Subgroups — #15682064

Order 2944.

392 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#67288	C₂	1	1	yes	maximal, characteristic
2	#13092479	C₂	1	2	yes	maximal, complemented
3	#13092475	C₂	1	2	yes	maximal, complemented
4	#13092473	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092470	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092469	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092439	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄⋊C₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092382	C₂	1	2	yes	maximal
10	#13092374	C₂	1	2	yes	maximal, complemented
11	C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#67288	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	C₄×C₂²⋊D₂₃	C₄	1	1	yes	characteristic
15	C₄×C₂²⋊D₂₃	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊D₂₃	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₂₃	C₄	1	1	yes	characteristic
18	#10625097	C₂²	1	1	yes	characteristic
19	C₂×#67262	C₂²	1	2	yes	—
20	C₂×C₈×D₂₃	C₂²	1	2	yes	—
21	C₄×D₉₂	C₄	1	1	yes	characteristic
22	C₄×D₉₂	C₄	1	1	yes	characteristic
23	#10625043	C₂²	1	1	yes	characteristic
24	#10625043	C₂²	1	1	yes	characteristic
25	C₄²×D₂₃	C₂²	1	1	yes	characteristic
26	C₄²×D₂₃	C₂²	1	1	yes	characteristic
27	C₄×Dic₄₆	C₄	1	1	yes	characteristic
28	C₄×Dic₄₆	C₄	1	1	yes	characteristic
29	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₂₃×#97	C₂²	1	2	yes	—
33	C₂²⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
34	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
35	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
36	#10624997	C₂²	1	2	yes	—
37	Dic₂₃⋊C₈	C₂²	1	2	yes	—
38	Dic₂₃⋊C₈	C₂²	1	2	yes	—
39	C₈×Dic₂₃	C₂²	1	2	yes	—
40	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
41	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
42	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₂₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₂₃	C₂×C₄	1	1	yes	characteristic
48	#67288	—	2	2	no	—
49	#67288	—	2	2	no	—
50	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
57	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×C₂₃	C₂³	1	2	yes	—
59	C₂×C₈×C₂₃	C₂³	1	2	yes	—
60	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
61	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
62	C₄²×C₂₃	C₂³	1	1	yes	characteristic
63	C₄²×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
65	C₂²⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
66	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
67	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
68	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
69	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
70	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
71	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
72	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
73	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
74	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
75	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
76	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
81	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
82	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
83	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
84	#67262	—	2	4	no	—
85	C₈×D₂₃	—	2	4	no	—
86	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	1	no	—
92	C₂×C₄×C₂₃	—	4	1	no	—
93	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
99	C₂²⋊D₂₃	—	2	2	no	—
100	C₂²⋊D₂₃	—	2	2	no	—
101	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
107	D₉₂	—	2	2	no	—
108	C₄×D₂₃	—	2	2	no	—
109	C₄×D₂₃	—	2	2	no	—
110	C₄×D₂₃	—	2	1	no	—
111	C₄×D₂₃	—	2	1	no	—
112	C₄×D₂₃	—	4	1	no	—
113	C₄×D₂₃	—	4	1	no	—
114	C₄×D₂₃	#41	1	2	yes	—
115	C₄×D₂₃	#41	1	2	yes	—
116	Dic₄₆	—	2	2	no	—
117	C₈×C₂₃	—	2	2	no	—
118	C₈×C₂₃	—	2	2	no	—
119	C₂₃⋊C₈	—	2	2	no	—
120	C₂₃⋊C₈	—	2	2	no	—
121	#2925	—	23	1	no	maximal, Sylow
122	C₂²×C₂₃	—	2	1	no	—
123	C₂²×C₂₃	—	2	1	no	—
124	C₂²×C₂₃	—	2	1	no	—
125	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×D₂₃	—	2	1	no	—
127	C₂×D₂₃	—	2	1	no	—
128	C₂×D₂₃	—	2	1	no	—
129	C₂×D₂₃	—	2	1	no	—
130	C₂×D₂₃	—	2	1	no	—
131	C₂×D₂₃	#117	1	2	yes	—
132	C₄×C₂₃	—	2	1	no	—
133	C₄×C₂₃	—	2	1	no	—
134	C₄×C₂₃	—	2	1	no	—
135	C₄×C₂₃	—	2	1	no	—
136	C₄×C₂₃	#131	1	2	yes	—
137	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
138	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
139	Dic₂₃	—	2	1	no	—
140	Dic₂₃	—	2	1	no	—
141	Dic₂₃	—	2	1	no	—
142	Dic₂₃	—	2	1	no	—
143	Dic₂₃	—	2	1	no	—
144	Dic₂₃	#117	1	2	yes	—
145	C₄×#41	—	23	1	no	—
146	#433	—	23	2	no	—
147	#433	—	23	2	no	—
148	#432	—	23	1	no	—
149	#432	D₂₃	1	1	yes	characteristic, complemented
150	#431	—	23	1	no	—
151	#431	—	23	1	no	—
152	#423	—	23	1	no	—
153	C₂×C₄⋊C₈	—	23	1	no	—
154	#408	—	23	2	no	—
155	#405	—	23	2	no	—
156	C₂×C₂₃	—	4	1	no	—
157	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
158	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
159	C₂×C₂₃	#568	1	1	yes	characteristic
160	D₂₃	—	2	2	no	—
161	D₂₃	—	4	1	no	—
162	C₂×#41	—	23	1	no	—
163	C₂×M₁₆	—	23	2	no	—
164	C₂²×C₈	—	23	2	no	—
165	C₄×Q₈	—	23	1	no	—
166	C₄×Q₈	—	23	1	no	—
167	C₄×D₄	—	23	1	no	—
168	C₄×D₄	—	23	1	no	—
169	C₄×D₄	—	23	1	no	—
170	C₄×D₄	—	23	1	no	—
171	C₄×D₄	—	23	1	no	—
172	C₄×D₄	—	23	1	no	—
173	#117	—	23	1	no	—
174	#117	—	23	1	no	—
175	#117	—	23	1	no	—
176	C₂×C₄²	—	23	1	no	—
177	C₂×C₄²	—	23	1	no	—
178	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
179	C₄⋊C₈	—	23	2	no	—
180	C₄⋊C₈	—	23	2	no	—
181	C₄⋊C₈	—	23	1	no	—
182	C₄⋊C₈	—	23	1	no	—
183	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
184	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
185	C₂²⋊C₈	—	23	2	no	—
186	C₂²⋊C₈	—	23	2	no	—
187	C₂²⋊C₈	—	23	2	no	—
188	C₂²⋊C₈	C₂×D₂₃	1	2	yes	—
189	#97	—	23	2	no	—
190	#97	C₂×D₂₃	1	2	yes	—
191	C₄×C₈	—	23	2	no	—
192	C₄×C₈	—	23	2	no	—
193	C₂₃	#2925	1	1	yes	characteristic, complemented, Sylow
194	#41	—	46	2	no	—
195	#41	—	46	2	no	—
196	C₂×Q₈	—	23	1	no	—
197	C₂×D₄	—	23	1	no	—
198	C₂×D₄	—	23	1	no	—
199	C₂×D₄	—	23	1	no	—
200	C₂²×C₄	—	23	1	no	—
201	C₂²×C₄	—	23	1	no	—
202	C₂²×C₄	—	23	1	no	—
203	C₂²×C₄	—	23	1	no	—
204	C₂²×C₄	—	23	1	no	—
205	C₂²×C₄	—	23	1	no	—
206	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
207	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
209	M₁₆	—	46	4	no	—
210	C₂×C₈	—	23	2	no	—
211	C₂×C₈	—	23	2	no	—
212	C₂×C₈	—	46	4	no	—
213	C₂×C₈	C₂²×D₂₃	1	2	yes	—
214	C₂×C₈	C₂²×D₂₃	1	2	yes	—
215	C₄⋊C₄	—	23	1	no	—
216	C₄⋊C₄	—	23	1	no	—
217	C₄⋊C₄	—	23	1	no	—
218	C₄⋊C₄	—	23	1	no	—
219	C₄⋊C₄	—	23	1	no	—
220	C₄⋊C₄	—	23	1	no	—
221	C₂²⋊C₄	—	23	1	no	—
222	C₂²⋊C₄	—	23	1	no	—
223	C₂²⋊C₄	—	23	1	no	—
224	C₂²⋊C₄	—	23	1	no	—
225	C₂²⋊C₄	—	23	1	no	—
226	C₂²⋊C₄	—	23	1	no	—
227	C₄²	—	23	1	no	—
228	C₄²	—	23	1	no	—
229	C₄²	—	23	1	no	—
230	C₄²	—	23	1	no	—
231	C₄²	—	23	1	no	—
232	C₄²	—	23	1	no	—
233	C₄²	C₄×D₂₃	1	1	yes	characteristic
234	C₄²	C₄×D₂₃	1	1	yes	characteristic
235	C₄²	C₂²×D₂₃	1	1	yes	characteristic
236	C₄²	C₂²×D₂₃	1	1	yes	characteristic
237	C₂³	—	23	1	no	—
238	C₂³	—	23	1	no	—
239	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
240	Q₈	—	46	2	no	—
241	D₄	—	46	2	no	—
242	D₄	—	46	2	no	—
243	D₄	—	46	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	23	1	no	—
249	C₂×C₄	—	23	2	no	—
250	C₂×C₄	—	23	2	no	—
251	C₂×C₄	—	23	1	no	—
252	C₂×C₄	—	23	1	no	—
253	C₂×C₄	—	23	1	no	—
254	C₂×C₄	—	23	1	no	—
255	C₂×C₄	—	23	1	no	—
256	C₂×C₄	—	46	2	no	—
257	C₂×C₄	—	46	2	no	—
258	C₂×C₄	—	46	1	no	—
259	C₂×C₄	—	46	1	no	—
260	C₂×C₄	—	92	1	no	—
261	C₂×C₄	—	92	1	no	—
262	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
267	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	2	2	no	—
270	C₈	—	46	2	no	—
271	C₈	—	46	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	23	2	no	—
276	C₂²	—	46	1	no	—
277	C₂²	—	46	1	no	—
278	C₂²	—	46	1	no	—
279	C₂²	—	46	1	no	—
280	C₂²	—	46	1	no	—
281	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	23	2	no	—
287	C₄	—	46	1	no	—
288	C₄	—	46	1	no	—
289	C₄	—	46	1	no	—
290	C₄	—	46	1	no	—
291	C₄	—	46	1	no	—
292	C₄	#10625078	1	2	yes	—
293	C₄	#41×D₂₃	1	1	yes	characteristic
294	C₄	#41×D₂₃	1	1	yes	characteristic
295	C₂	—	4	1	no	—
296	C₂	—	46	2	no	—
297	C₂	—	92	1	no	—
298	C₂	#117×D₂₃	1	1	yes	characteristic
299	C₂	C₂×#10625078	1	1	yes	characteristic
300	C₂	#13093183	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#432	C₂
prime radical	C₂×#67288	C₂²
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#432	D₂₃
hypocenter	C₂₃	#2925
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #13092479	C₂	semidirect
3. #13092475	C₂	semidirect
4. #13092473	C₂	semidirect
5. #13092470	C₂	semidirect
6. #13092469	C₂	semidirect
7. #13092439	C₂	semidirect
8. C₄⋊C₈×D₂₃	C₂	semidirect
10. #13092374	C₂	semidirect
11. C₂₃×#432	C₂	semidirect
149. #432	D₂₃	semidirect
193. C₂₃	#2925	semidirect

Maximal subgroups

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

subgroup	index
C₄×#67288	2
#13092479	2
#13092475	2
#13092473	2
#13092470	2
#13092469	2
#13092439	2
C₄⋊C₈×D₂₃	2
#13092382	2
#13092374	2
C₂₃×#432	2
#2925	23