Subgroups — #26245359

Order 12032.

874 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689161	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689149	C₂	1	1	yes	maximal, characteristic
3	#20246006	C₂	1	2	yes	maximal
4	#20244888	C₂	1	4	yes	maximal, complemented
5	#20244883	C₂	1	2	yes	maximal, complemented
6	#20244878	C₂	1	2	yes	maximal, complemented
7	#20243107	C₂	1	2	yes	maximal
8	C₂×C₄₇×#340	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625329	C₂²	1	1	yes	characteristic
10	C₂×#13093983	C₂²	1	2	yes	—
11	C₂×#13093983	C₂²	1	1	yes	characteristic
12	#15689161	—	2	4	no	—
13	C₂×#13093958	C₂²	1	1	yes	characteristic
14	C₂×#13093958	C₂²	1	1	yes	characteristic
15	C₂×#13093956	C₂²	1	2	yes	—
16	#15689149	—	2	4	no	—
17	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
18	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
19	#15689022	—	2	2	no	—
20	#15689022	—	2	2	no	—
21	#15689022	C₂²	1	4	yes	—
22	#15689022	C₂²	1	4	yes	—
23	#15688983	—	2	4	no	—
24	#15688885	C₂²	1	4	yes	—
25	#15688885	C₂²	1	4	yes	—
26	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
28	C₄₇×#340	C₂²	1	2	yes	—
29	C₄₇×#340	C₂²	1	2	yes	—
30	C₂×#10625329	—	2	2	no	—
31	C₂×#10625329	—	2	4	no	—
32	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
33	#13093983	—	2	2	no	—
34	#13093983	—	2	2	no	—
35	#13093983	—	2	4	no	—
36	#13093983	—	2	4	no	—
37	C₂×C₄⋊Dic₄₇	—	2	4	no	—
38	C₂×C₄⋊Dic₄₇	—	2	4	no	—
39	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
42	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
43	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
44	C₂×C₂²⋊Dic₄₇	D₄	1	2	yes	—
45	C₂×C₄×Dic₄₇	—	2	2	no	—
46	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
47	#13093958	—	2	2	no	—
48	#13093958	—	2	2	no	—
49	#13093956	—	2	4	no	—
50	Dic₄₇⋊C₈	—	2	4	no	—
51	Dic₄₇⋊C₈	—	2	4	no	—
52	C₄₇⋊#95	—	2	4	no	—
53	C₄₇⋊#95	—	2	4	no	—
54	C₂³×C₄×C₄₇	—	2	2	no	—
55	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	—	2	4	no	—
57	C₂²×C₈×C₄₇	D₄	1	4	yes	—
58	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
62	C₂×C₄⋊C₄×C₄₇	Q₈	1	8	yes	—
63	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
64	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	—	2	8	no	—
67	C₂²×C₄×C₄₇	—	8	2	no	—
68	C₂²×C₄×C₄₇	—	8	2	no	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₄₇	#41	1	2	yes	—
73	C₂²×C₄×C₄₇	#41	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₄₇	—	2	2	no	—
76	C₂²×Dic₄₇	—	2	2	no	—
77	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
78	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₄₇	—	2	8	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
84	C₄⋊C₄×C₄₇	—	2	8	no	—
85	C₄⋊C₄×C₄₇	—	2	4	no	—
86	C₄⋊C₄×C₄₇	C₂×Q₈	1	8	yes	—
87	C₂²⋊Dic₄₇	—	2	4	no	—
88	C₂²⋊Dic₄₇	—	4	4	no	—
89	C₄⋊Dic₄₇	—	2	2	no	—
90	C₄⋊Dic₄₇	—	2	2	no	—
91	C₄⋊Dic₄₇	—	2	2	no	—
92	C₄⋊Dic₄₇	—	2	2	no	—
93	C₄⋊Dic₄₇	—	2	2	no	—
94	C₄⋊Dic₄₇	—	2	2	no	—
95	C₄⋊Dic₄₇	—	8	4	no	—
96	C₄⋊Dic₄₇	—	8	4	no	—
97	C₄×Dic₄₇	—	2	2	no	—
98	C₄×Dic₄₇	—	2	2	no	—
99	C₄×Dic₄₇	—	4	4	no	—
100	#10625329	—	4	4	no	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	C₂³×C₄₇	—	2	1	no	—
104	C₂³×C₄₇	—	2	2	no	—
105	C₂³×C₄₇	—	2	1	no	—
106	C₂³×C₄₇	#121	1	2	yes	—
107	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
108	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	—	2	8	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	8	8	no	—
113	C₂×C₄×C₄₇	#121	1	2	yes	—
114	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
115	C₂×C₄×C₄₇	#123	1	4	yes	—
116	C₂×C₄×C₄₇	#125	1	8	yes	—
117	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
118	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
122	C₂×Dic₄₇	—	2	2	no	—
123	C₂×Dic₄₇	—	2	2	no	—
124	C₂×Dic₄₇	—	2	2	no	—
125	C₂×Dic₄₇	—	2	4	no	—
126	C₂×Dic₄₇	—	2	2	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	4	2	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₈×C₄₇	—	2	2	no	—
134	C₈×C₄₇	—	2	2	no	—
135	C₄₇⋊C₈	—	4	2	no	—
136	#23669	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	—	2	4	no	—
138	C₂²×C₄₇	—	2	4	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	#398	1	4	yes	—
142	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
144	C₂²×C₄₇	#576	1	1	yes	characteristic
145	C₂²×C₄₇	#577	1	2	yes	—
146	C₂²×C₄₇	#578	1	1	yes	characteristic
147	C₄×C₄₇	—	4	8	no	—
148	C₄×C₄₇	#398	1	4	yes	—
149	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
150	C₄×C₄₇	C₂×#125	1	2	yes	—
151	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
152	Dic₄₇	—	4	2	no	—
153	Dic₄₇	—	4	2	no	—
154	Dic₄₇	—	8	2	no	—
155	Dic₄₇	—	8	2	no	—
156	C₂×#427	—	47	1	no	—
157	C₂×#424	—	47	1	no	—
158	#2316	—	47	2	no	—
159	#2088	—	47	4	no	—
160	#2087	—	47	2	no	—
161	#2086	—	47	2	no	—
162	#1736	—	47	2	no	—
163	C₂×#340	D₄₇	1	1	yes	characteristic
164	C₂×C₄₇	—	2	4	no	—
165	C₂×C₄₇	C₂×#398	1	1	yes	characteristic
166	C₂×C₄₇	#3060	1	1	yes	characteristic
167	C₂×C₄₇	#3061	1	1	yes	characteristic
168	C₂×C₄₇	#3081	1	2	yes	—
169	C₂×C₄₇	#3082	1	2	yes	—
170	C₂²×M₁₆	—	47	1	no	—
171	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
172	C₂×#117	—	47	1	no	—
173	C₂×#117	—	47	2	no	—
174	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
175	#427	—	94	4	no	—
176	C₂×#107	—	47	1	no	—
177	C₂×#107	—	47	1	no	—
178	C₂×#106	—	47	2	no	—
179	#424	—	94	4	no	—
180	C₂×C₄⋊C₈	—	47	2	no	—
181	C₂×C₄⋊C₈	—	47	2	no	—
182	#389	—	47	4	no	—
183	#389	—	47	4	no	—
184	#389	—	94	2	no	—
185	#389	—	94	2	no	—
186	#380	—	94	4	no	—
187	#340	—	47	4	no	—
188	#340	—	47	4	no	—
189	#340	C₂×D₄₇	1	2	yes	—
190	#340	C₂×D₄₇	1	2	yes	—
191	C₄₇	#23669	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₄	—	2	2	no	—
193	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
194	C₂×M₁₆	—	94	2	no	—
195	C₂×M₁₆	—	94	4	no	—
196	C₂²×C₈	—	2	4	no	—
197	C₂²×C₈	—	47	2	no	—
198	C₂²×C₈	D₁₈₈	1	4	yes	—
199	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
201	#117	—	94	2	no	—
202	#117	—	94	2	no	—
203	#117	—	94	4	no	—
204	#117	—	94	4	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	47	1	no	—
208	C₂×C₄⋊C₄	—	47	1	no	—
209	C₂×C₄⋊C₄	—	47	2	no	—
210	C₂×C₄⋊C₄	—	47	2	no	—
211	C₂×C₄⋊C₄	—	94	4	no	—
212	C₂×C₄⋊C₄	—	94	4	no	—
213	C₂×C₄⋊C₄	Dic₉₄	1	8	yes	—
214	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
215	C₂×C₂²⋊C₄	—	47	2	no	—
216	C₂×C₂²⋊C₄	—	94	2	no	—
217	C₂×C₄²	—	47	2	no	—
218	C₂×C₄²	—	94	2	no	—
219	#107	—	94	2	no	—
220	#107	—	94	2	no	—
221	#106	—	94	4	no	—
222	C₄⋊C₈	—	94	4	no	—
223	C₄⋊C₈	—	94	4	no	—
224	#95	—	94	4	no	—
225	#95	—	94	4	no	—
226	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	8	no	—
229	C₂²×C₄	—	8	2	no	—
230	C₂²×C₄	—	8	2	no	—
231	C₂²×C₄	—	47	2	no	—
232	C₂²×C₄	—	47	2	no	—
233	C₂²×C₄	—	94	2	no	—
234	C₂²×C₄	—	94	2	no	—
235	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
236	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
237	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
238	C₂²×C₄	#10625352	1	4	yes	—
239	C₂²×C₄	#10625354	1	2	yes	—
240	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
241	M₁₆	—	188	4	no	—
242	C₂×C₈	—	2	8	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	94	2	no	—
246	C₂×C₈	—	94	4	no	—
247	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
248	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	94	2	no	—
252	C₄⋊C₄	—	94	2	no	—
253	C₄⋊C₄	—	94	2	no	—
254	C₄⋊C₄	—	94	2	no	—
255	C₄⋊C₄	—	94	2	no	—
256	C₄⋊C₄	—	94	2	no	—
257	C₄⋊C₄	—	376	4	no	—
258	C₄⋊C₄	—	376	4	no	—
259	C₄⋊C₄	C₂×Dic₉₄	1	8	yes	—
260	C₂²⋊C₄	—	94	4	no	—
261	C₂²⋊C₄	—	188	4	no	—
262	C₄²	—	94	2	no	—
263	C₄²	—	94	2	no	—
264	C₄²	—	188	4	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	1	no	—
268	C₂³	#13093999	1	2	yes	—
269	C₂³	#13094001	1	4	yes	—
270	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	8	8	no	—
275	C₂×C₄	—	94	2	no	—
276	C₂×C₄	—	94	2	no	—
277	C₂×C₄	—	94	2	no	—
278	C₂×C₄	—	94	4	no	—
279	C₂×C₄	—	94	2	no	—
280	C₂×C₄	—	188	2	no	—
281	C₂×C₄	—	188	2	no	—
282	C₂×C₄	—	188	2	no	—
283	C₂×C₄	—	188	2	no	—
284	C₂×C₄	—	188	2	no	—
285	C₂×C₄	—	188	2	no	—
286	C₂×C₄	#13093999	1	2	yes	—
287	C₂×C₄	#13094001	1	4	yes	—
288	C₂×C₄	#13094008	1	4	yes	—
289	C₂×C₄	#13094015	1	8	yes	—
290	C₂×C₄	#13094024	1	4	yes	—
291	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
292	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
293	C₂×C₄	C₂×#10625352	1	2	yes	—
294	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	188	2	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	#15689055	1	4	yes	—
303	C₂²	C₂×#13093999	1	1	yes	characteristic
304	C₂²	C₂×#13094001	1	2	yes	—
305	C₂²	#15689924	1	1	yes	characteristic
306	C₂²	#15689929	1	2	yes	—
307	C₂²	#15689936	1	1	yes	characteristic
308	C₄	—	4	8	no	—
309	C₄	—	376	2	no	—
310	C₄	—	376	2	no	—
311	C₄	—	188	2	no	—
312	C₄	—	188	2	no	—
313	C₄	#15689055	1	4	yes	—
314	C₄	C₂×#13094008	1	1	yes	characteristic
315	C₄	C₂×#13094015	1	2	yes	—
316	C₄	C₂×#13094024	1	1	yes	characteristic
317	C₂	—	2	4	no	—
318	C₂	C₂×#15689055	1	1	yes	characteristic
319	C₂	#20253051	1	1	yes	characteristic
320	C₂	#20253060	1	1	yes	characteristic
321	C₂	#20253215	1	2	yes	—
322	C₂	#20253226	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#340	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂×#340	D₄₇
hypocenter	C₄₇	#23669
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#340

Complements

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

N	H ≅ G/N	type
4. #20244888	C₂	semidirect
5. #20244883	C₂	semidirect
6. #20244878	C₂	semidirect
191. C₄₇	#23669	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689161	2
C₂×#15689149	2
#20246006	2
#20244888	2
#20244883	2
#20244878	2
#20243107	2
C₂×C₄₇×#340	2
#23669	47