Subgroups — #26107048

Order 12032.

518 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689247	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×#13093947	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688855	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#20241222	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1279	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
9	#15689247	—	2	2	no	—
10	#15689247	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×Dic₄₇⋊C₈	—	2	1	no	—
13	C₂×#13093947	C₄	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₄₇	—	2	1	no	—
18	#15688885	C₄	1	2	yes	—
19	#15688878	—	2	1	no	—
20	#15688855	C₄	1	4	yes	complemented
21	C₈×C₄₇⋊C₈	—	2	2	no	—
22	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄₇⋊C₈	—	2	1	no	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
30	C₄×Dic₉₄	—	2	1	no	—
31	C₄×Dic₉₄	—	2	1	no	—
32	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
33	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×Dic₄₇⋊C₄	—	2	1	no	—
36	C₂×C₄×Dic₄₇	—	2	1	no	—
37	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
38	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
39	Dic₄₇⋊C₈	—	2	4	no	—
40	#13093947	C₈	1	8	yes	complemented
41	C₄×C₄₇⋊C₈	—	2	1	no	—
42	C₄×C₄₇⋊C₈	—	2	1	no	—
43	C₄×C₄₇⋊C₈	—	4	2	no	—
44	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
45	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
46	C₈×Dic₄₇	—	2	4	no	—
47	C₂²×C₈×C₄₇	—	2	1	no	—
48	C₂²×C₈×C₄₇	—	2	1	no	—
49	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₇	—	2	1	no	—
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₄₇	—	2	1	no	—
53	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₄₇	—	2	1	no	—
55	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
56	#95×C₄₇	—	2	1	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
61	C₂²×Dic₄₇	—	2	1	no	—
62	C₂²×Dic₄₇	—	2	1	no	—
63	C₂×Dic₉₄	—	2	1	no	—
64	C₂×Dic₉₄	—	2	2	no	—
65	C₂×Dic₉₄	C₄²	1	2	yes	—
66	C₂×Dic₉₄	C₂×C₈	1	4	yes	—
67	C₂×C₈×C₄₇	—	2	4	no	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	4	1	no	—
71	C₂×C₈×C₄₇	—	4	1	no	—
72	C₄⋊C₄×C₄₇	—	2	2	no	—
73	C₄⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
74	C₄²×C₄₇	—	4	1	no	—
75	C₄²×C₄₇	—	4	1	no	—
76	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₄₇	#41	1	1	yes	characteristic
80	C₄⋊Dic₄₇	—	2	1	no	—
81	C₄⋊Dic₄₇	C₄²	1	2	yes	—
82	Dic₄₇⋊C₄	—	2	2	no	—
83	Dic₄₇⋊C₄	—	2	2	no	—
84	C₄×Dic₄₇	—	2	2	no	—
85	C₄×Dic₄₇	—	2	2	no	—
86	C₂×C₄₇⋊C₈	—	2	2	no	—
87	C₂×C₄₇⋊C₈	—	2	2	no	—
88	C₂×C₄₇⋊C₈	—	2	1	no	—
89	C₂×C₄₇⋊C₈	—	2	1	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	—	4	1	no	—
92	C₂×C₄₇⋊C₈	—	4	1	no	—
93	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
94	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
95	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
96	C₂×C₄₇⋊C₈	SD₁₆	1	2	yes	—
97	C₂×C₄₇⋊C₈	Q₁₆	1	2	yes	—
98	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	4	2	no	—
102	C₂×C₄×C₄₇	—	4	2	no	—
103	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₄₇	#131	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
115	C₂×Dic₄₇	—	2	2	no	—
116	C₂×Dic₄₇	—	2	4	no	—
117	C₂×Dic₄₇	—	2	2	no	—
118	C₂×Dic₄₇	—	4	2	no	—
119	Dic₉₄	—	2	2	no	—
120	Dic₉₄	—	2	4	no	—
121	Dic₉₄	C₄×C₈	1	4	yes	complemented
122	C₈×C₄₇	—	2	4	no	—
123	C₈×C₄₇	—	4	2	no	—
124	C₄₇⋊C₈	—	2	2	no	—
125	C₄₇⋊C₈	—	2	4	no	—
126	C₄₇⋊C₈	—	4	2	no	—
127	C₄₇⋊C₈	#103	1	4	yes	complemented
128	C₈×#103	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₄₇	C₈×D₄	1	2	yes	—
131	C₂²×C₄₇	C₄×SD₁₆	1	1	yes	characteristic
132	C₂²×C₄₇	C₄×Q₁₆	1	1	yes	characteristic
133	C₂²×C₄₇	#443	1	2	yes	—
134	C₄×C₄₇	—	2	1	no	—
135	C₄×C₄₇	—	2	1	no	—
136	C₄×C₄₇	—	4	4	no	—
137	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
138	C₄×C₄₇	#405	1	1	yes	characteristic
139	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
140	C₄×C₄₇	#416	1	1	yes	characteristic
141	C₄×C₄₇	C₈×D₄	1	2	yes	—
142	C₄×C₄₇	C₄×SD₁₆	1	1	yes	characteristic
143	C₄×C₄₇	C₄×Q₁₆	1	1	yes	characteristic
144	Dic₄₇	—	2	4	no	—
145	Dic₄₇	—	4	2	no	—
146	C₂×C₈×Q₈	—	47	1	no	—
147	C₈×C₄⋊C₄	—	47	1	no	—
148	C₄×#103	—	47	1	no	—
149	C₂×Q₈⋊C₈	—	47	1	no	—
150	C₂×C₈²	—	47	1	no	—
151	#1279	D₄₇	1	1	yes	characteristic
152	#1277	—	47	1	no	—
153	C₂×C₄₇	C₈×SD₁₆	1	2	yes	—
154	C₂×C₄₇	C₈×Q₁₆	1	2	yes	—
155	C₂×C₄₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₄₇	C₄×#103	1	1	yes	characteristic
157	C₂×C₄₇	#1764	1	1	yes	characteristic
158	C₂×C₄×Q₈	—	47	1	no	—
159	C₈×Q₈	—	47	4	no	—
160	C₈×Q₈	—	94	2	no	—
161	C₂×C₄⋊C₈	—	47	1	no	—
162	C₂×C₄⋊C₈	—	94	1	no	—
163	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
164	C₂×#103	—	47	2	no	—
165	C₂×C₄×C₈	—	47	1	no	—
166	C₂×C₄×C₈	—	47	1	no	—
167	C₂×C₄×C₈	—	94	1	no	—
168	C₂×C₄×C₈	—	94	1	no	—
169	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
170	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
171	#340	—	47	2	no	—
172	#336	—	94	1	no	—
173	Q₈⋊C₈	—	47	4	no	—
174	C₈²	—	94	2	no	—
175	C₄₇	C₈×#103	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	—	47	1	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	47	1	no	—
180	C₂²×C₈	—	47	2	no	—
181	C₂²×C₈	—	47	1	no	—
182	C₂²×C₈	—	94	1	no	—
183	C₄×Q₈	—	47	2	no	—
184	C₄×Q₈	—	47	2	no	—
185	C₄×Q₈	—	94	1	no	—
186	C₄×Q₈	—	94	1	no	—
187	C₂×C₄⋊C₄	—	47	1	no	—
188	C₂×C₄⋊C₄	—	94	1	no	—
189	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	—	94	1	no	—
192	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	47	2	no	—
195	C₄⋊C₈	—	94	4	no	—
196	C₄⋊C₈	—	94	1	no	—
197	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
198	#103	—	47	8	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	47	2	no	—
201	C₄×C₈	—	47	2	no	—
202	C₄×C₈	—	94	4	no	—
203	C₄×C₈	—	94	1	no	—
204	C₄×C₈	—	94	1	no	—
205	C₄×C₈	—	188	2	no	—
206	C₄×C₈	C₄×D₄₇	1	2	yes	—
207	#95	—	2	1	no	—
208	C₂×Q₈	—	47	2	no	—
209	C₂×Q₈	—	47	4	no	—
210	C₂×Q₈	—	94	1	no	—
211	C₂×Q₈	—	94	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	94	1	no	—
214	C₂²×C₄	—	94	1	no	—
215	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
217	C₂²×C₄	#10625352	1	1	yes	characteristic
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	47	2	no	—
224	C₂×C₈	—	47	2	no	—
225	C₂×C₈	—	47	2	no	—
226	C₂×C₈	—	47	2	no	—
227	C₂×C₈	—	47	4	no	—
228	C₂×C₈	—	94	2	no	—
229	C₂×C₈	—	94	2	no	—
230	C₂×C₈	—	94	1	no	—
231	C₂×C₈	—	94	1	no	—
232	C₂×C₈	—	94	2	no	—
233	C₂×C₈	—	188	1	no	—
234	C₂×C₈	—	188	1	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	47	2	no	—
237	C₄⋊C₄	—	94	2	no	—
238	C₄⋊C₄	—	94	2	no	—
239	C₄⋊C₄	—	94	1	no	—
240	C₄⋊C₄	C₈×D₄₇	1	4	yes	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	94	2	no	—
244	C₄²	—	94	2	no	—
245	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
246	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
247	C₄²	D₄×D₄₇	1	1	yes	characteristic
248	C₄²	#10625354	1	1	yes	characteristic
249	C₂³	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
250	Q₈	—	47	4	no	—
251	Q₈	—	94	2	no	—
252	Q₈	—	94	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	94	2	no	—
258	C₂×C₄	—	94	4	no	—
259	C₂×C₄	—	94	2	no	—
260	C₂×C₄	—	188	2	no	—
261	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
262	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
263	C₂×C₄	#13093982	1	1	yes	characteristic
264	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
265	C₂×C₄	#13093986	1	1	yes	characteristic
266	C₂×C₄	#13093986	1	1	yes	characteristic
267	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
268	C₂×C₄	#13094032	1	2	yes	—
269	C₂×C₄	SD₁₆×D₄₇	1	1	yes	characteristic
270	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
271	C₂×C₄	#13094043	1	1	yes	characteristic
272	C₂×C₄	#13094045	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	—	4	2	no	—
275	C₈	—	47	4	no	—
276	C₈	—	94	2	no	—
277	C₈	—	94	4	no	—
278	C₈	—	188	2	no	—
279	C₂²	#15688970	1	1	yes	characteristic
280	C₂²	(C₂²⋊D₄₇)⋊C₈	1	2	yes	—
281	C₂²	#15689214	1	1	yes	characteristic
282	C₂²	#15689215	1	1	yes	characteristic
283	C₂²	#15689237	1	2	yes	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	4	no	—
287	C₄	—	94	4	no	—
288	C₄	—	188	2	no	—
289	C₄	C₄×C₈×D₄₇	1	1	yes	characteristic
290	C₄	#15689080	1	1	yes	characteristic
291	C₄	#103×D₄₇	1	1	yes	characteristic
292	C₄	#15689116	1	1	yes	characteristic
293	C₄	#15689188	1	2	yes	—
294	C₄	#15689210	1	1	yes	characteristic
295	C₄	#15689217	1	1	yes	characteristic
296	C₂	#20242191	1	2	yes	—
297	C₂	#20242192	1	2	yes	—
298	C₂	#20243158	1	1	yes	characteristic
299	C₂	#20243207	1	1	yes	characteristic
300	C₂	#20243222	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1279	C₂
socle, prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1279	D₄₇
hypocenter	C₄₇	C₈×#103
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1279

Complements

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

N	H ≅ G/N	type
10. #15689247	C₄	semidirect
20. #15688855	C₄	semidirect
40. #13093947	C₈	semidirect
121. Dic₉₄	C₄×C₈	semidirect
127. C₄₇⋊C₈	#103	semidirect
175. C₄₇	C₈×#103	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689247	2
C₄⋊C₄×C₄₇⋊C₈	2
C₄×#13093947	2
C₂×#15688855	2
C₂×C₈×C₄₇⋊C₈	2
#20241222	2
C₄₇×#1279	2
C₈×#103	47