Subgroups — #27080688

Order 12032.

1810 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252337	C₂	1	16	yes	maximal, complemented
2	#20252329	C₂	1	4	yes	maximal, complemented
3	#20252323	C₂	1	4	yes	maximal, complemented
4	#20251952	C₂	1	1	yes	maximal, characteristic
5	C₂×#15689150	C₂	1	2	yes	maximal
6	#20251807	C₂	1	2	yes	maximal
7	C₂×#15689079	C₂	1	1	yes	maximal, characteristic
8	C₄₇×#3458	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625329	C₂²	1	2	yes	—
10	C₂²×#10625329	C₂²	1	1	yes	characteristic
11	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	#15689254	C₂²	1	16	yes	complemented
13	#15689254	C₂²	1	32	yes	complemented
14	Q₈×C₄₇⋊C₈	C₂²	1	16	yes	complemented
15	#15689150	C₂²	1	4	yes	—
16	#15689150	C₂²	1	4	yes	—
17	#15689150	C₂²	1	8	yes	—
18	#15689150	C₂²	1	8	yes	—
19	#15689147	C₂²	1	4	yes	—
20	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
21	#15689079	C₂²	1	8	yes	—
22	#15689079	C₂²	1	8	yes	—
23	#15689079	C₂²	1	8	yes	—
24	C₄×#10625329	C₂²	1	8	yes	—
25	C₂×#13093935	C₂²	1	2	yes	—
26	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
27	C₄₇×#544	C₄	1	8	yes	—
28	C₄₇×#544	C₄	1	4	yes	—
29	C₄₇×#543	C₄	1	4	yes	—
30	C₄₇×#533	C₂²	1	4	yes	—
31	C₄₇×#533	C₂²	1	8	yes	—
32	C₂×#117×C₄₇	C₂²	1	2	yes	—
33	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×#10625329	—	2	8	no	—
35	C₂×#10625329	Q₈	1	8	yes	complemented
36	C₂×#10625329	C₂³	1	4	yes	—
37	C₂×#10625329	C₂³	1	4	yes	—
38	C₂×#10625329	C₂³	1	4	yes	—
39	C₂²×C₄₇⋊C₈	—	2	4	no	—
40	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
43	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
44	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
45	C₄⋊(C₄₇⋊C₈)	C₂³	1	16	yes	—
46	C₄⋊(C₄₇⋊C₈)	C₂³	1	16	yes	—
47	C₄⋊(C₄₇⋊C₈)	C₂³	1	16	yes	—
48	#13093935	C₂³	1	8	yes	—
49	#13093935	C₂³	1	16	yes	—
50	C₄×C₄₇⋊C₈	C₂³	1	8	yes	—
51	C₄×C₄₇⋊C₈	C₂³	1	16	yes	—
52	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	—
54	C₄⋊Q₈×C₄₇	C₂×C₄	1	16	yes	—
55	#125×C₄₇	C₂×C₄	1	8	yes	—
56	#125×C₄₇	C₂×C₄	1	16	yes	—
57	C₂²⋊Q₈×C₄₇	C₂×C₄	1	16	yes	—
58	C₂²⋊Q₈×C₄₇	C₂×C₄	1	32	yes	—
59	C₄×Q₈×C₄₇	C₂³	1	16	yes	—
60	#117×C₄₇	C₂×C₄	1	4	yes	—
61	#117×C₄₇	C₂×C₄	1	4	yes	—
62	#117×C₄₇	C₂×C₄	1	4	yes	—
63	#117×C₄₇	C₂³	1	4	yes	—
64	#117×C₄₇	C₂³	1	4	yes	—
65	#117×C₄₇	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
67	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
69	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
70	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
71	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
72	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
79	C₂²×C₄×C₄₇	#41	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
82	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
83	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
84	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
85	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
86	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
87	C₂²⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
88	C₂²⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
89	C₂²⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
90	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
91	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
92	C₄²×C₄₇	C₂⁴	1	4	yes	—
93	C₄²×C₄₇	C₂⁴	1	8	yes	—
94	#10625329	—	2	8	no	—
95	#10625329	—	2	8	no	—
96	#10625329	C₂×Q₈	1	16	yes	—
97	C₂×C₄₇⋊C₈	—	2	8	no	—
98	C₂×C₄₇⋊C₈	—	2	2	no	—
99	C₂×C₄₇⋊C₈	—	2	2	no	—
100	C₂×C₄₇⋊C₈	—	2	8	no	—
101	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
102	C₂×C₄₇⋊C₈	C₂⁴	1	2	yes	—
103	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
104	C₂×C₄₇⋊C₈	C₂⁴	1	2	yes	—
105	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
106	C₂³×C₄₇	—	4	2	no	—
107	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
108	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
109	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
110	Q₈×C₄₇	—	4	16	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	4	8	no	—
113	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
114	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
115	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
116	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
117	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
120	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
121	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
122	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
123	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
124	C₄₇⋊C₈	—	2	2	no	—
125	C₄₇⋊C₈	—	2	2	no	—
126	C₄₇⋊C₈	—	2	8	no	—
127	C₄₇⋊C₈	C₂²×Q₈	1	8	yes	—
128	#60193	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	—	4	4	no	—
132	C₂²×C₄₇	C₂×C₄×Q₈	1	2	yes	—
133	C₂²×C₄₇	C₂×C₄×Q₈	1	4	yes	—
134	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
135	C₄×C₄₇	—	2	8	no	—
136	C₄×C₄₇	—	2	16	no	—
137	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
138	C₄×C₄₇	C₂×C₄×Q₈	1	4	yes	—
139	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
140	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
141	#3458	D₄₇	1	1	yes	characteristic
142	#2962	—	47	16	no	—
143	#2961	—	47	4	no	—
144	#2960	—	47	4	no	—
145	#2905	—	47	1	no	—
146	C₂×#424	—	47	2	no	—
147	#2873	—	47	2	no	—
148	C₂×#405	—	47	1	no	—
149	C₂×C₄₇	—	2	2	no	—
150	C₂×C₄₇	—	2	4	no	—
151	C₂×C₄₇	C₂²×C₄×Q₈	1	1	yes	characteristic
152	C₂×C₄₇	#3573	1	2	yes	—
153	C₂²×M₁₆	—	47	1	no	—
154	C₂²×M₁₆	—	47	2	no	—
155	C₂³×C₈	—	47	1	no	—
156	#544	Dic₄₇	1	4	yes	—
157	#544	Dic₄₇	1	8	yes	—
158	#543	Dic₄₇	1	4	yes	—
159	#533	C₂×D₄₇	1	4	yes	—
160	#533	C₂×D₄₇	1	8	yes	—
161	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
162	C₂×#117	C₂×D₄₇	1	2	yes	—
163	#446	—	47	16	no	—
164	#446	—	47	32	no	—
165	C₈×Q₈	—	47	16	no	—
166	#424	—	47	4	no	—
167	#424	—	47	4	no	—
168	#424	—	47	8	no	—
169	#424	—	47	8	no	—
170	#423	—	47	4	no	—
171	C₂×C₄⋊C₈	—	47	8	no	—
172	#405	—	47	8	no	—
173	#405	—	47	8	no	—
174	#405	—	47	8	no	—
175	C₄×M₁₆	—	47	8	no	—
176	C₂×#97	—	47	2	no	—
177	C₂×C₄×C₈	—	47	2	no	—
178	C₄₇	#60193	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
180	C₂×M₁₆	—	47	8	no	—
181	C₂×M₁₆	—	47	4	no	—
182	C₂×M₁₆	—	47	4	no	—
183	C₂×M₁₆	—	47	4	no	—
184	C₂×M₁₆	—	94	8	no	—
185	C₂²×C₈	—	47	2	no	—
186	C₂²×C₈	—	47	1	no	—
187	C₂²×C₈	—	47	1	no	—
188	C₂²×C₈	—	47	4	no	—
189	C₂²×C₈	—	47	4	no	—
190	C₂²×C₈	—	94	4	no	—
191	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	—
192	C₄⋊Q₈	C₂×Dic₄₇	1	16	yes	—
193	#125	C₂×Dic₄₇	1	8	yes	—
194	#125	C₂×Dic₄₇	1	16	yes	—
195	C₂²⋊Q₈	C₂×Dic₄₇	1	16	yes	—
196	C₂²⋊Q₈	C₂×Dic₄₇	1	32	yes	—
197	C₄×Q₈	C₂²×D₄₇	1	16	yes	—
198	#117	C₂×Dic₄₇	1	4	yes	—
199	#117	C₂×Dic₄₇	1	4	yes	—
200	#117	C₂×Dic₄₇	1	4	yes	—
201	#117	C₂²×D₄₇	1	4	yes	—
202	#117	C₂²×D₄₇	1	4	yes	—
203	#117	C₂²×D₄₇	1	4	yes	—
204	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
205	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
206	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
207	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
208	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
209	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
210	C₄⋊C₈	—	47	16	no	—
211	C₄⋊C₈	—	47	16	no	—
212	C₄⋊C₈	—	47	16	no	—
213	#97	—	47	8	no	—
214	#97	—	47	16	no	—
215	C₄×C₈	—	47	8	no	—
216	C₄×C₈	—	47	16	no	—
217	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
218	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
219	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
220	C₂²×C₄	#10625354	1	4	yes	—
221	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
222	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
223	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
224	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
225	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
226	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
227	M₁₆	—	47	16	no	—
228	M₁₆	—	94	8	no	—
229	M₁₆	—	94	8	no	—
230	C₂×C₈	—	47	8	no	—
231	C₂×C₈	—	47	4	no	—
232	C₂×C₈	—	47	2	no	—
233	C₂×C₈	—	47	8	no	—
234	C₂×C₈	—	47	2	no	—
235	C₂×C₈	—	94	8	no	—
236	C₂×C₈	—	94	2	no	—
237	C₂×C₈	—	94	2	no	—
238	C₂×C₈	—	94	8	no	—
239	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
240	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
241	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
242	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
243	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
244	C₂²⋊C₄	C₂²×Dic₄₇	1	8	yes	—
245	C₂²⋊C₄	C₂²×Dic₄₇	1	8	yes	—
246	C₂²⋊C₄	C₂²×Dic₄₇	1	8	yes	—
247	C₄²	C₂²×Dic₄₇	1	4	yes	—
248	C₄²	C₂²×Dic₄₇	1	8	yes	—
249	C₄²	C₂³×D₄₇	1	4	yes	—
250	C₄²	C₂³×D₄₇	1	8	yes	—
251	C₂³	—	4	2	no	—
252	C₂³	Q₈×Dic₄₇	1	4	yes	—
253	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
254	C₂³	C₂³×Dic₄₇	1	2	yes	—
255	Q₈	—	4	16	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	Q₈×Dic₄₇	1	8	yes	—
259	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
260	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
261	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
262	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
263	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
264	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
265	C₂×C₄	C₂×Q₈×D₄₇	1	4	yes	—
266	C₂×C₄	C₂×#10625354	1	4	yes	—
267	C₂×C₄	C₂×#10625354	1	2	yes	—
268	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
269	C₈	—	47	8	no	—
270	C₈	—	94	2	no	—
271	C₈	—	94	2	no	—
272	C₈	—	94	8	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	4	4	no	—
276	C₂²	C₂×Q₈×Dic₄₇	1	2	yes	—
277	C₂²	C₂×Q₈×Dic₄₇	1	4	yes	—
278	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
279	C₄	—	2	8	no	—
280	C₄	—	2	16	no	—
281	C₄	C₂×Q₈×Dic₄₇	1	2	yes	—
282	C₄	C₂×Q₈×Dic₄₇	1	4	yes	—
283	C₄	C₂²×Q₈×D₄₇	1	1	yes	characteristic
284	C₄	C₂²×#10625354	1	1	yes	characteristic
285	C₂	—	2	2	no	—
286	C₂	—	2	4	no	—
287	C₂	C₂²×Q₈×Dic₄₇	1	1	yes	characteristic
288	C₂	#20258100	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁴×C₄
Fitting	C₄₇×#3458	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	#3458	D₄₇
hypocenter	C₄₇	#60193
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #3458

Complements

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

N	H ≅ G/N	type
1. #20252337	C₂	semidirect
2. #20252329	C₂	semidirect
3. #20252323	C₂	semidirect
12. #15689254	C₂²	semidirect
13. #15689254	C₂²	semidirect
14. Q₈×C₄₇⋊C₈	C₂²	semidirect
35. C₂×#10625329	Q₈	semidirect
178. C₄₇	#60193	semidirect

Maximal subgroups

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

subgroup	index
#20252337	2
#20252329	2
#20252323	2
#20251952	2
C₂×#15689150	2
#20251807	2
C₂×#15689079	2
C₄₇×#3458	2
#60193	47