Subgroups — #26369706

Order 12032.

642 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689555	C₂	1	1	yes	maximal, characteristic
2	#20252306	C₂	1	2	yes	maximal, complemented
3	#20245275	C₂	1	4	yes	maximal, complemented
4	#20245270	C₂	1	4	yes	maximal, complemented
5	C₂×#15688953	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245117	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094046	C₂²	1	2	yes	—
9	#15689889	—	2	2	no	—
10	#15689555	C₄	1	2	yes	complemented
11	#15689555	C₄	1	2	yes	complemented
12	#15689555	C₂²	1	2	yes	—
13	#15689555	C₂²	1	2	yes	—
14	C₂×Dic₃₇₆	C₄	1	1	yes	characteristic, complemented
15	C₂×#13093969	C₄	1	2	yes	complemented
16	C₂×D₇₅₂	C₄	1	1	yes	characteristic, complemented
17	#15689246	—	2	4	no	—
18	#15689241	C₂²	1	8	yes	complemented
19	C₂×#13093960	C₂²	1	2	yes	—
20	#15689141	—	2	2	no	—
21	#15688953	C₂²	1	2	yes	complemented
22	#15688953	C₂²	1	2	yes	complemented
23	#15688931	C₂²	1	8	yes	complemented
24	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
26	C₄₇×#405	C₂²	1	1	yes	characteristic
27	C₄₇×#346	C₂²	1	2	yes	complemented
28	C₄×C₁₆×C₄₇	C₂²	1	2	yes	complemented
29	C₂×#10625350	—	2	2	no	—
30	#13094046	—	2	2	no	—
31	#13094046	—	2	2	no	—
32	#13094046	C₂×C₄	1	4	yes	—
33	#13094046	C₂³	1	4	yes	—
34	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
35	C₂×#10625325	—	2	2	no	—
36	C₂×D₃₇₆	C₂×C₄	1	2	yes	—
37	#13094032	—	4	4	no	—
38	#13094030	—	2	8	no	—
39	C₂×#10625329	—	2	2	no	—
40	M₁₆×D₄₇	—	4	4	no	—
41	Dic₃₇₆	C₂×C₄	1	2	yes	—
42	Dic₃₇₆	C₂×C₄	1	2	yes	—
43	#13093969	C₂×C₄	1	4	yes	—
44	#13093969	C₂×C₄	1	4	yes	—
45	D₇₅₂	C₂×C₄	1	2	yes	—
46	D₇₅₂	C₂×C₄	1	2	yes	—
47	#13093960	—	2	2	no	—
48	#13093960	C₂³	1	4	yes	—
49	#13093952	—	2	8	no	—
50	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	#117×C₄₇	D₄	1	1	yes	characteristic
53	M₃₂×C₄₇	—	2	1	no	—
54	M₃₂×C₄₇	—	2	1	no	—
55	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
60	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
62	#97×C₄₇	D₄	1	2	yes	—
63	C₄×C₈×C₄₇	C₂³	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₂²⋊D₄₇	—	4	2	no	—
66	#10625350	—	2	4	no	—
67	#10625350	—	2	4	no	—
68	#10625350	—	4	2	no	—
69	#10625350	—	4	2	no	—
70	C₂×D₁₈₈	—	2	2	no	—
71	C₂×C₄×D₄₇	—	4	2	no	—
72	C₂×Dic₉₄	—	2	2	no	—
73	M₁₆×C₄₇	D₈	1	2	yes	—
74	M₁₆×C₄₇	D₈	1	2	yes	—
75	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
80	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
81	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₄₇	—	2	1	no	—
83	C₂²⋊C₄×C₄₇	—	2	1	no	—
84	C₄²×C₄₇	C₂×D₄	1	2	yes	—
85	#10625329	—	2	4	no	—
86	#10625329	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	Dic₁₈₈	—	2	2	no	—
89	Dic₁₈₈	C₂²×C₄	1	4	yes	—
90	D₃₇₆	—	2	2	no	—
91	D₃₇₆	C₂²×C₄	1	4	yes	—
92	#10625325	—	2	4	no	—
93	#10625325	—	2	4	no	—
94	#10625324	—	4	8	no	—
95	C₈×D₄₇	—	4	8	no	—
96	C₁₆×C₄₇	—	2	2	no	—
97	C₁₆×C₄₇	C₂²×C₄	1	1	yes	characteristic
98	C₁₆×C₄₇	C₂²×C₄	1	1	yes	characteristic
99	C₁₆×C₄₇	C₂²×C₄	1	1	yes	characteristic
100	C₁₆×C₄₇	C₂²×C₄	1	1	yes	characteristic
101	C₂³×C₄₇	#135	1	1	yes	characteristic
102	C₂²×D₄₇	—	4	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
110	C₂²⋊D₄₇	—	4	4	no	—
111	C₂²⋊D₄₇	—	4	4	no	—
112	C₂×Dic₄₇	—	4	2	no	—
113	D₁₈₈	—	2	4	no	—
114	D₁₈₈	—	4	2	no	—
115	C₄×D₄₇	—	4	4	no	—
116	C₄×D₄₇	—	4	4	no	—
117	Dic₉₄	—	2	4	no	—
118	Dic₉₄	—	4	2	no	—
119	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
120	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
121	C₈×C₄₇	C₂×D₈	1	4	yes	—
122	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
123	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
124	C₄₇⋊C₈	—	4	4	no	—
125	#33465	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	4	1	no	—
127	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
128	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
129	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
130	C₂×D₄₇	—	4	4	no	—
131	C₂×D₄₇	—	16	2	no	—
132	C₄×C₄₇	—	4	2	no	—
133	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
134	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
135	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
137	Dic₄₇	—	4	4	no	—
138	C₂×#508	—	47	1	no	—
139	#2957	—	47	2	no	—
140	#2179	—	47	4	no	—
141	#2178	—	47	4	no	—
142	C₂×#368	—	47	1	no	—
143	#2143	—	47	2	no	—
144	#2108	D₄₇	1	1	yes	characteristic, complemented
145	C₂×C₄₇	—	2	1	no	—
146	C₂×C₄₇	—	2	1	no	—
147	C₂×C₄₇	—	2	1	no	—
148	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
149	D₄₇	—	8	4	no	—
150	C₂×#135	—	47	2	no	—
151	#568	—	94	2	no	—
152	#508	—	47	2	no	—
153	#508	—	47	2	no	—
154	#508	—	47	2	no	—
155	#508	—	47	2	no	—
156	C₂×Q₃₂	—	47	1	no	—
157	C₂×SD₃₂	—	47	2	no	—
158	C₂×D₁₆	—	47	1	no	—
159	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
160	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
161	#444	—	94	4	no	—
162	#443	—	47	8	no	—
163	C₂×#108	—	47	2	no	—
164	#421	—	94	2	no	—
165	#405	C₂×D₄₇	1	1	yes	characteristic
166	#368	—	47	2	no	—
167	#368	—	47	2	no	—
168	#359	—	47	8	no	—
169	#346	C₂×D₄₇	1	2	yes	complemented
170	C₄×C₁₆	C₂×D₄₇	1	2	yes	complemented
171	C₄₇	#33465	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	94	2	no	—
173	#135	—	47	4	no	—
174	#135	—	47	4	no	—
175	#135	—	94	2	no	—
176	#135	—	94	2	no	—
177	C₂×Q₁₆	—	47	2	no	—
178	C₂×SD₁₆	—	94	2	no	—
179	C₂×D₈	—	47	2	no	—
180	#131	—	94	8	no	—
181	#131	—	188	4	no	—
182	C₂×M₁₆	—	94	2	no	—
183	C₂×M₁₆	—	188	4	no	—
184	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
186	#117	D₁₈₈	1	1	yes	characteristic
187	Q₃₂	—	47	2	no	—
188	Q₃₂	—	47	2	no	—
189	SD₃₂	—	47	4	no	—
190	SD₃₂	—	47	4	no	—
191	D₁₆	—	47	2	no	—
192	D₁₆	—	47	2	no	—
193	M₃₂	—	2	1	no	—
194	M₃₂	—	2	1	no	—
195	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
196	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
197	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
198	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
199	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
200	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
201	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
202	#108	—	47	4	no	—
203	#108	—	94	2	no	—
204	#104	—	94	8	no	—
205	#97	D₁₈₈	1	2	yes	—
206	C₄×C₈	C₂²×D₄₇	1	2	yes	—
207	#41	—	94	4	no	—
208	#41	—	94	4	no	—
209	#41	—	188	2	no	—
210	#41	—	188	2	no	—
211	C₂×Q₈	—	94	2	no	—
212	C₂×D₄	—	94	2	no	—
213	C₂×D₄	—	188	2	no	—
214	C₂²×C₄	—	188	2	no	—
215	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
216	Q₁₆	—	47	4	no	—
217	Q₁₆	—	94	2	no	—
218	SD₁₆	—	94	4	no	—
219	SD₁₆	—	94	4	no	—
220	D₈	—	47	4	no	—
221	D₈	—	94	2	no	—
222	M₁₆	—	94	4	no	—
223	M₁₆	—	188	2	no	—
224	M₁₆	—	188	8	no	—
225	M₁₆	D₃₇₆	1	2	yes	—
226	M₁₆	D₃₇₆	1	2	yes	—
227	C₂×C₈	—	188	2	no	—
228	C₂×C₈	—	188	8	no	—
229	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
231	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
232	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
233	C₂×C₈	#10625350	1	1	yes	characteristic
234	C₂×C₈	#10625350	1	1	yes	characteristic
235	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	2	1	no	—
238	C₄²	C₂×D₁₈₈	1	2	yes	—
239	C₁₆	—	2	2	no	—
240	C₁₆	C₂×C₄×D₄₇	1	1	yes	characteristic
241	C₁₆	C₂×C₄×D₄₇	1	1	yes	characteristic
242	C₁₆	C₂×C₄×D₄₇	1	1	yes	characteristic
243	C₁₆	C₂×C₄×D₄₇	1	1	yes	characteristic
244	C₂³	—	188	2	no	—
245	C₂³	#13094046	1	1	yes	characteristic
246	Q₈	—	94	4	no	—
247	Q₈	—	188	2	no	—
248	D₄	—	94	4	no	—
249	D₄	—	188	4	no	—
250	D₄	—	188	4	no	—
251	D₄	—	188	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	188	2	no	—
254	C₂×C₄	—	188	4	no	—
255	C₂×C₄	—	188	4	no	—
256	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
257	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
258	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
259	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
260	C₂×C₄	#13094046	1	1	yes	characteristic
261	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
262	C₈	—	188	4	no	—
263	C₈	C₄×D₁₈₈	1	1	yes	characteristic
264	C₈	C₄×D₁₈₈	1	1	yes	characteristic
265	C₈	C₂×D₃₇₆	1	4	yes	—
266	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
267	C₈	C₂×#10625350	1	1	yes	characteristic
268	C₂²	—	4	1	no	—
269	C₂²	—	188	4	no	—
270	C₂²	—	752	2	no	—
271	C₂²	C₄×D₃₇₆	1	1	yes	characteristic
272	C₂²	C₄×D₃₇₆	1	1	yes	characteristic
273	C₂²	C₂×#13094046	1	1	yes	characteristic
274	C₄	—	4	2	no	—
275	C₄	—	188	4	no	—
276	C₄	C₄×D₃₇₆	1	1	yes	characteristic
277	C₄	C₄×D₃₇₆	1	1	yes	characteristic
278	C₄	C₂×C₄×D₁₈₈	1	1	yes	characteristic
279	C₄	C₂²×D₃₇₆	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	376	4	no	—
284	C₂	C₂×C₄×D₃₇₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₃₇₆
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2108	C₂
socle	C₂×C₄₇	C₂×C₄×D₈
prime radical	C₂×#15689555	C₂
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2108	D₄₇
hypocenter	C₄₇	#33465
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #20252306	C₂	semidirect
3. #20245275	C₂	semidirect
4. #20245270	C₂	semidirect
5. C₂×#15688953	C₂	semidirect
6. #20245117	C₂	semidirect
7. C₄₇×#2108	C₂	semidirect
10. #15689555	C₄	semidirect
11. #15689555	C₄	semidirect
14. C₂×Dic₃₇₆	C₄	semidirect
15. C₂×#13093969	C₄	semidirect
16. C₂×D₇₅₂	C₄	semidirect
18. #15689241	C₂²	semidirect
21. #15688953	C₂²	semidirect
22. #15688953	C₂²	semidirect
23. #15688931	C₂²	semidirect
27. C₄₇×#346	C₂²	semidirect
28. C₄×C₁₆×C₄₇	C₂²	semidirect
144. #2108	D₄₇	semidirect
169. #346	C₂×D₄₇	semidirect
170. C₄×C₁₆	C₂×D₄₇	semidirect
171. C₄₇	#33465	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689555	2
#20252306	2
#20245275	2
#20245270	2
C₂×#15688953	2
#20245117	2
C₄₇×#2108	2
#33465	47