Subgroups — #30607344

Order 15616.

414 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840683	C₂	1	4	yes	maximal
2	C₂×C₄⋊#11758187	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11758187	C₂²	1	4	yes	—
5	#16840683	—	2	8	no	—
6	C₄⋊#11758187	—	2	4	no	—
7	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
9	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
10	C₄²×Dic₆₁	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
12	C₂×#11758187	—	2	4	no	—
13	C₂×#11758187	—	2	8	no	—
14	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
15	C₂²×#92418	C₂³	1	1	yes	characteristic
16	C₂×C₄×Dic₆₁	C₈	1	1	yes	characteristic
17	C₂×C₄×Dic₆₁	C₈	1	2	yes	—
18	C₂×C₄×Dic₆₁	C₈	1	1	yes	characteristic
19	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
20	C₄×#92418	D₄	1	2	yes	—
21	C₄×#92418	Q₈	1	2	yes	—
22	C₈×Dic₆₁	—	2	4	no	—
23	C₈×Dic₆₁	D₄	1	4	yes	—
24	C₈×Dic₆₁	Q₈	1	4	yes	—
25	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₆₁	—	2	2	no	—
28	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
31	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
32	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
33	C₂×C₈×C₆₁	—	2	2	no	—
34	C₂×C₈×C₆₁	—	2	2	no	—
35	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
36	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
37	C₄²×C₆₁	M₁₆	1	2	yes	—
38	C₄×Dic₆₁	—	2	2	no	—
39	C₄×Dic₆₁	—	2	4	no	—
40	C₄×Dic₆₁	C₁₆	1	4	yes	complemented
41	C₄×Dic₆₁	C₄⋊C₄	1	4	yes	—
42	C₄×Dic₆₁	M₁₆	1	4	yes	—
43	C₂×#92418	—	2	2	no	—
44	C₂×#92418	C₄⋊C₄	1	2	yes	—
45	C₂×#92418	C₂×D₄	1	2	yes	—
46	C₂×#92418	C₂×D₄	1	1	yes	characteristic
47	C₂×#92418	C₂×Q₈	1	2	yes	—
48	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
49	#11758187	—	4	4	no	—
50	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	—	2	2	no	—
52	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
53	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₆₁	C₂×C₁₆	1	2	yes	—
55	C₂×C₄×C₆₁	M₃₂	1	2	yes	—
56	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
60	C₂×Dic₆₁	—	2	2	no	—
61	C₂×Dic₆₁	—	2	2	no	—
62	C₂×Dic₆₁	C₄⋊C₈	1	4	yes	—
63	C₂×Dic₆₁	C₄⋊C₈	1	4	yes	—
64	C₂×Dic₆₁	C₂×C₁₆	1	4	yes	—
65	C₂×Dic₆₁	M₃₂	1	4	yes	—
66	C₈×C₆₁	—	2	4	no	—
67	#92418	—	2	2	no	—
68	#92418	#127	1	1	yes	characteristic
69	#92418	C₄⋊Q₈	1	2	yes	—
70	#92418	C₄⋊Q₈	1	1	yes	characteristic
71	#12898	—	61	1	no	maximal, Sylow
72	C₂²×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
73	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
74	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
75	C₂²×C₆₁	C₂×M₃₂	1	2	yes	—
76	C₂²×C₆₁	C₂×M₃₂	1	1	yes	characteristic
77	C₄×C₆₁	—	2	2	no	—
78	C₄×C₆₁	C₄⋊C₁₆	1	4	yes	complemented
79	C₄×C₆₁	#384	1	1	yes	characteristic
80	C₄×C₆₁	#423	1	2	yes	—
81	C₄×C₆₁	#423	1	1	yes	characteristic
82	Dic₆₁	—	2	4	no	—
83	Dic₆₁	C₄⋊C₁₆	1	8	yes	complemented
84	C₂×C₄⋊C₁₆	—	61	2	no	—
85	C₂×C₄⋊C₁₆	—	61	4	no	—
86	C₄²×C₈	—	61	1	no	—
87	C₂×C₆₁	#1842	1	1	yes	characteristic
88	C₂×C₆₁	C₂×C₄⋊C₁₆	1	2	yes	—
89	C₂×C₆₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
90	C₂×C₆₁	#2150	1	2	yes	—
91	C₂×C₆₁	#2150	1	1	yes	characteristic
92	C₂²×C₁₆	—	61	4	no	—
93	C₂×C₄×C₈	—	61	2	no	—
94	C₂×C₄×C₈	—	61	1	no	—
95	C₂×C₄×C₈	—	61	2	no	—
96	C₂×C₄×C₈	#6923	1	1	yes	characteristic
97	C₄³	—	61	1	no	—
98	C₄⋊C₁₆	—	122	4	no	—
99	C₄⋊C₁₆	—	122	8	no	—
100	C₆₁	#12898	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₈	—	61	1	no	—
102	C₂²×C₈	—	61	1	no	—
103	C₂²×C₈	C₂×#6923	1	2	yes	—
104	C₂×C₄²	—	61	1	no	—
105	C₂×C₄²	—	61	2	no	—
106	C₂×C₄²	—	61	2	no	—
107	C₂×C₄²	—	61	1	no	—
108	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
109	C₂×C₁₆	—	122	4	no	—
110	C₂×C₁₆	—	122	8	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	—	61	4	no	—
113	C₄×C₈	—	61	4	no	—
114	C₄×C₈	—	61	2	no	—
115	C₄×C₈	—	61	2	no	—
116	C₄×C₈	—	122	4	no	—
117	C₂²×C₄	—	61	2	no	—
118	C₂²×C₄	—	61	2	no	—
119	C₂²×C₄	D₆₁⋊C₈	1	1	yes	characteristic
120	C₂²×C₄	D₆₁⋊C₈	1	1	yes	characteristic
121	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	61	2	no	—
125	C₂×C₈	—	61	1	no	—
126	C₂×C₈	—	61	2	no	—
127	C₂×C₈	—	61	1	no	—
128	C₂×C₈	—	61	2	no	—
129	C₂×C₈	—	122	2	no	—
130	C₂×C₈	C₄⋊#6923	1	4	yes	—
131	C₄²	—	61	4	no	—
132	C₄²	—	61	4	no	—
133	C₄²	—	61	4	no	—
134	C₄²	—	122	4	no	—
135	C₄²	—	122	2	no	—
136	C₄²	C₄⋊#6923	1	2	yes	—
137	C₄²	#11758217	1	2	yes	—
138	C₁₆	—	244	4	no	—
139	C₂³	C₂×D₆₁⋊C₈	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	61	4	no	—
142	C₂×C₄	—	61	4	no	—
143	C₂×C₄	—	61	4	no	—
144	C₂×C₄	—	61	4	no	—
145	C₂×C₄	—	122	2	no	—
146	C₂×C₄	—	122	2	no	—
147	C₂×C₄	#14519844	1	2	yes	—
148	C₂×C₄	#14519844	1	2	yes	—
149	C₂×C₄	D₆₁⋊C₁₆	1	2	yes	—
150	C₂×C₄	#14519852	1	2	yes	—
151	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
152	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
153	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
154	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
155	C₈	—	2	4	no	—
156	C₈	—	61	1	no	—
157	C₈	—	61	2	no	—
158	C₈	—	61	1	no	—
159	C₈	—	122	2	no	—
160	C₂²	D₆₁⋊(C₄⋊C₈)	1	2	yes	—
161	C₂²	C₂×#14519844	1	1	yes	characteristic
162	C₂²	C₂×D₆₁⋊C₁₆	1	1	yes	characteristic
163	C₂²	D₆₁⋊M₃₂	1	2	yes	—
164	C₂²	C₂×#14519852	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	61	8	no	—
167	C₄	—	122	4	no	—
168	C₄	#16840683	1	4	yes	complemented
169	C₄	#16840708	1	1	yes	characteristic
170	C₄	#16840765	1	2	yes	—
171	C₄	#16840767	1	1	yes	characteristic
172	C₂	#21461752	1	1	yes	characteristic
173	C₂	D₆₁⋊(C₄⋊C₁₆)	1	2	yes	—
174	C₂	C₂×#16840683	1	1	yes	characteristic
175	C₂	#21462152	1	2	yes	—
176	C₂	#21462153	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6923
commutator	C₂²×C₆₁	C₂²×C₁₆
Fitting	C₂×C₄×C₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂²×#92418	C₂³
hypercenter	C₂×C₄×C₈	#6923
hypocenter	C₆₁	#12898
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
40. C₄×Dic₆₁	C₁₆	semidirect
78. C₄×C₆₁	C₄⋊C₁₆	semidirect
83. Dic₆₁	C₄⋊C₁₆	semidirect
100. C₆₁	#12898	semidirect
168. C₄	#16840683	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840683	2
C₂×C₄⋊#11758187	2
C₄×C₈×Dic₆₁	2
#12898	61