Subgroups — #30624422

Order 15616.

614 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462203	C₂	1	8	yes	maximal, complemented
2	#21462176	C₂	1	2	yes	maximal, complemented
3	#21462046	C₂	1	4	yes	maximal, complemented
4	C₂×D₄×#92418	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519851	C₂²	1	8	yes	complemented
6	C₄⋊#11758187	C₂²	1	4	yes	complemented
7	C₂²⋊#11758187	C₂²	1	4	yes	—
8	C₂²⋊#11758187	C₂²	1	8	yes	complemented
9	C₄×#11758187	C₂²	1	4	yes	complemented
10	C₂³×#92418	C₂²	1	2	yes	—
11	D₄×#92418	C₄	1	4	yes	—
12	D₄×#92418	C₂²	1	4	yes	—
13	C₂×C₄⋊#92418	C₄	1	1	yes	characteristic
14	C₂×C₂²⋊#92418	C₄	1	2	yes	—
15	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₆₁	C₄	1	1	yes	characteristic
17	#14519851	—	4	8	no	—
18	C₂×#11758187	C₂³	1	8	yes	—
19	C₂²×#92418	—	4	4	no	—
20	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
21	C₂²×#92418	C₂×C₄	1	2	yes	—
22	C₂²×#92418	C₂×C₄	1	4	yes	—
23	C₂²×#92418	C₂³	1	1	yes	characteristic
24	C₂²×#92418	C₂³	1	4	yes	—
25	C₄⋊#92418	C₂×C₄	1	4	yes	—
26	C₂²⋊#92418	C₂×C₄	1	8	yes	—
27	C₄×#92418	C₂×C₄	1	2	yes	—
28	C₄×#92418	C₂³	1	2	yes	—
29	C₂²×D₄×C₆₁	C₈	1	1	yes	characteristic
30	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
31	C₄×D₄×C₆₁	C₈	1	4	yes	—
32	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₆₁	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₆₁	C₈	1	2	yes	—
35	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂⁴×C₆₁	C₂×C₈	1	2	yes	—
37	C₂×D₄×C₆₁	—	2	4	no	—
38	C₂×D₄×C₆₁	C₂×C₈	1	4	yes	—
39	C₂²×C₄×C₆₁	—	4	2	no	—
40	C₂²×C₄×C₆₁	—	4	2	no	—
41	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
47	C₄⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
48	C₂²⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
49	C₄²×C₆₁	C₂×C₈	1	2	yes	—
50	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
51	C₂×#92418	—	2	8	no	—
52	C₂×#92418	—	2	2	no	—
53	C₂×#92418	—	4	4	no	—
54	C₂×#92418	—	4	2	no	—
55	C₂×#92418	—	4	1	no	—
56	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
57	C₂×#92418	C₂²×C₄	1	2	yes	—
58	C₂×#92418	C₂²×C₄	1	4	yes	—
59	C₂×#92418	C₂⁴	1	1	yes	characteristic
60	#11758187	—	2	8	no	—
61	C₂³×C₆₁	—	2	4	no	—
62	C₂³×C₆₁	—	4	2	no	—
63	C₂³×C₆₁	—	4	2	no	—
64	C₂³×C₆₁	C₂²×C₈	1	4	yes	—
65	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
66	D₄×C₆₁	—	2	8	no	—
67	C₂×C₄×C₆₁	—	2	1	no	—
68	C₂×C₄×C₆₁	—	2	4	no	—
69	C₂×C₄×C₆₁	—	2	1	no	—
70	C₂×C₄×C₆₁	—	2	4	no	—
71	C₂×C₄×C₆₁	—	4	4	no	—
72	C₂×C₄×C₆₁	—	4	2	no	—
73	C₂×C₄×C₆₁	—	4	1	no	—
74	C₂×C₄×C₆₁	M₃₂	1	4	yes	complemented
75	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
78	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
79	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
80	#92418	—	2	1	no	—
81	#92418	—	2	4	no	—
82	#92418	—	2	2	no	—
83	#92418	H(5,𝔽₂)	1	2	yes	—
84	#33489	—	61	1	no	maximal, Sylow
85	C₂²×C₆₁	—	2	1	no	—
86	C₂²×C₆₁	—	2	1	no	—
87	C₂²×C₆₁	—	2	4	no	—
88	C₂²×C₆₁	—	2	4	no	—
89	C₂²×C₆₁	—	2	4	no	—
90	C₂²×C₆₁	—	4	8	no	—
91	C₂²×C₆₁	C₂×M₃₂	1	2	yes	—
92	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
93	C₄×C₆₁	—	2	2	no	—
94	C₄×C₆₁	—	2	1	no	—
95	C₄×C₆₁	—	2	4	no	—
96	C₄×C₆₁	C₂×M₃₂	1	4	yes	—
97	C₄×C₆₁	#518	1	1	yes	characteristic
98	C₄×C₆₁	#568	1	1	yes	characteristic
99	C₂×C₈×D₄	—	61	1	no	—
100	#2169	—	61	8	no	—
101	#2162	—	61	2	no	—
102	#2112	—	61	4	no	—
103	C₂×C₆₁	—	2	2	no	—
104	C₂×C₆₁	—	4	4	no	—
105	C₂×C₆₁	#2972	1	1	yes	characteristic
106	C₂×C₆₁	C₂²×M₃₂	1	1	yes	characteristic
107	C₂×C₆₁	#3407	1	1	yes	characteristic
108	C₂³×C₈	—	61	2	no	—
109	C₂×C₄×D₄	#6923	1	1	yes	characteristic
110	C₂×M₃₂	—	61	8	no	—
111	C₈×D₄	—	61	4	no	—
112	C₈×D₄	—	61	4	no	—
113	C₂×C₄⋊C₈	—	61	1	no	—
114	C₂×C₂²⋊C₈	—	61	2	no	—
115	C₂×C₄×C₈	—	61	1	no	—
116	C₄⋊C₁₆	—	61	4	no	—
117	C₂²⋊C₁₆	—	61	4	no	—
118	C₂²⋊C₁₆	—	61	8	no	—
119	C₄×C₁₆	—	61	4	no	—
120	C₆₁	#33489	1	1	yes	characteristic, complemented, Sylow
121	C₂²×D₄	#92420	1	1	yes	characteristic
122	C₂³×C₄	C₂×#6923	1	2	yes	—
123	C₂²×C₈	—	61	1	no	—
124	C₂²×C₈	—	61	4	no	—
125	C₂²×C₈	—	61	2	no	—
126	C₂²×C₈	—	61	4	no	—
127	C₂²×C₈	—	61	1	no	—
128	C₂²×C₈	—	244	4	no	—
129	C₄×D₄	#92420	1	4	yes	—
130	C₄×D₄	C₂×#6923	1	4	yes	—
131	C₂×C₄⋊C₄	#92420	1	1	yes	characteristic
132	C₂×C₂²⋊C₄	#92420	1	2	yes	—
133	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
134	M₃₂	—	244	8	no	—
135	C₂×C₁₆	—	61	8	no	—
136	C₄⋊C₈	—	61	4	no	—
137	C₂²⋊C₈	—	61	8	no	—
138	C₄×C₈	—	61	2	no	—
139	C₄×C₈	—	61	2	no	—
140	C₂⁴	C₂×#92420	1	2	yes	—
141	C₂×D₄	—	2	4	no	—
142	C₂×D₄	C₂×#92420	1	4	yes	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
146	C₂²×C₄	C₂×#92420	1	2	yes	—
147	C₂²×C₄	C₂×#92420	1	4	yes	—
148	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
149	C₂²×C₄	C₂²×#6923	1	4	yes	—
150	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
151	C₂×C₈	—	61	1	no	—
152	C₂×C₈	—	61	2	no	—
153	C₂×C₈	—	61	1	no	—
154	C₂×C₈	—	61	4	no	—
155	C₂×C₈	—	122	8	no	—
156	C₂×C₈	—	122	2	no	—
157	C₂×C₈	—	244	1	no	—
158	C₂×C₈	—	244	2	no	—
159	C₂×C₈	—	244	4	no	—
160	C₄⋊C₄	C₂×#92420	1	4	yes	—
161	C₂²⋊C₄	C₂×#92420	1	8	yes	—
162	C₄²	C₂×#92420	1	2	yes	—
163	C₄²	C₂²×#6923	1	2	yes	—
164	C₁₆	—	122	8	no	—
165	C₂³	—	2	4	no	—
166	C₂³	—	4	2	no	—
167	C₂³	—	4	2	no	—
168	C₂³	C₂²×#92420	1	4	yes	—
169	C₂³	C₂²×#92420	1	1	yes	characteristic
170	D₄	—	2	8	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	#14519851	1	4	yes	complemented
179	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
180	C₂×C₄	C₂²×#92420	1	2	yes	—
181	C₂×C₄	C₂²×#92420	1	4	yes	—
182	C₂×C₄	C₂²×#92420	1	2	yes	—
183	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
184	C₈	—	61	2	no	—
185	C₈	—	122	4	no	—
186	C₈	—	122	1	no	—
187	C₈	—	122	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	—	4	8	no	—
194	C₂²	C₂×#14519851	1	2	yes	—
195	C₂²	C₂³×#92420	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	4	no	—
199	C₄	C₂×#14519851	1	4	yes	—
200	C₄	#16840829	1	1	yes	characteristic
201	C₄	#16840840	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	4	4	no	—
204	C₂	#21462666	1	1	yes	characteristic
205	C₂	C₂²×#14519851	1	1	yes	characteristic
206	C₂	#21462786	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₂×C₄×D₄×C₆₁	C₄
prime radical	C₂²×D₄×C₆₁	C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	C₂×C₄×D₄	#6923
hypocenter	C₆₁	#33489
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #21462203	C₂	semidirect
2. #21462176	C₂	semidirect
3. #21462046	C₂	semidirect
5. C₂×#14519851	C₂²	semidirect
6. C₄⋊#11758187	C₂²	semidirect
8. C₂²⋊#11758187	C₂²	semidirect
9. C₄×#11758187	C₂²	semidirect
74. C₂×C₄×C₆₁	M₃₂	semidirect
120. C₆₁	#33489	semidirect
178. C₂×C₄	#14519851	semidirect

Maximal subgroups

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

subgroup	index
#21462203	2
#21462176	2
#21462046	2
C₂×D₄×#92418	2
#33489	61