Subgroups — #26371787

Order 12032.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253927	C₂	1	2	yes	maximal
2	#20245674	C₂	1	4	yes	maximal, complemented
3	#20245667	C₂	1	2	yes	maximal, complemented
4	#20245663	C₂	1	2	yes	maximal, complemented
5	#20245176	C₂	1	2	yes	maximal
6	#20245172	C₂	1	1	yes	maximal, characteristic
7	C₂×#15688949	C₂	1	1	yes	maximal, characteristic
8	C₄₇×#2108	C₂	1	1	yes	maximal, characteristic
9	#15689751	—	2	2	no	—
10	#15689532	—	2	4	no	—
11	#15689528	C₂²	1	4	yes	—
12	#15689525	C₂²	1	4	yes	—
13	#15689165	—	2	2	no	—
14	#15689161	C₂²	1	2	yes	—
15	C₂×#13093958	C₂²	1	2	yes	—
16	#15688951	C₂²	1	4	yes	—
17	#15688951	C₂²	1	2	yes	—
18	#15688949	C₂²	1	1	yes	characteristic
19	#15688949	C₂²	1	2	yes	—
20	#15688949	C₂²	1	4	yes	—
21	#15688949	C₂²	1	1	yes	characteristic
22	#15688949	C₂²	1	2	yes	—
23	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#405	C₂²	1	1	yes	characteristic
26	C₄₇×#346	C₂²	1	2	yes	—
27	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
28	#13094024	—	2	4	no	—
29	#13094023	—	4	2	no	—
30	#13094016	—	2	4	no	—
31	#13094015	—	4	2	no	—
32	#13094001	—	4	4	no	—
33	#13093983	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	—	2	2	no	—
35	#13093958	—	2	4	no	—
36	#13093958	—	2	2	no	—
37	#13093958	C₂³	1	2	yes	—
38	#13093958	C₂³	1	4	yes	—
39	#13093958	C₂³	1	2	yes	—
40	#13093956	—	2	4	no	—
41	#13093956	—	2	2	no	—
42	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
44	#117×C₄₇	D₄	1	1	yes	characteristic
45	M₃₂×C₄₇	Q₈	1	4	yes	complemented
46	C₂×C₁₆×C₄₇	Q₈	1	4	yes	complemented
47	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
50	#97×C₄₇	D₄	1	2	yes	—
51	C₄×C₈×C₄₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₄₇	—	4	2	no	—
54	C₂×Dic₉₄	—	4	4	no	—
55	M₁₆×C₄₇	Q₁₆	1	4	yes	—
56	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
62	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
63	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
64	C₄²×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²⋊Dic₄₇	—	4	2	no	—
66	C₄⋊Dic₄₇	—	2	2	no	—
67	C₄⋊Dic₄₇	—	2	4	no	—
68	C₄⋊Dic₄₇	—	2	2	no	—
69	C₄⋊Dic₄₇	—	4	2	no	—
70	C₄⋊Dic₄₇	—	4	4	no	—
71	Dic₄₇⋊C₄	—	4	4	no	—
72	Dic₄₇⋊C₄	—	4	4	no	—
73	C₄×Dic₄₇	—	4	2	no	—
74	C₁₆×C₄₇	C₂×Q₈	1	2	yes	—
75	C₁₆×C₄₇	C₂×Q₈	1	2	yes	—
76	C₁₆×C₄₇	C₂×Q₈	1	4	yes	—
77	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
79	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₄₇	—	4	2	no	—
85	C₂×Dic₄₇	—	4	2	no	—
86	C₂×Dic₄₇	—	4	4	no	—
87	C₂×Dic₄₇	—	16	2	no	—
88	Dic₉₄	—	16	4	no	—
89	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
90	C₈×C₄₇	C₂×Q₁₆	1	4	yes	—
91	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
92	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
93	#33729	—	47	1	no	maximal, Sylow
94	C₂²×C₄₇	—	2	1	no	—
95	C₂²×C₄₇	—	2	1	no	—
96	C₂²×C₄₇	#500	1	2	yes	—
97	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₄₇	—	2	2	no	—
99	C₄×C₄₇	—	2	2	no	—
100	C₄×C₄₇	#500	1	2	yes	—
101	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
103	Dic₄₇	—	8	2	no	—
104	Dic₄₇	—	8	4	no	—
105	Dic₄₇	—	8	2	no	—
106	#3164	—	47	2	no	—
107	#2255	—	47	4	no	—
108	#2253	—	47	2	no	—
109	#2252	—	47	2	no	—
110	#2159	—	47	2	no	—
111	#2158	—	47	1	no	—
112	C₂×#366	—	47	1	no	—
113	#2108	D₄₇	1	1	yes	characteristic
114	C₂×C₄₇	—	2	2	no	—
115	C₂×C₄₇	C₂×#500	1	1	yes	characteristic
116	C₂×C₄₇	#3415	1	1	yes	characteristic
117	C₂×C₄₇	#3417	1	1	yes	characteristic
118	#544	—	94	2	no	—
119	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
120	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
121	#501	—	94	4	no	—
122	#500	—	47	4	no	—
123	#499	—	47	4	no	—
124	#428	—	94	2	no	—
125	#427	—	47	2	no	—
126	C₂×#107	—	47	2	no	—
127	#405	C₂×D₄₇	1	1	yes	characteristic
128	#367	—	47	2	no	—
129	#367	—	47	4	no	—
130	#366	—	47	2	no	—
131	#366	—	47	2	no	—
132	#366	—	47	1	no	—
133	#366	—	47	1	no	—
134	#366	—	47	4	no	—
135	#346	C₂×D₄₇	1	2	yes	—
136	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
137	C₄₇	#33729	1	1	yes	characteristic, complemented, Sylow
138	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
139	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
140	C₄⋊Q₈	—	94	4	no	—
141	C₄⋊Q₈	—	188	2	no	—
142	#125	—	94	4	no	—
143	#125	—	188	2	no	—
144	C₂²⋊Q₈	—	188	4	no	—
145	#117	—	94	2	no	—
146	#117	D₁₈₈	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	94	2	no	—
148	M₃₂	Dic₉₄	1	4	yes	complemented
149	C₂×C₁₆	Dic₉₄	1	4	yes	complemented
150	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
151	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
152	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
153	#107	—	47	4	no	—
154	#107	—	47	2	no	—
155	#107	—	47	2	no	—
156	#107	—	94	2	no	—
157	#107	—	94	4	no	—
158	#106	—	94	2	no	—
159	#106	—	94	4	no	—
160	#97	D₁₈₈	1	2	yes	—
161	C₄×C₈	C₂²×D₄₇	1	2	yes	—
162	C₂×Q₈	—	188	4	no	—
163	C₂²×C₄	—	188	2	no	—
164	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
165	M₁₆	Dic₁₈₈	1	4	yes	—
166	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
167	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
168	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
169	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
170	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
171	C₄⋊C₄	—	94	2	no	—
172	C₄⋊C₄	—	94	4	no	—
173	C₄⋊C₄	—	94	2	no	—
174	C₄⋊C₄	—	188	4	no	—
175	C₄⋊C₄	—	188	4	no	—
176	C₄⋊C₄	—	188	2	no	—
177	C₄⋊C₄	—	188	4	no	—
178	C₄⋊C₄	D₃₇₆	1	1	yes	characteristic
179	C₄⋊C₄	D₃₇₆	1	1	yes	characteristic
180	C₂²⋊C₄	—	188	2	no	—
181	C₂²⋊C₄	D₃₇₆	1	2	yes	—
182	C₄²	—	188	2	no	—
183	C₄²	C₂×D₁₈₈	1	2	yes	—
184	C₁₆	C₂×Dic₉₄	1	2	yes	—
185	C₁₆	C₂×Dic₉₄	1	2	yes	—
186	C₁₆	C₂×Dic₉₄	1	4	yes	—
187	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
188	Q₈	—	752	4	no	—
189	C₂×C₄	—	188	2	no	—
190	C₂×C₄	—	188	2	no	—
191	C₂×C₄	—	188	4	no	—
192	C₂×C₄	—	752	2	no	—
193	C₂×C₄	#13094024	1	2	yes	—
194	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
195	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
196	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
197	C₂×C₄	C₂×Dic₁₈₈	1	2	yes	—
198	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
199	C₈	#13094024	1	2	yes	—
200	C₈	C₂×Dic₁₈₈	1	4	yes	—
201	C₈	C₂²×Dic₉₄	1	1	yes	characteristic
202	C₈	C₂²×Dic₉₄	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#15689528	1	2	yes	—
206	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	376	4	no	—
210	C₄	—	376	2	no	—
211	C₄	—	376	2	no	—
212	C₄	#15689528	1	2	yes	—
213	C₄	C₂×#13094024	1	1	yes	characteristic
214	C₄	C₂²×Dic₁₈₈	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×#15689528	1	1	yes	characteristic
217	C₂	#20256529	1	1	yes	characteristic
218	C₂	#20256541	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2108	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂³×C₄₇	C₂×D₈
hypercenter	#2108	D₄₇
hypocenter	C₄₇	#33729
LCS	G ▷ C₂×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. #20245674	C₂	semidirect
3. #20245667	C₂	semidirect
4. #20245663	C₂	semidirect
45. M₃₂×C₄₇	Q₈	semidirect
46. C₂×C₁₆×C₄₇	Q₈	semidirect
137. C₄₇	#33729	semidirect
148. M₃₂	Dic₉₄	semidirect
149. C₂×C₁₆	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20253927	2
#20245674	2
#20245667	2
#20245663	2
#20245176	2
#20245172	2
C₂×#15688949	2
C₄₇×#2108	2
#33729	47