Subgroups — #15682615

Order 2944.

764 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625121	C₂	1	2	yes	maximal, complemented
2	#13092928	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092579	C₂	1	2	yes	maximal, complemented
4	#13092573	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092568	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092562	C₂	1	2	yes	maximal, complemented
7	#13092556	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092551	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092413	C₂	1	2	yes	maximal, complemented
10	C₂×C₂²⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	—	2	8	no	—
13	C₂×#41×C₂₃	—	2	1	no	—
14	C₂×#41×C₂₃	—	2	1	no	—
15	C₂×#41×C₂₃	C₂²	1	2	yes	—
16	C₂×#41×C₂₃	C₂²	1	2	yes	—
17	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×D₄×C₂₃	—	2	1	no	—
19	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂³×C₄×C₂₃	—	2	1	no	—
21	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×#67288	C₂²	1	1	yes	characteristic, complemented
23	#10625124	—	2	1	no	—
24	#10625121	—	2	8	no	—
25	C₂×#67275	C₂²	1	2	yes	complemented
26	C₂×#67274	C₂²	1	2	yes	complemented
27	C₂×#67273	C₂²	1	2	yes	complemented
28	C₂×#67272	C₂²	1	2	yes	complemented
29	#10625101	C₂²	1	1	yes	characteristic
30	#10625101	C₂²	1	1	yes	characteristic, complemented
31	#10625100	—	2	1	no	—
32	C₄×C₂²⋊D₂₃	—	2	1	no	—
33	C₄×C₂²⋊D₂₃	—	2	1	no	—
34	#10625097	C₂²	1	1	yes	characteristic, complemented
35	#10625095	C₂²	1	1	yes	characteristic
36	#10625095	C₂²	1	1	yes	characteristic, complemented
37	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	complemented
38	#10625017	C₂²	1	2	yes	complemented
39	#10625017	C₂²	1	2	yes	complemented
40	#10625014	C₂²	1	2	yes	complemented
41	#10625014	C₂²	1	2	yes	complemented
42	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
43	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
44	C₂²⋊(C₂₃⋊C₈)	C₂²	1	2	yes	complemented
45	C₂⁴×C₂₃	—	2	1	no	—
46	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
47	#41×C₂₃	—	2	8	no	—
48	#41×C₂₃	—	2	8	no	—
49	#41×C₂₃	—	2	4	no	—
50	#41×C₂₃	—	2	4	no	—
51	#41×C₂₃	—	2	4	no	—
52	#41×C₂₃	D₄	1	8	yes	complemented
53	C₂×Q₈×C₂₃	—	2	4	no	—
54	C₂×Q₈×C₂₃	—	2	1	no	—
55	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
56	C₂×D₄×C₂₃	—	2	8	no	—
57	C₂×D₄×C₂₃	—	2	2	no	—
58	C₂×D₄×C₂₃	—	2	2	no	—
59	C₂×D₄×C₂₃	—	2	4	no	—
60	C₂×D₄×C₂₃	—	2	1	no	—
61	C₂×D₄×C₂₃	C₂³	1	2	yes	—
62	C₂²×C₄×C₂₃	—	2	8	no	—
63	C₂²×C₄×C₂₃	—	2	1	no	—
64	C₂²×C₄×C₂₃	—	2	2	no	—
65	C₂²×C₄×C₂₃	—	2	2	no	—
66	C₂²×C₄×C₂₃	—	2	4	no	—
67	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
73	C₂×C₂²⋊D₂₃	—	2	1	no	—
74	C₂×C₂²⋊D₂₃	—	2	1	no	—
75	C₂×C₂²⋊D₂₃	—	2	1	no	—
76	#67288	—	2	4	no	—
77	C₂×D₉₂	C₂³	1	1	yes	characteristic
78	C₂×C₄×D₂₃	—	2	1	no	—
79	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
80	C₂²⋊Dic₂₃	—	2	1	no	—
81	C₂²⋊Dic₂₃	—	2	1	no	—
82	C₂²⋊Dic₂₃	—	2	1	no	—
83	#67275	—	2	4	no	—
84	#67274	—	2	4	no	—
85	#67273	—	2	4	no	—
86	#67272	—	2	4	no	—
87	(C₂×D₂₃)⋊C₄	—	2	1	no	—
88	(C₂×D₂₃)⋊C₄	—	2	1	no	—
89	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
91	Dic₂₃⋊C₄	—	2	1	no	—
92	Dic₂₃⋊C₄	—	2	1	no	—
93	C₄×Dic₂₃	—	2	1	no	—
94	C₂×C₂₃⋊C₈	—	2	4	no	—
95	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
96	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
97	C₂³×C₂₃	—	2	4	no	—
98	C₂³×C₂₃	—	2	2	no	—
99	C₂³×C₂₃	—	2	2	no	—
100	C₂³×C₂₃	—	2	2	no	—
101	C₂³×C₂₃	—	4	1	no	—
102	C₂³×C₂₃	—	4	2	no	—
103	C₂³×C₂₃	—	4	2	no	—
104	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
105	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₂₃	—	2	1	no	—
108	Q₈×C₂₃	—	2	4	no	—
109	Q₈×C₂₃	—	2	2	no	—
110	Q₈×C₂₃	C₂×D₄	1	4	yes	—
111	D₄×C₂₃	—	2	8	no	—
112	D₄×C₂₃	—	2	4	no	—
113	D₄×C₂₃	—	2	8	no	—
114	D₄×C₂₃	—	2	2	no	—
115	D₄×C₂₃	C₂×D₄	1	4	yes	—
116	C₂×C₄×C₂₃	—	2	4	no	—
117	C₂×C₄×C₂₃	—	2	8	no	—
118	C₂×C₄×C₂₃	—	2	2	no	—
119	C₂×C₄×C₂₃	—	2	2	no	—
120	C₂×C₄×C₂₃	—	2	2	no	—
121	C₂×C₄×C₂₃	—	2	8	no	—
122	C₂×C₄×C₂₃	—	2	2	no	—
123	C₂×C₄×C₂₃	—	2	2	no	—
124	C₂×C₄×C₂₃	—	2	2	no	—
125	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
128	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₂₃	—	4	2	no	—
131	C₂²⋊D₂₃	—	4	2	no	—
132	C₂²⋊D₂₃	—	4	2	no	—
133	C₂×Dic₂₃	—	2	1	no	—
134	C₂×Dic₂₃	—	2	1	no	—
135	C₂×Dic₂₃	—	2	1	no	—
136	D₉₂	—	2	2	no	—
137	C₄×D₂₃	—	4	2	no	—
138	Dic₄₆	—	2	2	no	—
139	C₂₃⋊C₈	—	2	2	no	—
140	C₂₃⋊C₈	—	2	2	no	—
141	#3002	—	23	1	no	maximal, Sylow
142	C₂²×C₂₃	—	2	1	no	—
143	C₂²×C₂₃	—	2	1	no	—
144	C₂²×C₂₃	—	2	1	no	—
145	C₂²×C₂₃	—	2	1	no	—
146	C₂²×C₂₃	—	2	1	no	—
147	C₂²×C₂₃	—	2	4	no	—
148	C₂²×C₂₃	—	2	1	no	—
149	C₂²×C₂₃	—	2	1	no	—
150	C₂²×C₂₃	—	4	4	no	—
151	C₂²×C₂₃	—	4	8	no	—
152	C₂²×C₂₃	—	4	2	no	—
153	C₂²×C₂₃	—	4	2	no	—
154	C₂²×C₂₃	#120	1	2	yes	—
155	C₂²×C₂₃	#135	1	2	yes	complemented
156	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×D₂₃	—	4	1	no	—
158	C₂×D₂₃	—	4	1	no	—
159	C₂×D₂₃	—	4	1	no	—
160	C₄×C₂₃	—	2	1	no	—
161	C₄×C₂₃	—	2	4	no	—
162	C₄×C₂₃	—	2	1	no	—
163	C₄×C₂₃	#120	1	2	yes	—
164	C₄×C₂₃	C₂²×D₄	1	2	yes	—
165	Dic₂₃	—	4	1	no	—
166	Dic₂₃	—	4	1	no	—
167	Dic₂₃	—	4	1	no	—
168	C₂²×#41	D₂₃	1	1	yes	characteristic, complemented
169	C₂×#135	—	23	2	no	—
170	#525	—	23	1	no	—
171	#452	—	23	2	no	—
172	#451	—	23	1	no	—
173	#450	—	23	1	no	—
174	#449	—	23	2	no	—
175	#448	—	23	1	no	—
176	#447	—	23	1	no	—
177	#416	—	23	2	no	—
178	C₂×C₂²⋊C₈	—	23	1	no	—
179	C₂×C₂₃	—	2	2	no	—
180	C₂×C₂₃	—	2	2	no	—
181	C₂×C₂₃	—	4	1	no	—
182	C₂×C₂₃	—	4	4	no	—
183	C₂×C₂₃	C₂×#120	1	1	yes	characteristic
184	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
185	C₂×C₂₃	#575	1	1	yes	characteristic
186	D₂₃	—	8	1	no	—
187	C₂×#41	—	2	8	no	—
188	C₂×#41	—	2	1	no	—
189	C₂×#41	—	2	1	no	—
190	C₂×#41	—	23	1	no	—
191	C₂×#41	C₂×D₂₃	1	2	yes	—
192	C₂×#41	C₂×D₂₃	1	2	yes	—
193	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
194	C₂²×D₄	—	2	1	no	—
195	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
196	C₂³×C₄	—	2	1	no	—
197	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
198	#135	—	46	8	no	—
199	C₂×Q₁₆	—	23	2	no	—
200	C₂×SD₁₆	—	23	2	no	—
201	C₂×SD₁₆	—	23	2	no	—
202	C₂×D₈	—	23	2	no	—
203	C₂²×C₈	—	23	2	no	—
204	#123	—	46	1	no	—
205	C₂²⋊Q₈	—	23	1	no	—
206	C₂²⋊Q₈	—	23	1	no	—
207	#121	—	23	1	no	—
208	#121	—	23	1	no	—
209	#120	—	46	1	no	—
210	C₄×D₄	—	46	1	no	—
211	C₄×D₄	—	46	1	no	—
212	#117	—	23	1	no	—
213	#103	—	23	2	no	—
214	#103	—	23	2	no	—
215	#102	—	23	2	no	—
216	#102	—	23	2	no	—
217	C₂²⋊C₈	—	23	1	no	—
218	C₂²⋊C₈	—	23	1	no	—
219	C₂²⋊C₈	—	23	2	no	—
220	C₂₃	#3002	1	1	yes	characteristic, complemented, Sylow
221	C₂⁴	—	2	1	no	—
222	C₂⁴	C₂²⋊D₂₃	1	1	yes	characteristic
223	#41	—	2	8	no	—
224	#41	—	2	8	no	—
225	#41	—	2	4	no	—
226	#41	—	2	4	no	—
227	#41	—	2	4	no	—
228	#41	—	46	4	no	—
229	#41	C₂²⋊D₂₃	1	8	yes	complemented
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	—	23	1	no	—
233	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
234	C₂×D₄	—	2	8	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	23	1	no	—
240	C₂×D₄	—	46	1	no	—
241	C₂×D₄	—	46	1	no	—
242	C₂×D₄	—	46	1	no	—
243	C₂×D₄	C₂²×D₂₃	1	2	yes	—
244	C₂²×C₄	—	2	8	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	46	1	no	—
250	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
252	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
254	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
255	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
256	Q₁₆	—	46	4	no	—
257	SD₁₆	—	46	4	no	—
258	SD₁₆	—	46	4	no	—
259	D₈	—	46	4	no	—
260	C₂×C₈	—	23	2	no	—
261	C₂×C₈	—	23	2	no	—
262	C₂×C₈	—	46	4	no	—
263	C₄⋊C₄	—	23	1	no	—
264	C₄⋊C₄	—	23	1	no	—
265	C₄⋊C₄	—	46	1	no	—
266	C₄⋊C₄	—	46	1	no	—
267	C₂²⋊C₄	—	46	1	no	—
268	C₂²⋊C₄	—	46	1	no	—
269	C₂²⋊C₄	—	46	1	no	—
270	C₂²⋊C₄	—	46	1	no	—
271	C₂²⋊C₄	—	46	1	no	—
272	C₄²	—	46	1	no	—
273	C₂³	—	2	4	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	4	1	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	46	1	no	—
281	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
282	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
283	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
284	Q₈	—	2	4	no	—
285	Q₈	—	2	2	no	—
286	Q₈	—	46	2	no	—
287	Q₈	C₂×C₂²⋊D₂₃	1	4	yes	—
288	D₄	—	2	8	no	—
289	D₄	—	2	4	no	—
290	D₄	—	2	8	no	—
291	D₄	—	2	2	no	—
292	D₄	—	46	2	no	—
293	D₄	—	92	2	no	—
294	D₄	—	92	2	no	—
295	D₄	—	92	2	no	—
296	D₄	C₂×C₂²⋊D₂₃	1	4	yes	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	8	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	46	1	no	—
307	C₂×C₄	—	46	1	no	—
308	C₂×C₄	—	46	1	no	—
309	C₂×C₄	—	92	2	no	—
310	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
311	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
312	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
313	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
314	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
315	C₈	—	46	2	no	—
316	C₈	—	46	2	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	4	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	4	4	no	—
326	C₂²	—	4	8	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	92	1	no	—
330	C₂²	—	92	1	no	—
331	C₂²	—	92	1	no	—
332	C₂²	#10625121	1	2	yes	complemented
333	C₂²	#10625124	1	2	yes	—
334	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
335	C₄	—	2	1	no	—
336	C₄	—	2	4	no	—
337	C₄	—	2	1	no	—
338	C₄	—	92	1	no	—
339	C₄	—	92	1	no	—
340	C₄	—	92	1	no	—
341	C₄	#10625124	1	2	yes	—
342	C₄	C₂²×C₂²⋊D₂₃	1	2	yes	—
343	C₂	—	2	2	no	—
344	C₂	—	2	2	no	—
345	C₂	—	4	1	no	—
346	C₂	—	4	4	no	—
347	C₂	—	184	1	no	—
348	C₂	C₂×#10625124	1	1	yes	characteristic
349	C₂	C₂×#10625121	1	1	yes	characteristic
350	C₂	#13093215	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂²×#41×C₂₃	C₂
socle	C₂²×C₂₃	C₂²×D₄
hypercenter	C₂²×#41	D₂₃
hypocenter	C₂₃	#3002
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#10625121	C₂	semidirect
2. #13092928	C₂	semidirect
3. #13092579	C₂	semidirect
4. #13092573	C₂	semidirect
5. #13092568	C₂	semidirect
6. #13092562	C₂	semidirect
7. #13092556	C₂	semidirect
8. #13092551	C₂	semidirect
9. #13092413	C₂	semidirect
10. C₂×C₂²⋊(C₂₃⋊C₈)	C₂	semidirect
11. C₂²×#41×C₂₃	C₂	semidirect
22. C₂×#67288	C₂²	semidirect
25. C₂×#67275	C₂²	semidirect
26. C₂×#67274	C₂²	semidirect
27. C₂×#67273	C₂²	semidirect
28. C₂×#67272	C₂²	semidirect
30. #10625101	C₂²	semidirect
34. #10625097	C₂²	semidirect
36. #10625095	C₂²	semidirect
37. C₂²×C₂₃⋊C₈	C₂²	semidirect
38. #10625017	C₂²	semidirect
39. #10625017	C₂²	semidirect
40. #10625014	C₂²	semidirect
41. #10625014	C₂²	semidirect
44. C₂²⋊(C₂₃⋊C₈)	C₂²	semidirect
52. #41×C₂₃	D₄	semidirect
155. C₂²×C₂₃	#135	semidirect
168. C₂²×#41	D₂₃	semidirect
220. C₂₃	#3002	semidirect
229. #41	C₂²⋊D₂₃	semidirect
332. C₂²	#10625121	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625121	2
#13092928	2
#13092579	2
#13092573	2
#13092568	2
#13092562	2
#13092556	2
#13092551	2
#13092413	2
C₂×C₂²⋊(C₂₃⋊C₈)	2
C₂²×#41×C₂₃	2
#3002	23