Subgroups — #26125205

Order 12032.

1050 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	C₂×#15688882	C₂	1	4	yes	maximal
2	#20243082	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄₇⋊#95	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₂²	1	2	yes	—
6	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
7	C₂×C₄₇⋊#95	—	2	2	no	—
8	#15688882	—	2	8	no	—
9	#15688878	—	2	2	no	—
10	C₄₇×#533	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₄₇	C₄	1	2	yes	complemented
12	C₂×#125×C₄₇	C₄	1	2	yes	complemented
13	C₂×C₂²⋊Q₈×C₄₇	—	2	2	no	—
14	C₂×C₄×Q₈×C₄₇	—	2	2	no	—
15	C₂×#117×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂³×Dic₄₇	—	2	2	no	—
18	C₂×#10625329	—	2	2	no	—
19	C₂×#10625329	—	2	4	no	—
20	C₂×#10625329	—	2	4	no	—
21	C₂×#10625329	Q₈	1	4	yes	complemented
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂×C₄×Dic₄₇	—	2	4	no	—
24	C₂×C₄×Dic₄₇	—	2	4	no	—
25	C₂×C₄×Dic₄₇	—	2	2	no	—
26	C₂×C₄×Dic₄₇	Q₈	1	4	yes	complemented
27	C₄₇⋊#95	—	2	8	no	—
28	C₂²×Q₈×C₄₇	—	2	2	no	—
29	C₂³×C₄×C₄₇	D₄	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₄₇	—	2	8	no	—
32	#125×C₄₇	—	2	8	no	—
33	C₂²⋊Q₈×C₄₇	—	2	16	no	—
34	C₄×Q₈×C₄₇	—	2	16	no	—
35	#117×C₄₇	—	2	4	no	—
36	#117×C₄₇	D₄	1	8	yes	—
37	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	—	2	4	no	—
42	C₂×C₄²×C₄₇	—	2	2	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	2	yes	—
46	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂×Q₈×C₄₇	—	2	8	no	—
48	C₂×Q₈×C₄₇	—	4	4	no	—
49	C₂×Q₈×C₄₇	—	4	4	no	—
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	—	4	2	no	—
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	2	yes	—
63	C₂²×Dic₄₇	—	2	8	no	—
64	C₂²×Dic₄₇	—	2	4	no	—
65	C₂²×Dic₄₇	—	4	4	no	—
66	C₂²×Dic₄₇	—	4	2	no	—
67	C₂²×Dic₄₇	—	4	2	no	—
68	C₄⋊C₄×C₄₇	—	2	16	no	—
69	C₄⋊C₄×C₄₇	—	2	8	no	—
70	C₄⋊C₄×C₄₇	—	2	4	no	—
71	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
72	C₂²⋊C₄×C₄₇	—	2	8	no	—
73	C₄²×C₄₇	—	2	2	no	—
74	C₄²×C₄₇	—	2	2	no	—
75	C₄²×C₄₇	—	2	4	no	—
76	C₄²×C₄₇	—	2	8	no	—
77	C₄×Dic₄₇	—	2	8	no	—
78	C₄×Dic₄₇	—	4	4	no	—
79	#10625329	—	2	8	no	—
80	#10625329	—	4	4	no	—
81	C₂×C₄₇⋊C₈	—	2	4	no	—
82	C₂×C₄₇⋊C₈	—	4	2	no	—
83	C₂×C₄₇⋊C₈	—	4	2	no	—
84	C₂³×C₄₇	—	2	2	no	—
85	C₂³×C₄₇	—	2	2	no	—
86	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂³×C₄₇	#120	1	2	yes	—
88	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
89	C₂³×C₄₇	#124	1	1	yes	characteristic
90	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
91	Q₈×C₄₇	—	4	16	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	8	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	4	8	no	—
99	C₂×C₄×C₄₇	—	4	4	no	—
100	C₂×C₄×C₄₇	—	4	4	no	—
101	C₂×C₄×C₄₇	#104	1	4	yes	—
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
105	C₂×C₄×C₄₇	#120	1	2	yes	—
106	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
107	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
108	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₇	#124	1	4	yes	—
110	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
112	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
113	C₂×Dic₄₇	—	2	8	no	—
114	C₂×Dic₄₇	—	4	8	no	—
115	C₂×Dic₄₇	—	4	4	no	—
116	C₂×Dic₄₇	—	4	8	no	—
117	C₂×Dic₄₇	—	4	4	no	—
118	C₄₇⋊C₈	—	4	4	no	—
119	#11386	—	47	1	no	maximal, Sylow
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	2	2	no	—
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	2	4	no	—
125	C₂²×C₄₇	#379	1	1	yes	characteristic
126	C₂²×C₄₇	#391	1	1	yes	characteristic
127	C₂²×C₄₇	#391	1	1	yes	characteristic
128	C₂²×C₄₇	#393	1	2	yes	—
129	C₂²×C₄₇	#395	1	2	yes	—
130	C₂²×C₄₇	C₂×#104	1	2	yes	—
131	C₂²×C₄₇	#421	1	2	yes	—
132	C₄×C₄₇	—	2	4	no	—
133	C₄×C₄₇	—	4	2	no	—
134	C₄×C₄₇	—	4	8	no	—
135	C₄×C₄₇	#391	1	4	yes	—
136	C₄×C₄₇	#393	1	2	yes	—
137	C₄×C₄₇	#395	1	2	yes	—
138	Dic₄₇	—	4	8	no	—
139	C₂×#533	D₄₇	1	1	yes	characteristic
140	C₂×#339	—	47	4	no	—
141	#1728	—	47	1	no	—
142	C₄×#95	—	47	1	no	—
143	C₂×C₄₇	—	2	2	no	—
144	C₂×C₄₇	—	2	2	no	—
145	C₂×C₄₇	#1446	1	1	yes	characteristic
146	C₂×C₄₇	#1789	1	2	yes	—
147	C₂×C₄₇	#1986	1	2	yes	—
148	C₂×C₄₇	#1994	1	1	yes	characteristic
149	C₂×C₄₇	#1995	1	1	yes	characteristic
150	C₂²×M₁₆	—	47	2	no	—
151	#533	—	2	8	no	—
152	C₂×C₄⋊Q₈	Dic₄₇	1	2	yes	complemented
153	C₂×#125	Dic₄₇	1	2	yes	complemented
154	C₂×C₂²⋊Q₈	—	2	2	no	—
155	C₂×C₄×Q₈	—	2	2	no	—
156	C₂×#117	C₂×D₄₇	1	2	yes	—
157	C₂²×C₄²	—	47	2	no	—
158	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
159	C₂×#95	—	94	2	no	—
160	#339	—	94	8	no	—
161	#336	—	94	2	no	—
162	C₄₇	#11386	1	1	yes	characteristic, complemented, Sylow
163	C₂²×Q₈	—	2	2	no	—
164	C₂³×C₄	—	94	2	no	—
165	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
166	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
167	C₂×M₁₆	—	47	4	no	—
168	C₂×M₁₆	—	94	2	no	—
169	C₂×M₁₆	—	94	4	no	—
170	C₂×M₁₆	—	94	4	no	—
171	C₂²×C₈	—	94	2	no	—
172	C₄⋊Q₈	—	2	8	no	—
173	#125	—	2	8	no	—
174	C₂²⋊Q₈	—	2	16	no	—
175	C₄×Q₈	—	2	16	no	—
176	#117	—	2	4	no	—
177	#117	C₂²⋊D₄₇	1	8	yes	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
181	C₂×C₂²⋊C₄	—	2	2	no	—
182	C₂×C₄²	—	2	4	no	—
183	C₂×C₄²	—	2	2	no	—
184	C₂×C₄²	—	47	4	no	—
185	C₂×C₄²	—	94	4	no	—
186	C₂×C₄²	—	94	4	no	—
187	C₂×C₄²	—	94	2	no	—
188	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
189	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
191	#95	—	94	8	no	—
192	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
193	C₂×Q₈	—	2	8	no	—
194	C₂×Q₈	—	4	4	no	—
195	C₂×Q₈	—	4	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	94	8	no	—
201	C₂²×C₄	—	94	4	no	—
202	C₂²×C₄	—	188	4	no	—
203	C₂²×C₄	—	188	2	no	—
204	C₂²×C₄	—	188	2	no	—
205	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
206	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
207	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
208	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
209	C₂²×C₄	#10625354	1	2	yes	—
210	C₂²×C₄	#10625354	1	2	yes	—
211	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
213	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
214	M₁₆	—	94	8	no	—
215	M₁₆	—	188	4	no	—
216	C₂×C₈	—	94	4	no	—
217	C₂×C₈	—	188	2	no	—
218	C₂×C₈	—	188	2	no	—
219	C₄⋊C₄	—	2	16	no	—
220	C₄⋊C₄	—	2	8	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	C₂²⋊Dic₄₇	1	8	yes	—
223	C₂²⋊C₄	—	2	8	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	4	no	—
227	C₄²	—	2	8	no	—
228	C₄²	—	94	8	no	—
229	C₄²	—	188	4	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
233	C₂³	#13093993	1	2	yes	—
234	C₂³	#13094000	1	2	yes	—
235	C₂³	#13094013	1	1	yes	characteristic
236	C₂³	#13094022	1	1	yes	characteristic
237	Q₈	—	4	16	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	8	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	94	8	no	—
248	C₂×C₄	—	188	8	no	—
249	C₂×C₄	—	188	8	no	—
250	C₂×C₄	—	188	4	no	—
251	C₂×C₄	—	188	4	no	—
252	C₂×C₄	#13093951	1	4	yes	—
253	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
254	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
255	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
256	C₂×C₄	#13093993	1	2	yes	—
257	C₂×C₄	#13094000	1	4	yes	—
258	C₂×C₄	#13094000	1	4	yes	—
259	C₂×C₄	#13094000	1	2	yes	—
260	C₂×C₄	#13094013	1	4	yes	—
261	C₂×C₄	#13094013	1	1	yes	characteristic
262	C₂×C₄	#13094022	1	4	yes	—
263	C₂×C₄	#13094022	1	1	yes	characteristic
264	C₈	—	188	4	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	#15688979	1	1	yes	characteristic
271	C₂²	#15689030	1	1	yes	characteristic
272	C₂²	#15689030	1	1	yes	characteristic
273	C₂²	#15689034	1	2	yes	—
274	C₂²	#15689042	1	2	yes	—
275	C₂²	C₂×#13093951	1	2	yes	—
276	C₂²	#15689140	1	2	yes	—
277	C₄	—	2	4	no	—
278	C₄	—	4	2	no	—
279	C₄	—	4	8	no	—
280	C₄	—	188	8	no	—
281	C₄	#15689030	1	4	yes	—
282	C₄	#15689034	1	2	yes	—
283	C₄	#15689042	1	2	yes	—
284	C₂	—	2	2	no	—
285	C₂	—	2	2	no	—
286	C₂	#20241659	1	1	yes	characteristic
287	C₂	#20243332	1	2	yes	—
288	C₂	#20244333	1	2	yes	—
289	C₂	#20244366	1	1	yes	characteristic
290	C₂	#20244370	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#533	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	#11386
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
11. C₂×C₄⋊Q₈×C₄₇	C₄	semidirect
12. C₂×#125×C₄₇	C₄	semidirect
21. C₂×#10625329	Q₈	semidirect
26. C₂×C₄×Dic₄₇	Q₈	semidirect
152. C₂×C₄⋊Q₈	Dic₄₇	semidirect
153. C₂×#125	Dic₄₇	semidirect
162. C₄₇	#11386	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688882	2
#20243082	2
C₄×C₄₇⋊#95	2
C₂×C₄₇×#533	2
#11386	47