Subgroups — #26220772

Order 12032.

1106 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	#20251807	C₂	1	1	yes	maximal, characteristic
2	#20245872	C₂	1	1	yes	maximal, characteristic
3	#20244333	C₂	1	8	yes	maximal, complemented
4	#20243097	C₂	1	2	yes	maximal
5	C₂×#15688882	C₂	1	2	yes	maximal
6	C₄₇×#3458	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂×#13093983	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
10	#15689079	—	2	4	no	—
11	C₄×#10625329	C₂²	1	4	yes	—
12	#15689022	—	2	2	no	—
13	#15688998	—	2	2	no	—
14	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
15	#15688882	C₂²	1	8	yes	—
16	#15688882	C₂²	1	8	yes	—
17	C₄₇×#544	—	2	2	no	—
18	C₄₇×#544	C₄	1	8	yes	complemented
19	C₄₇×#543	—	2	2	no	—
20	C₄₇×#533	—	2	4	no	—
21	C₄₇×#533	C₂²	1	4	yes	—
22	C₂×#117×C₄₇	C₂²	1	2	yes	—
23	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
24	C₂³×Dic₄₇	—	2	1	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	4	no	—
27	C₂×#10625329	Q₈	1	8	yes	complemented
28	C₂×#10625329	C₂³	1	4	yes	—
29	C₂²×C₄₇⋊C₈	—	2	2	no	—
30	#13093983	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	—	2	1	no	—
33	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
34	C₂×C₄×Dic₄₇	—	2	4	no	—
35	C₂×C₄×Dic₄₇	—	2	1	no	—
36	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
37	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
38	#13093935	—	2	4	no	—
39	C₄×C₄₇⋊C₈	—	2	4	no	—
40	C₄₇⋊#95	—	2	4	no	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈×C₄₇	—	2	8	no	—
43	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	complemented
44	#125×C₄₇	—	2	8	no	—
45	#125×C₄₇	C₂×C₄	1	8	yes	complemented
46	C₂²⋊Q₈×C₄₇	—	2	8	no	—
47	C₂²⋊Q₈×C₄₇	—	2	16	no	—
48	C₄×Q₈×C₄₇	—	2	8	no	—
49	#117×C₄₇	—	2	2	no	—
50	#117×C₄₇	—	2	2	no	—
51	#117×C₄₇	C₂×C₄	1	4	yes	—
52	#117×C₄₇	C₂×C₄	1	4	yes	—
53	#117×C₄₇	D₄	1	4	yes	—
54	#117×C₄₇	C₂³	1	4	yes	—
55	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
56	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
57	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
58	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
59	C₂×C₄²×C₄₇	—	2	2	no	—
60	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
61	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂×Q₈×C₄₇	—	2	8	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₄₇	#41	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	2	no	—
72	C₂²×Dic₄₇	—	2	2	no	—
73	C₂²×Dic₄₇	—	4	4	no	—
74	C₂²×Dic₄₇	—	4	2	no	—
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄⋊C₄×C₄₇	—	2	8	no	—
77	C₄⋊C₄×C₄₇	—	2	16	no	—
78	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
79	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
80	C₂²⋊C₄×C₄₇	—	2	4	no	—
81	C₂²⋊C₄×C₄₇	—	2	4	no	—
82	C₂²⋊C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
83	C₄²×C₄₇	—	2	4	no	—
84	C₄²×C₄₇	—	2	4	no	—
85	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
86	C₄²×C₄₇	C₂×D₄	1	4	yes	—
87	C₂²⋊Dic₄₇	—	4	2	no	—
88	C₄⋊Dic₄₇	—	4	8	no	—
89	C₄⋊Dic₄₇	—	4	2	no	—
90	C₄×Dic₄₇	—	2	4	no	—
91	C₄×Dic₄₇	—	2	4	no	—
92	C₄×Dic₄₇	—	4	2	no	—
93	#10625329	—	2	8	no	—
94	#10625329	—	2	4	no	—
95	#10625329	C₂×Q₈	1	8	yes	—
96	C₂×C₄₇⋊C₈	—	2	8	no	—
97	C₂×C₄₇⋊C₈	—	2	4	no	—
98	C₂³×C₄₇	—	4	2	no	—
99	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂³×C₄₇	#124	1	2	yes	—
101	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
102	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
103	Q₈×C₄₇	—	8	8	no	—
104	C₂×C₄×C₄₇	—	2	8	no	—
105	C₂×C₄×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	—	8	4	no	—
107	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
108	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
109	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
110	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
111	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
112	C₂×C₄×C₄₇	#124	1	2	yes	—
113	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
114	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
118	C₂×Dic₄₇	—	2	4	no	—
119	C₂×Dic₄₇	—	4	2	no	—
120	C₂×Dic₄₇	—	4	8	no	—
121	C₂×Dic₄₇	—	4	2	no	—
122	C₂×Dic₄₇	—	4	4	no	—
123	C₂×Dic₄₇	—	4	2	no	—
124	C₄₇⋊C₈	—	2	8	no	—
125	#21307	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	—	4	2	no	—
129	C₂²×C₄₇	—	4	2	no	—
130	C₂²×C₄₇	#391	1	4	yes	—
131	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
133	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
134	C₄×C₄₇	—	2	8	no	—
135	C₄×C₄₇	—	4	8	no	—
136	C₄×C₄₇	#391	1	4	yes	—
137	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
138	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
139	Dic₄₇	—	4	4	no	—
140	Dic₄₇	—	8	2	no	—
141	#3458	D₄₇	1	1	yes	characteristic
142	#2873	—	47	1	no	—
143	#2296	—	47	1	no	—
144	#1986	—	47	8	no	—
145	#1733	—	47	2	no	—
146	C₂×#339	—	47	2	no	—
147	C₂×C₄₇	—	2	1	no	—
148	C₂×C₄₇	—	2	1	no	—
149	C₂×C₄₇	—	2	4	no	—
150	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
151	C₂×C₄₇	#2901	1	2	yes	—
152	C₂²×M₁₆	—	47	2	no	—
153	#544	—	2	2	no	—
154	#544	Dic₄₇	1	8	yes	complemented
155	#543	—	2	2	no	—
156	#533	—	2	4	no	—
157	#533	C₂×D₄₇	1	4	yes	—
158	C₂×#117	—	47	1	no	—
159	C₂×#117	C₂×D₄₇	1	2	yes	—
160	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
161	C₂²×C₄²	—	47	1	no	—
162	#405	—	94	4	no	—
163	C₄×M₁₆	—	47	4	no	—
164	#389	—	94	2	no	—
165	#385	—	94	2	no	—
166	C₄×C₄⋊C₄	—	47	4	no	—
167	#339	—	47	8	no	—
168	#339	—	47	8	no	—
169	C₄₇	#21307	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	—	94	1	no	—
171	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
172	C₂×M₁₆	—	47	8	no	—
173	C₂×M₁₆	—	47	4	no	—
174	C₂×M₁₆	—	94	4	no	—
175	C₂×M₁₆	—	94	2	no	—
176	C₂²×C₈	—	94	2	no	—
177	C₄⋊Q₈	—	2	8	no	—
178	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	complemented
179	#125	—	2	8	no	—
180	#125	C₂×Dic₄₇	1	8	yes	complemented
181	C₂²⋊Q₈	—	2	8	no	—
182	C₂²⋊Q₈	—	2	16	no	—
183	C₄×Q₈	—	2	8	no	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	94	4	no	—
187	#117	C₂×Dic₄₇	1	4	yes	—
188	#117	C₂×Dic₄₇	1	4	yes	—
189	#117	C₂²⋊D₄₇	1	4	yes	—
190	#117	C₂²×D₄₇	1	4	yes	—
191	C₂×C₄⋊C₄	—	94	1	no	—
192	C₂×C₄⋊C₄	—	94	4	no	—
193	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
194	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	—	94	1	no	—
197	C₂×C₂²⋊C₄	C₂²⋊D₄₇	1	2	yes	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	47	2	no	—
200	C₂×C₄²	—	47	2	no	—
201	C₂×C₄²	—	94	4	no	—
202	C₂×C₄²	—	94	1	no	—
203	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
204	#97	—	94	4	no	—
205	C₄×C₈	—	94	4	no	—
206	#95	—	94	4	no	—
207	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
208	C₂×Q₈	—	2	8	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	94	2	no	—
211	C₂²×C₄	—	94	2	no	—
212	C₂²×C₄	—	188	4	no	—
213	C₂²×C₄	—	188	2	no	—
214	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
215	C₂²×C₄	#10625354	1	4	yes	—
216	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
217	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
218	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
220	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
221	M₁₆	—	47	8	no	—
222	M₁₆	—	94	8	no	—
223	M₁₆	—	94	4	no	—
224	C₂×C₈	—	94	8	no	—
225	C₂×C₈	—	94	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	2	16	no	—
229	C₄⋊C₄	—	188	8	no	—
230	C₄⋊C₄	—	188	2	no	—
231	C₄⋊C₄	C₂²⋊Dic₄₇	1	8	yes	—
232	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₂²⋊C₄	—	188	2	no	—
236	C₂²⋊C₄	C₂²⋊Dic₄₇	1	8	yes	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	94	4	no	—
240	C₄²	—	94	4	no	—
241	C₄²	—	188	2	no	—
242	C₄²	C₂²×Dic₄₇	1	4	yes	—
243	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
244	C₂³	—	4	2	no	—
245	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
246	C₂³	#13094013	1	2	yes	—
247	C₂³	#13094022	1	2	yes	—
248	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
249	Q₈	—	8	8	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	8	4	no	—
253	C₂×C₄	—	94	4	no	—
254	C₂×C₄	—	188	4	no	—
255	C₂×C₄	—	188	2	no	—
256	C₂×C₄	—	188	2	no	—
257	C₂×C₄	—	188	8	no	—
258	C₂×C₄	—	188	2	no	—
259	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	8	yes	—
260	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
261	C₂×C₄	Q₈×Dic₄₇	1	8	yes	—
262	C₂×C₄	#13094000	1	4	yes	—
263	C₂×C₄	#13094000	1	4	yes	—
264	C₂×C₄	#13094013	1	2	yes	—
265	C₂×C₄	#13094022	1	2	yes	—
266	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
268	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
269	C₂×C₄	C₂×#10625354	1	2	yes	—
270	C₈	—	94	8	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	#15689030	1	4	yes	—
276	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
277	C₂²	C₂×#13094013	1	1	yes	characteristic
278	C₂²	C₂×#13094022	1	1	yes	characteristic
279	C₄	—	2	8	no	—
280	C₄	—	4	8	no	—
281	C₄	—	376	2	no	—
282	C₄	—	188	4	no	—
283	C₄	#15689030	1	4	yes	—
284	C₄	C₂×Q₈×Dic₄₇	1	2	yes	—
285	C₄	C₂×#13094000	1	2	yes	—
286	C₂	—	2	1	no	—
287	C₂	—	2	4	no	—
288	C₂	—	2	1	no	—
289	C₂	C₂×#15689030	1	1	yes	characteristic
290	C₂	#20251935	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#3458	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#3458	D₄₇
hypocenter	C₄₇	#21307
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ #3458

Complements

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

N	H ≅ G/N	type
3. #20244333	C₂	semidirect
18. C₄₇×#544	C₄	semidirect
27. C₂×#10625329	Q₈	semidirect
43. C₄⋊Q₈×C₄₇	C₂×C₄	semidirect
45. #125×C₄₇	C₂×C₄	semidirect
154. #544	Dic₄₇	semidirect
169. C₄₇	#21307	semidirect
178. C₄⋊Q₈	C₂×Dic₄₇	semidirect
180. #125	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20251807	2
#20245872	2
#20244333	2
#20243097	2
C₂×#15688882	2
C₄₇×#3458	2
#21307	47