C₂×C₂₃×#117	C₂²	1	2	yes	—
26	C₂₃×#400	C₂²	1	4	yes	complemented
27	C₂₃×#399	C₂²	1	4	yes	complemented
28	C₂₃×#398	C₂²	1	4	yes	complemented
29	C₂₃×#398	C₂²	1	4	yes	complemented
30	C₂₃×#397	C₂²	1	8	yes	—
31	C₂₃×#397	C₂²	1	4	yes	—
32	C₂₃×#397	C₂²	1	4	yes	complemented
33	C₂₃×#396	C₂²	1	8	yes	complemented
34	C₂₃×#395	C₂²	1	4	yes	complemented
35	C₂₃×#394	C₂²	1	8	yes	—
36	C₂₃×#394	C₂²	1	8	yes	—
37	C₂₃×#394	C₂²	1	8	yes	—
38	C₂₃×#394	C₂²	1	4	yes	complemented
39	C₂₃×#394	C₂²	1	4	yes	complemented
40	C₂₃×#393	C₂²	1	4	yes	complemented
41	C₂₃×#392	C₂²	1	8	yes	—
42	C₂₃×#392	C₂²	1	4	yes	—
43	C₂₃×#392	C₂²	1	4	yes	complemented
44	C₂₃×#381	C₂²	1	1	yes	characteristic
45	C₂₃×#381	C₂²	1	1	yes	characteristic
46	C₂₃×#380	C₂²	1	4	yes	—
47	C₂₃×#379	C₂²	1	2	yes	—
48	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	complemented
49	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
50	C₄×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	complemented
51	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×#41×C₂₃	—	2	8	no	—
53	C₂×#41×C₂₃	—	4	8	no	—
54	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
55	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
56	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
57	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
58	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
59	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
60	C₂³×C₄×C₂₃	—	4	4	no	—
61	C₂³×C₄×C₂₃	D₄	1	8	yes	complemented
62	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
63	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
64	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
66	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂₃×#123	—	2	16	no	—
68	C₂₃×C₂²⋊Q₈	—	2	8	no	—
69	C₂₃×C₂²⋊Q₈	—	2	8	no	—
70	C₂₃×#121	—	2	8	no	—
71	C₂₃×#121	—	2	8	no	—
72	C₂₃×#120	—	2	16	no	—
73	C₄×D₄×C₂₃	—	2	16	no	—
74	C₄×D₄×C₂₃	—	2	16	no	—
75	C₄×D₄×C₂₃	—	4	8	no	—
76	C₂₃×#117	—	2	8	no	—
77	C₂₃×#117	—	4	4	no	—
78	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
83	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
85	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
86	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
87	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
91	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
92	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
93	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
94	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
95	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
96	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
97	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
98	C₂₃×#95	C₂³	1	4	yes	—
99	C₂₃×#95	C₂³	1	4	yes	—
100	C₂₃×#95	C₂³	1	8	yes	—
101	C₂₃×#95	C₂³	1	8	yes	—
102	C₂₃×#95	C₂³	1	4	yes	—
103	C₂₃×#95	C₂³	1	4	yes	—
104	C₂⁴×C₂₃	—	4	2	no	—
105	C₂⁴×C₂₃	—	4	4	no	—
106	C₂⁴×C₂₃	C₂×D₄	1	4	yes	—
107	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
108	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
109	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
110	#41×C₂₃	—	4	32	no	—
111	C₂×Q₈×C₂₃	—	2	4	no	—
112	C₂×Q₈×C₂₃	—	4	4	no	—
113	C₂×D₄×C₂₃	—	2	8	no	—
114	C₂×D₄×C₂₃	—	2	8	no	—
115	C₂×D₄×C₂₃	—	2	8	no	—
116	C₂×D₄×C₂₃	—	2	4	no	—
117	C₂×D₄×C₂₃	—	4	4	no	—
118	C₂×D₄×C₂₃	—	4	4	no	—
119	C₂×D₄×C₂₃	—	4	8	no	—
120	C₂×D₄×C₂₃	—	4	8	no	—
121	C₂×D₄×C₂₃	—	4	4	no	—
122	C₂×D₄×C₂₃	—	4	4	no	—
123	C₂×D₄×C₂₃	—	4	8	no	—
124	C₂²×C₄×C₂₃	—	2	4	no	—
125	C₂²×C₄×C₂₃	—	2	8	no	—
126	C₂²×C₄×C₂₃	—	2	8	no	—
127	C₂²×C₄×C₂₃	—	2	16	no	—
128	C₂²×C₄×C₂₃	—	4	16	no	—
129	C₂²×C₄×C₂₃	—	4	4	no	—
130	C₂²×C₄×C₂₃	—	4	2	no	—
131	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
132	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
133	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
134	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
135	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
136	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
137	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
138	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
139	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
140	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
142	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
143	C₄⋊C₄×C₂₃	—	2	4	no	—
144	C₄⋊C₄×C₂₃	—	2	4	no	—
145	C₄⋊C₄×C₂₃	—	2	8	no	—
146	C₄⋊C₄×C₂₃	—	2	8	no	—
147	C₄⋊C₄×C₂₃	—	4	8	no	—
148	C₄⋊C₄×C₂₃	—	4	4	no	—
149	C₄⋊C₄×C₂₃	—	4	4	no	—
150	C₂²⋊C₄×C₂₃	—	2	8	no	—
151	C₂²⋊C₄×C₂₃	—	2	8	no	—
152	C₂²⋊C₄×C₂₃	—	2	8	no	—
153	C₂²⋊C₄×C₂₃	—	2	8	no	—
154	C₂²⋊C₄×C₂₃	—	2	8	no	—
155	C₂²⋊C₄×C₂₃	—	4	8	no	—
156	C₂²⋊C₄×C₂₃	—	4	8	no	—
157	C₂²⋊C₄×C₂₃	—	4	8	no	—
158	C₂²⋊C₄×C₂₃	—	4	2	no	—
159	C₂²⋊C₄×C₂₃	—	4	2	no	—
160	C₂²⋊C₄×C₂₃	—	4	8	no	—
161	C₄²×C₂₃	—	2	8	no	—
162	C₄²×C₂₃	—	2	8	no	—
163	#36652	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂³×C₂₃	—	2	2	no	—
165	C₂³×C₂₃	—	2	4	no	—
166	C₂³×C₂₃	—	2	4	no	—
167	C₂³×C₂₃	—	2	4	no	—
168	C₂³×C₂₃	—	2	4	no	—
169	C₂³×C₂₃	—	2	4	no	—
170	C₂³×C₂₃	—	2	8	no	—
171	C₂³×C₂₃	—	2	4	no	—
172	C₂³×C₂₃	—	2	8	no	—
173	C₂³×C₂₃	—	2	4	no	—
174	C₂³×C₂₃	—	4	4	no	—
175	C₂³×C₂₃	—	4	8	no	—
176	C₂³×C₂₃	—	4	16	no	—
177	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
178	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
179	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
180	Q₈×C₂₃	—	4	8	no	—
181	D₄×C₂₃	—	4	8	no	—
182	D₄×C₂₃	—	4	16	no	—
183	D₄×C₂₃	—	4	16	no	—
184	D₄×C₂₃	—	4	16	no	—
185	D₄×C₂₃	—	4	16	no	—
186	C₂×C₄×C₂₃	—	2	2	no	—
187	C₂×C₄×C₂₃	—	2	4	no	—
188	C₂×C₄×C₂₃	—	2	8	no	—
189	C₂×C₄×C₂₃	—	2	16	no	—
190	C₂×C₄×C₂₃	—	2	4	no	—
191	C₂×C₄×C₂₃	—	2	4	no	—
192	C₂×C₄×C₂₃	—	2	4	no	—
193	C₂×C₄×C₂₃	—	2	4	no	—
194	C₂×C₄×C₂₃	—	2	4	no	—
195	C₂×C₄×C₂₃	—	2	8	no	—
196	C₂×C₄×C₂₃	—	2	8	no	—
197	C₂×C₄×C₂₃	—	2	4	no	—
198	C₂×C₄×C₂₃	—	2	8	no	—
199	C₂×C₄×C₂₃	—	2	8	no	—
200	C₂×C₄×C₂₃	—	2	4	no	—
201	C₂×C₄×C₂₃	—	4	8	no	—
202	C₂×C₄×C₂₃	—	4	4	no	—
203	C₂×C₄×C₂₃	—	4	16	no	—
204	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
205	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
206	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
207	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
208	C₂×#525	C₂×C₂₃	1	2	yes	complemented
209	#2817	C₂×C₂₃	1	1	yes	characteristic, complemented
210	#2816	C₂×C₂₃	1	2	yes	complemented
211	#2815	C₂×C₂₃	1	1	yes	characteristic, complemented
212	#2814	C₂×C₂₃	1	4	yes	complemented
213	#2813	C₂×C₂₃	1	4	yes	complemented
214	#2812	C₂×C₂₃	1	4	yes	complemented
215	#2395	C₂×C₂₃	1	4	yes	complemented
216	#2392	C₂×C₂₃	1	4	yes	complemented
217	#2388	C₂×C₂₃	1	4	yes	complemented
218	#2280	C₂×C₂₃	1	1	yes	characteristic, complemented
219	C₂²×C₂₃	—	2	4	no	—
220	C₂²×C₂₃	—	2	2	no	—
221	C₂²×C₂₃	—	2	4	no	—
222	C₂²×C₂₃	—	2	4	no	—
223	C₂²×C₂₃	—	2	4	no	—
224	C₂²×C₂₃	—	4	16	no	—
225	C₂²×C₂₃	—	4	4	no	—
226	C₂²×C₂₃	—	4	8	no	—
227	C₂²×C₂₃	—	4	4	no	—
228	C₂²×C₂₃	—	4	4	no	—
229	C₂²×C₂₃	—	4	4	no	—
230	C₂²×C₂₃	—	4	8	no	—
231	C₂²×C₂₃	—	4	8	no	—
232	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
233	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
234	C₂²×C₂₃	#585	1	4	yes	—
235	C₄×C₂₃	—	2	2	no	—
236	C₄×C₂₃	—	2	4	no	—
237	C₄×C₂₃	—	4	4	no	—
238	C₄×C₂₃	—	4	4	no	—
239	C₄×C₂₃	—	4	4	no	—
240	C₄×C₂₃	—	4	4	no	—
241	C₄×C₂₃	—	4	4	no	—
242	C₄×C₂₃	#534	1	4	yes	—
243	C₂²×#41	C₂²×C₂₃	1	2	yes	—
244	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
245	#525	—	2	16	no	—
246	C₂×#123	C₂²×C₂₃	1	4	yes	—
247	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
248	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
249	C₂×#121	C₂²×C₂₃	1	2	yes	—
250	C₂×#121	C₂²×C₂₃	1	2	yes	—
251	C₂×#120	C₂²×C₂₃	1	4	yes	—
252	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
253	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
254	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	complemented
255	C₂×#117	C₂²×C₂₃	1	2	yes	—
256	C₂×#117	C₂²×C₂₃	1	2	yes	—
257	#400	C₂²×C₂₃	1	4	yes	complemented
258	#399	C₂²×C₂₃	1	4	yes	complemented
259	#398	C₂²×C₂₃	1	4	yes	complemented
260	#398	C₂²×C₂₃	1	4	yes	complemented
261	#397	C₂²×C₂₃	1	4	yes	—
262	#397	C₂²×C₂₃	1	8	yes	—
263	#397	C₂²×C₂₃	1	4	yes	complemented
264	#396	C₂²×C₂₃	1	8	yes	complemented
265	#395	C₂²×C₂₃	1	4	yes	complemented
266	#394	C₂²×C₂₃	1	8	yes	—
267	#394	C₂²×C₂₃	1	8	yes	—
268	#394	C₂²×C₂₃	1	8	yes	—
269	#394	C₂²×C₂₃	1	4	yes	complemented
270	#394	C₂²×C₂₃	1	4	yes	complemented
271	#393	C₂²×C₂₃	1	4	yes	complemented
272	#392	C₂²×C₂₃	1	8	yes	—
273	#392	C₂²×C₂₃	1	4	yes	—
274	#392	C₂²×C₂₃	1	4	yes	complemented
275	#381	C₂²×C₂₃	1	1	yes	characteristic
276	#381	C₂²×C₂₃	1	1	yes	characteristic
277	#380	C₂²×C₂₃	1	4	yes	—
278	#379	C₂²×C₂₃	1	2	yes	—
279	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	complemented
280	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
281	C₄×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	complemented
282	C₂×C₂₃	—	4	4	no	—
283	C₂×C₂₃	—	4	2	no	—
284	C₂×C₂₃	—	8	4	no	—
285	C₂×C₂₃	C₂×#534	1	1	yes	characteristic
286	C₂×C₂₃	#3449	1	2	yes	—
287	C₂×C₂₃	#3529	1	4	yes	—
288	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
289	C₂×#41	—	2	8	no	—
290	C₂×#41	—	4	8	no	—
291	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
292	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
293	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
294	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
295	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
296	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
297	C₂³×C₄	—	4	4	no	—
298	C₂³×C₄	D₄×C₂₃	1	8	yes	complemented
299	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
300	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
301	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
302	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
303	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
304	#123	—	2	16	no	—
305	C₂²⋊Q₈	—	2	8	no	—
306	C₂²⋊Q₈	—	2	8	no	—
307	#121	—	2	8	no	—
308	#121	—	2	8	no	—
309	#120	—	2	16	no	—
310	C₄×D₄	—	2	16	no	—
311	C₄×D₄	—	2	16	no	—
312	C₄×D₄	—	4	8	no	—
313	#117	—	2	8	no	—
314	#117	—	4	4	no	—
315	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
316	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
317	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
318	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
319	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
320	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
321	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
322	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
323	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
324	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
325	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
326	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
327	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
328	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
329	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
330	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
331	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
332	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
333	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
334	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
335	#95	C₂³×C₂₃	1	4	yes	—
336	#95	C₂³×C₂₃	1	4	yes	—
337	#95	C₂³×C₂₃	1	8	yes	—
338	#95	C₂³×C₂₃	1	8	yes	—
339	#95	C₂³×C₂₃	1	4	yes	—
340	#95	C₂³×C₂₃	1	4	yes	—
341	C₂₃	#36652	1	1	yes	characteristic, direct product factor, Sylow
342	C₂⁴	—	4	2	no	—
343	C₂⁴	—	4	4	no	—
344	C₂⁴	C₂×D₄×C₂₃	1	4	yes	—
345	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
346	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
347	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
348	#41	—	4	32	no	—
349	C₂×Q₈	—	2	4	no	—
350	C₂×Q₈	—	4	4	no	—
351	C₂×D₄	—	2	8	no	—
352	C₂×D₄	—	2	8	no	—
353	C₂×D₄	—	2	8	no	—
354	C₂×D₄	—	2	4	no	—
355	C₂×D₄	—	4	4	no	—
356	C₂×D₄	—	4	4	no	—
357	C₂×D₄	—	4	8	no	—
358	C₂×D₄	—	4	4	no	—
359	C₂×D₄	—	4	8	no	—
360	C₂×D₄	—	4	8	no	—
361	C₂×D₄	—	4	4	no	—
362	C₂²×C₄	—	2	4	no	—
363	C₂²×C₄	—	2	8	no	—
364	C₂²×C₄	—	2	8	no	—
365	C₂²×C₄	—	2	16	no	—
366	C₂²×C₄	—	4	16	no	—
367	C₂²×C₄	—	4	4	no	—
368	C₂²×C₄	—	4	2	no	—
369	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
370	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
371	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
372	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
373	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
374	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
375	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
376	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
377	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
378	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
379	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
380	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
381	C₄⋊C₄	—	2	4	no	—
382	C₄⋊C₄	—	2	4	no	—
383	C₄⋊C₄	—	2	8	no	—
384	C₄⋊C₄	—	2	8	no	—
385	C₄⋊C₄	—	4	4	no	—
386	C₄⋊C₄	—	4	4	no	—
387	C₄⋊C₄	—	4	8	no	—
388	C₂²⋊C₄	—	2	8	no	—
389	C₂²⋊C₄	—	2	8	no	—
390	C₂²⋊C₄	—	2	8	no	—
391	C₂²⋊C₄	—	2	8	no	—
392	C₂²⋊C₄	—	2	8	no	—
393	C₂²⋊C₄	—	4	2	no	—
394	C₂²⋊C₄	—	4	2	no	—
395	C₂²⋊C₄	—	4	8	no	—
396	C₂²⋊C₄	—	4	8	no	—
397	C₂²⋊C₄	—	4	8	no	—
398	C₂²⋊C₄	—	4	8	no	—
399	C₄²	—	2	8	no	—
400	C₄²	—	2	8	no	—
401	C₂³	—	2	2	no	—
402	C₂³	—	2	4	no	—
403	C₂³	—	2	4	no	—
404	C₂³	—	2	4	no	—
405	C₂³	—	2	4	no	—
406	C₂³	—	2	4	no	—
407	C₂³	—	2	8	no	—
408	C₂³	—	2	4	no	—
409	C₂³	—	2	8	no	—
410	C₂³	—	2	4	no	—
411	C₂³	—	4	4	no	—
412	C₂³	—	4	8	no	—
413	C₂³	—	4	16	no	—
414	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
415	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
416	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
417	Q₈	—	4	8	no	—
418	D₄	—	4	8	no	—
419	D₄	—	4	16	no	—
420	D₄	—	4	16	no	—
421	D₄	—	4	16	no	—
422	D₄	—	4	16	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	16	no	—
426	C₂×C₄	—	2	8	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	2	8	no	—
433	C₂×C₄	—	2	8	no	—
434	C₂×C₄	—	2	4	no	—
435	C₂×C₄	—	2	8	no	—
436	C₂×C₄	—	2	8	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	4	4	no	—
439	C₂×C₄	—	4	8	no	—
440	C₂×C₄	—	4	16	no	—
441	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
442	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
443	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
444	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
445	C₂²	—	2	4	no	—
446	C₂²	—	2	2	no	—
447	C₂²	—	2	4	no	—
448	C₂²	—	2	4	no	—
449	C₂²	—	2	4	no	—
450	C₂²	—	4	16	no	—
451	C₂²	—	4	4	no	—
452	C₂²	—	4	8	no	—
453	C₂²	—	4	4	no	—
454	C₂²	—	4	4	no	—
455	C₂²	—	4	4	no	—
456	C₂²	—	4	8	no	—
457	C₂²	—	4	8	no	—
458	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
459	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
460	C₂²	C₂₃×#585	1	4	yes	—
461	C₄	—	2	4	no	—
462	C₄	—	2	2	no	—
463	C₄	—	4	4	no	—
464	C₄	—	4	4	no	—
465	C₄	—	4	4	no	—
466	C₄	—	4	4	no	—
467	C₄	—	4	4	no	—
468	C₄	C₂₃×#534	1	4	yes	—
469	C₂	—	4	4	no	—
470	C₂	—	4	2	no	—
471	C₂	—	8	4	no	—
472	C₂	C₂×C₂₃×#534	1	1	yes	characteristic
473	C₂	C₂₃×#3449	1	2	yes	—
474	C₂	C₂₃×#3529	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#525	C₂	semidirect
2. C₂₃×#2817	C₂	semidirect
3. C₂₃×#2816	C₂	semidirect
4. C₂₃×#2815	C₂	semidirect
5. C₂₃×#2814	C₂	semidirect
6. C₂₃×#2813	C₂	semidirect
7. C₂₃×#2812	C₂	semidirect
8. C₂₃×#2395	C₂	semidirect
9. C₂₃×#2392	C₂	semidirect
10. C₂₃×#2388	C₂	semidirect
11. C₂₃×#2280	C₂	semidirect
23. C₂×C₄×D₄×C₂₃	C₂²	semidirect
26. C₂₃×#400	C₂²	semidirect
27. C₂₃×#399	C₂²	semidirect
28. C₂₃×#398	C₂²	semidirect
29. C₂₃×#398	C₂²	semidirect
32. C₂₃×#397	C₂²	semidirect
33. C₂₃×#396	C₂²	semidirect
34. C₂₃×#395	C₂²	semidirect
38. C₂₃×#394	C₂²	semidirect
39. C₂₃×#394	C₂²	semidirect
40. C₂₃×#393	C₂²	semidirect
43. C₂₃×#392	C₂²	semidirect
48. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
50. C₄×C₂²⋊C₄×C₂₃	C₂²	semidirect
61. C₂³×C₄×C₂₃	D₄	semidirect
163. #36652	C₂₃	direct (N × H)
208. C₂×#525	C₂×C₂₃	semidirect
209. #2817	C₂×C₂₃	semidirect
210. #2816	C₂×C₂₃	semidirect
211. #2815	C₂×C₂₃	semidirect
212. #2814	C₂×C₂₃	semidirect
213. #2813	C₂×C₂₃	semidirect
214. #2812	C₂×C₂₃	semidirect
215. #2395	C₂×C₂₃	semidirect
216. #2392	C₂×C₂₃	semidirect
217. #2388	C₂×C₂₃	semidirect
218. #2280	C₂×C₂₃	semidirect
254. C₂×C₄×D₄	C₂²×C₂₃	semidirect
257. #400	C₂²×C₂₃	semidirect
258. #399	C₂²×C₂₃	semidirect
259. #398	C₂²×C₂₃	semidirect
260. #398	C₂²×C₂₃	semidirect
263. #397	C₂²×C₂₃	semidirect
264. #396	C₂²×C₂₃	semidirect
265. #395	C₂²×C₂₃	semidirect
269. #394	C₂²×C₂₃	semidirect
270. #394	C₂²×C₂₃	semidirect
271. #393	C₂²×C₂₃	semidirect
274. #392	C₂²×C₂₃	semidirect
279. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
281. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
298. C₂³×C₄	D₄×C₂₃	semidirect
341. C₂₃	#36652	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#525	2
C₂₃×#2817	2
C₂₃×#2816	2
C₂₃×#2815	2
C₂₃×#2814	2
C₂₃×#2813	2
C₂₃×#2812	2
C₂₃×#2395	2
C₂₃×#2392	2
C₂₃×#2388	2
C₂₃×#2280	2
#36652	23

Subgroups — C23×#36652 #19184278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#36652 #19184278