Subgroups — #30603678

Order 15616.

330 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14519851	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11758187	C₂	1	1	yes	maximal, characteristic
3	#21460731	C₂	1	2	yes	maximal, complemented
4	C₈×#11758187	C₂	1	2	yes	maximal, complemented
5	#21444103	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519851	C₄	1	2	yes	—
7	C₂²×#11758187	C₄	1	2	yes	—
8	#16840650	C₄	1	2	yes	—
9	C₄×#11758187	C₄	1	2	yes	—
10	C₄×#11758187	C₂²	1	2	yes	—
11	C₄×#11758187	C₂²	1	1	yes	characteristic
12	C₄×#11758187	C₂²	1	1	yes	characteristic
13	#16839815	C₄	1	1	yes	characteristic
14	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
15	#16839484	C₄	1	2	yes	—
16	C₈×#92418	C₂²	1	2	yes	—
17	C₆₁×#431	C₄	1	1	yes	characteristic
18	#14519851	C₈	1	8	yes	complemented
19	C₂×#11758187	C₈	1	8	yes	complemented
20	C₂×#11758187	C₂×C₄	1	2	yes	—
21	C₂×#11758187	C₂×C₄	1	2	yes	—
22	C₂×#11758187	C₂×C₄	1	4	yes	—
23	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
24	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
25	C₄⋊#92418	C₈	1	2	yes	—
26	C₂²⋊#92418	C₈	1	2	yes	—
27	C₄×#92418	C₂×C₄	1	1	yes	characteristic
28	C₄×#92418	C₂×C₄	1	1	yes	characteristic
29	C₄×#92418	C₂×C₄	1	1	yes	characteristic
30	C₄×#92418	C₂×C₄	1	2	yes	—
31	C₄×#92418	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₈×C₆₁	C₈	1	2	yes	—
34	C₂²⋊C₈×C₆₁	C₈	1	2	yes	—
35	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
40	C₄²×C₆₁	C₄²	1	1	yes	characteristic
41	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×#92418	C₄²	1	1	yes	characteristic
45	C₂×#92418	C₄²	1	1	yes	characteristic
46	C₂×#92418	C₂×C₈	1	4	yes	—
47	C₂×#92418	C₂×C₈	1	4	yes	—
48	C₂×#92418	C₂×C₈	1	4	yes	—
49	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
50	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
51	#11758187	C₂×C₈	1	4	yes	—
52	#11758187	C₂×C₈	1	4	yes	—
53	#11758187	C₂×C₈	1	8	yes	—
54	C₂³×C₆₁	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₆₁	—	2	2	no	—
56	C₂×C₄×C₆₁	—	2	2	no	—
57	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
61	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
62	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
65	C₈×C₆₁	—	2	4	no	—
66	#92418	—	2	4	no	—
67	#92418	C₄×C₈	1	2	yes	—
68	#92418	C₄×C₈	1	2	yes	—
69	#92418	C₂²×C₈	1	2	yes	—
70	#92418	C₂²×C₈	1	2	yes	—
71	#11399	—	61	1	no	maximal, Sylow
72	C₂²×C₆₁	—	2	1	no	—
73	C₂²×C₆₁	—	2	1	no	—
74	C₂²×C₆₁	C₈²	1	2	yes	—
75	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
76	C₄×C₆₁	—	2	1	no	—
77	C₄×C₆₁	—	2	1	no	—
78	C₄×C₆₁	C₈²	1	2	yes	—
79	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
80	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
81	C₄×C₆₁	#504	1	2	yes	—
82	C₄×C₆₁	#504	1	2	yes	—
83	C₄×M₃₂	—	61	1	no	—
84	C₂×C₄×C₁₆	—	61	1	no	—
85	#1450	—	61	1	no	—
86	#1311	—	61	2	no	—
87	C₈×C₁₆	—	61	2	no	—
88	C₂×C₆₁	—	2	2	no	—
89	C₂×C₆₁	C₂×C₈²	1	1	yes	characteristic
90	C₂×C₆₁	#2108	1	1	yes	characteristic
91	C₂×C₆₁	#2108	1	1	yes	characteristic
92	C₂×M₃₂	—	61	2	no	—
93	C₂²×C₁₆	—	61	2	no	—
94	#431	—	61	1	no	—
95	#431	#6923	1	1	yes	characteristic
96	C₂×C₄×C₈	—	61	1	no	—
97	#346	—	61	2	no	—
98	C₄×C₁₆	—	61	1	no	—
99	C₄×C₁₆	—	61	2	no	—
100	C₄×C₁₆	—	61	1	no	—
101	C₄×C₁₆	—	61	2	no	—
102	#322	—	61	2	no	—
103	C₈²	—	61	2	no	—
104	C₆₁	#11399	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₈	—	61	1	no	—
106	C₂²×C₈	—	61	1	no	—
107	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
108	M₃₂	—	61	8	no	—
109	C₂×C₁₆	—	61	8	no	—
110	C₂×C₁₆	—	61	2	no	—
111	C₂×C₁₆	—	61	2	no	—
112	C₂×C₁₆	—	61	4	no	—
113	C₄⋊C₈	—	61	2	no	—
114	C₄⋊C₈	#92420	1	2	yes	—
115	C₂²⋊C₈	—	61	2	no	—
116	C₂²⋊C₈	#92420	1	2	yes	—
117	C₄×C₈	—	61	1	no	—
118	C₄×C₈	—	61	1	no	—
119	C₄×C₈	—	61	1	no	—
120	C₄×C₈	—	61	2	no	—
121	C₄×C₈	—	61	1	no	—
122	C₄×C₈	C₂×#6923	1	2	yes	—
123	C₂²×C₄	D₆₁⋊C₈	1	1	yes	characteristic
124	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
125	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
126	C₂×C₈	—	61	4	no	—
127	C₂×C₈	—	61	4	no	—
128	C₂×C₈	—	61	1	no	—
129	C₂×C₈	—	61	1	no	—
130	C₂×C₈	—	61	1	no	—
131	C₂×C₈	—	61	1	no	—
132	C₂×C₈	—	61	4	no	—
133	C₂×C₈	C₂×#92420	1	4	yes	—
134	C₄²	D₆₁⋊C₈	1	1	yes	characteristic
135	C₄²	C₄×#6923	1	1	yes	characteristic
136	C₄²	C₂×#92420	1	1	yes	characteristic
137	C₄²	C₂²×#6923	1	1	yes	characteristic
138	C₁₆	—	61	4	no	—
139	C₁₆	—	61	4	no	—
140	C₁₆	—	61	8	no	—
141	C₂³	C₄×#92420	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	C₈×#6923	1	2	yes	—
145	C₂×C₄	C₈×#6923	1	2	yes	—
146	C₂×C₄	C₄×#92420	1	1	yes	characteristic
147	C₂×C₄	C₄×#92420	1	1	yes	characteristic
148	C₂×C₄	C₄×#92420	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
150	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
151	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
152	C₈	—	2	4	no	—
153	C₈	—	61	2	no	—
154	C₈	—	61	2	no	—
155	C₈	—	61	2	no	—
156	C₈	—	61	2	no	—
157	C₈	—	122	4	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₈×#92420	1	2	yes	—
161	C₂²	C₂×C₄×#92420	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	C₈×#92420	1	2	yes	—
165	C₄	C₂×C₈×#6923	1	1	yes	characteristic
166	C₄	C₂×C₈×#6923	1	1	yes	characteristic
167	C₄	#16840820	1	2	yes	—
168	C₄	#16840821	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	C₂×C₈×#92420	1	1	yes	characteristic
171	C₂	#21462028	1	1	yes	characteristic
172	C₂	#21462030	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×#6923
commutator	C₂×C₆₁	C₂×C₈²
Fitting	C₆₁×#431	C₄
socle	C₂²×C₆₁	C₂×C₄×C₈
prime radical	C₂³×C₆₁	C₄×C₈
cosocle	C₄×#92418	C₂³
hypercenter	#431	#6923
hypocenter	C₆₁	#11399
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
3. #21460731	C₂	semidirect
4. C₈×#11758187	C₂	semidirect
18. #14519851	C₈	semidirect
19. C₂×#11758187	C₈	semidirect
104. C₆₁	#11399	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519851	2
C₂×C₄×#11758187	2
#21460731	2
C₈×#11758187	2
#21444103	2
#11399	61