Subgroups — #26235264

Order 12032.

714 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689253	C₂	1	1	yes	maximal, characteristic
2	#20246346	C₂	1	2	yes	maximal
3	#20244156	C₂	1	2	yes	maximal
4	#20244153	C₂	1	2	yes	maximal
5	#20244143	C₂	1	2	yes	maximal
6	#20243520	C₂	1	2	yes	maximal
7	#20243519	C₂	1	1	yes	maximal, characteristic
8	#20243284	C₂	1	2	yes	maximal
9	C₄×#107×C₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
11	#15689253	C₄	1	8	yes	complemented
12	C₂×#13093957	C₂²	1	2	yes	—
13	C₂×#13093955	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
16	C₂×#13093934	C₂²	1	2	yes	—
17	#15689027	—	2	2	no	—
18	#15689007	—	2	2	no	—
19	#15689006	—	2	2	no	—
20	#15688992	C₂²	1	2	yes	—
21	#15688989	C₂²	1	2	yes	—
22	#15688988	C₂²	1	2	yes	—
23	C₄⋊C₄×Dic₄₇	C₂²	1	2	yes	—
24	C₄⋊C₄×Dic₄₇	C₂²	1	2	yes	—
25	C₄×Dic₄₇⋊C₄	—	2	2	no	—
26	#15688886	C₂²	1	2	yes	—
27	#15688886	C₂²	1	1	yes	characteristic
28	#15688886	C₂²	1	1	yes	characteristic
29	#15688885	C₂²	1	2	yes	—
30	#15688885	C₂²	1	2	yes	—
31	C₂×#107×C₄₇	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
34	C₄₇×#340	C₂²	1	1	yes	characteristic
35	C₄₇×#340	C₂²	1	1	yes	characteristic
36	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
37	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
38	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
39	C₄×Dic₉₄	Q₈	1	4	yes	—
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×Dic₄₇⋊C₄	—	2	2	no	—
44	C₂×Dic₄₇⋊C₄	—	2	2	no	—
45	C₂×Dic₄₇⋊C₄	D₄	1	2	yes	—
46	C₂×C₄×Dic₄₇	—	2	2	no	—
47	C₂×C₄×Dic₄₇	—	2	2	no	—
48	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
49	#13093957	—	2	4	no	—
50	#13093955	—	2	4	no	—
51	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
52	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	—
53	#13093934	C₂×C₄	1	8	yes	—
54	C₄₇⋊#95	—	2	2	no	—
55	C₄₇⋊#95	—	2	2	no	—
56	C₄₇⋊#95	—	2	2	no	—
57	C₄₇⋊#95	—	2	2	no	—
58	C₄₇⋊#95	—	2	2	no	—
59	C₄₇⋊#95	—	2	2	no	—
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₄₇	—	2	2	no	—
65	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
66	#107×C₄₇	—	2	4	no	—
67	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
68	#95×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	2	no	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	2	2	no	—
75	C₂²×Dic₄₇	—	2	2	no	—
76	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
77	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
78	C₂×Dic₉₄	—	4	2	no	—
79	C₂×Dic₉₄	C₄⋊C₄	1	4	yes	—
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	C₄⋊C₄×C₄₇	—	2	2	no	—
85	C₄⋊C₄×C₄₇	—	2	2	no	—
86	C₄⋊C₄×C₄₇	—	2	2	no	—
87	C₄⋊C₄×C₄₇	#41	1	4	yes	—
88	C₄²×C₄₇	—	8	2	no	—
89	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
90	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
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	C₄⋊C₄	1	4	yes	—
96	Dic₄₇⋊C₄	—	4	4	no	—
97	Dic₄₇⋊C₄	—	8	2	no	—
98	Dic₄₇⋊C₄	C₄⋊C₄	1	8	yes	—
99	C₄×Dic₄₇	—	4	4	no	—
100	C₄×Dic₄₇	—	8	2	no	—
101	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
102	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
103	C₂×C₄₇⋊C₈	—	2	2	no	—
104	C₂×C₄₇⋊C₈	—	2	2	no	—
105	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
106	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
121	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
122	C₂×Dic₄₇	—	2	2	no	—
123	C₂×Dic₄₇	—	2	2	no	—
124	C₂×Dic₄₇	—	2	2	no	—
125	C₂×Dic₄₇	—	2	2	no	—
126	C₂×Dic₄₇	—	4	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	C₂×C₄⋊C₄	1	4	yes	—
133	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
134	Dic₉₄	—	4	4	no	—
135	C₈×C₄₇	—	2	2	no	—
136	C₈×C₄₇	—	2	2	no	—
137	C₄₇⋊C₈	—	2	4	no	—
138	#22744	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₄₇	#548	1	2	yes	—
141	C₂²×C₄₇	#575	1	2	yes	—
142	C₂²×C₄₇	#576	1	1	yes	characteristic
143	C₂²×C₄₇	#578	1	1	yes	characteristic
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	4	4	no	—
147	C₄×C₄₇	—	8	2	no	—
148	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
149	C₄×C₄₇	#519	1	1	yes	characteristic
150	C₄×C₄₇	#557	1	2	yes	—
151	Dic₄₇	—	2	4	no	—
152	Dic₄₇	—	2	4	no	—
153	Dic₄₇	—	8	2	no	—
154	Dic₄₇	—	8	2	no	—
155	C₂×#446	—	47	1	no	—
156	#2355	—	47	2	no	—
157	#1949	—	47	2	no	—
158	#1949	—	47	2	no	—
159	#1947	—	47	2	no	—
160	#1836	—	47	2	no	—
161	#1835	—	47	1	no	—
162	#1776	—	47	2	no	—
163	C₄×#107	D₄₇	1	1	yes	characteristic
164	C₂×C₄₇	#2351	1	1	yes	characteristic
165	C₂×C₄₇	#2911	1	1	yes	characteristic
166	C₂×C₄₇	#2913	1	1	yes	characteristic
167	C₂×C₄₇	#3370	1	2	yes	—
168	C₂×C₄₇	#3377	1	2	yes	—
169	C₂×C₄×Q₈	—	47	1	no	—
170	#446	—	47	8	no	—
171	C₂×#107	—	47	2	no	—
172	C₂×#107	C₂×D₄₇	1	2	yes	—
173	C₂×#106	—	47	2	no	—
174	C₂×C₄⋊C₈	—	47	2	no	—
175	C₂×C₄⋊C₈	—	47	1	no	—
176	C₂×#97	—	47	2	no	—
177	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
178	#391	—	94	2	no	—
179	#387	—	94	2	no	—
180	#387	—	94	2	no	—
181	#384	—	47	2	no	—
182	#382	—	47	2	no	—
183	#382	—	47	2	no	—
184	C₄×C₄⋊C₄	—	47	2	no	—
185	C₄×C₄⋊C₄	—	47	2	no	—
186	C₄×C₄⋊C₄	—	94	2	no	—
187	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
188	#340	—	47	2	no	—
189	#340	—	47	1	no	—
190	#340	—	47	1	no	—
191	#340	—	47	2	no	—
192	#340	—	47	2	no	—
193	#340	C₂×D₄₇	1	1	yes	characteristic
194	#340	C₂×D₄₇	1	1	yes	characteristic
195	C₄₇	#22744	1	1	yes	characteristic, complemented, Sylow
196	C₂²×Q₈	—	47	1	no	—
197	C₂²×C₈	—	47	2	no	—
198	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
200	C₄×Q₈	—	47	4	no	—
201	C₄×Q₈	—	47	4	no	—
202	C₂×C₄⋊C₄	—	47	2	no	—
203	C₂×C₄⋊C₄	—	47	1	no	—
204	C₂×C₄⋊C₄	—	47	2	no	—
205	C₂×C₄⋊C₄	—	47	2	no	—
206	C₂×C₄⋊C₄	—	94	2	no	—
207	C₂×C₄⋊C₄	—	94	2	no	—
208	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
209	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	—	47	2	no	—
212	C₂×C₄²	—	94	2	no	—
213	C₂×C₄²	—	94	2	no	—
214	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
215	#107	—	2	4	no	—
216	#107	—	94	4	no	—
217	#106	—	94	4	no	—
218	C₄⋊C₈	—	47	8	no	—
219	C₄⋊C₈	—	47	4	no	—
220	#97	—	47	8	no	—
221	C₄×C₈	C₄×D₄₇	1	4	yes	—
222	#95	—	2	2	no	—
223	#95	—	94	2	no	—
224	#95	—	94	2	no	—
225	#95	—	94	2	no	—
226	#95	—	94	2	no	—
227	#95	—	94	2	no	—
228	#95	—	94	2	no	—
229	C₂×Q₈	—	47	4	no	—
230	C₂×Q₈	—	188	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	47	2	no	—
234	C₂²×C₄	—	47	2	no	—
235	C₂²×C₄	—	94	2	no	—
236	C₂²×C₄	—	94	2	no	—
237	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
238	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	47	4	no	—
243	C₂×C₈	—	94	2	no	—
244	C₂×C₈	—	94	2	no	—
245	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
246	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	47	8	no	—
251	C₄⋊C₄	—	47	4	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	—	376	2	no	—
257	C₄⋊C₄	—	188	4	no	—
258	C₄⋊C₄	#10625350	1	4	yes	—
259	C₄²	—	8	2	no	—
260	C₄²	—	47	4	no	—
261	C₄²	—	47	4	no	—
262	C₄²	—	376	2	no	—
263	C₄²	—	188	4	no	—
264	C₄²	C₂×C₄×D₄₇	1	2	yes	—
265	C₄²	Q₈×D₄₇	1	2	yes	—
266	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
267	Q₈	—	188	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	47	4	no	—
278	C₂×C₄	—	47	4	no	—
279	C₂×C₄	—	94	2	no	—
280	C₂×C₄	—	94	2	no	—
281	C₂×C₄	—	94	2	no	—
282	C₂×C₄	—	94	2	no	—
283	C₂×C₄	—	188	2	no	—
284	C₂×C₄	—	188	2	no	—
285	C₂×C₄	—	188	2	no	—
286	C₂×C₄	—	188	2	no	—
287	C₂×C₄	—	188	2	no	—
288	C₂×C₄	—	188	2	no	—
289	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
290	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
291	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
293	C₂×C₄	C₂×#10625350	1	2	yes	—
294	C₂×C₄	#13094071	1	2	yes	—
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	94	4	no	—
298	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
299	C₂²	#15689782	1	2	yes	—
300	C₂²	#15689920	1	2	yes	—
301	C₂²	#15689924	1	1	yes	characteristic
302	C₂²	#15689936	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	4	4	no	—
306	C₄	—	8	2	no	—
307	C₄	—	94	4	no	—
308	C₄	—	94	4	no	—
309	C₄	—	376	2	no	—
310	C₄	—	376	2	no	—
311	C₄	C₄×#10625350	1	1	yes	characteristic
312	C₄	#15689600	1	1	yes	characteristic
313	C₄	#15689851	1	2	yes	—
314	C₂	#20246308	1	1	yes	characteristic
315	C₂	#20251976	1	1	yes	characteristic
316	C₂	#20251984	1	1	yes	characteristic
317	C₂	#20256137	1	2	yes	—
318	C₂	#20256226	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#107×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#107	D₄₇
hypocenter	C₄₇	#22744
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
11. #15689253	C₄	semidirect
195. C₄₇	#22744	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689253	2
#20246346	2
#20244156	2
#20244153	2
#20244143	2
#20243520	2
#20243519	2
#20243284	2
C₄×#107×C₄₇	2
#22744	47