Subgroups — #26104259

Order 12032.

516 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13094024	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#20241213	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₉₄	C₄	1	2	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
9	#15689029	C₂²	1	2	yes	—
10	#15689021	—	2	1	no	—
11	#15689021	—	2	1	no	—
12	#15688993	C₄	1	1	yes	characteristic
13	C₄×C₄⋊Dic₄₇	—	2	1	no	—
14	C₄×C₄⋊Dic₄₇	—	2	1	no	—
15	#15688878	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
17	C₂×C₄×C₈×C₄₇	—	2	1	no	—
18	C₄³×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#336	C₂²	1	2	yes	—
20	C₂²×Dic₉₄	C₂×C₄	1	2	yes	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	#13094024	—	2	2	no	—
23	#13094024	C₈	1	4	yes	complemented
24	C₄×Dic₉₄	—	4	4	no	—
25	C₂×C₄⋊Dic₄₇	—	2	1	no	—
26	C₂×C₄⋊Dic₄₇	—	2	1	no	—
27	C₂×C₄⋊Dic₄₇	—	2	1	no	—
28	C₂×C₄⋊Dic₄₇	—	2	1	no	—
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
30	C₂×Dic₄₇⋊C₄	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
33	C₄×C₄₇⋊C₈	—	4	2	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₂²×C₈×C₄₇	—	2	2	no	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₄₇	—	2	4	no	—
45	C₄×C₈×C₄₇	—	4	2	no	—
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₄₇	—	2	2	no	—
54	C₂²×Dic₄₇	—	2	2	no	—
55	C₂×Dic₉₄	—	2	2	no	—
56	C₂×Dic₉₄	—	4	2	no	—
57	C₂×Dic₉₄	—	4	2	no	—
58	C₂×Dic₉₄	C₂×C₈	1	4	yes	—
59	C₂×C₈×C₄₇	—	2	4	no	—
60	C₂×C₈×C₄₇	—	4	2	no	—
61	C₂×C₈×C₄₇	—	4	2	no	—
62	C₄²×C₄₇	—	2	2	no	—
63	C₄²×C₄₇	—	2	2	no	—
64	C₄²×C₄₇	—	2	2	no	—
65	C₄²×C₄₇	—	4	1	no	—
66	C₄²×C₄₇	C₂×C₈	1	2	yes	—
67	C₄²×C₄₇	M₁₆	1	2	yes	—
68	C₄²×C₄₇	SD₁₆	1	2	yes	—
69	C₄²×C₄₇	SD₁₆	1	2	yes	—
70	C₄²×C₄₇	Q₁₆	1	2	yes	—
71	C₄²×C₄₇	Q₁₆	1	2	yes	—
72	C₄⋊Dic₄₇	—	2	4	no	—
73	C₄⋊Dic₄₇	—	2	4	no	—
74	C₄⋊Dic₄₇	—	4	2	no	—
75	C₄⋊Dic₄₇	—	4	2	no	—
76	C₄⋊Dic₄₇	—	4	2	no	—
77	Dic₄₇⋊C₄	—	4	4	no	—
78	C₄×Dic₄₇	—	4	4	no	—
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂×C₄₇⋊C₈	—	4	2	no	—
81	C₂×C₄₇⋊C₈	—	4	2	no	—
82	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	2	1	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
93	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
94	C₂×C₄×C₄₇	#100	1	2	yes	—
95	C₂×C₄×C₄₇	#103	1	2	yes	—
96	C₂×C₄×C₄₇	#103	1	2	yes	—
97	C₂×C₄×C₄₇	#103	1	2	yes	—
98	C₂×C₄×C₄₇	#103	1	2	yes	—
99	C₂×C₄×C₄₇	#104	1	2	yes	—
100	C₂×C₄×C₄₇	#104	1	2	yes	—
101	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×Dic₄₇	—	2	4	no	—
108	C₂×Dic₄₇	—	4	2	no	—
109	C₂×Dic₄₇	—	4	2	no	—
110	C₂×Dic₄₇	—	4	2	no	—
111	C₂×Dic₄₇	—	4	2	no	—
112	C₂×Dic₄₇	—	4	2	no	—
113	Dic₉₄	—	4	4	no	—
114	Dic₉₄	—	4	4	no	—
115	C₈×C₄₇	—	4	4	no	—
116	C₄₇⋊C₈	—	4	4	no	—
117	#7571	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₄₇	#410	1	1	yes	characteristic
120	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
122	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
123	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
124	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
125	C₄×C₄₇	—	2	1	no	—
126	C₄×C₄₇	—	2	1	no	—
127	C₄×C₄₇	—	2	1	no	—
128	C₄×C₄₇	—	2	1	no	—
129	C₄×C₄₇	—	2	1	no	—
130	C₄×C₄₇	—	2	1	no	—
131	C₄×C₄₇	—	4	1	no	—
132	C₄×C₄₇	Q₈⋊C₈	1	4	yes	—
133	C₄×C₄₇	Q₈⋊C₈	1	4	yes	—
134	C₄×C₄₇	#333	1	4	yes	—
135	Dic₄₇	—	4	4	no	—
136	Dic₄₇	—	8	2	no	—
137	C₄×C₄⋊Q₈	—	47	1	no	—
138	C₄×C₄⋊C₈	—	47	1	no	—
139	C₄×C₄⋊C₈	D₄₇	1	1	yes	characteristic
140	#1272	—	47	4	no	—
141	C₂×C₄₇	#1460	1	1	yes	characteristic
142	C₂×C₄₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
143	C₂×C₄₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
144	C₂×C₄₇	#1503	1	1	yes	characteristic
145	C₂×C₄₇	#1503	1	1	yes	characteristic
146	C₂×C₄₇	#1525	1	1	yes	characteristic
147	C₂×C₄₇	C₂×#333	1	1	yes	characteristic
148	C₂×C₄⋊Q₈	—	47	1	no	—
149	C₂×C₄×Q₈	—	47	2	no	—
150	C₂×C₄⋊C₈	—	2	1	no	—
151	C₂×C₄⋊C₈	—	94	1	no	—
152	C₂×C₄×C₈	—	2	1	no	—
153	C₂×C₄×C₈	—	94	1	no	—
154	#391	—	47	2	no	—
155	#389	—	94	1	no	—
156	#389	—	94	1	no	—
157	#384	—	47	1	no	—
158	C₄×C₄⋊C₄	—	94	1	no	—
159	C₄×C₄⋊C₄	—	94	1	no	—
160	C₄³	C₂×D₄₇	1	1	yes	characteristic
161	#336	—	47	2	no	—
162	#336	C₂×D₄₇	1	2	yes	—
163	C₄₇	#7571	1	1	yes	characteristic, complemented, Sylow
164	C₂²×Q₈	—	47	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	94	2	no	—
167	C₄⋊Q₈	—	47	4	no	—
168	C₄⋊Q₈	—	94	2	no	—
169	C₄×Q₈	—	188	4	no	—
170	C₂×C₄⋊C₄	—	47	2	no	—
171	C₂×C₄⋊C₄	—	94	1	no	—
172	C₂×C₄⋊C₄	—	94	1	no	—
173	C₂×C₄⋊C₄	—	94	2	no	—
174	C₂×C₄⋊C₄	—	94	1	no	—
175	C₂×C₄⋊C₄	—	94	1	no	—
176	C₂×C₄²	—	94	2	no	—
177	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
179	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
180	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
184	C₄⋊C₈	—	2	4	no	—
185	C₄⋊C₈	—	94	4	no	—
186	C₄×C₈	—	4	2	no	—
187	C₄×C₈	—	188	2	no	—
188	#95	—	94	2	no	—
189	#95	—	94	2	no	—
190	C₂×Q₈	—	47	4	no	—
191	C₂×Q₈	—	94	2	no	—
192	C₂×Q₈	—	188	2	no	—
193	C₂×Q₈	—	188	2	no	—
194	C₂²×C₄	—	94	2	no	—
195	C₂²×C₄	—	94	2	no	—
196	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	94	4	no	—
207	C₂×C₈	—	188	2	no	—
208	C₂×C₈	—	188	2	no	—
209	C₄⋊C₄	—	94	4	no	—
210	C₄⋊C₄	—	94	4	no	—
211	C₄⋊C₄	—	188	2	no	—
212	C₄⋊C₄	—	188	4	no	—
213	C₄⋊C₄	—	188	2	no	—
214	C₄⋊C₄	—	188	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	4	1	no	—
219	C₄²	—	188	4	no	—
220	C₄²	C₈×D₄₇	1	2	yes	—
221	C₄²	#10625324	1	2	yes	—
222	C₄²	#10625325	1	2	yes	—
223	C₄²	Dic₁₈₈	1	2	yes	—
224	C₄²	#10625335	1	2	yes	—
225	C₄²	#10625337	1	2	yes	—
226	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
227	Q₈	—	188	4	no	—
228	Q₈	—	188	4	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	94	4	no	—
239	C₂×C₄	—	188	2	no	—
240	C₂×C₄	—	188	2	no	—
241	C₂×C₄	—	188	2	no	—
242	C₂×C₄	—	188	2	no	—
243	C₂×C₄	—	188	2	no	—
244	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
245	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
246	C₂×C₄	#13093940	1	2	yes	—
247	C₂×C₄	#13093947	1	2	yes	—
248	C₂×C₄	#13093947	1	2	yes	—
249	C₂×C₄	#13093949	1	2	yes	—
250	C₂×C₄	#13093949	1	2	yes	—
251	C₂×C₄	#13093950	1	2	yes	—
252	C₂×C₄	#13093952	1	2	yes	—
253	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
254	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
255	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
256	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
257	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
258	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
259	C₈	—	4	4	no	—
260	C₈	—	188	4	no	—
261	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
262	C₂²	#15689096	1	1	yes	characteristic
263	C₂²	C₂×#13093940	1	1	yes	characteristic
264	C₂²	C₂×#13093947	1	1	yes	characteristic
265	C₂²	C₂×#13093949	1	1	yes	characteristic
266	C₂²	C₂×#13093950	1	1	yes	characteristic
267	C₂²	C₂×#13093952	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	376	2	no	—
276	C₄	—	188	4	no	—
277	C₄	#15688853	1	4	yes	—
278	C₄	#15688855	1	4	yes	—
279	C₄	#15688871	1	4	yes	—
280	C₂	#20241708	1	1	yes	characteristic
281	C₂	C₂×#15688853	1	1	yes	characteristic
282	C₂	C₂×#15688855	1	1	yes	characteristic
283	C₂	#20241886	1	1	yes	characteristic
284	C₂	#20241888	1	1	yes	characteristic
285	C₂	#20241979	1	1	yes	characteristic
286	C₂	C₂×#15688871	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×C₄⋊C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#7571
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
23. #13094024	C₈	semidirect
163. C₄₇	#7571	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13094024	2
C₄×C₄⋊(C₄₇⋊C₈)	2
#20241213	2
C₄×C₄⋊C₈×C₄₇	2
#7571	47