Subgroups — #26236104

Order 12032.

850 conjugacy classes of subgroups.

The subgroup lattice has 312 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	#20246363	C₂	1	2	yes	maximal, complemented
3	#20244262	C₂	1	2	yes	maximal
4	#20244007	C₂	1	4	yes	maximal, complemented
5	#20243395	C₂	1	2	yes	maximal, complemented
6	#20243288	C₂	1	2	yes	maximal, complemented
7	#20243255	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₈₈	C₂²	1	2	yes	—
11	C₄×D₃₇₆	C₄	1	8	yes	complemented
12	C₂×#13093958	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×#13093946	C₂²	1	2	yes	—
15	C₂×#13093944	C₂²	1	4	yes	—
16	C₂×#13093934	C₂²	1	2	yes	—
17	#15689024	C₂²	1	4	yes	—
18	#15689015	—	2	4	no	—
19	#15688985	—	2	2	no	—
20	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
21	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
22	#15688970	—	2	2	no	—
23	#15688885	C₂²	1	4	yes	—
24	#15688877	C₂²	1	2	yes	—
25	C₂×#107×C₄₇	C₂²	1	2	yes	—
26	C₂×#106×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#382	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	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	C₄×D₁₈₈	—	2	4	no	—
36	C₄×D₁₈₈	C₂×C₄	1	8	yes	—
37	C₂×C₄⋊Dic₄₇	D₄	1	2	yes	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
39	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
40	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
41	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
42	C₂×C₄×Dic₄₇	—	2	4	no	—
43	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
44	#13093958	C₂×C₄	1	4	yes	—
45	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
46	#13093946	C₂×C₄	1	8	yes	—
47	#13093944	—	4	4	no	—
48	#13093934	—	2	4	no	—
49	C₄₇⋊#95	—	2	4	no	—
50	C₄₇⋊#95	D₄	1	2	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	#107×C₄₇	—	4	2	no	—
56	#106×C₄₇	—	2	4	no	—
57	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
58	#95×C₄₇	—	2	2	no	—
59	#95×C₄₇	—	2	2	no	—
60	C₂³×D₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₄₇	—	2	2	no	—
66	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
67	C₂×D₁₈₈	—	2	2	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	C₂²×C₄	1	4	yes	—
73	C₂×C₄×D₄₇	—	2	8	no	—
74	C₂×C₄×D₄₇	—	8	4	no	—
75	C₂×C₈×C₄₇	—	2	1	no	—
76	C₂×C₈×C₄₇	—	2	1	no	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₇	#41	1	2	yes	—
80	C₂×C₈×C₄₇	#41	1	2	yes	—
81	C₄⋊C₄×C₄₇	—	2	4	no	—
82	C₄⋊C₄×C₄₇	—	2	4	no	—
83	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
84	C₄²×C₄₇	#41	1	2	yes	—
85	(C₂×D₄₇)⋊C₄	—	2	8	no	—
86	(C₂×D₄₇)⋊C₄	—	4	8	no	—
87	(C₂×D₄₇)⋊C₄	—	8	4	no	—
88	C₄⋊Dic₄₇	—	2	2	no	—
89	C₄⋊Dic₄₇	—	4	2	no	—
90	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
91	C₄×Dic₄₇	—	2	2	no	—
92	C₄×Dic₄₇	—	2	2	no	—
93	C₄×Dic₄₇	—	8	4	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	D₃₇₆	—	2	8	no	—
98	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
99	C₂²×D₄₇	—	2	4	no	—
100	C₂²×D₄₇	—	4	2	no	—
101	C₂²×D₄₇	—	4	2	no	—
102	C₂²×D₄₇	—	4	4	no	—
103	C₂²×D₄₇	—	4	4	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	4	4	no	—
107	C₂×C₄×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	—	4	4	no	—
109	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
117	C₂×Dic₄₇	—	2	4	no	—
118	C₂×Dic₄₇	—	2	4	no	—
119	C₂×Dic₄₇	—	2	4	no	—
120	C₂×Dic₄₇	—	2	4	no	—
121	C₂×Dic₄₇	—	4	4	no	—
122	D₁₈₈	—	2	4	no	—
123	D₁₈₈	—	2	4	no	—
124	D₁₈₈	—	4	2	no	—
125	D₁₈₈	—	4	2	no	—
126	C₄×D₄₇	—	8	8	no	—
127	C₈×C₄₇	—	2	2	no	—
128	C₈×C₄₇	#117	1	4	yes	—
129	C₄₇⋊C₈	—	4	2	no	—
130	#22816	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂²×C₄₇	#547	1	1	yes	characteristic
133	C₂²×C₄₇	#550	1	1	yes	characteristic
134	C₂²×C₄₇	#575	1	2	yes	—
135	C₂²×C₄₇	#576	1	2	yes	—
136	C₂×D₄₇	—	4	4	no	—
137	C₂×D₄₇	—	4	4	no	—
138	C₂×D₄₇	—	4	4	no	—
139	C₂×D₄₇	—	8	4	no	—
140	C₂×D₄₇	—	8	4	no	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	8	4	no	—
144	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
145	C₄×C₄₇	#520	1	1	yes	characteristic
146	C₄×C₄₇	#555	1	1	yes	characteristic
147	C₄×C₄₇	#556	1	1	yes	characteristic
148	Dic₄₇	—	4	4	no	—
149	Dic₄₇	—	4	4	no	—
150	C₂×C₄×D₈	—	47	1	no	—
151	#2357	—	47	2	no	—
152	#1972	—	47	2	no	—
153	#1925	—	47	4	no	—
154	#1846	D₄₇	1	1	yes	characteristic, complemented
155	#1804	—	47	2	no	—
156	#1776	—	47	2	no	—
157	#1770	—	47	1	no	—
158	C₂×C₄₇	#2354	1	1	yes	characteristic
159	C₂×C₄₇	#2943	1	1	yes	characteristic
160	C₂×C₄₇	#2948	1	1	yes	characteristic
161	C₂×C₄₇	#3355	1	2	yes	—
162	C₂×C₄₇	#3403	1	2	yes	—
163	D₄₇	—	8	4	no	—
164	C₂²×D₈	—	47	1	no	—
165	C₂×C₄×D₄	—	47	2	no	—
166	C₄×D₈	—	47	8	no	—
167	C₂×#107	—	47	1	no	—
168	C₂×#107	C₂×D₄₇	1	2	yes	—
169	C₂×#106	C₂×D₄₇	1	2	yes	—
170	C₂×C₄⋊C₈	—	47	1	no	—
171	C₂×#102	—	47	2	no	—
172	C₂×#102	—	47	4	no	—
173	C₂×#97	—	47	2	no	—
174	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
175	#390	—	47	4	no	—
176	#388	—	94	4	no	—
177	#382	C₂×D₄₇	1	2	yes	—
178	#381	—	94	2	no	—
179	C₄×C₄⋊C₄	—	47	1	no	—
180	C₄×C₄⋊C₄	—	47	4	no	—
181	C₄×C₂²⋊C₄	—	94	2	no	—
182	#340	—	47	4	no	—
183	#336	—	47	2	no	—
184	C₄₇	#22816	1	1	yes	characteristic, complemented, Sylow
185	C₂²×D₄	—	47	2	no	—
186	C₂³×C₄	—	94	2	no	—
187	C₂×D₈	—	47	4	no	—
188	C₂×D₈	—	94	2	no	—
189	C₂×D₈	—	94	2	no	—
190	C₂²×C₈	—	47	2	no	—
191	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
193	C₄×D₄	—	47	8	no	—
194	C₄×D₄	—	94	4	no	—
195	C₂×C₄⋊C₄	—	47	2	no	—
196	C₂×C₄⋊C₄	—	47	2	no	—
197	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
198	C₂×C₂²⋊C₄	—	94	2	no	—
199	C₂×C₂²⋊C₄	—	94	4	no	—
200	C₂×C₂²⋊C₄	—	94	4	no	—
201	C₂×C₄²	—	47	2	no	—
202	C₂×C₄²	—	94	4	no	—
203	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
204	#107	—	4	2	no	—
205	#107	—	47	4	no	—
206	#106	—	2	4	no	—
207	C₄⋊C₈	—	94	2	no	—
208	#102	—	47	8	no	—
209	#102	—	188	4	no	—
210	#97	—	94	4	no	—
211	C₄×C₈	C₄×D₄₇	1	4	yes	—
212	#95	—	2	2	no	—
213	#95	—	2	2	no	—
214	#95	—	47	2	no	—
215	#95	—	94	4	no	—
216	C₂⁴	—	94	2	no	—
217	C₂×D₄	—	47	4	no	—
218	C₂×D₄	—	94	2	no	—
219	C₂×D₄	—	94	2	no	—
220	C₂×D₄	—	94	2	no	—
221	C₂×D₄	—	94	2	no	—
222	C₂×D₄	—	94	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	47	4	no	—
225	C₂²×C₄	—	94	2	no	—
226	C₂²×C₄	—	94	8	no	—
227	C₂²×C₄	—	376	4	no	—
228	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
229	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
231	D₈	—	94	8	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	94	2	no	—
235	C₂×C₈	—	94	2	no	—
236	C₂×C₈	—	94	2	no	—
237	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
238	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
239	C₂×C₈	#10625352	1	2	yes	—
240	C₂×C₈	#10625354	1	2	yes	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	47	4	no	—
244	C₄⋊C₄	—	94	2	no	—
245	C₄⋊C₄	—	188	2	no	—
246	C₂²⋊C₄	—	94	8	no	—
247	C₂²⋊C₄	—	376	4	no	—
248	C₂²⋊C₄	—	188	8	no	—
249	C₄²	—	94	2	no	—
250	C₄²	—	94	2	no	—
251	C₄²	—	376	4	no	—
252	C₄²	C₂×C₄×D₄₇	1	2	yes	—
253	C₄²	#10625354	1	2	yes	—
254	C₂³	—	94	4	no	—
255	C₂³	—	188	2	no	—
256	C₂³	—	188	2	no	—
257	C₂³	—	188	4	no	—
258	C₂³	—	188	4	no	—
259	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
260	D₄	—	94	4	no	—
261	D₄	—	94	4	no	—
262	D₄	—	188	2	no	—
263	D₄	—	188	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	94	4	no	—
270	C₂×C₄	—	94	4	no	—
271	C₂×C₄	—	94	4	no	—
272	C₂×C₄	—	94	4	no	—
273	C₂×C₄	—	376	8	no	—
274	C₂×C₄	—	188	4	no	—
275	C₂×C₄	#13093986	1	2	yes	—
276	C₂×C₄	#13093986	1	2	yes	—
277	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
278	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
279	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
280	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
281	C₂×C₄	#13094069	1	1	yes	characteristic
282	C₂×C₄	#13094072	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	188	2	no	—
285	C₈	#13093982	1	4	yes	—
286	C₂²	—	376	4	no	—
287	C₂²	—	376	4	no	—
288	C₂²	—	188	4	no	—
289	C₂²	—	188	4	no	—
290	C₂²	—	188	4	no	—
291	C₂²	C₂×#13093986	1	1	yes	characteristic
292	C₂²	#15689776	1	1	yes	characteristic
293	C₂²	#15689793	1	1	yes	characteristic
294	C₂²	#15689920	1	2	yes	—
295	C₂²	#15689925	1	2	yes	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	8	4	no	—
299	C₄	—	188	4	no	—
300	C₄	—	188	4	no	—
301	C₄	C₂×#13093982	1	1	yes	characteristic
302	C₄	#15689604	1	1	yes	characteristic
303	C₄	#15689839	1	1	yes	characteristic
304	C₄	#15689847	1	1	yes	characteristic
305	C₂	—	376	4	no	—
306	C₂	#20246339	1	1	yes	characteristic
307	C₂	#20252188	1	1	yes	characteristic
308	C₂	#20252239	1	1	yes	characteristic
309	C₂	#20255992	1	2	yes	—
310	C₂	#20256479	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1846	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₃₇₆	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1846	D₄₇
hypocenter	C₄₇	#22816
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
2. #20246363	C₂	semidirect
4. #20244007	C₂	semidirect
5. #20243395	C₂	semidirect
6. #20243288	C₂	semidirect
7. #20243255	C₂	semidirect
8. C₄₇×#1846	C₂	semidirect
11. C₄×D₃₇₆	C₄	semidirect
154. #1846	D₄₇	semidirect
184. C₄₇	#22816	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
#20246363	2
#20244262	2
#20244007	2
#20243395	2
#20243288	2
#20243255	2
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