Subgroups — #26316187

Order 12032.

710 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251750	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20244537	C₂	1	4	yes	maximal
4	#20244209	C₂	1	1	yes	maximal, characteristic
5	#20243522	C₂	1	1	yes	maximal, characteristic
6	#20243521	C₂	1	1	yes	maximal, characteristic
7	#20243391	C₂	1	4	yes	maximal
8	#20243242	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1960	C₂	1	1	yes	maximal, characteristic
10	C₂×#13094015	C₂²	1	2	yes	—
11	C₂×Q₈×Dic₄₇	C₂²	1	2	yes	—
12	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
13	C₂×#13093948	C₂²	1	4	yes	—
14	C₂×#13093935	C₂²	1	1	yes	characteristic
15	#15689030	C₂²	1	2	yes	—
16	#15689022	—	2	2	no	—
17	#15689020	—	2	2	no	—
18	#15689020	—	2	2	no	—
19	#15688993	C₂²	1	1	yes	characteristic
20	#15688992	C₂²	1	2	yes	—
21	#15688989	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×Dic₄₇	—	2	2	no	—
23	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
24	#15688886	C₂²	1	4	yes	—
25	#15688885	C₂²	1	4	yes	—
26	#15688877	C₂²	1	2	yes	—
27	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#103×C₄₇	C₂²	1	4	yes	—
29	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#382	C₂²	1	1	yes	characteristic
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	#13094015	—	4	4	no	—
33	Q₈×Dic₄₇	—	2	8	no	—
34	C₂×C₄⋊Dic₄₇	—	2	2	no	—
35	C₂×C₄⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄⋊Dic₄₇	—	2	2	no	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
39	C₂×Dic₄₇⋊C₄	—	2	2	no	—
40	C₂×Dic₄₇⋊C₄	—	2	2	no	—
41	C₂×Dic₄₇⋊C₄	—	2	2	no	—
42	C₂×C₄×Dic₄₇	—	2	2	no	—
43	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
44	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
45	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
47	Dic₄₇⋊C₈	—	2	4	no	—
48	#13093948	—	4	4	no	—
49	#13093935	—	2	2	no	—
50	C₄₇⋊#95	—	2	2	no	—
51	C₄₇⋊#95	—	2	4	no	—
52	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
54	C₄⋊Q₈×C₄₇	—	2	2	no	—
55	C₄⋊Q₈×C₄₇	Q₈	1	4	yes	—
56	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	#103×C₄₇	—	4	4	no	—
60	#97×C₄₇	—	2	2	no	—
61	#95×C₄₇	—	2	2	no	—
62	C₂×Q₈×C₄₇	—	2	2	no	—
63	C₂×Q₈×C₄₇	—	2	4	no	—
64	C₂×Q₈×C₄₇	—	2	4	no	—
65	C₂×Q₈×C₄₇	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₄₇	—	2	2	no	—
71	C₂²×Dic₄₇	—	2	2	no	—
72	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
73	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₄⋊C₄×C₄₇	—	2	8	no	—
79	C₄⋊C₄×C₄₇	—	2	4	no	—
80	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
81	C₄²×C₄₇	#41	1	2	yes	—
82	C₄⋊Dic₄₇	—	2	4	no	—
83	C₄⋊Dic₄₇	—	2	4	no	—
84	C₄⋊Dic₄₇	—	4	4	no	—
85	C₄⋊Dic₄₇	—	4	4	no	—
86	C₄⋊Dic₄₇	—	4	4	no	—
87	Dic₄₇⋊C₄	—	4	4	no	—
88	Dic₄₇⋊C₄	—	4	4	no	—
89	Dic₄₇⋊C₄	—	4	4	no	—
90	C₄×Dic₄₇	—	2	2	no	—
91	C₄×Dic₄₇	—	2	4	no	—
92	C₄×Dic₄₇	—	4	2	no	—
93	C₄×Dic₄₇	—	4	4	no	—
94	C₄×Dic₄₇	#41	1	4	yes	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
99	Q₈×C₄₇	—	2	8	no	—
100	Q₈×C₄₇	—	4	4	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	4	2	no	—
105	C₂×C₄×C₄₇	—	4	2	no	—
106	C₂×C₄×C₄₇	—	4	2	no	—
107	C₂×C₄×C₄₇	—	4	2	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
110	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
114	C₂×Dic₄₇	—	2	2	no	—
115	C₂×Dic₄₇	—	2	2	no	—
116	C₂×Dic₄₇	—	2	2	no	—
117	C₂×Dic₄₇	—	2	2	no	—
118	C₂×Dic₄₇	—	4	2	no	—
119	C₂×Dic₄₇	—	4	2	no	—
120	C₂×Dic₄₇	—	4	2	no	—
121	C₂×Dic₄₇	—	4	2	no	—
122	C₂×Dic₄₇	—	4	4	no	—
123	C₂×Dic₄₇	#124	1	2	yes	—
124	C₂×Dic₄₇	#124	1	2	yes	—
125	C₂×Dic₄₇	#137	1	4	yes	—
126	C₈×C₄₇	—	4	2	no	—
127	C₄₇⋊C₈	—	4	2	no	—
128	#29150	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
130	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
131	C₂²×C₄₇	#535	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×#137	1	2	yes	—
133	C₂²×C₄₇	#575	1	2	yes	—
134	C₄×C₄₇	—	2	4	no	—
135	C₄×C₄₇	—	4	4	no	—
136	C₄×C₄₇	—	8	2	no	—
137	C₄×C₄₇	#554	1	2	yes	—
138	C₄×C₄₇	#555	1	2	yes	—
139	Dic₄₇	—	2	2	no	—
140	Dic₄₇	—	4	2	no	—
141	Dic₄₇	—	8	2	no	—
142	Dic₄₇	—	8	2	no	—
143	Dic₄₇	#491	1	4	yes	—
144	#2859	—	47	1	no	—
145	C₄×C₄⋊Q₈	—	47	1	no	—
146	#2026	—	47	4	no	—
147	#1960	—	47	1	no	—
148	#1960	D₄₇	1	1	yes	characteristic
149	#1836	—	47	1	no	—
150	#1836	—	47	1	no	—
151	#1803	—	47	4	no	—
152	#1767	—	47	1	no	—
153	C₂×C₄₇	#2563	1	1	yes	characteristic
154	C₂×C₄₇	#3078	1	2	yes	—
155	C₂×C₄₇	C₂×#491	1	1	yes	characteristic
156	C₂×C₄₇	#3142	1	1	yes	characteristic
157	C₂×C₄₇	#3216	1	2	yes	—
158	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
159	C₂×#125	—	47	2	no	—
160	C₂×C₄×Q₈	—	47	2	no	—
161	C₂×C₄⋊C₈	—	47	2	no	—
162	C₂×#103	—	47	4	no	—
163	C₂×#103	C₂×D₄₇	1	4	yes	—
164	C₂×#97	—	47	1	no	—
165	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
166	#391	—	47	2	no	—
167	#389	—	94	2	no	—
168	#389	—	94	2	no	—
169	#389	—	94	2	no	—
170	#384	—	47	2	no	—
171	#384	—	47	1	no	—
172	#382	—	47	1	no	—
173	#382	C₂×D₄₇	1	1	yes	characteristic
174	C₄×C₄⋊C₄	—	94	2	no	—
175	C₄³	—	47	1	no	—
176	#340	—	47	4	no	—
177	#340	—	47	4	no	—
178	#336	—	47	2	no	—
179	C₄₇	#29150	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
181	C₂²×C₈	—	47	2	no	—
182	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
183	C₄⋊Q₈	—	2	2	no	—
184	C₄⋊Q₈	Dic₉₄	1	4	yes	—
185	#125	—	188	4	no	—
186	C₄×Q₈	—	94	8	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	47	2	no	—
189	C₂×C₄⋊C₄	—	47	2	no	—
190	C₂×C₄⋊C₄	—	94	2	no	—
191	C₂×C₄⋊C₄	—	94	2	no	—
192	C₂×C₄⋊C₄	—	94	2	no	—
193	C₂×C₄⋊C₄	—	94	2	no	—
194	C₂×C₄⋊C₄	—	94	2	no	—
195	C₂×C₄⋊C₄	—	94	2	no	—
196	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
197	C₂×C₄²	—	47	1	no	—
198	C₂×C₄²	—	47	2	no	—
199	C₂×C₄²	—	47	1	no	—
200	C₂×C₄²	—	47	2	no	—
201	C₂×C₄²	—	94	2	no	—
202	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
203	C₄⋊C₈	—	94	4	no	—
204	#103	—	4	4	no	—
205	#103	—	188	4	no	—
206	#97	—	2	2	no	—
207	#97	—	94	2	no	—
208	#95	—	2	2	no	—
209	#95	—	94	4	no	—
210	#95	—	94	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	4	no	—
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	C₂×Dic₉₄	1	4	yes	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	47	1	no	—
218	C₂²×C₄	—	47	1	no	—
219	C₂²×C₄	—	47	2	no	—
220	C₂²×C₄	—	94	2	no	—
221	C₂²×C₄	—	94	2	no	—
222	C₂²×C₄	D₄×D₄₇	1	2	yes	—
223	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	94	2	no	—
228	C₂×C₈	—	94	2	no	—
229	C₂×C₈	—	94	2	no	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	94	4	no	—
233	C₄⋊C₄	—	94	4	no	—
234	C₄⋊C₄	—	188	4	no	—
235	C₄⋊C₄	—	188	4	no	—
236	C₄⋊C₄	—	188	4	no	—
237	C₄⋊C₄	—	188	4	no	—
238	C₄⋊C₄	—	188	4	no	—
239	C₄⋊C₄	—	188	4	no	—
240	C₄²	—	47	4	no	—
241	C₄²	—	94	4	no	—
242	C₄²	—	94	2	no	—
243	C₄²	—	188	2	no	—
244	C₄²	—	188	4	no	—
245	C₄²	C₂×Dic₉₄	1	2	yes	—
246	C₄²	#10625352	1	2	yes	—
247	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
248	Q₈	—	2	8	no	—
249	Q₈	—	4	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	47	2	no	—
259	C₂×C₄	—	47	4	no	—
260	C₂×C₄	—	47	2	no	—
261	C₂×C₄	—	94	2	no	—
262	C₂×C₄	—	94	2	no	—
263	C₂×C₄	—	94	2	no	—
264	C₂×C₄	—	94	2	no	—
265	C₂×C₄	—	188	2	no	—
266	C₂×C₄	—	188	2	no	—
267	C₂×C₄	—	188	4	no	—
268	C₂×C₄	—	188	2	no	—
269	C₂×C₄	—	188	2	no	—
270	C₂×C₄	#13094001	1	4	yes	—
271	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
272	C₂×C₄	#41×D₄₇	1	2	yes	—
273	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
274	C₂×C₄	#13094069	1	2	yes	—
275	C₈	—	4	2	no	—
276	C₈	—	188	2	no	—
277	C₂²	C₂×#13094001	1	1	yes	characteristic
278	C₂²	#124×D₄₇	1	1	yes	characteristic
279	C₂²	#15689686	1	1	yes	characteristic
280	C₂²	#137×D₄₇	1	2	yes	—
281	C₂²	#15689922	1	2	yes	—
282	C₄	—	2	4	no	—
283	C₄	—	4	4	no	—
284	C₄	—	8	2	no	—
285	C₄	—	47	4	no	—
286	C₄	—	94	2	no	—
287	C₄	—	376	2	no	—
288	C₄	—	376	2	no	—
289	C₄	—	188	2	no	—
290	C₄	#15689832	1	2	yes	—
291	C₄	#15689837	1	2	yes	—
292	C₂	#20248559	1	1	yes	characteristic
293	C₂	#20253190	1	2	yes	—
294	C₂	#491×D₄₇	1	1	yes	characteristic
295	C₂	#20253790	1	1	yes	characteristic
296	C₂	#20254394	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1960	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	#1960	D₄₇
hypocenter	C₄₇	#29150
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1960

Complements

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

N	H ≅ G/N	type
179. C₄₇	#29150	semidirect

Maximal subgroups

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

subgroup	index
#20251750	2
C₄⋊Q₈×Dic₄₇	2
#20244537	2
#20244209	2
#20243522	2
#20243521	2
#20243391	2
#20243242	2
C₄₇×#1960	2
#29150	47