Subgroups — #30599501

Order 15616.

462 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	#21461751	C₂	1	2	yes	maximal
2	C₂×#16840625	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#92420	—	2	4	no	—
5	C₂×C₄⋊#92420	C₂²	1	4	yes	—
6	#16840625	—	2	8	no	—
7	C₂×C₄×#92418	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
9	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
10	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
12	C₂²×#92420	—	2	4	no	—
13	C₄⋊#92420	—	2	8	no	—
14	C₄⋊#92420	—	4	8	no	—
15	C₂²×#92418	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
18	C₂×C₄×Dic₆₁	Q₈	1	2	yes	—
19	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
20	C₄×#92418	—	2	2	no	—
21	C₈×Dic₆₁	—	2	4	no	—
22	C₈×Dic₆₁	D₄	1	4	yes	—
23	C₈×Dic₆₁	Q₈	1	4	yes	—
24	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₆₁	C₈	1	4	yes	complemented
27	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
29	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
30	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
31	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
32	C₂×#92420	—	4	4	no	—
33	C₂×#92420	—	4	8	no	—
34	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
37	C₄²×C₆₁	C₂×C₈	1	2	yes	—
38	C₄²×C₆₁	M₁₆	1	2	yes	—
39	C₄×Dic₆₁	—	2	4	no	—
40	C₄×Dic₆₁	—	2	2	no	—
41	C₄×Dic₆₁	D₈	1	4	yes	—
42	C₄×Dic₆₁	Q₁₆	1	4	yes	—
43	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
44	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
45	C₂×#92418	—	2	2	no	—
46	C₂×#92418	—	2	2	no	—
47	C₂×#92418	C₄⋊C₄	1	4	yes	—
48	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
50	C₂×C₄×C₆₁	#107	1	4	yes	—
51	C₂×C₄×C₆₁	#108	1	4	yes	—
52	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
53	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
55	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
56	C₂×Dic₆₁	—	2	2	no	—
57	C₂×Dic₆₁	—	2	2	no	—
58	C₂×Dic₆₁	—	2	2	no	—
59	C₂×Dic₆₁	#107	1	4	yes	—
60	C₂×Dic₆₁	#127	1	1	yes	characteristic
61	C₂×Dic₆₁	C₄⋊Q₈	1	2	yes	—
62	C₂×Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
64	C₂×Dic₆₁	C₂×Q₁₆	1	2	yes	—
65	#92420	—	8	4	no	—
66	C₈×C₆₁	C₄⋊C₈	1	8	yes	complemented
67	#92418	—	2	4	no	—
68	#8806	—	61	1	no	maximal, Sylow
69	C₂²×C₆₁	#384	1	1	yes	characteristic
70	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₂²×C₆₁	#423	1	1	yes	characteristic
72	C₂²×C₆₁	C₂×#107	1	2	yes	—
73	C₂²×C₆₁	C₂×#108	1	2	yes	—
74	C₄×C₆₁	#335	1	8	yes	complemented
75	C₄×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
76	C₄×C₆₁	#423	1	2	yes	—
77	Dic₆₁	—	2	4	no	—
78	Dic₆₁	—	2	2	no	—
79	Dic₆₁	#493	1	1	yes	characteristic
80	Dic₆₁	#494	1	1	yes	characteristic
81	Dic₆₁	#500	1	2	yes	—
82	#1842	—	61	2	no	—
83	C₄²×C₈	—	61	1	no	—
84	C₂×#335	—	61	4	no	—
85	C₂×C₆₁	C₂×#335	1	2	yes	—
86	C₂×C₆₁	#1567	1	2	yes	—
87	C₂×C₆₁	#1842	1	1	yes	characteristic
88	C₂×C₆₁	#1845	1	1	yes	characteristic
89	C₂×C₆₁	#1848	1	1	yes	characteristic
90	C₂×C₄⋊C₈	—	61	4	no	—
91	C₂×C₄⋊C₈	—	122	4	no	—
92	C₂×C₄×C₈	—	61	2	no	—
93	C₂×C₄×C₈	—	61	2	no	—
94	C₂×C₄×C₈	—	61	1	no	—
95	C₂×C₄×C₈	#6923	1	1	yes	characteristic
96	C₄³	—	61	1	no	—
97	#335	—	122	8	no	—
98	C₆₁	#8806	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	—	61	2	no	—
100	C₂²×C₈	—	122	4	no	—
101	C₂²×C₈	C₂×#6923	1	2	yes	—
102	C₂×C₄²	—	61	2	no	—
103	C₂×C₄²	—	61	1	no	—
104	C₂×C₄²	—	61	2	no	—
105	C₂×C₄²	—	61	1	no	—
106	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
107	C₄⋊C₈	—	122	8	no	—
108	C₄⋊C₈	—	244	8	no	—
109	C₄×C₈	—	61	4	no	—
110	C₄×C₈	—	61	4	no	—
111	C₄×C₈	—	122	4	no	—
112	C₄×C₈	—	122	2	no	—
113	C₄×C₈	#92420	1	4	yes	complemented
114	C₂²×C₄	—	61	1	no	—
115	C₂²×C₄	—	61	2	no	—
116	C₂²×C₄	—	61	1	no	—
117	C₂²×C₄	C₄⋊#6923	1	2	yes	—
118	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
119	C₂×C₈	—	61	4	no	—
120	C₂×C₈	—	122	2	no	—
121	C₂×C₈	—	122	2	no	—
122	C₂×C₈	—	244	4	no	—
123	C₂×C₈	—	244	8	no	—
124	C₂×C₈	#11758213	1	4	yes	—
125	C₂×C₈	C₄⋊#6923	1	4	yes	—
126	C₂×C₈	C₂×#92420	1	4	yes	—
127	C₄²	—	61	4	no	—
128	C₄²	—	61	4	no	—
129	C₄²	—	61	2	no	—
130	C₄²	—	61	2	no	—
131	C₄²	—	122	4	no	—
132	C₄²	—	122	2	no	—
133	C₄²	#11758213	1	2	yes	—
134	C₄²	C₂×#92420	1	2	yes	—
135	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
136	C₂×C₄	—	61	1	no	—
137	C₂×C₄	—	61	1	no	—
138	C₂×C₄	—	61	2	no	—
139	C₂×C₄	—	61	2	no	—
140	C₂×C₄	—	61	2	no	—
141	C₂×C₄	—	61	4	no	—
142	C₂×C₄	—	122	2	no	—
143	C₂×C₄	—	122	2	no	—
144	C₂×C₄	—	122	2	no	—
145	C₂×C₄	C₄⋊#92420	1	4	yes	—
146	C₂×C₄	#14519846	1	4	yes	—
147	C₂×C₄	#14519848	1	4	yes	—
148	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
149	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
150	C₂×C₄	C₂×#11758213	1	2	yes	—
151	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
152	C₈	—	122	4	no	—
153	C₈	—	488	4	no	—
154	C₈	C₄⋊#92420	1	8	yes	complemented
155	C₂²	#16840708	1	1	yes	characteristic
156	C₂²	C₂×C₄⋊#92420	1	1	yes	characteristic
157	C₂²	#16840766	1	1	yes	characteristic
158	C₂²	C₂×#14519846	1	2	yes	—
159	C₂²	C₂×#14519848	1	2	yes	—
160	C₄	—	61	1	no	—
161	C₄	—	61	1	no	—
162	C₄	—	61	2	no	—
163	C₄	—	122	2	no	—
164	C₄	—	122	4	no	—
165	C₄	#16840625	1	8	yes	complemented
166	C₄	C₂×C₄⋊#92420	1	2	yes	—
167	C₄	#16840766	1	2	yes	—
168	C₂	C₂×#16840625	1	2	yes	—
169	C₂	#21461377	1	2	yes	—
170	C₂	#21461751	1	1	yes	characteristic
171	C₂	#21461755	1	1	yes	characteristic
172	C₂	#21461760	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×C₄×C₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×C₄×C₈	#6923
hypocenter	C₆₁	#8806
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
26. C₄×C₈×C₆₁	C₈	semidirect
66. C₈×C₆₁	C₄⋊C₈	semidirect
74. C₄×C₆₁	#335	semidirect
98. C₆₁	#8806	semidirect
113. C₄×C₈	#92420	semidirect
154. C₈	C₄⋊#92420	semidirect
165. C₄	#16840625	semidirect

Maximal subgroups

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

subgroup	index
#21461751	2
C₂×#16840625	2
C₄×C₈×Dic₆₁	2
#8806	61