Subgroups — #26369649

Order 12032.

682 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093969	C₂	1	1	yes	maximal, characteristic
2	#20252257	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252241	C₂	1	1	yes	maximal, characteristic
4	#20245254	C₂	1	4	yes	maximal, complemented
5	C₄×#13093969	C₂	1	4	yes	maximal, complemented
6	#20245172	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245109	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245105	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#2108	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
11	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
12	#15689600	—	2	1	no	—
13	#15689597	—	2	1	no	—
14	C₂×#13093969	C₄	1	8	yes	complemented
15	C₂×#13093969	C₂²	1	2	yes	—
16	C₂×#13093969	C₂²	1	2	yes	—
17	#15689236	—	2	2	no	—
18	#15689231	—	2	2	no	—
19	C₄×Dic₁₈₈	C₂²	1	4	yes	complemented
20	C₄×D₃₇₆	C₂²	1	4	yes	—
21	#15689161	C₂²	1	1	yes	characteristic
22	#15689161	C₂²	1	1	yes	characteristic
23	#15689133	—	2	1	no	—
24	#15689129	—	2	1	no	—
25	#15688951	C₂²	1	2	yes	complemented
26	#15688949	C₂²	1	2	yes	complemented
27	#15688928	C₂²	1	4	yes	complemented
28	#15688925	C₂²	1	4	yes	—
29	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
30	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#405	C₂²	1	1	yes	characteristic
32	C₄₇×#346	C₂²	1	2	yes	complemented
33	C₄×C₁₆×C₄₇	C₂²	1	2	yes	complemented
34	C₂²×Dic₉₄	—	2	1	no	—
35	C₂²×D₁₈₈	—	2	1	no	—
36	C₂×Dic₁₈₈	—	2	1	no	—
37	C₂×Dic₁₈₈	—	2	2	no	—
38	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	complemented
39	C₂×Dic₁₈₈	C₂³	1	2	yes	—
40	C₂×D₃₇₆	—	2	1	no	—
41	C₂×D₃₇₆	—	2	2	no	—
42	C₂×D₃₇₆	C₂×C₄	1	4	yes	—
43	C₂×D₃₇₆	C₂³	1	2	yes	—
44	C₄×Dic₉₄	—	2	4	no	—
45	#13093989	—	4	2	no	—
46	C₄×D₁₈₈	—	2	4	no	—
47	#13093986	—	4	2	no	—
48	#13093983	—	2	1	no	—
49	#13093983	—	2	1	no	—
50	#13093980	—	4	2	no	—
51	C₂²⋊C₄×D₄₇	—	4	2	no	—
52	#13093969	C₂×C₄	1	8	yes	complemented
53	#13093969	C₂×C₄	1	8	yes	complemented
54	#13093958	C₂³	1	2	yes	—
55	#13093958	C₂³	1	2	yes	—
56	#13093956	—	2	1	no	—
57	#13093956	—	2	1	no	—
58	#13093949	—	2	4	no	—
59	#13093946	—	2	4	no	—
60	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	#117×C₄₇	D₄	1	1	yes	characteristic
63	M₃₂×C₄₇	—	2	2	no	—
64	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
65	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
66	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
68	#97×C₄₇	D₄	1	2	yes	—
69	C₄×C₈×C₄₇	C₂³	1	2	yes	—
70	C₂³×D₄₇	—	4	1	no	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	4	1	no	—
73	C₂×D₁₈₈	—	2	4	no	—
74	C₂×D₁₈₈	—	2	2	no	—
75	C₂×D₁₈₈	—	4	1	no	—
76	C₂×D₁₈₈	—	4	2	no	—
77	C₂×C₄×D₄₇	—	4	4	no	—
78	C₂×Dic₉₄	—	2	4	no	—
79	C₂×Dic₉₄	—	2	2	no	—
80	C₂×Dic₉₄	—	4	1	no	—
81	C₂×Dic₉₄	—	4	2	no	—
82	M₁₆×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₇	#41	1	2	yes	—
87	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₄₇	D₈	1	2	yes	—
89	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
90	C₄²×C₄₇	C₂×D₄	1	2	yes	—
91	C₂²⋊Dic₄₇	—	4	1	no	—
92	C₂²⋊Dic₄₇	—	4	1	no	—
93	(C₂×D₄₇)⋊C₄	—	4	4	no	—
94	C₄⋊Dic₄₇	—	2	2	no	—
95	C₄⋊Dic₄₇	—	2	2	no	—
96	Dic₄₇⋊C₄	—	4	4	no	—
97	C₄×Dic₄₇	—	4	4	no	—
98	C₄×Dic₄₇	—	4	1	no	—
99	C₄×Dic₄₇	—	4	1	no	—
100	Dic₁₈₈	—	2	4	no	—
101	Dic₁₈₈	—	2	2	no	—
102	Dic₁₈₈	C₂²×C₄	1	4	yes	—
103	D₃₇₆	—	2	4	no	—
104	D₃₇₆	—	2	2	no	—
105	D₃₇₆	C₂²×C₄	1	4	yes	—
106	C₁₆×C₄₇	—	2	2	no	—
107	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
108	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
109	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
110	C₂²×D₄₇	—	4	2	no	—
111	C₂²×D₄₇	—	4	4	no	—
112	C₂²×D₄₇	—	16	2	no	—
113	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
115	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#135	1	2	yes	—
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×Dic₄₇	—	4	2	no	—
119	C₂×Dic₄₇	—	4	4	no	—
120	C₂×Dic₄₇	—	4	2	no	—
121	C₂×Dic₄₇	—	4	2	no	—
122	D₁₈₈	—	2	4	no	—
123	D₁₈₈	—	4	2	no	—
124	D₁₈₈	—	4	4	no	—
125	C₄×D₄₇	—	16	4	no	—
126	Dic₉₄	—	2	4	no	—
127	Dic₉₄	—	4	2	no	—
128	Dic₉₄	—	4	4	no	—
129	C₈×C₄₇	—	2	2	no	—
130	C₈×C₄₇	C₄×D₄	1	2	yes	—
131	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
132	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
133	#33457	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	C₄×D₈	1	2	yes	—
136	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
137	C₂×D₄₇	—	4	4	no	—
138	C₂×D₄₇	—	8	4	no	—
139	C₂×D₄₇	—	16	4	no	—
140	C₄×C₄₇	—	2	4	no	—
141	C₄×C₄₇	C₄×D₈	1	2	yes	—
142	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
143	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
144	Dic₄₇	—	4	4	no	—
145	Dic₄₇	—	8	2	no	—
146	Dic₄₇	—	8	2	no	—
147	C₂²×SD₃₂	—	47	1	no	—
148	#2950	—	47	1	no	—
149	#2948	—	47	1	no	—
150	#2175	—	47	4	no	—
151	C₄×SD₃₂	—	47	4	no	—
152	#2158	—	47	1	no	—
153	#2141	—	47	1	no	—
154	#2140	—	47	1	no	—
155	#2108	D₄₇	1	1	yes	characteristic, complemented
156	C₂×C₄₇	—	2	2	no	—
157	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
158	C₂×C₄₇	#3416	1	2	yes	—
159	D₄₇	—	8	4	no	—
160	C₂²×Q₁₆	—	47	1	no	—
161	C₂²×D₈	—	47	1	no	—
162	#519	—	94	1	no	—
163	#518	—	94	1	no	—
164	C₂×SD₃₂	—	47	8	no	—
165	C₂×SD₃₂	—	47	2	no	—
166	C₂×SD₃₂	—	47	2	no	—
167	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
168	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
169	#442	—	94	2	no	—
170	#441	—	94	2	no	—
171	C₄×Q₁₆	—	47	4	no	—
172	C₄×D₈	—	47	4	no	—
173	#427	—	47	1	no	—
174	#427	—	47	1	no	—
175	#419	—	94	1	no	—
176	#418	—	94	1	no	—
177	#405	C₂×D₄₇	1	1	yes	characteristic
178	#367	—	47	2	no	—
179	#366	—	47	2	no	—
180	#358	—	47	4	no	—
181	#357	—	47	4	no	—
182	#346	C₂×D₄₇	1	2	yes	complemented
183	C₄×C₁₆	C₂×D₄₇	1	2	yes	complemented
184	C₄₇	#33457	1	1	yes	characteristic, complemented, Sylow
185	C₂²×Q₈	—	94	1	no	—
186	C₂²×D₄	—	94	1	no	—
187	C₂×Q₁₆	—	47	4	no	—
188	C₂×Q₁₆	—	47	2	no	—
189	C₂×Q₁₆	—	94	1	no	—
190	C₂×Q₁₆	—	94	2	no	—
191	C₂×D₈	—	47	4	no	—
192	C₂×D₈	—	47	2	no	—
193	C₂×D₈	—	94	1	no	—
194	C₂×D₈	—	94	2	no	—
195	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
197	C₄×Q₈	—	94	4	no	—
198	C₄×Q₈	—	188	2	no	—
199	C₄×D₄	—	94	4	no	—
200	C₄×D₄	—	188	2	no	—
201	#117	—	94	1	no	—
202	#117	—	94	1	no	—
203	#117	—	188	2	no	—
204	#117	D₁₈₈	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	—	188	2	no	—
206	SD₃₂	—	47	8	no	—
207	SD₃₂	—	47	8	no	—
208	M₃₂	—	2	2	no	—
209	C₂×C₁₆	C₄×D₄₇	1	4	yes	complemented
210	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
211	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
212	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
213	#107	—	47	2	no	—
214	#107	—	47	2	no	—
215	#106	—	94	1	no	—
216	#106	—	94	1	no	—
217	#103	—	94	4	no	—
218	#102	—	94	4	no	—
219	#97	D₁₈₈	1	2	yes	—
220	C₄×C₈	C₂²×D₄₇	1	2	yes	—
221	C₂⁴	—	188	1	no	—
222	C₂×Q₈	—	94	4	no	—
223	C₂×Q₈	—	94	2	no	—
224	C₂×Q₈	—	188	1	no	—
225	C₂×Q₈	—	188	2	no	—
226	C₂×D₄	—	94	4	no	—
227	C₂×D₄	—	94	2	no	—
228	C₂×D₄	—	188	1	no	—
229	C₂×D₄	—	188	2	no	—
230	C₂²×C₄	—	188	1	no	—
231	C₂²×C₄	—	188	4	no	—
232	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
233	Q₁₆	—	47	4	no	—
234	Q₁₆	—	94	4	no	—
235	Q₁₆	—	94	2	no	—
236	D₈	—	47	4	no	—
237	D₈	—	94	4	no	—
238	D₈	—	94	2	no	—
239	M₁₆	—	2	2	no	—
240	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
241	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
242	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
243	C₂×C₈	#10625350	1	2	yes	—
244	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
245	C₄⋊C₄	—	94	2	no	—
246	C₄⋊C₄	—	94	2	no	—
247	C₄⋊C₄	—	188	4	no	—
248	C₄⋊C₄	D₃₇₆	1	2	yes	—
249	C₂²⋊C₄	—	188	1	no	—
250	C₂²⋊C₄	—	188	1	no	—
251	C₂²⋊C₄	—	188	4	no	—
252	C₂²⋊C₄	D₃₇₆	1	2	yes	—
253	C₄²	—	188	1	no	—
254	C₄²	—	188	1	no	—
255	C₄²	—	188	4	no	—
256	C₄²	C₂×D₁₈₈	1	2	yes	—
257	C₁₆	—	2	2	no	—
258	C₁₆	C₂×C₄×D₄₇	1	2	yes	—
259	C₁₆	C₂×C₄×D₄₇	1	2	yes	—
260	C₂³	—	188	2	no	—
261	C₂³	—	188	4	no	—
262	C₂³	—	752	2	no	—
263	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
264	Q₈	—	94	4	no	—
265	Q₈	—	188	2	no	—
266	Q₈	—	188	4	no	—
267	D₄	—	94	4	no	—
268	D₄	—	188	2	no	—
269	D₄	—	188	4	no	—
270	C₂×C₄	—	188	2	no	—
271	C₂×C₄	—	188	2	no	—
272	C₂×C₄	—	188	2	no	—
273	C₂×C₄	—	188	4	no	—
274	C₂×C₄	—	752	4	no	—
275	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
276	C₂×C₄	C₂×D₃₇₆	1	4	yes	—
277	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
278	C₂×C₄	#13094046	1	2	yes	—
279	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
280	C₈	—	2	2	no	—
281	C₈	C₄×D₁₈₈	1	2	yes	—
282	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
283	C₈	C₂×#10625350	1	1	yes	characteristic
284	C₂²	—	2	2	no	—
285	C₂²	—	376	4	no	—
286	C₂²	—	188	4	no	—
287	C₂²	—	752	4	no	—
288	C₂²	C₄×D₃₇₆	1	2	yes	—
289	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
290	C₄	—	2	4	no	—
291	C₄	—	376	2	no	—
292	C₄	—	376	2	no	—
293	C₄	—	188	4	no	—
294	C₄	C₄×D₃₇₆	1	2	yes	—
295	C₄	C₂×C₄×D₁₈₈	1	1	yes	characteristic
296	C₄	C₂×#13094046	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	376	4	no	—
299	C₂	C₂×C₄×D₃₇₆	1	1	yes	characteristic
300	C₂	#20256536	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₃₇₆
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2108	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂²×D₃₇₆	C₂²
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2108	D₄₇
hypocenter	C₄₇	#33457
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #20252257	C₂	semidirect
4. #20245254	C₂	semidirect
5. C₄×#13093969	C₂	semidirect
6. #20245172	C₂	semidirect
7. #20245109	C₂	semidirect
9. C₄₇×#2108	C₂	semidirect
14. C₂×#13093969	C₄	semidirect
19. C₄×Dic₁₈₈	C₂²	semidirect
25. #15688951	C₂²	semidirect
26. #15688949	C₂²	semidirect
27. #15688928	C₂²	semidirect
32. C₄₇×#346	C₂²	semidirect
33. C₄×C₁₆×C₄₇	C₂²	semidirect
38. C₂×Dic₁₈₈	C₂×C₄	semidirect
52. #13093969	C₂×C₄	semidirect
53. #13093969	C₂×C₄	semidirect
64. C₂×C₁₆×C₄₇	C₂×C₄	semidirect
155. #2108	D₄₇	semidirect
182. #346	C₂×D₄₇	semidirect
183. C₄×C₁₆	C₂×D₄₇	semidirect
184. C₄₇	#33457	semidirect
209. C₂×C₁₆	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093969	2
#20252257	2
#20252241	2
#20245254	2
C₄×#13093969	2
#20245172	2
#20245109	2
#20245105	2
C₄₇×#2108	2
#33457	47