Subgroups — #26383659

Order 12032.

2110 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689680	C₂	1	2	yes	maximal
2	#20249507	C₂	1	8	yes	maximal, complemented
3	#20249502	C₂	1	8	yes	maximal
4	#20246681	C₂	1	8	yes	maximal, complemented
5	#20245795	C₂	1	4	yes	maximal
6	C₂×C₄₇×#376	C₂	1	1	yes	maximal, characteristic
7	#15689680	—	2	16	no	—
8	C₂×#13094024	C₂²	1	4	yes	—
9	C₂×#13094016	C₂²	1	4	yes	—
10	C₂×#13094003	C₂²	1	8	yes	—
11	C₂×C₄×Dic₉₄	C₂²	1	8	yes	—
12	C₂×#13093983	C₂²	1	4	yes	—
13	#15689055	C₂²	1	32	yes	—
14	#15689044	C₂²	1	32	yes	—
15	#15688993	C₂²	1	8	yes	—
16	#15688989	C₂²	1	8	yes	—
17	#15688983	C₂²	1	16	yes	—
18	C₄×C₄⋊Dic₄₇	C₂²	1	16	yes	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
21	C₄₇×#376	C₂²	1	4	yes	—
22	C₄₇×#376	C₂²	1	4	yes	—
23	C₂×#95×C₄₇	C₂²	1	4	yes	—
24	C₂²×Dic₉₄	C₂³	1	8	yes	—
25	#13094024	—	2	16	no	—
26	#13094016	—	2	16	no	—
27	#13094003	—	2	32	no	—
28	C₄×Dic₉₄	—	2	32	no	—
29	#13093983	—	2	16	no	—
30	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	32	yes	—
33	C₂×Dic₄₇⋊C₄	C₂³	1	16	yes	—
34	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
35	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
36	C₄₇⋊#95	C₂³	1	32	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	D₄	1	8	yes	—
41	C₂×C₄²×C₄₇	Q₈	1	16	yes	complemented
42	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
46	#95×C₄₇	C₂³	1	16	yes	—
47	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	—	4	8	no	—
49	C₂²×C₄×C₄₇	—	4	4	no	—
50	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
51	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
52	C₂²×C₄×C₄₇	C₂×Q₈	1	16	yes	—
53	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
54	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
56	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
59	C₂×Dic₉₄	—	2	16	no	—
60	C₂×Dic₉₄	—	4	16	no	—
61	C₄²×C₄₇	C₂×D₄	1	8	yes	—
62	C₄²×C₄₇	C₂×D₄	1	8	yes	—
63	C₄²×C₄₇	C₂×Q₈	1	16	yes	—
64	C₄²×C₄₇	C₂×Q₈	1	16	yes	—
65	C₂²⋊Dic₄₇	—	2	16	no	—
66	C₄⋊Dic₄₇	—	2	16	no	—
67	C₄⋊Dic₄₇	—	2	16	no	—
68	C₄⋊Dic₄₇	—	4	32	no	—
69	Dic₄₇⋊C₄	—	2	32	no	—
70	C₄×Dic₄₇	—	2	16	no	—
71	C₂³×C₄₇	—	2	4	no	—
72	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
73	C₂³×C₄₇	C₂²×Q₈	1	4	yes	—
74	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
75	C₂×C₄×C₄₇	—	2	16	no	—
76	C₂×C₄×C₄₇	—	2	8	no	—
77	C₂×C₄×C₄₇	—	4	8	no	—
78	C₂×C₄×C₄₇	—	4	16	no	—
79	C₂×C₄×C₄₇	C₄⋊Q₈	1	16	yes	—
80	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
81	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
82	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
83	C₂×C₄×C₄₇	C₂²×Q₈	1	16	yes	—
84	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
85	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
86	C₂×Dic₄₇	—	2	16	no	—
87	C₂×Dic₄₇	—	2	32	no	—
88	Dic₉₄	—	4	32	no	—
89	#34642	—	47	1	no	maximal, Sylow
90	C₂²×C₄₇	—	2	4	no	—
91	C₂²×C₄₇	—	2	8	no	—
92	C₂²×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
93	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
94	C₂²×C₄₇	C₂³×Q₈	1	2	yes	—
95	C₂²×C₄₇	#585	1	4	yes	—
96	C₄×C₄₇	—	2	8	no	—
97	C₄×C₄₇	—	2	16	no	—
98	C₄×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
99	C₄×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
100	Dic₄₇	—	4	16	no	—
101	C₂×#533	—	47	2	no	—
102	#2660	—	47	8	no	—
103	#2659	—	47	8	no	—
104	#2399	—	47	8	no	—
105	#2285	—	47	4	no	—
106	C₂×#376	D₄₇	1	1	yes	characteristic
107	C₂×C₄₇	—	2	4	no	—
108	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
109	C₂×C₄₇	#3449	1	2	yes	—
110	C₂×C₄₇	#3458	1	4	yes	—
111	#533	—	94	16	no	—
112	C₂×C₄⋊Q₈	—	47	4	no	—
113	C₂×#125	—	47	4	no	—
114	C₂×C₂²⋊Q₈	—	47	8	no	—
115	C₂×C₄×Q₈	—	47	8	no	—
116	C₂×#117	—	47	4	no	—
117	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
118	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
119	#398	—	47	32	no	—
120	#395	—	47	32	no	—
121	#384	—	47	8	no	—
122	#382	—	47	8	no	—
123	#380	—	47	16	no	—
124	C₄×C₄⋊C₄	—	47	16	no	—
125	#376	C₂×D₄₇	1	4	yes	—
126	#376	C₂×D₄₇	1	4	yes	—
127	C₂×#95	C₂×D₄₇	1	4	yes	—
128	C₄₇	#34642	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	47	8	no	—
130	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
131	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
132	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
133	C₄⋊Q₈	—	94	16	no	—
134	#125	—	94	16	no	—
135	C₂²⋊Q₈	—	94	32	no	—
136	C₄×Q₈	—	94	32	no	—
137	#117	—	94	16	no	—
138	C₂×C₄⋊C₄	—	47	8	no	—
139	C₂×C₄⋊C₄	—	47	16	no	—
140	C₂×C₄⋊C₄	—	47	8	no	—
141	C₂×C₄⋊C₄	—	47	32	no	—
142	C₂×C₂²⋊C₄	—	47	8	no	—
143	C₂×C₄²	—	47	8	no	—
144	C₂×C₄²	Dic₉₄	1	16	yes	complemented
145	C₂×C₄²	D₁₈₈	1	8	yes	—
146	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
147	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
148	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
149	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
150	#95	—	47	32	no	—
151	#95	C₂²×D₄₇	1	16	yes	—
152	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
153	C₂×Q₈	—	94	16	no	—
154	C₂×Q₈	—	188	16	no	—
155	C₂²×C₄	—	4	8	no	—
156	C₂²×C₄	—	4	4	no	—
157	C₂²×C₄	—	47	16	no	—
158	C₂²×C₄	C₂×Dic₉₄	1	16	yes	—
159	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
160	C₂²×C₄	C₂×D₁₈₈	1	8	yes	—
161	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
162	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
163	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
164	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
165	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
166	C₄⋊C₄	—	94	32	no	—
167	C₄⋊C₄	—	94	16	no	—
168	C₄⋊C₄	—	94	16	no	—
169	C₄⋊C₄	—	188	32	no	—
170	C₂²⋊C₄	—	94	16	no	—
171	C₄²	—	94	16	no	—
172	C₄²	C₂×Dic₉₄	1	16	yes	—
173	C₄²	C₂×Dic₉₄	1	16	yes	—
174	C₄²	C₂×D₁₈₈	1	8	yes	—
175	C₄²	C₂×D₁₈₈	1	8	yes	—
176	C₂³	—	2	4	no	—
177	C₂³	C₂²×D₁₈₈	1	2	yes	—
178	C₂³	C₂²×Dic₉₄	1	4	yes	—
179	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
180	Q₈	—	188	32	no	—
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	—	2	8	no	—
183	C₂×C₄	—	4	16	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	—	94	16	no	—
186	C₂×C₄	—	94	32	no	—
187	C₂×C₄	#13094024	1	16	yes	—
188	C₂×C₄	C₂²×D₁₈₈	1	8	yes	—
189	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
190	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
191	C₂×C₄	C₂²×Dic₉₄	1	16	yes	—
192	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
193	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
194	C₂²	—	2	8	no	—
195	C₂²	—	2	4	no	—
196	C₂²	C₂×#13094024	1	4	yes	—
197	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
198	C₂²	C₂³×Dic₉₄	1	2	yes	—
199	C₂²	#15689959	1	4	yes	—
200	C₄	—	2	16	no	—
201	C₄	—	2	8	no	—
202	C₄	—	188	16	no	—
203	C₄	C₂×#13094024	1	4	yes	—
204	C₄	C₂×#13094024	1	4	yes	—
205	C₂	—	2	4	no	—
206	C₂	C₂²×#13094024	1	1	yes	characteristic
207	C₂	#20256721	1	2	yes	—
208	C₂	#20256793	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₇
commutator, socle, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#376	C₂
prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#376	D₄₇
hypocenter	C₄₇	#34642
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
2. #20249507	C₂	semidirect
4. #20246681	C₂	semidirect
41. C₂×C₄²×C₄₇	Q₈	semidirect
128. C₄₇	#34642	semidirect
144. C₂×C₄²	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689680	2
#20249507	2
#20249502	2
#20246681	2
#20245795	2
C₂×C₄₇×#376	2
#34642	47