Subgroups — #26224811

Order 12032.

1014 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689503	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689231	C₂	1	2	yes	maximal
3	C₂×C₄×Dic₁₈₈	C₂	1	2	yes	maximal
4	#20246107	C₂	1	2	yes	maximal
5	#20244259	C₂	1	2	yes	maximal
6	#20244249	C₂	1	2	yes	maximal
7	#20244208	C₂	1	2	yes	maximal
8	#20243552	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1726	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₁₈₈	C₂²	1	2	yes	—
11	C₂²×Dic₁₈₈	C₂²	1	2	yes	—
12	C₂×#13094024	C₂²	1	2	yes	—
13	C₂×C₄×Dic₉₄	C₂²	1	4	yes	—
14	#15689503	C₄	1	8	yes	complemented
15	#15689231	—	2	8	no	—
16	C₄×Dic₁₈₈	—	2	8	no	—
17	C₂×#13093958	C₂²	1	2	yes	—
18	C₂×#13093956	C₂²	1	2	yes	—
19	C₂×#13093949	C₂²	1	4	yes	—
20	C₂×#13093949	C₂²	1	4	yes	—
21	#15689029	C₂²	1	4	yes	—
22	#15689008	—	2	8	no	—
23	#15688989	C₂²	1	2	yes	—
24	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#376	C₂²	1	1	yes	characteristic
31	C₂²×Dic₉₄	C₂³	1	4	yes	—
32	C₂×Dic₁₈₈	—	2	4	no	—
33	C₂×Dic₁₈₈	—	2	4	no	—
34	C₂×Dic₁₈₈	—	2	4	no	—
35	C₂×Dic₁₈₈	—	2	4	no	—
36	C₂×Dic₁₈₈	C₂×C₄	1	8	yes	—
37	C₂×Dic₁₈₈	C₂×C₄	1	8	yes	—
38	#13094024	—	2	4	no	—
39	#13094024	C₂×C₄	1	8	yes	—
40	C₄×Dic₉₄	—	2	8	no	—
41	C₄×Dic₉₄	—	2	8	no	—
42	C₂×C₄⋊Dic₄₇	—	2	4	no	—
43	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
44	C₂×Dic₄₇⋊C₄	—	2	4	no	—
45	C₂×C₄×Dic₄₇	—	2	4	no	—
46	#13093958	—	2	4	no	—
47	#13093956	—	2	4	no	—
48	#13093949	—	2	8	no	—
49	#13093949	—	2	8	no	—
50	C₄₇⋊#95	—	2	4	no	—
51	C₄₇⋊#95	—	2	8	no	—
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×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	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	#97×C₄₇	—	2	2	no	—
60	#97×C₄₇	—	2	2	no	—
61	#97×C₄₇	D₄	1	4	yes	—
62	C₄×C₈×C₄₇	—	2	2	no	—
63	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
64	C₄×C₈×C₄₇	D₄	1	4	yes	—
65	#95×C₄₇	D₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂×Dic₉₄	—	2	4	no	—
73	C₂×Dic₉₄	—	2	4	no	—
74	C₂×Dic₉₄	—	2	4	no	—
75	C₂×Dic₉₄	—	2	4	no	—
76	C₂×Dic₉₄	—	2	4	no	—
77	C₂×Dic₉₄	C₂²×C₄	1	8	yes	—
78	C₂×C₈×C₄₇	—	2	1	no	—
79	C₂×C₈×C₄₇	—	2	1	no	—
80	C₂×C₈×C₄₇	—	2	1	no	—
81	C₂×C₈×C₄₇	—	2	1	no	—
82	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₄₇	#41	1	2	yes	—
89	C₂×C₈×C₄₇	#41	1	2	yes	—
90	C₄²×C₄₇	—	2	1	no	—
91	C₄²×C₄₇	—	2	1	no	—
92	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
93	C₄²×C₄₇	C₂×D₄	1	2	yes	—
94	C₄²×C₄₇	C₂×D₄	1	2	yes	—
95	C₄²×C₄₇	#41	1	2	yes	—
96	C₄⋊Dic₄₇	—	2	16	no	—
97	C₄⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	—	2	4	no	—
99	Dic₄₇⋊C₄	—	4	8	no	—
100	C₄×Dic₄₇	—	4	8	no	—
101	Dic₁₈₈	—	2	16	no	—
102	Dic₁₈₈	—	2	16	no	—
103	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
109	C₂×C₄×C₄₇	#127	1	4	yes	—
110	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
116	C₂×Dic₄₇	—	2	8	no	—
117	C₂×Dic₄₇	—	4	4	no	—
118	C₂×Dic₄₇	—	4	4	no	—
119	C₂×Dic₄₇	—	4	4	no	—
120	C₂×Dic₄₇	—	4	8	no	—
121	Dic₉₄	—	2	8	no	—
122	Dic₉₄	—	2	8	no	—
123	Dic₉₄	—	4	4	no	—
124	Dic₉₄	—	4	4	no	—
125	C₈×C₄₇	—	2	2	no	—
126	C₈×C₄₇	—	2	2	no	—
127	C₈×C₄₇	C₄×D₄	1	4	yes	—
128	C₈×C₄₇	C₄×D₄	1	4	yes	—
129	#21698	—	47	1	no	maximal, Sylow
130	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
132	C₂²×C₄₇	#532	1	1	yes	characteristic
133	C₂²×C₄₇	#578	1	4	yes	—
134	C₄×C₄₇	—	2	4	no	—
135	C₄×C₄₇	—	2	4	no	—
136	C₄×C₄₇	—	4	2	no	—
137	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
138	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
139	C₄×C₄₇	#532	1	1	yes	characteristic
140	C₄×C₄₇	#532	1	1	yes	characteristic
141	Dic₄₇	—	4	8	no	—
142	Dic₄₇	—	8	4	no	—
143	C₂×#494	—	47	1	no	—
144	C₂×#441	—	47	2	no	—
145	C₂×C₄×Q₁₆	—	47	2	no	—
146	#2327	—	47	2	no	—
147	#1971	—	47	2	no	—
148	#1969	—	47	2	no	—
149	#1960	—	47	2	no	—
150	#1846	—	47	1	no	—
151	#1726	D₄₇	1	1	yes	characteristic
152	C₂×C₄₇	C₄×#127	1	1	yes	characteristic
153	C₂×C₄₇	#2950	1	2	yes	—
154	C₂×C₄₇	#3156	1	2	yes	—
155	C₂×C₄₇	#3177	1	2	yes	—
156	C₂²×Q₁₆	—	47	2	no	—
157	C₂²×Q₁₆	—	47	2	no	—
158	C₂×C₄⋊Q₈	—	47	2	no	—
159	C₂×C₄×Q₈	—	47	4	no	—
160	#494	—	47	8	no	—
161	#441	—	94	8	no	—
162	C₄×Q₁₆	—	94	8	no	—
163	C₂×#107	—	47	2	no	—
164	C₂×#106	—	47	2	no	—
165	C₂×#103	—	47	4	no	—
166	C₂×#103	—	47	4	no	—
167	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
168	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
169	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
173	#391	—	47	4	no	—
174	#387	—	94	8	no	—
175	#382	—	47	2	no	—
176	#376	C₂×D₄₇	1	1	yes	characteristic
177	C₄₇	#21698	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₈	—	47	4	no	—
179	C₂×Q₁₆	—	47	8	no	—
180	C₂×Q₁₆	—	47	8	no	—
181	C₂×Q₁₆	—	94	4	no	—
182	C₂×Q₁₆	—	94	4	no	—
183	C₂×Q₁₆	—	94	4	no	—
184	C₂×Q₁₆	—	94	4	no	—
185	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
189	C₄⋊Q₈	—	47	8	no	—
190	C₄⋊Q₈	—	94	4	no	—
191	C₄×Q₈	—	94	8	no	—
192	C₄×Q₈	—	94	8	no	—
193	C₂×C₄⋊C₄	—	47	4	no	—
194	C₂×C₄⋊C₄	—	94	4	no	—
195	C₂×C₄⋊C₄	—	94	4	no	—
196	C₂×C₄²	—	94	4	no	—
197	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
199	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
200	#107	—	94	4	no	—
201	#106	—	94	4	no	—
202	#103	—	94	8	no	—
203	#103	—	94	8	no	—
204	#97	—	2	2	no	—
205	#97	—	2	2	no	—
206	#97	D₁₈₈	1	4	yes	—
207	C₄×C₈	—	2	2	no	—
208	C₄×C₈	C₄×D₄₇	1	4	yes	—
209	C₄×C₈	D₁₈₈	1	4	yes	—
210	#95	—	94	4	no	—
211	#95	—	94	8	no	—
212	#95	D₁₈₈	1	4	yes	—
213	C₂×Q₈	—	47	8	no	—
214	C₂×Q₈	—	94	4	no	—
215	C₂×Q₈	—	94	4	no	—
216	C₂×Q₈	—	94	4	no	—
217	C₂×Q₈	—	94	4	no	—
218	C₂×Q₈	—	94	4	no	—
219	C₂²×C₄	—	94	4	no	—
220	C₂²×C₄	—	94	4	no	—
221	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
222	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
223	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
224	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
225	Q₁₆	—	94	16	no	—
226	Q₁₆	—	94	16	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
232	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
233	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
234	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
235	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
236	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
237	C₂×C₈	#10625350	1	2	yes	—
238	C₂×C₈	#10625350	1	2	yes	—
239	C₄⋊C₄	—	94	16	no	—
240	C₄⋊C₄	—	94	4	no	—
241	C₄⋊C₄	—	94	4	no	—
242	C₄⋊C₄	—	188	8	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	188	8	no	—
246	C₄²	C₂×C₄×D₄₇	1	2	yes	—
247	C₄²	C₂×D₁₈₈	1	2	yes	—
248	C₄²	C₂×D₁₈₈	1	2	yes	—
249	C₄²	#10625350	1	2	yes	—
250	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
251	Q₈	—	94	8	no	—
252	Q₈	—	94	8	no	—
253	Q₈	—	188	4	no	—
254	Q₈	—	188	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	94	8	no	—
260	C₂×C₄	—	188	4	no	—
261	C₂×C₄	—	188	8	no	—
262	C₂×C₄	—	188	4	no	—
263	C₂×C₄	—	188	4	no	—
264	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
265	C₂×C₄	#13094021	1	4	yes	—
266	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
267	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
268	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
269	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
270	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
271	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	2	2	no	—
274	C₈	C₄×D₁₈₈	1	4	yes	—
275	C₈	C₄×D₁₈₈	1	4	yes	—
276	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
277	C₂²	C₂×#13094021	1	1	yes	characteristic
278	C₂²	#15689673	1	1	yes	characteristic
279	C₂²	#15689940	1	4	yes	—
280	C₄	—	2	4	no	—
281	C₄	—	2	4	no	—
282	C₄	—	4	2	no	—
283	C₄	—	376	4	no	—
284	C₄	—	188	8	no	—
285	C₄	C₂×C₄×D₁₈₈	1	1	yes	characteristic
286	C₄	C₂×C₄×D₁₈₈	1	1	yes	characteristic
287	C₄	#15689673	1	1	yes	characteristic
288	C₄	#15689673	1	1	yes	characteristic
289	C₂	C₄×#13094021	1	1	yes	characteristic
290	C₂	#20252257	1	2	yes	—
291	C₂	#20253882	1	2	yes	—
292	C₂	#20254047	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1726	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1726	D₄₇
hypocenter	C₄₇	#21698
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
14. #15689503	C₄	semidirect
177. C₄₇	#21698	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689503	2
C₂×#15689231	2
C₂×C₄×Dic₁₈₈	2
#20246107	2
#20244259	2
#20244249	2
#20244208	2
#20243552	2
C₄₇×#1726	2
#21698	47