Subgroups — #26219638

Order 12032.

982 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689161	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093956	C₂	1	1	yes	maximal, characteristic
3	#20246008	C₂	1	2	yes	maximal
4	#20244126	C₂	1	4	yes	maximal, complemented
5	#20244116	C₂	1	4	yes	maximal, complemented
6	#20243108	C₂	1	2	yes	maximal
7	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
8	C₂×#13093983	C₂²	1	2	yes	—
9	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	#15689161	C₄	1	8	yes	complemented
11	C₂×#13093958	C₂²	1	2	yes	—
12	C₂×#13093956	—	2	4	no	—
13	C₂×#13093956	C₂²	1	2	yes	—
14	C₂×#13093956	C₂²	1	2	yes	—
15	C₂×#13093956	C₂²	1	2	yes	—
16	#15689021	C₂²	1	8	yes	—
17	#15688995	—	2	4	no	—
18	#15688992	—	2	2	no	—
19	#15688988	—	2	2	no	—
20	#15688886	C₂²	1	8	yes	—
21	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
22	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#405	—	2	4	no	—
25	C₂×#97×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
27	C₂³×Dic₄₇	—	2	2	no	—
28	#13093983	—	2	4	no	—
29	#13093983	C₂×C₄	1	8	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×Dic₄₇⋊C₄	—	2	8	no	—
36	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	—	2	2	no	—
38	#13093958	C₂×C₄	1	8	yes	complemented
39	#13093956	—	2	4	no	—
40	#13093956	—	2	4	no	—
41	#13093956	C₂×C₄	1	8	yes	complemented
42	C₄₇⋊#95	—	2	8	no	—
43	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₄₇	—	2	2	no	—
45	C₂×M₁₆×C₄₇	—	2	2	no	—
46	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
47	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₄₇	D₄	1	4	yes	—
49	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
50	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	#117×C₄₇	—	2	2	no	—
54	#117×C₄₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
56	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
58	#97×C₄₇	—	2	4	no	—
59	C₄×C₈×C₄₇	—	2	4	no	—
60	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₇	#41	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	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	C₂²×Dic₄₇	—	4	4	no	—
73	C₂²×Dic₄₇	—	8	4	no	—
74	M₁₆×C₄₇	—	2	8	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	C₄⋊C₄	1	16	yes	complemented
78	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
83	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
85	C₄⋊C₄×C₄₇	—	2	4	no	—
86	C₂²⋊C₄×C₄₇	—	2	4	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	—	2	2	no	—
89	C₂²⋊Dic₄₇	—	2	8	no	—
90	C₄⋊Dic₄₇	—	2	4	no	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	C₄⋊Dic₄₇	—	4	4	no	—
94	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
95	Dic₄₇⋊C₄	—	8	8	no	—
96	C₄×Dic₄₇	—	2	8	no	—
97	C₂³×C₄₇	—	2	4	no	—
98	C₂³×C₄₇	C₄×D₄	1	2	yes	—
99	C₂³×C₄₇	#121	1	2	yes	—
100	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
101	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	8	no	—
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
105	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
107	C₂×C₄×C₄₇	#121	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
109	C₂×C₄×C₄₇	#125	1	4	yes	—
110	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
111	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
117	C₂×Dic₄₇	—	2	8	no	—
118	C₂×Dic₄₇	—	4	4	no	—
119	C₂×Dic₄₇	—	4	4	no	—
120	C₂×Dic₄₇	—	4	4	no	—
121	C₂×Dic₄₇	—	4	8	no	—
122	C₂×Dic₄₇	—	8	8	no	—
123	C₈×C₄₇	—	2	4	no	—
124	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
125	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
126	#21155	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	—	2	4	no	—
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	#389	1	4	yes	—
131	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
133	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
134	C₂²×C₄₇	#577	1	4	yes	—
135	C₄×C₄₇	—	4	4	no	—
136	C₄×C₄₇	#389	1	4	yes	—
137	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
139	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
140	C₄×C₄₇	C₂×#125	1	1	yes	characteristic
141	Dic₄₇	—	4	8	no	—
142	Dic₄₇	—	8	4	no	—
143	C₂×#427	—	47	1	no	—
144	C₂²×#106	—	47	1	no	—
145	C₂×#405	D₄₇	1	1	yes	characteristic
146	#2317	—	47	2	no	—
147	#1943	—	47	4	no	—
148	#1941	—	47	4	no	—
149	#1736	—	47	2	no	—
150	C₂×C₄₇	—	2	4	no	—
151	C₂×C₄₇	C₂×#389	1	1	yes	characteristic
152	C₂×C₄₇	#2952	1	2	yes	—
153	C₂×C₄₇	#3057	1	2	yes	—
154	C₂×C₄₇	#3163	1	2	yes	—
155	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
156	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
157	C₂×#117	—	47	2	no	—
158	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
159	C₂²×C₄⋊C₄	—	47	2	no	—
160	#427	—	47	8	no	—
161	C₂×#107	—	47	2	no	—
162	C₂×#106	—	47	2	no	—
163	C₂×#106	—	47	2	no	—
164	C₂×#106	—	47	2	no	—
165	C₂×#106	—	94	4	no	—
166	#405	—	2	4	no	—
167	C₂×#97	C₂×D₄₇	1	2	yes	—
168	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
169	#389	—	47	8	no	—
170	#385	—	94	4	no	—
171	#384	—	94	2	no	—
172	#382	—	94	2	no	—
173	#340	—	47	8	no	—
174	C₄₇	#21155	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₄	—	94	2	no	—
176	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	Dic₉₄	1	4	yes	—
180	C₂²×C₈	Dic₉₄	1	4	yes	—
181	C₂²×C₈	C₄×D₄₇	1	4	yes	—
182	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
183	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
184	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	—	47	8	no	—
189	#117	—	94	4	no	—
190	C₂×C₄⋊C₄	—	47	4	no	—
191	C₂×C₄⋊C₄	—	47	4	no	—
192	C₂×C₄⋊C₄	—	94	4	no	—
193	C₂×C₄⋊C₄	—	94	4	no	—
194	C₂×C₄⋊C₄	—	94	8	no	—
195	C₂×C₄⋊C₄	—	94	2	no	—
196	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
197	C₂×C₂²⋊C₄	—	94	2	no	—
198	C₂×C₂²⋊C₄	D₁₈₈	1	2	yes	—
199	C₂×C₄²	—	94	2	no	—
200	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
201	#107	—	47	8	no	—
202	#106	—	47	8	no	—
203	#106	—	94	4	no	—
204	#106	—	94	4	no	—
205	#97	—	2	4	no	—
206	C₄×C₈	—	2	4	no	—
207	#95	—	94	8	no	—
208	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
209	C₂²×C₄	—	94	4	no	—
210	C₂²×C₄	—	94	4	no	—
211	C₂²×C₄	—	376	4	no	—
212	C₂²×C₄	—	188	4	no	—
213	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
214	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
215	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
216	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
217	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
218	C₂²×C₄	#10625350	1	2	yes	—
219	C₂²×C₄	#10625350	1	2	yes	—
220	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
221	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
222	M₁₆	—	2	8	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	Dic₄₇⋊C₄	1	16	yes	complemented
226	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
227	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
228	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
229	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
230	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
231	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
232	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	47	8	no	—
235	C₄⋊C₄	—	94	4	no	—
236	C₄⋊C₄	—	94	4	no	—
237	C₄⋊C₄	—	94	4	no	—
238	C₄⋊C₄	—	376	8	no	—
239	C₄⋊C₄	—	188	4	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	94	8	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	94	8	no	—
245	C₂³	—	2	4	no	—
246	C₂³	C₄×D₁₈₈	1	2	yes	—
247	C₂³	#13093998	1	2	yes	—
248	C₂³	#13094024	1	2	yes	—
249	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	94	8	no	—
253	C₂×C₄	—	376	8	no	—
254	C₂×C₄	—	188	4	no	—
255	C₂×C₄	—	188	8	no	—
256	C₂×C₄	—	188	4	no	—
257	C₂×C₄	—	188	4	no	—
258	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
259	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
260	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
261	C₂×C₄	#13093998	1	2	yes	—
262	C₂×C₄	#13094003	1	4	yes	—
263	C₂×C₄	#13094016	1	4	yes	—
264	C₂×C₄	#13094024	1	2	yes	—
265	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
267	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
268	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
269	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
270	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
271	C₈	—	2	4	no	—
272	C₈	C₂×Dic₄₇⋊C₄	1	4	yes	—
273	C₈	C₂×Dic₄₇⋊C₄	1	4	yes	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	4	no	—
277	C₂²	#15689021	1	4	yes	—
278	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
279	C₂²	C₂×#13093998	1	1	yes	characteristic
280	C₂²	C₂×#13094024	1	1	yes	characteristic
281	C₂²	#15689935	1	4	yes	—
282	C₄	—	4	4	no	—
283	C₄	—	376	4	no	—
284	C₄	—	188	8	no	—
285	C₄	#15689021	1	4	yes	—
286	C₄	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
287	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
288	C₄	C₂×#13094003	1	1	yes	characteristic
289	C₄	C₂×#13094016	1	1	yes	characteristic
290	C₂	—	2	4	no	—
291	C₂	C₂×#15689021	1	1	yes	characteristic
292	C₂	#20252273	1	2	yes	—
293	C₂	#20253027	1	2	yes	—
294	C₂	#20253919	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#405	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	#21155
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #20244126	C₂	semidirect
5. #20244116	C₂	semidirect
10. #15689161	C₄	semidirect
38. #13093958	C₂×C₄	semidirect
41. #13093956	C₂×C₄	semidirect
77. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
174. C₄₇	#21155	semidirect
225. C₂×C₈	Dic₄₇⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689161	2
C₂²×#13093956	2
#20246008	2
#20244126	2
#20244116	2
#20243108	2
C₂×C₄₇×#405	2
#21155	47