Subgroups — #26115625

Order 12032.

514 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#20242136	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241883	C₂	1	2	yes	maximal, complemented
4	#20241787	C₂	1	2	yes	maximal, complemented
5	C₄₇×#3147	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₄₇	—	2	1	no	—
9	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
10	#15688877	—	2	1	no	—
11	#15688876	—	2	1	no	—
12	#15688874	—	2	1	no	—
13	#15688857	—	2	4	no	—
14	D₄⋊(C₄₇⋊C₈)	—	2	4	no	—
15	C₂×#135×C₄₇	—	2	1	no	—
16	C₂×#135×C₄₇	—	2	1	no	—
17	C₄₇×#532	C₂²	1	2	yes	—
18	C₄₇×#495	—	2	1	no	—
19	C₄₇×#495	—	2	1	no	—
20	C₄₇×#494	C₄	1	1	yes	characteristic, complemented
21	C₄₇×#493	C₄	1	1	yes	characteristic, complemented
22	C₄₇×#492	C₄	1	2	yes	complemented
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄₇⋊C₈	—	2	1	no	—
25	C₂×C₄⋊Dic₄₇	—	2	1	no	—
26	C₂×C₄×Dic₄₇	—	2	1	no	—
27	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
28	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
29	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
30	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
31	C₄×C₄₇⋊C₈	—	2	2	no	—
32	C₈×Dic₄₇	—	4	2	no	—
33	C₄₇⋊#95	D₄	1	2	yes	—
34	C₂×#41×C₄₇	—	2	2	no	—
35	#135×C₄₇	—	4	2	no	—
36	#135×C₄₇	—	4	2	no	—
37	#135×C₄₇	—	4	2	no	—
38	#135×C₄₇	—	4	2	no	—
39	C₂×Q₁₆×C₄₇	—	2	1	no	—
40	C₂×Q₁₆×C₄₇	—	2	1	no	—
41	C₂×SD₁₆×C₄₇	—	2	2	no	—
42	C₂×SD₁₆×C₄₇	—	2	2	no	—
43	C₂×D₈×C₄₇	—	2	1	no	—
44	C₂×D₈×C₄₇	—	2	1	no	—
45	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
47	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
48	#127×C₄₇	C₂×C₄	1	2	yes	—
49	#124×C₄₇	—	2	2	no	—
50	#121×C₄₇	—	2	2	no	—
51	#121×C₄₇	—	2	2	no	—
52	C₄×D₄×C₄₇	—	2	2	no	—
53	C₄×D₄×C₄₇	—	2	2	no	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
58	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
59	#41×C₄₇	—	4	2	no	—
60	#41×C₄₇	—	4	2	no	—
61	#41×C₄₇	—	4	2	no	—
62	#41×C₄₇	—	4	2	no	—
63	C₂×Q₈×C₄₇	—	2	2	no	—
64	C₂×D₄×C₄₇	—	2	2	no	—
65	C₂×D₄×C₄₇	—	2	2	no	—
66	C₂×D₄×C₄₇	—	2	2	no	—
67	C₂×D₄×C₄₇	—	4	2	no	—
68	C₂²×C₄×C₄₇	—	4	2	no	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	2	2	no	—
73	Q₁₆×C₄₇	—	4	2	no	—
74	Q₁₆×C₄₇	—	4	2	no	—
75	SD₁₆×C₄₇	—	4	4	no	—
76	SD₁₆×C₄₇	—	4	4	no	—
77	D₈×C₄₇	—	4	2	no	—
78	D₈×C₄₇	—	4	2	no	—
79	C₂×C₈×C₄₇	—	2	1	no	—
80	C₂×C₈×C₄₇	—	2	1	no	—
81	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₇	D₈	1	2	yes	—
86	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
87	C₄⋊C₄×C₄₇	—	2	2	no	—
88	C₄⋊C₄×C₄₇	—	2	2	no	—
89	C₂²⋊C₄×C₄₇	—	4	2	no	—
90	C₂²⋊C₄×C₄₇	—	4	2	no	—
91	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₇	#41	1	1	yes	characteristic
95	C₄⋊Dic₄₇	—	2	4	no	—
96	C₄×Dic₄₇	—	4	1	no	—
97	C₄×Dic₄₇	—	4	1	no	—
98	C₂×C₄₇⋊C₈	—	2	2	no	—
99	C₂×C₄₇⋊C₈	—	4	1	no	—
100	C₂×C₄₇⋊C₈	—	4	1	no	—
101	C₂×C₄₇⋊C₈	—	4	2	no	—
102	C₂³×C₄₇	—	4	2	no	—
103	C₂³×C₄₇	#121	1	1	yes	characteristic
104	Q₈×C₄₇	—	4	2	no	—
105	Q₈×C₄₇	—	4	2	no	—
106	D₄×C₄₇	—	4	2	no	—
107	D₄×C₄₇	—	4	2	no	—
108	D₄×C₄₇	—	4	4	no	—
109	D₄×C₄₇	—	8	2	no	—
110	D₄×C₄₇	—	8	2	no	—
111	D₄×C₄₇	—	8	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	4	1	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	8	2	no	—
116	C₂×C₄×C₄₇	—	8	2	no	—
117	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	4	2	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₈×C₄₇	—	2	1	no	—
131	C₈×C₄₇	—	2	1	no	—
132	C₈×C₄₇	#102	1	2	yes	—
133	C₈×C₄₇	#102	1	2	yes	—
134	C₄₇⋊C₈	—	4	2	no	—
135	C₄₇⋊C₈	—	8	2	no	—
136	#9750	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	—	2	1	no	—
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	—	8	2	no	—
140	C₂²×C₄₇	—	8	2	no	—
141	C₂²×C₄₇	—	8	2	no	—
142	C₂²×C₄₇	#390	1	1	yes	characteristic
143	C₂²×C₄₇	#459	1	1	yes	characteristic
144	C₂²×C₄₇	#460	1	1	yes	characteristic
145	C₄×C₄₇	—	2	1	no	—
146	C₄×C₄₇	—	2	1	no	—
147	C₄×C₄₇	—	2	1	no	—
148	C₄×C₄₇	—	8	2	no	—
149	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
150	C₄×C₄₇	#418	1	1	yes	characteristic
151	C₄×C₄₇	#443	1	2	yes	—
152	C₄×C₄₇	#459	1	1	yes	characteristic
153	C₄×C₄₇	#460	1	1	yes	characteristic
154	Dic₄₇	—	4	4	no	—
155	#3147	D₄₇	1	1	yes	characteristic
156	C₈×C₄⋊C₄	—	47	1	no	—
157	#1564	—	47	1	no	—
158	#1502	—	47	2	no	—
159	#1480	—	47	2	no	—
160	C₂×C₄₇	—	2	1	no	—
161	C₂×C₄₇	—	2	1	no	—
162	C₂×C₄₇	—	16	2	no	—
163	C₂×C₄₇	#1974	1	1	yes	characteristic
164	C₂×C₄₇	#1976	1	1	yes	characteristic
165	C₂×C₄₇	#1982	1	1	yes	characteristic
166	C₂×#135	—	2	1	no	—
167	C₂×#135	—	2	1	no	—
168	#532	C₂×D₄₇	1	2	yes	—
169	#495	—	2	1	no	—
170	#495	—	2	1	no	—
171	#494	Dic₄₇	1	1	yes	characteristic, complemented
172	#493	Dic₄₇	1	1	yes	characteristic, complemented
173	#492	Dic₄₇	1	2	yes	complemented
174	#431	—	47	2	no	—
175	C₂×C₄⋊C₈	—	47	1	no	—
176	C₂×C₄×C₈	—	94	1	no	—
177	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
178	C₄×C₄⋊C₄	—	47	1	no	—
179	#336	—	94	1	no	—
180	#335	—	94	1	no	—
181	#334	—	94	1	no	—
182	#327	—	94	4	no	—
183	D₄⋊C₈	—	94	4	no	—
184	C₄₇	#9750	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	2	2	no	—
186	#135	—	4	2	no	—
187	#135	—	4	2	no	—
188	#135	—	4	2	no	—
189	#135	—	4	2	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×SD₁₆	—	2	2	no	—
193	C₂×SD₁₆	—	2	2	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂²×C₈	—	94	1	no	—
197	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
198	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
199	C₄⋊Q₈	C₂×Dic₄₇	1	2	yes	—
200	#127	C₂×Dic₄₇	1	2	yes	—
201	#124	—	2	2	no	—
202	#121	—	2	2	no	—
203	#121	—	2	2	no	—
204	C₄×D₄	—	2	2	no	—
205	C₄×D₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	94	1	no	—
207	C₂×C₄²	—	94	1	no	—
208	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
209	C₄⋊C₈	—	47	2	no	—
210	C₄⋊C₈	—	94	1	no	—
211	C₄⋊C₈	—	94	2	no	—
212	C₂²⋊C₈	—	94	2	no	—
213	C₄×C₈	—	94	2	no	—
214	C₄×C₈	—	188	2	no	—
215	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
216	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
217	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
218	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
219	#95	—	47	2	no	—
220	#41	—	4	2	no	—
221	#41	—	4	2	no	—
222	#41	—	4	2	no	—
223	#41	—	4	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	94	2	no	—
231	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
232	C₂²×C₄	#10625352	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
234	Q₁₆	—	4	2	no	—
235	Q₁₆	—	4	2	no	—
236	SD₁₆	—	4	4	no	—
237	SD₁₆	—	4	4	no	—
238	D₈	—	4	2	no	—
239	D₈	—	4	2	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	94	2	no	—
243	C₂×C₈	—	188	2	no	—
244	C₂×C₈	—	188	1	no	—
245	C₂×C₈	—	188	1	no	—
246	C₂×C₈	#10625334	1	2	yes	—
247	C₂×C₈	#10625335	1	2	yes	—
248	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
251	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	94	4	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	188	1	no	—
258	C₄²	—	188	1	no	—
259	C₄²	D₄×D₄₇	1	1	yes	characteristic
260	C₄²	#10625352	1	1	yes	characteristic
261	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
262	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
263	C₂³	—	4	2	no	—
264	C₂³	#13093995	1	1	yes	characteristic
265	Q₈	—	4	2	no	—
266	Q₈	—	4	2	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	2	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	D₄	—	8	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	8	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	94	4	no	—
279	C₂×C₄	—	188	2	no	—
280	C₂×C₄	—	188	2	no	—
281	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
282	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
283	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
284	C₂×C₄	#13093995	1	1	yes	characteristic
285	C₂×C₄	#13094011	1	1	yes	characteristic
286	C₂×C₄	#13094020	1	1	yes	characteristic
287	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
288	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
289	C₂×C₄	#13094050	1	1	yes	characteristic
290	C₂×C₄	#13094050	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	376	2	no	—
294	C₈	—	188	2	no	—
295	C₈	#13093945	1	2	yes	—
296	C₈	#13093945	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	#15689026	1	1	yes	characteristic
303	C₂²	#15689315	1	1	yes	characteristic
304	C₂²	#15689320	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	8	2	no	—
309	C₄	—	188	4	no	—
310	C₄	C₂×#13093945	1	1	yes	characteristic
311	C₄	#15689128	1	1	yes	characteristic
312	C₄	#15689240	1	2	yes	—
313	C₄	#15689315	1	1	yes	characteristic
314	C₄	#15689320	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	16	2	no	—
318	C₂	#20244277	1	1	yes	characteristic
319	C₂	#20244287	1	1	yes	characteristic
320	C₂	#20244316	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093995
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting, prime radical	C₄₇×#3147	C₂
socle	C₂²×C₄₇	#390
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#3147	D₄₇
hypocenter	C₄₇	#9750
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₈×C₄⋊Dic₄₇	C₂	semidirect
2. #20242136	C₂	semidirect
3. #20241883	C₂	semidirect
4. #20241787	C₂	semidirect
20. C₄₇×#494	C₄	semidirect
21. C₄₇×#493	C₄	semidirect
22. C₄₇×#492	C₄	semidirect
171. #494	Dic₄₇	semidirect
172. #493	Dic₄₇	semidirect
173. #492	Dic₄₇	semidirect
184. C₄₇	#9750	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₄⋊Dic₄₇	2
#20242136	2
#20241883	2
#20241787	2
C₄₇×#3147	2
#9750	47