Subgroups — #26216713

Order 12032.

1102 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689165	C₂	1	4	yes	maximal
2	#20245795	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688989	C₂	1	1	yes	maximal, characteristic
4	#20243556	C₂	1	4	yes	maximal, complemented
5	#20243552	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#13093983	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	#15689165	—	2	16	no	—
10	C₂×#13093958	C₂²	1	8	yes	—
11	C₂×#13093956	C₂²	1	8	yes	—
12	#15688993	C₂²	1	2	yes	—
13	#15688989	—	2	2	no	—
14	#15688989	C₂²	1	2	yes	—
15	#15688989	C₂²	1	2	yes	—
16	#15688989	C₂²	1	2	yes	—
17	#15688983	—	2	2	no	—
18	C₄×C₄⋊Dic₄₇	—	2	2	no	—
19	C₂×C₄₇⋊#95	—	2	2	no	—
20	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×M₁₆×C₄₇	C₄	1	8	yes	complemented
23	C₂×#97×C₄₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
25	C₂³×Dic₄₇	—	2	2	no	—
26	#13093983	—	2	8	no	—
27	C₂×C₄⋊Dic₄₇	—	2	8	no	—
28	C₂×C₄⋊Dic₄₇	—	2	4	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
32	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	2	no	—
34	#13093958	—	2	16	no	—
35	#13093956	—	2	16	no	—
36	C₄₇⋊#95	—	2	4	no	—
37	C₄₇⋊#95	—	2	4	no	—
38	C₄₇⋊#95	—	2	4	no	—
39	C₂³×C₄×C₄₇	D₄	1	2	yes	—
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
42	C₂×M₁₆×C₄₇	D₄	1	8	yes	—
43	C₂×M₁₆×C₄₇	Q₈	1	8	yes	—
44	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	#97×C₄₇	C₂×C₄	1	8	yes	complemented
51	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
52	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	4	4	no	—
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
63	C₂²×Dic₄₇	—	2	4	no	—
64	C₂²×Dic₄₇	—	2	4	no	—
65	C₂²×Dic₄₇	—	4	8	no	—
66	C₂²×Dic₄₇	—	4	4	no	—
67	M₁₆×C₄₇	C₄⋊C₄	1	32	yes	complemented
68	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
69	C₂×C₈×C₄₇	C₂×D₄	1	8	yes	—
70	C₂×C₈×C₄₇	C₂×Q₈	1	8	yes	—
71	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
72	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
73	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
74	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
75	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
76	C₂²⋊Dic₄₇	—	4	4	no	—
77	C₄⋊Dic₄₇	—	2	4	no	—
78	C₄⋊Dic₄₇	—	2	8	no	—
79	C₄⋊Dic₄₇	—	2	4	no	—
80	C₄⋊Dic₄₇	—	4	8	no	—
81	C₄⋊Dic₄₇	—	4	4	no	—
82	C₄×Dic₄₇	—	4	4	no	—
83	C₂³×C₄₇	—	2	4	no	—
84	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂³×C₄₇	#127	1	2	yes	—
86	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
87	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	—	2	4	no	—
89	C₂×C₄×C₄₇	—	2	4	no	—
90	C₂×C₄×C₄₇	—	4	8	no	—
91	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
93	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
94	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₄₇	#127	1	2	yes	—
96	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
97	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
98	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
100	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
101	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×Dic₄₇	—	4	4	no	—
103	C₂×Dic₄₇	—	4	4	no	—
104	C₂×Dic₄₇	—	4	8	no	—
105	C₂×Dic₄₇	—	4	8	no	—
106	C₂×Dic₄₇	—	8	8	no	—
107	C₈×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
108	#20801	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	—	2	4	no	—
110	C₂²×C₄₇	—	2	4	no	—
111	C₂²×C₄₇	—	2	4	no	—
112	C₂²×C₄₇	#384	1	4	yes	—
113	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
115	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₂²×C₄₇	#575	1	4	yes	—
117	C₄×C₄₇	—	2	4	no	—
118	C₄×C₄₇	—	2	4	no	—
119	C₄×C₄₇	#384	1	4	yes	—
120	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
121	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
122	Dic₄₇	—	8	4	no	—
123	Dic₄₇	—	8	4	no	—
124	C₂×#428	—	47	4	no	—
125	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
126	#2285	—	47	1	no	—
127	C₂×#382	—	47	1	no	—
128	#1847	—	47	4	no	—
129	#1846	—	47	4	no	—
130	C₂×C₄₇	—	2	4	no	—
131	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
132	C₂×C₄₇	#2911	1	2	yes	—
133	C₂×C₄₇	#3153	1	2	yes	—
134	C₂×C₄₇	#3166	1	2	yes	—
135	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
136	C₂×#117	—	47	2	no	—
137	C₂²×C₄⋊C₄	—	47	2	no	—
138	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
139	#428	—	94	16	no	—
140	C₂×#107	—	47	8	no	—
141	C₂×#106	—	47	8	no	—
142	C₄×M₁₆	Dic₄₇	1	8	yes	complemented
143	C₂×#97	C₂×D₄₇	1	2	yes	—
144	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
145	#384	—	47	2	no	—
146	#382	—	47	2	no	—
147	#382	—	47	2	no	—
148	#382	—	47	2	no	—
149	#382	—	94	2	no	—
150	#380	—	94	2	no	—
151	C₄×C₄⋊C₄	—	94	2	no	—
152	C₂×#95	—	94	2	no	—
153	C₄₇	#20801	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₄	—	94	2	no	—
155	C₂³×C₄	D₁₈₈	1	2	yes	—
156	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
157	C₂×M₁₆	Dic₉₄	1	8	yes	—
158	C₂×M₁₆	D₁₈₈	1	8	yes	—
159	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
160	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
161	#117	—	94	8	no	—
162	C₂×C₄⋊C₄	—	47	4	no	—
163	C₂×C₄⋊C₄	—	47	4	no	—
164	C₂×C₄⋊C₄	—	94	8	no	—
165	C₂×C₄⋊C₄	—	94	2	no	—
166	C₂×C₄⋊C₄	—	94	4	no	—
167	C₂×C₂²⋊C₄	—	94	2	no	—
168	C₂×C₄²	—	94	2	no	—
169	C₂×C₄²	Dic₉₄	1	4	yes	—
170	C₂×C₄²	D₁₈₈	1	2	yes	—
171	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
172	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
174	#107	—	94	16	no	—
175	#106	—	94	16	no	—
176	#97	C₂×Dic₄₇	1	8	yes	complemented
177	C₄×C₈	C₂×Dic₄₇	1	8	yes	complemented
178	#95	—	94	4	no	—
179	#95	—	94	4	no	—
180	#95	—	94	4	no	—
181	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
182	C₂²×C₄	—	4	4	no	—
183	C₂²×C₄	—	94	4	no	—
184	C₂²×C₄	—	94	4	no	—
185	C₂²×C₄	—	188	8	no	—
186	C₂²×C₄	—	188	4	no	—
187	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
188	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
189	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
190	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
191	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
192	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
194	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
195	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
196	M₁₆	C₄⋊Dic₄₇	1	32	yes	complemented
197	C₂×C₈	C₂×Dic₉₄	1	8	yes	—
198	C₂×C₈	C₂×D₁₈₈	1	8	yes	—
199	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
200	C₄⋊C₄	—	94	4	no	—
201	C₄⋊C₄	—	94	8	no	—
202	C₄⋊C₄	—	94	4	no	—
203	C₄⋊C₄	—	188	8	no	—
204	C₄⋊C₄	—	188	4	no	—
205	C₂²⋊C₄	—	188	4	no	—
206	C₄²	—	188	4	no	—
207	C₄²	C₄⋊Dic₄₇	1	8	yes	—
208	C₄²	C₂×Dic₉₄	1	2	yes	—
209	C₄²	C₂×Dic₉₄	1	2	yes	—
210	C₄²	C₂²×Dic₄₇	1	2	yes	—
211	C₄²	C₂²×Dic₄₇	1	2	yes	—
212	C₂³	—	2	4	no	—
213	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
214	C₂³	#13094021	1	2	yes	—
215	C₂³	#13094024	1	2	yes	—
216	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	4	8	no	—
220	C₂×C₄	—	376	8	no	—
221	C₂×C₄	—	188	4	no	—
222	C₂×C₄	—	188	4	no	—
223	C₂×C₄	—	188	8	no	—
224	C₂×C₄	—	188	8	no	—
225	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
226	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
227	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
228	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
229	C₂×C₄	#13094021	1	2	yes	—
230	C₂×C₄	#13094024	1	2	yes	—
231	C₂×C₄	#13094024	1	8	yes	—
232	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
233	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
234	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
235	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
236	C₈	C₂×C₄⋊Dic₄₇	1	16	yes	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	4	no	—
240	C₂²	#15688993	1	4	yes	—
241	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
242	C₂²	C₂×#13094021	1	1	yes	characteristic
243	C₂²	C₂×#13094024	1	1	yes	characteristic
244	C₂²	#15689923	1	4	yes	—
245	C₄	—	2	4	no	—
246	C₄	—	2	4	no	—
247	C₄	—	376	4	no	—
248	C₄	—	376	4	no	—
249	C₄	#15688993	1	4	yes	—
250	C₄	C₂²×C₄⋊Dic₄₇	1	2	yes	—
251	C₄	C₂×#13094024	1	2	yes	—
252	C₂	—	2	4	no	—
253	C₂	C₂×#15688993	1	1	yes	characteristic
254	C₂	#20251978	1	2	yes	—
255	C₂	#20253854	1	2	yes	—
256	C₂	#20253940	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	#20801
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #20243556	C₂	semidirect
5. #20243552	C₂	semidirect
22. C₄×M₁₆×C₄₇	C₄	semidirect
50. #97×C₄₇	C₂×C₄	semidirect
51. C₄×C₈×C₄₇	C₂×C₄	semidirect
67. M₁₆×C₄₇	C₄⋊C₄	semidirect
142. C₄×M₁₆	Dic₄₇	semidirect
153. C₄₇	#20801	semidirect
176. #97	C₂×Dic₄₇	semidirect
177. C₄×C₈	C₂×Dic₄₇	semidirect
196. M₁₆	C₄⋊Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689165	2
#20245795	2
C₂×#15688989	2
#20243556	2
#20243552	2
C₂×C₄×M₁₆×C₄₇	2
#20801	47