Subgroups — #26316004

Order 12032.

674 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	#20251775	C₂	1	1	yes	maximal, characteristic
2	#20244950	C₂	1	2	yes	maximal
3	#20244631	C₂	1	2	yes	maximal
4	#20244081	C₂	1	4	yes	maximal
5	#20243403	C₂	1	2	yes	maximal
6	#20243403	C₂	1	2	yes	maximal
7	#20243241	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×#13093948	C₂²	1	4	yes	—
11	C₂×#13093948	C₂²	1	4	yes	—
12	C₂×#13093935	C₂²	1	2	yes	—
13	#15689021	C₂²	1	2	yes	—
14	#15689021	C₂²	1	2	yes	—
15	#15689007	—	2	2	no	—
16	#15689007	—	2	2	no	—
17	#15688989	C₂²	1	1	yes	characteristic
18	#15688989	C₂²	1	1	yes	characteristic
19	#15688885	C₂²	1	4	yes	—
20	#15688885	C₂²	1	4	yes	—
21	#15688878	C₂²	1	2	yes	—
22	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
24	C₄₇×#391	C₂²	1	2	yes	—
25	C₄₇×#389	—	2	2	no	—
26	C₄₇×#384	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
28	C₄³×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
30	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
32	C₂×Dic₄₇⋊C₄	—	2	2	no	—
33	C₂×Dic₄₇⋊C₄	—	2	2	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
35	#13093948	—	2	8	no	—
36	#13093948	—	4	4	no	—
37	#13093935	—	4	2	no	—
38	C₄₇⋊#95	—	2	1	no	—
39	C₄₇⋊#95	—	2	1	no	—
40	C₄₇⋊#95	—	2	1	no	—
41	C₄₇⋊#95	—	2	1	no	—
42	C₄₇⋊#95	—	2	2	no	—
43	C₄₇⋊#95	—	2	2	no	—
44	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
45	C₄⋊Q₈×C₄₇	—	2	4	no	—
46	C₄×Q₈×C₄₇	—	2	4	no	—
47	C₄×Q₈×C₄₇	—	2	4	no	—
48	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₄₇	—	2	2	no	—
53	C₂×C₄²×C₄₇	D₄	1	2	yes	—
54	C₂×C₄²×C₄₇	D₄	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	#95×C₄₇	—	2	4	no	—
59	C₂×Q₈×C₄₇	—	2	2	no	—
60	C₂×Q₈×C₄₇	—	2	2	no	—
61	C₂×Q₈×C₄₇	—	2	2	no	—
62	C₂×Q₈×C₄₇	—	2	2	no	—
63	C₂×Q₈×C₄₇	—	2	2	no	—
64	C₂×Q₈×C₄₇	#41	1	4	yes	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	2	2	no	—
73	C₂²×Dic₄₇	—	2	2	no	—
74	C₄⋊C₄×C₄₇	—	2	8	no	—
75	C₄⋊C₄×C₄₇	—	2	2	no	—
76	C₄⋊C₄×C₄₇	—	2	2	no	—
77	C₄⋊C₄×C₄₇	—	4	4	no	—
78	C₄⋊C₄×C₄₇	—	4	4	no	—
79	C₄²×C₄₇	—	2	4	no	—
80	C₄²×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	—	2	1	no	—
82	C₄²×C₄₇	—	2	1	no	—
83	C₄²×C₄₇	—	4	2	no	—
84	C₄²×C₄₇	—	4	4	no	—
85	C₄²×C₄₇	#41	1	4	yes	—
86	C₄⋊Dic₄₇	—	2	4	no	—
87	C₄⋊Dic₄₇	—	2	2	no	—
88	C₄⋊Dic₄₇	—	2	2	no	—
89	Dic₄₇⋊C₄	—	4	4	no	—
90	Dic₄₇⋊C₄	—	4	4	no	—
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
95	Q₈×C₄₇	—	2	4	no	—
96	Q₈×C₄₇	—	2	4	no	—
97	Q₈×C₄₇	—	4	2	no	—
98	Q₈×C₄₇	—	4	2	no	—
99	C₂×C₄×C₄₇	—	2	2	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	4	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	4	2	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	#124	1	2	yes	—
112	C₂×C₄×C₄₇	#124	1	2	yes	—
113	C₂×C₄×C₄₇	#137	1	4	yes	—
114	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
116	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
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	2	no	—
123	C₂×Dic₄₇	—	4	2	no	—
124	C₄₇⋊C₈	—	4	4	no	—
125	#29129	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	#525	1	1	yes	characteristic
127	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
128	C₂²×C₄₇	#537	1	1	yes	characteristic
129	C₂²×C₄₇	C₂×#137	1	2	yes	—
130	C₂²×C₄₇	#575	1	2	yes	—
131	C₄×C₄₇	—	2	4	no	—
132	C₄×C₄₇	—	2	2	no	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	4	4	no	—
135	C₄×C₄₇	—	4	2	no	—
136	C₄×C₄₇	—	8	2	no	—
137	C₄×C₄₇	#552	1	2	yes	—
138	C₄×C₄₇	#553	1	2	yes	—
139	Dic₄₇	—	8	2	no	—
140	Dic₄₇	—	8	2	no	—
141	#2864	—	47	1	no	—
142	C₄×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
143	#2100	—	47	2	no	—
144	#2043	—	47	2	no	—
145	#1935	—	47	4	no	—
146	#1805	—	47	2	no	—
147	#1805	—	47	2	no	—
148	#1767	—	47	1	no	—
149	C₂×C₄₇	#2562	1	1	yes	characteristic
150	C₂×C₄₇	#3117	1	2	yes	—
151	C₂×C₄₇	#3135	1	1	yes	characteristic
152	C₂×C₄₇	#3139	1	1	yes	characteristic
153	C₂×C₄₇	#3252	1	2	yes	—
154	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
155	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
156	C₂×C₄⋊C₈	—	47	2	no	—
157	C₂×#103	—	47	4	no	—
158	C₂×#103	—	47	4	no	—
159	C₂×#97	—	47	2	no	—
160	#391	C₂×D₄₇	1	2	yes	—
161	#389	—	2	2	no	—
162	#389	—	47	2	no	—
163	#389	—	47	2	no	—
164	#387	—	94	2	no	—
165	#387	—	94	2	no	—
166	#384	C₂×D₄₇	1	1	yes	characteristic
167	#382	—	47	1	no	—
168	#382	—	47	1	no	—
169	C₄×C₄⋊C₄	—	2	2	no	—
170	C₄³	C₂×D₄₇	1	1	yes	characteristic
171	#340	—	47	4	no	—
172	#340	—	47	4	no	—
173	#336	—	47	2	no	—
174	C₄₇	#29129	1	1	yes	characteristic, complemented, Sylow
175	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
176	C₂²×C₈	—	47	4	no	—
177	C₄⋊Q₈	—	2	4	no	—
178	C₄×Q₈	—	2	4	no	—
179	C₄×Q₈	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	47	2	no	—
184	C₂×C₄⋊C₄	—	47	2	no	—
185	C₂×C₄⋊C₄	—	94	2	no	—
186	C₂×C₄⋊C₄	—	94	2	no	—
187	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
190	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
193	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
194	C₄⋊C₈	—	94	4	no	—
195	#103	—	94	8	no	—
196	#103	—	188	4	no	—
197	#97	—	188	2	no	—
198	#95	—	2	4	no	—
199	#95	—	94	1	no	—
200	#95	—	94	1	no	—
201	#95	—	94	1	no	—
202	#95	—	94	1	no	—
203	#95	—	94	2	no	—
204	#95	—	94	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	#10625350	1	4	yes	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	94	2	no	—
214	C₂²×C₄	—	94	2	no	—
215	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
216	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
219	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
220	C₂×C₈	—	94	4	no	—
221	C₂×C₈	—	94	4	no	—
222	C₂×C₈	—	94	4	no	—
223	C₄⋊C₄	—	2	8	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	94	4	no	—
229	C₄⋊C₄	—	94	2	no	—
230	C₄⋊C₄	—	94	2	no	—
231	C₄⋊C₄	—	188	4	no	—
232	C₄⋊C₄	—	188	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	4	2	no	—
238	C₄²	—	4	4	no	—
239	C₄²	#10625352	1	4	yes	—
240	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	4	no	—
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	188	2	no	—
258	C₂×C₄	—	188	2	no	—
259	C₂×C₄	—	188	2	no	—
260	C₂×C₄	—	188	2	no	—
261	C₂×C₄	—	188	2	no	—
262	C₂×C₄	—	188	2	no	—
263	C₂×C₄	#13094011	1	2	yes	—
264	C₂×C₄	#13094011	1	2	yes	—
265	C₂×C₄	#13094055	1	4	yes	—
266	C₂×C₄	C₂×#10625352	1	2	yes	—
267	C₂×C₄	C₂×#10625350	1	2	yes	—
268	C₂×C₄	#13094070	1	1	yes	characteristic
269	C₂×C₄	#13094071	1	1	yes	characteristic
270	C₈	—	188	4	no	—
271	C₂²	#15689634	1	1	yes	characteristic
272	C₂²	C₂×#13094011	1	1	yes	characteristic
273	C₂²	#15689701	1	1	yes	characteristic
274	C₂²	C₂×#13094055	1	2	yes	—
275	C₂²	#15689921	1	2	yes	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	4	no	—
279	C₄	—	4	2	no	—
280	C₄	—	4	4	no	—
281	C₄	—	8	2	no	—
282	C₄	—	376	2	no	—
283	C₄	—	376	2	no	—
284	C₄	#15689813	1	2	yes	—
285	C₄	#15689819	1	2	yes	—
286	C₂	#20248554	1	1	yes	characteristic
287	C₂	#20253559	1	2	yes	—
288	C₂	#20253733	1	1	yes	characteristic
289	C₂	#20253768	1	1	yes	characteristic
290	C₂	#20254755	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	C₄×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#29129
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
174. C₄₇	#29129	semidirect

Maximal subgroups

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

subgroup	index
#20251775	2
#20244950	2
#20244631	2
#20244081	2
#20243403	2
#20243241	2
C₄×C₄⋊Q₈×C₄₇	2
#29129	47