Subgroups — #30607296

Order 15616.

366 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11758187	C₂	1	2	yes	maximal
2	#21460857	C₂	1	4	yes	maximal
3	C₄²×#92418	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11758187	C₂²	1	4	yes	—
5	C₄⋊#11758187	—	2	4	no	—
6	C₂×C₄×#92418	C₄	1	2	yes	—
7	C₂×C₄×#92418	C₄	1	1	yes	characteristic
8	C₂×C₄×#92418	C₂²	1	2	yes	—
9	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
10	C₄³×C₆₁	C₄	1	1	yes	characteristic
11	C₂×#11758187	—	2	4	no	—
12	C₂×#11758187	—	2	8	no	—
13	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
14	C₂²×#92418	C₂×C₄	1	2	yes	—
15	C₂²×#92418	C₂³	1	1	yes	characteristic
16	C₄×#92418	—	2	2	no	—
17	C₄×#92418	—	2	4	no	—
18	C₄×#92418	—	2	4	no	—
19	C₄×#92418	D₄	1	2	yes	—
20	C₄×#92418	Q₈	1	2	yes	—
21	C₂×C₄²×C₆₁	C₈	1	2	yes	—
22	C₂×C₄²×C₆₁	C₈	1	1	yes	characteristic
23	C₂×C₄²×C₆₁	C₈	1	1	yes	characteristic
24	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
28	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
29	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
31	C₄²×C₆₁	—	2	4	no	—
32	C₄²×C₆₁	—	2	2	no	—
33	C₄²×C₆₁	—	2	2	no	—
34	C₄²×C₆₁	—	2	2	no	—
35	C₄²×C₆₁	C₁₆	1	4	yes	complemented
36	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
37	C₄²×C₆₁	M₁₆	1	2	yes	—
38	C₂×#92418	—	2	2	no	—
39	C₂×#92418	—	2	2	no	—
40	C₂×#92418	—	2	2	no	—
41	C₂×#92418	C₄⋊C₄	1	2	yes	—
42	C₂×#92418	M₁₆	1	4	yes	—
43	C₂×#92418	C₂×D₄	1	1	yes	characteristic
44	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
45	C₂×#92418	#41	1	4	yes	—
46	#11758187	—	4	4	no	—
47	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₆₁	—	2	2	no	—
49	C₂×C₄×C₆₁	—	2	2	no	—
50	C₂×C₄×C₆₁	—	2	2	no	—
51	C₂×C₄×C₆₁	—	2	2	no	—
52	C₂×C₄×C₆₁	—	2	2	no	—
53	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₆₁	C₂×C₁₆	1	4	yes	—
56	C₂×C₄×C₆₁	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₆₁	M₃₂	1	4	yes	—
58	C₂×C₄×C₆₁	M₃₂	1	2	yes	—
59	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₁	#117	1	2	yes	—
61	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
62	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
63	#92418	—	2	2	no	—
64	#92418	—	2	4	no	—
65	#92418	#124	1	2	yes	—
66	#92418	#125	1	2	yes	—
67	#12879	—	61	1	no	maximal, Sylow
68	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₆₁	#431	1	2	yes	—
70	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
71	C₂²×C₆₁	C₂×M₃₂	1	1	yes	characteristic
72	C₂²×C₆₁	C₂×M₃₂	1	2	yes	—
73	C₄×C₆₁	—	2	2	no	—
74	C₄×C₆₁	—	2	4	no	—
75	C₄×C₆₁	—	2	4	no	—
76	C₄×C₆₁	C₄⋊C₁₆	1	4	yes	—
77	C₄×C₆₁	#382	1	1	yes	characteristic
78	C₄×C₆₁	#423	1	1	yes	characteristic
79	C₄×C₆₁	#432	1	2	yes	—
80	C₂×C₄⋊C₁₆	—	61	2	no	—
81	C₄²×C₈	—	61	1	no	—
82	#1374	—	61	4	no	—
83	C₂×C₆₁	#1831	1	1	yes	characteristic
84	C₂×C₆₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
85	C₂×C₆₁	#2150	1	1	yes	characteristic
86	C₂×C₆₁	#2162	1	2	yes	—
87	C₂×C₆₁	#2163	1	2	yes	—
88	C₂²×C₁₆	—	61	4	no	—
89	C₂×C₄×C₈	—	61	1	no	—
90	C₂×C₄×C₈	—	61	2	no	—
91	C₂×C₄×C₈	—	61	1	no	—
92	C₂×C₄×C₈	—	61	2	no	—
93	C₄³	#6923	1	1	yes	characteristic
94	C₄⋊C₁₆	—	122	4	no	—
95	C₆₁	#12879	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	61	1	no	—
97	C₂²×C₈	—	61	1	no	—
98	C₂²×C₈	—	61	2	no	—
99	C₂×C₄²	#92420	1	2	yes	—
100	C₂×C₄²	#92420	1	1	yes	characteristic
101	C₂×C₄²	#92420	1	1	yes	characteristic
102	C₂×C₄²	C₂×#6923	1	2	yes	—
103	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
104	C₂×C₁₆	—	122	4	no	—
105	C₂×C₁₆	—	122	8	no	—
106	C₄×C₈	—	61	2	no	—
107	C₄×C₈	—	61	2	no	—
108	C₄×C₈	—	122	4	no	—
109	C₄×C₈	—	122	4	no	—
110	C₄×C₈	—	122	2	no	—
111	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
112	C₂²×C₄	C₂×#92420	1	2	yes	—
113	C₂²×C₄	C₂×#92420	1	2	yes	—
114	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
115	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
116	C₂×C₈	—	61	1	no	—
117	C₂×C₈	—	61	1	no	—
118	C₂×C₈	—	61	4	no	—
119	C₂×C₈	—	61	2	no	—
120	C₂×C₈	—	61	4	no	—
121	C₂×C₈	—	122	2	no	—
122	C₂×C₈	—	122	2	no	—
123	C₂×C₈	—	122	2	no	—
124	C₄²	—	2	4	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	2	2	no	—
128	C₄²	#11758187	1	4	yes	complemented
129	C₄²	#11758213	1	2	yes	—
130	C₄²	C₄⋊#6923	1	2	yes	—
131	C₁₆	—	244	4	no	—
132	C₂³	C₂²×#92420	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₄⋊#92420	1	2	yes	—
139	C₂×C₄	C₄⋊#92420	1	2	yes	—
140	C₂×C₄	C₂×#11758187	1	4	yes	—
141	C₂×C₄	C₂×#11758187	1	2	yes	—
142	C₂×C₄	#14519851	1	4	yes	—
143	C₂×C₄	#14519851	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
145	C₂×C₄	#14519856	1	2	yes	—
146	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
147	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
148	C₈	—	61	2	no	—
149	C₈	—	61	2	no	—
150	C₈	—	122	2	no	—
151	C₈	—	122	4	no	—
152	C₂²	C₂×C₄⋊#92420	1	1	yes	characteristic
153	C₂²	#16840778	1	2	yes	—
154	C₂²	C₂²×#11758187	1	1	yes	characteristic
155	C₂²	C₂×#14519851	1	1	yes	characteristic
156	C₂²	C₂×#14519851	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	4	no	—
159	C₄	—	2	4	no	—
160	C₄	C₄⋊#11758187	1	4	yes	—
161	C₄	#16840706	1	1	yes	characteristic
162	C₄	#16840767	1	1	yes	characteristic
163	C₄	#16840784	1	2	yes	—
164	C₂	#21461734	1	1	yes	characteristic
165	C₂	C₂×C₄⋊#11758187	1	1	yes	characteristic
166	C₂	#21462151	1	1	yes	characteristic
167	C₂	#21462176	1	2	yes	—
168	C₂	#21462180	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6923
commutator	C₂²×C₆₁	C₂²×C₁₆
Fitting	C₄³×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂²×#92418	C₂³
hypercenter	C₄³	#6923
hypocenter	C₆₁	#12879
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
35. C₄²×C₆₁	C₁₆	semidirect
95. C₆₁	#12879	semidirect
128. C₄²	#11758187	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11758187	2
#21460857	2
C₄²×#92418	2
#12879	61