Subgroups — #26115341

Order 12032.

622 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243540	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15688869	C₂	1	2	yes	maximal, complemented
4	C₂×#15688867	C₂	1	2	yes	maximal
5	C₄₇×#2858	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
10	#15688878	—	2	1	no	—
11	#15688869	—	2	4	no	—
12	#15688867	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#124×C₄₇	—	2	2	no	—
16	C₄₇×#390	—	2	1	no	—
17	C₄₇×#390	—	2	1	no	—
18	C₄₇×#390	C₄	1	4	yes	—
19	C₄³×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
26	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
27	C₄×C₄₇⋊C₈	—	4	2	no	—
28	C₂²×Q₈×C₄₇	—	2	1	no	—
29	C₂²×D₄×C₄₇	—	2	1	no	—
30	C₂²×D₄×C₄₇	—	2	1	no	—
31	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₄⋊Q₈×C₄₇	—	2	2	no	—
33	C₄⋊Q₈×C₄₇	—	2	2	no	—
34	#127×C₄₇	—	2	2	no	—
35	#127×C₄₇	—	2	2	no	—
36	#124×C₄₇	—	4	4	no	—
37	#124×C₄₇	—	4	4	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
41	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₂⁴×C₄₇	—	2	2	no	—
48	C₂×Q₈×C₄₇	—	4	1	no	—
49	C₂×Q₈×C₄₇	—	4	1	no	—
50	C₂×Q₈×C₄₇	—	4	2	no	—
51	C₂×Q₈×C₄₇	—	4	2	no	—
52	C₂×D₄×C₄₇	—	2	4	no	—
53	C₂×D₄×C₄₇	—	4	4	no	—
54	C₂×D₄×C₄₇	—	4	2	no	—
55	C₂×D₄×C₄₇	—	4	1	no	—
56	C₂×D₄×C₄₇	—	4	1	no	—
57	C₂×D₄×C₄₇	—	4	2	no	—
58	C₂×D₄×C₄₇	—	4	2	no	—
59	C₂×D₄×C₄₇	—	4	2	no	—
60	C₂×D₄×C₄₇	—	4	1	no	—
61	C₂×D₄×C₄₇	—	4	1	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₄₇	—	2	4	no	—
69	C₄⋊C₄×C₄₇	—	4	2	no	—
70	C₂²⋊C₄×C₄₇	—	4	8	no	—
71	C₂²⋊C₄×C₄₇	—	4	8	no	—
72	C₄²×C₄₇	—	2	2	no	—
73	C₄²×C₄₇	—	2	1	no	—
74	C₄²×C₄₇	—	2	1	no	—
75	C₄²×C₄₇	—	4	2	no	—
76	C₄²×C₄₇	D₈	1	4	yes	—
77	C₄²×C₄₇	SD₁₆	1	4	yes	—
78	C₄²×C₄₇	C₂×D₄	1	2	yes	—
79	C₄²×C₄₇	#41	1	2	yes	—
80	C₂×C₄₇⋊C₈	—	2	4	no	—
81	C₂×C₄₇⋊C₈	—	4	2	no	—
82	C₂×C₄₇⋊C₈	—	4	2	no	—
83	C₂×C₄₇⋊C₈	—	4	2	no	—
84	C₂×C₄₇⋊C₈	—	4	4	no	—
85	C₂³×C₄₇	—	4	2	no	—
86	C₂³×C₄₇	—	4	4	no	—
87	C₂³×C₄₇	—	4	4	no	—
88	C₂³×C₄₇	—	4	2	no	—
89	C₂³×C₄₇	—	4	2	no	—
90	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	Q₈×C₄₇	—	8	2	no	—
92	Q₈×C₄₇	—	8	1	no	—
93	Q₈×C₄₇	—	8	1	no	—
94	D₄×C₄₇	—	8	4	no	—
95	D₄×C₄₇	—	8	2	no	—
96	D₄×C₄₇	—	8	2	no	—
97	D₄×C₄₇	—	8	2	no	—
98	D₄×C₄₇	—	8	1	no	—
99	D₄×C₄₇	—	8	1	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	4	2	no	—
106	C₂×C₄×C₄₇	—	4	4	no	—
107	C₂×C₄×C₄₇	#102	1	4	yes	—
108	C₂×C₄×C₄₇	#102	1	4	yes	—
109	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₄₇	#121	1	2	yes	—
111	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
115	C₂×C₄×C₄₇	#135	1	2	yes	—
116	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
118	C₄₇⋊C₈	—	4	4	no	—
119	C₄₇⋊C₈	—	8	2	no	—
120	#9698	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	—	8	4	no	—
122	C₂²×C₄₇	—	8	4	no	—
123	C₂²×C₄₇	—	8	2	no	—
124	C₂²×C₄₇	—	8	2	no	—
125	C₂²×C₄₇	—	8	2	no	—
126	C₂²×C₄₇	#390	1	1	yes	characteristic
127	C₂²×C₄₇	#413	1	2	yes	—
128	C₂²×C₄₇	C₂×#102	1	2	yes	—
129	C₂²×C₄₇	#416	1	1	yes	characteristic
130	C₂²×C₄₇	#417	1	1	yes	characteristic
131	C₄×C₄₇	—	2	4	no	—
132	C₄×C₄₇	—	2	4	no	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	8	2	no	—
135	C₄×C₄₇	#463	1	2	yes	—
136	C₄×C₄₇	#464	1	2	yes	—
137	C₄×C₄₇	#486	1	2	yes	—
138	C₄×C₄₇	#492	1	1	yes	characteristic
139	C₄×C₄₇	#493	1	1	yes	characteristic
140	#2858	D₄₇	1	1	yes	characteristic
141	#1842	—	47	1	no	—
142	C₄×C₄⋊C₈	—	47	1	no	—
143	C₂×#332	—	47	2	no	—
144	C₂×#331	—	47	2	no	—
145	C₂×C₄₇	—	16	2	no	—
146	C₂×C₄₇	#1540	1	2	yes	—
147	C₂×C₄₇	#1547	1	2	yes	—
148	C₂×C₄₇	#1957	1	1	yes	characteristic
149	C₂×C₄₇	#1967	1	1	yes	characteristic
150	C₂×C₄₇	#1983	1	1	yes	characteristic
151	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
152	C₂×#127	C₂×D₄₇	1	1	yes	characteristic
153	C₂×#124	—	2	2	no	—
154	C₂×C₄⋊C₈	—	47	2	no	—
155	C₂×C₄⋊C₈	—	47	2	no	—
156	C₂×C₄⋊C₈	—	94	2	no	—
157	C₂×C₄×C₈	—	94	1	no	—
158	#390	—	2	1	no	—
159	#390	—	2	1	no	—
160	#390	Dic₄₇	1	4	yes	—
161	C₄³	C₂×D₄₇	1	1	yes	characteristic
162	#336	—	94	1	no	—
163	#332	—	94	4	no	—
164	#331	—	94	4	no	—
165	C₄₇	#9698	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₈	—	2	1	no	—
167	C₂²×D₄	—	2	1	no	—
168	C₂²×D₄	—	2	1	no	—
169	C₂²×D₄	C₂×Dic₄₇	1	2	yes	—
170	C₂²×C₈	—	94	2	no	—
171	C₂²×C₈	—	94	2	no	—
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	—	2	2	no	—
174	#127	—	2	2	no	—
175	#127	—	2	2	no	—
176	#124	—	4	4	no	—
177	#124	—	4	4	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
180	C₂×C₂²⋊C₄	—	2	4	no	—
181	C₂×C₂²⋊C₄	—	2	4	no	—
182	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
183	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
184	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
186	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
187	C₄⋊C₈	—	47	4	no	—
188	C₄⋊C₈	—	94	2	no	—
189	C₄⋊C₈	—	94	2	no	—
190	C₄⋊C₈	—	94	2	no	—
191	C₄⋊C₈	—	188	4	no	—
192	C₄×C₈	—	188	2	no	—
193	C₂⁴	—	2	2	no	—
194	C₂×Q₈	—	4	1	no	—
195	C₂×Q₈	—	4	1	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	—	4	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	4	4	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	1	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
210	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
211	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
214	C₂×C₈	—	94	4	no	—
215	C₂×C₈	—	188	2	no	—
216	C₂×C₈	—	188	4	no	—
217	C₂×C₈	—	188	2	no	—
218	C₂×C₈	—	188	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	—	4	8	no	—
222	C₂²⋊C₄	—	4	8	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	4	2	no	—
227	C₄²	#10625334	1	4	yes	—
228	C₄²	#10625335	1	4	yes	—
229	C₄²	D₄×D₄₇	1	2	yes	—
230	C₄²	#10625352	1	2	yes	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	4	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	2	no	—
235	C₂³	—	4	2	no	—
236	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
237	Q₈	—	8	1	no	—
238	Q₈	—	8	1	no	—
239	Q₈	—	8	2	no	—
240	D₄	—	8	4	no	—
241	D₄	—	8	1	no	—
242	D₄	—	8	1	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	2	no	—
245	D₄	—	8	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	#13093945	1	4	yes	—
254	C₂×C₄	#13093945	1	4	yes	—
255	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
256	C₂×C₄	#13093995	1	2	yes	—
257	C₂×C₄	#13094011	1	1	yes	characteristic
258	C₂×C₄	#13094020	1	1	yes	characteristic
259	C₂×C₄	C₂×#10625334	1	2	yes	—
260	C₂×C₄	C₂×#10625335	1	2	yes	—
261	C₂×C₄	#13094044	1	2	yes	—
262	C₂×C₄	#13094049	1	1	yes	characteristic
263	C₂×C₄	#13094054	1	1	yes	characteristic
264	C₈	—	376	2	no	—
265	C₈	—	188	4	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	8	2	no	—
268	C₂²	—	8	4	no	—
269	C₂²	—	8	4	no	—
270	C₂²	—	8	2	no	—
271	C₂²	#15689026	1	1	yes	characteristic
272	C₂²	#15689107	1	2	yes	—
273	C₂²	C₂×#13093945	1	2	yes	—
274	C₂²	#15689119	1	1	yes	characteristic
275	C₂²	#15689124	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	4	no	—
279	C₄	—	8	2	no	—
280	C₄	#15689335	1	2	yes	—
281	C₄	#15689342	1	2	yes	—
282	C₄	#15689469	1	2	yes	—
283	C₄	#15689495	1	1	yes	characteristic
284	C₄	#15689499	1	1	yes	characteristic
285	C₂	—	16	2	no	—
286	C₂	#20242037	1	2	yes	—
287	C₂	#20242065	1	2	yes	—
288	C₂	#20244195	1	1	yes	characteristic
289	C₂	#20244238	1	1	yes	characteristic
290	C₂	#20244321	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2858	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₄₇	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2858	D₄₇
hypocenter	C₄₇	#9698
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2858

Complements

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

N	H ≅ G/N	type
1. #20243540	C₂	semidirect
2. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
3. C₂×#15688869	C₂	semidirect
165. C₄₇	#9698	semidirect

Maximal subgroups

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

subgroup	index
#20243540	2
C₄×C₄⋊(C₄₇⋊C₈)	2
C₂×#15688869	2
C₂×#15688867	2
C₄₇×#2858	2
#9698	47