Subgroups — #26236076

Order 12032.

922 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₃₇₆	C₂	1	1	yes	maximal, characteristic
2	#20246335	C₂	1	2	yes	maximal, complemented
3	#20244264	C₂	1	2	yes	maximal
4	#20244016	C₂	1	4	yes	maximal, complemented
5	#20243545	C₂	1	1	yes	maximal, characteristic, complemented
6	#20243389	C₂	1	2	yes	maximal, complemented
7	#20243285	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₈₈	C₂²	1	2	yes	—
11	C₂×#13093982	—	2	2	no	—
12	C₄×D₃₇₆	C₄	1	8	yes	complemented
13	C₂×#13093958	C₂²	1	1	yes	characteristic
14	C₂×#13093956	C₂²	1	2	yes	—
15	C₂×#13093956	C₂²	1	2	yes	—
16	C₂×#13093946	C₂²	1	4	yes	—
17	C₂×#13093946	C₂²	1	2	yes	—
18	#15689025	C₂²	1	4	yes	—
19	#15689015	—	2	4	no	—
20	#15688993	C₂²	1	2	yes	—
21	#15688980	—	2	2	no	—
22	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
23	#15688886	C₂²	1	4	yes	—
24	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#97×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#336	C₂²	1	2	yes	—
29	C₂²×D₁₈₈	C₂³	1	2	yes	—
30	C₂²×C₄×D₄₇	—	2	2	no	—
31	C₂×D₃₇₆	—	2	2	no	—
32	C₂×D₃₇₆	—	2	2	no	—
33	C₂×D₃₇₆	C₂×C₄	1	4	yes	—
34	C₄×D₁₈₈	—	2	4	no	—
35	C₄×D₁₈₈	C₂×C₄	1	8	yes	—
36	#13093982	—	4	8	no	—
37	C₂×C₄⋊Dic₄₇	—	2	2	no	—
38	C₂×C₄⋊Dic₄₇	—	2	2	no	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
40	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
41	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
42	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
43	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
44	C₂×C₄×Dic₄₇	—	2	4	no	—
45	#13093958	C₂×C₄	1	4	yes	—
46	#13093956	—	2	4	no	—
47	#13093956	—	4	2	no	—
48	#13093946	—	4	4	no	—
49	#13093946	C₂×C₄	1	8	yes	—
50	C₄₇⋊#95	—	2	4	no	—
51	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₄₇	—	2	2	no	—
58	#97×C₄₇	—	2	4	no	—
59	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
60	#95×C₄₇	D₄	1	2	yes	—
61	C₂³×D₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₄₇	—	2	2	no	—
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂×D₁₈₈	—	2	2	no	—
69	C₂×D₁₈₈	—	2	2	no	—
70	C₂×D₁₈₈	—	2	2	no	—
71	C₂×D₁₈₈	—	2	2	no	—
72	C₂×D₁₈₈	—	2	2	no	—
73	C₂×D₁₈₈	C₂²×C₄	1	4	yes	—
74	C₂×C₄×D₄₇	—	2	8	no	—
75	C₂×C₄×D₄₇	—	4	4	no	—
76	C₂×C₄×D₄₇	—	4	4	no	—
77	C₂×C₈×C₄₇	—	2	1	no	—
78	C₂×C₈×C₄₇	—	2	1	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₇	#41	1	2	yes	—
85	C₂×C₈×C₄₇	#41	1	2	yes	—
86	C₄⋊C₄×C₄₇	—	2	4	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	—	2	2	no	—
89	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
90	C₄²×C₄₇	#41	1	2	yes	—
91	(C₂×D₄₇)⋊C₄	—	2	8	no	—
92	(C₂×D₄₇)⋊C₄	—	4	8	no	—
93	(C₂×D₄₇)⋊C₄	—	8	4	no	—
94	C₄⋊Dic₄₇	—	2	2	no	—
95	C₄⋊Dic₄₇	—	2	4	no	—
96	C₄⋊Dic₄₇	—	2	4	no	—
97	C₄⋊Dic₄₇	—	4	4	no	—
98	C₄⋊Dic₄₇	—	8	2	no	—
99	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
100	C₄×Dic₄₇	—	4	8	no	—
101	D₃₇₆	—	2	8	no	—
102	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
103	C₂²×D₄₇	—	2	4	no	—
104	C₂²×D₄₇	—	4	2	no	—
105	C₂²×D₄₇	—	4	2	no	—
106	C₂²×D₄₇	—	4	2	no	—
107	C₂²×D₄₇	—	4	2	no	—
108	C₂²×D₄₇	—	4	2	no	—
109	C₂²×D₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
117	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂×Dic₄₇	—	2	4	no	—
124	C₂×Dic₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	4	4	no	—
127	C₂×Dic₄₇	—	4	4	no	—
128	C₂×Dic₄₇	—	4	4	no	—
129	D₁₈₈	—	2	4	no	—
130	D₁₈₈	—	2	4	no	—
131	D₁₈₈	—	4	2	no	—
132	D₁₈₈	—	4	2	no	—
133	C₄×D₄₇	—	4	16	no	—
134	C₈×C₄₇	—	2	2	no	—
135	C₈×C₄₇	—	4	2	no	—
136	C₈×C₄₇	#117	1	4	yes	complemented
137	#22813	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂²×C₄₇	#547	1	1	yes	characteristic
140	C₂²×C₄₇	#550	1	1	yes	characteristic
141	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
142	C₂²×C₄₇	#577	1	2	yes	—
143	C₂×D₄₇	—	4	4	no	—
144	C₂×D₄₇	—	4	4	no	—
145	C₂×D₄₇	—	4	4	no	—
146	C₂×D₄₇	—	8	2	no	—
147	C₂×D₄₇	—	8	2	no	—
148	C₂×D₄₇	—	8	2	no	—
149	C₂×D₄₇	—	8	2	no	—
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	—	4	4	no	—
152	C₄×C₄₇	#442	1	4	yes	—
153	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
154	C₄×C₄₇	#520	1	1	yes	characteristic
155	C₄×C₄₇	#555	1	1	yes	characteristic
156	C₄×C₄₇	#556	1	1	yes	characteristic
157	Dic₄₇	—	4	4	no	—
158	Dic₄₇	—	8	4	no	—
159	C₂×C₄×D₈	—	47	1	no	—
160	#2354	—	47	2	no	—
161	#1972	—	47	2	no	—
162	#1926	—	47	4	no	—
163	#1844	—	47	1	no	—
164	#1802	—	47	2	no	—
165	#1776	—	47	2	no	—
166	#1770	D₄₇	1	1	yes	characteristic, complemented
167	C₂×C₄₇	#2354	1	1	yes	characteristic
168	C₂×C₄₇	C₂×#442	1	1	yes	characteristic
169	C₂×C₄₇	#2952	1	1	yes	characteristic
170	C₂×C₄₇	#3354	1	2	yes	—
171	C₂×C₄₇	#3397	1	2	yes	—
172	D₄₇	—	8	4	no	—
173	C₂²×D₈	—	47	1	no	—
174	C₂×C₄×D₄	—	47	2	no	—
175	C₂×#117	—	94	2	no	—
176	C₄×D₈	—	47	8	no	—
177	C₂×#107	—	47	1	no	—
178	C₂×#106	—	47	2	no	—
179	C₂×#106	—	47	2	no	—
180	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
181	C₂×#102	—	47	2	no	—
182	C₂×#102	—	47	4	no	—
183	C₂×#97	C₂×D₄₇	1	2	yes	—
184	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
185	#390	—	47	4	no	—
186	#388	—	94	4	no	—
187	#384	—	47	2	no	—
188	#380	—	94	2	no	—
189	C₄×C₄⋊C₄	—	47	4	no	—
190	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
191	#340	—	47	4	no	—
192	#336	C₂×D₄₇	1	2	yes	—
193	C₄₇	#22813	1	1	yes	characteristic, complemented, Sylow
194	C₂²×D₄	—	47	2	no	—
195	C₂³×C₄	—	94	2	no	—
196	C₂×D₈	—	47	4	no	—
197	C₂×D₈	—	94	2	no	—
198	C₂×D₈	—	94	2	no	—
199	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
200	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
201	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
202	C₄×D₄	—	47	8	no	—
203	C₄×D₄	—	94	4	no	—
204	#117	—	188	8	no	—
205	C₂×C₄⋊C₄	—	47	2	no	—
206	C₂×C₄⋊C₄	—	47	4	no	—
207	C₂×C₄⋊C₄	—	94	2	no	—
208	C₂×C₄⋊C₄	—	94	2	no	—
209	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
210	C₂×C₂²⋊C₄	—	94	2	no	—
211	C₂×C₂²⋊C₄	—	94	4	no	—
212	C₂×C₂²⋊C₄	—	94	4	no	—
213	C₂×C₄²	—	94	4	no	—
214	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
215	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
216	#107	—	47	4	no	—
217	#106	—	94	4	no	—
218	#106	—	188	2	no	—
219	C₄⋊C₈	—	2	2	no	—
220	#102	—	47	8	no	—
221	#102	—	188	4	no	—
222	#97	—	2	4	no	—
223	C₄×C₈	C₄×D₄₇	1	4	yes	—
224	#95	—	94	4	no	—
225	#95	D₁₈₈	1	2	yes	—
226	C₂⁴	—	94	2	no	—
227	C₂×D₄	—	47	4	no	—
228	C₂×D₄	—	94	2	no	—
229	C₂×D₄	—	94	2	no	—
230	C₂×D₄	—	94	2	no	—
231	C₂×D₄	—	94	2	no	—
232	C₂×D₄	—	94	2	no	—
233	C₂²×C₄	—	94	2	no	—
234	C₂²×C₄	—	94	8	no	—
235	C₂²×C₄	—	94	4	no	—
236	C₂²×C₄	—	188	4	no	—
237	C₂²×C₄	—	188	4	no	—
238	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
239	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
241	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
242	D₈	—	94	8	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
249	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
250	C₂×C₈	#10625352	1	2	yes	—
251	C₂×C₈	#10625354	1	2	yes	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	47	4	no	—
254	C₄⋊C₄	—	94	2	no	—
255	C₄⋊C₄	—	94	4	no	—
256	C₄⋊C₄	—	94	4	no	—
257	C₄⋊C₄	—	376	2	no	—
258	C₄⋊C₄	—	188	4	no	—
259	C₂²⋊C₄	—	94	8	no	—
260	C₂²⋊C₄	—	376	4	no	—
261	C₂²⋊C₄	—	188	8	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	188	8	no	—
265	C₄²	C₂×C₄×D₄₇	1	2	yes	—
266	C₄²	#10625350	1	2	yes	—
267	C₂³	—	94	4	no	—
268	C₂³	—	188	2	no	—
269	C₂³	—	188	2	no	—
270	C₂³	—	188	2	no	—
271	C₂³	—	188	2	no	—
272	C₂³	—	188	2	no	—
273	C₂³	—	188	2	no	—
274	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
275	D₄	—	94	4	no	—
276	D₄	—	94	4	no	—
277	D₄	—	188	2	no	—
278	D₄	—	188	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	94	4	no	—
284	C₂×C₄	—	188	4	no	—
285	C₂×C₄	—	188	4	no	—
286	C₂×C₄	—	188	4	no	—
287	C₂×C₄	—	188	2	no	—
288	C₂×C₄	—	188	2	no	—
289	C₂×C₄	—	188	16	no	—
290	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
291	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
292	C₂×C₄	#13094052	1	4	yes	—
293	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
294	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
295	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
296	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
297	C₂×C₄	#13094070	1	1	yes	characteristic
298	C₂×C₄	#13094071	1	1	yes	characteristic
299	C₈	—	2	2	no	—
300	C₈	—	4	2	no	—
301	C₈	#13093982	1	4	yes	complemented
302	C₂²	—	376	2	no	—
303	C₂²	—	376	2	no	—
304	C₂²	—	376	2	no	—
305	C₂²	—	376	2	no	—
306	C₂²	—	188	4	no	—
307	C₂²	—	188	4	no	—
308	C₂²	—	188	4	no	—
309	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
310	C₂²	#15689777	1	1	yes	characteristic
311	C₂²	#15689795	1	1	yes	characteristic
312	C₂²	C₂×#13094052	1	2	yes	—
313	C₂²	#15689935	1	2	yes	—
314	C₄	—	2	2	no	—
315	C₄	—	4	4	no	—
316	C₄	—	376	4	no	—
317	C₄	—	188	4	no	—
318	C₄	#15689236	1	4	yes	—
319	C₄	C₂×#13093982	1	1	yes	characteristic
320	C₄	#15689603	1	1	yes	characteristic
321	C₄	#15689838	1	1	yes	characteristic
322	C₄	#15689846	1	1	yes	characteristic
323	C₂	—	376	4	no	—
324	C₂	#20246335	1	1	yes	characteristic
325	C₂	C₂×#15689236	1	1	yes	characteristic
326	C₂	#20252273	1	1	yes	characteristic
327	C₂	#20255984	1	2	yes	—
328	C₂	#20256410	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1770	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₃₇₆	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1770	D₄₇
hypocenter	C₄₇	#22813
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ #1770

Complements

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

N	H ≅ G/N	type
2. #20246335	C₂	semidirect
4. #20244016	C₂	semidirect
5. #20243545	C₂	semidirect
6. #20243389	C₂	semidirect
7. #20243285	C₂	semidirect
8. C₄₇×#1770	C₂	semidirect
12. C₄×D₃₇₆	C₄	semidirect
136. C₈×C₄₇	#117	semidirect
166. #1770	D₄₇	semidirect
193. C₄₇	#22813	semidirect
301. C₈	#13093982	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₃₇₆	2
#20246335	2
#20244264	2
#20244016	2
#20243545	2
#20243389	2
#20243285	2
C₄₇×#1770	2
#22813	47