Subgroups — #26422362

Order 12032.

1990 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251775	C₂	1	4	yes	maximal, complemented
2	#20251751	C₂	1	4	yes	maximal, complemented
3	#20251698	C₂	1	8	yes	maximal
4	C₄×#13094016	C₂	1	8	yes	maximal, complemented
5	#20245871	C₂	1	4	yes	maximal
6	#20245795	C₂	1	2	yes	maximal
7	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13094016	C₂²	1	8	yes	—
9	C₂×#13093983	C₂²	1	4	yes	—
10	#15689021	C₂²	1	16	yes	—
11	#15689021	C₂²	1	16	yes	—
12	#15689007	C₂²	1	32	yes	—
13	#15688996	C₂²	1	16	yes	—
14	#15688993	C₂²	1	4	yes	—
15	#15688989	C₂²	1	4	yes	—
16	#15688989	C₂²	1	8	yes	—
17	#15688983	C₂²	1	8	yes	—
18	C₄×Dic₄₇⋊C₄	C₂²	1	16	yes	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	8	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	4	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₄³×C₄₇	C₂²	1	2	yes	—
24	C₄³×C₄₇	C₂²	1	2	yes	—
25	C₄³×C₄₇	C₂²	1	4	yes	—
26	#13094016	—	2	32	no	—
27	#13093983	—	2	16	no	—
28	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
29	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
30	C₂×Dic₄₇⋊C₄	C₂³	1	32	yes	—
31	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
32	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
33	C₄₇⋊#95	C₂³	1	16	yes	—
34	C₄₇⋊#95	C₂³	1	32	yes	—
35	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
36	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	—	2	8	no	—
39	C₂×C₄²×C₄₇	—	2	16	no	—
40	C₂×C₄²×C₄₇	Q₈	1	8	yes	—
41	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
42	C₂×C₄²×C₄₇	C₂³	1	8	yes	—
43	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
49	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	2	8	no	—
51	C₂²×C₄×C₄₇	—	2	16	no	—
52	C₂²×C₄×C₄₇	—	2	8	no	—
53	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
54	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₄₇	#41	1	8	yes	—
56	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
58	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
62	C₄²×C₄₇	—	2	16	no	—
63	C₄²×C₄₇	—	2	8	no	—
64	C₄²×C₄₇	—	2	8	no	—
65	C₄²×C₄₇	—	2	8	no	—
66	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
67	C₄²×C₄₇	C₂×Q₈	1	8	yes	—
68	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
69	C₄²×C₄₇	#41	1	16	yes	—
70	C₂²⋊Dic₄₇	—	2	16	no	—
71	C₄⋊Dic₄₇	—	2	16	no	—
72	C₄⋊Dic₄₇	—	2	32	no	—
73	Dic₄₇⋊C₄	—	2	64	no	—
74	C₄×Dic₄₇	—	2	16	no	—
75	C₂³×C₄₇	—	2	4	no	—
76	C₂³×C₄₇	C₂²×Q₈	1	2	yes	—
77	C₂³×C₄₇	C₂×#41	1	4	yes	—
78	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
79	C₂×C₄×C₄₇	—	2	32	no	—
80	C₂×C₄×C₄₇	—	2	4	no	—
81	C₂×C₄×C₄₇	—	2	8	no	—
82	C₂×C₄×C₄₇	—	2	8	no	—
83	C₂×C₄×C₄₇	—	2	16	no	—
84	C₂×C₄×C₄₇	#125	1	16	yes	complemented
85	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
86	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
87	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
88	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
89	C₂×C₄×C₄₇	C₂×#41	1	16	yes	—
90	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
91	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
92	C₂×C₄×C₄₇	C₂×#41	1	8	yes	—
93	C₂×Dic₄₇	—	2	16	no	—
94	C₂×Dic₄₇	—	2	32	no	—
95	#36640	—	47	1	no	maximal, Sylow
96	C₂²×C₄₇	—	2	4	no	—
97	C₂²×C₄₇	—	2	8	no	—
98	C₂²×C₄₇	C₂×#125	1	4	yes	—
99	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
100	C₂²×C₄₇	C₂²×#41	1	2	yes	—
101	C₂²×C₄₇	C₂²×#41	1	4	yes	—
102	C₄×C₄₇	—	2	4	no	—
103	C₄×C₄₇	—	2	16	no	—
104	C₄×C₄₇	C₂×#125	1	4	yes	—
105	C₄×C₄₇	C₂×#125	1	4	yes	—
106	C₄×C₄₇	#533	1	8	yes	—
107	Dic₄₇	—	4	16	no	—
108	#2864	—	47	4	no	—
109	#2859	—	47	4	no	—
110	#2851	—	47	8	no	—
111	C₄×#125	—	47	8	no	—
112	#2296	—	47	4	no	—
113	#2285	—	47	2	no	—
114	C₂×C₄³	D₄₇	1	1	yes	characteristic
115	C₂×C₄₇	—	2	4	no	—
116	C₂×C₄₇	C₂²×#125	1	1	yes	characteristic
117	C₂×C₄₇	C₂×#533	1	2	yes	—
118	C₂×C₄₇	#3444	1	4	yes	—
119	C₂×#125	—	47	8	no	—
120	C₂×#117	—	47	4	no	—
121	C₂²×C₄²	C₂×D₄₇	1	4	yes	—
122	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
123	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
124	#389	—	47	16	no	—
125	#389	—	47	16	no	—
126	#387	—	47	32	no	—
127	#385	—	47	16	no	—
128	#384	—	47	4	no	—
129	#382	—	47	8	no	—
130	#382	—	47	4	no	—
131	#380	—	47	8	no	—
132	C₄×C₄⋊C₄	—	47	8	no	—
133	C₄×C₄⋊C₄	—	47	16	no	—
134	C₄³	C₂×D₄₇	1	4	yes	—
135	C₄³	C₂×D₄₇	1	2	yes	—
136	C₄³	C₂×D₄₇	1	2	yes	—
137	C₄₇	#36640	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
139	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
140	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
141	#125	—	94	32	no	—
142	#117	—	94	16	no	—
143	C₂×C₄⋊C₄	—	47	8	no	—
144	C₂×C₄⋊C₄	—	47	32	no	—
145	C₂×C₄⋊C₄	—	47	16	no	—
146	C₂×C₂²⋊C₄	—	47	8	no	—
147	C₂×C₄²	—	2	8	no	—
148	C₂×C₄²	—	2	16	no	—
149	C₂×C₄²	—	47	8	no	—
150	C₂×C₄²	Dic₉₄	1	8	yes	—
151	C₂×C₄²	C₂²×D₄₇	1	8	yes	—
152	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
153	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
154	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
155	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
156	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
157	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
158	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
159	#95	—	47	16	no	—
160	#95	—	47	32	no	—
161	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	2	16	no	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	—	47	16	no	—
166	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
167	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
168	C₂²×C₄	#10625350	1	8	yes	—
169	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
170	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
171	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
172	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
173	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
174	C₄⋊C₄	—	94	64	no	—
175	C₄⋊C₄	—	94	16	no	—
176	C₄⋊C₄	—	94	32	no	—
177	C₂²⋊C₄	—	94	16	no	—
178	C₄²	—	2	16	no	—
179	C₄²	—	2	8	no	—
180	C₄²	—	2	8	no	—
181	C₄²	—	2	8	no	—
182	C₄²	—	94	16	no	—
183	C₄²	C₂×Dic₉₄	1	4	yes	—
184	C₄²	C₂×Dic₉₄	1	8	yes	—
185	C₄²	C₂×Dic₉₄	1	4	yes	—
186	C₄²	#10625350	1	16	yes	—
187	C₂³	—	2	4	no	—
188	C₂³	C₂²×Dic₉₄	1	2	yes	—
189	C₂³	C₂×#10625350	1	4	yes	—
190	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
191	C₂×C₄	—	2	32	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	94	16	no	—
197	C₂×C₄	—	94	32	no	—
198	C₂×C₄	#13094016	1	16	yes	complemented
199	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
200	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
201	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
202	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
203	C₂×C₄	C₂×#10625350	1	16	yes	—
204	C₂×C₄	C₂×#10625350	1	4	yes	—
205	C₂×C₄	C₂×#10625350	1	4	yes	—
206	C₂×C₄	C₂×#10625350	1	8	yes	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	8	no	—
209	C₂²	C₂×#13094016	1	4	yes	—
210	C₂²	C₂³×Dic₉₄	1	1	yes	characteristic
211	C₂²	C₂²×#10625350	1	2	yes	—
212	C₂²	C₂²×#10625350	1	4	yes	—
213	C₄	—	2	4	no	—
214	C₄	—	2	16	no	—
215	C₄	—	188	16	no	—
216	C₄	C₂×#13094016	1	4	yes	—
217	C₄	C₂×#13094016	1	4	yes	—
218	C₄	#15689680	1	8	yes	—
219	C₂	—	2	4	no	—
220	C₂	C₂²×#13094016	1	1	yes	characteristic
221	C₂	C₂×#15689680	1	2	yes	—
222	C₂	#20256684	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₇
commutator, socle, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂×C₄³×C₄₇	C₂
prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×C₄³	D₄₇
hypocenter	C₄₇	#36640
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
1. #20251775	C₂	semidirect
2. #20251751	C₂	semidirect
4. C₄×#13094016	C₂	semidirect
84. C₂×C₄×C₄₇	#125	semidirect
137. C₄₇	#36640	semidirect
198. C₂×C₄	#13094016	semidirect

Maximal subgroups

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

subgroup	index
#20251775	2
#20251751	2
#20251698	2
C₄×#13094016	2
#20245871	2
#20245795	2
C₂×C₄³×C₄₇	2
#36640	47