Subgroups — #30624351

Order 15616.

490 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462236	C₂	1	8	yes	maximal, complemented
2	#21462156	C₂	1	2	yes	maximal
3	#21462151	C₂	1	1	yes	maximal, characteristic
4	#21462030	C₂	1	2	yes	maximal
5	C₂×#16840650	C₂	1	1	yes	maximal, characteristic
6	#21454749	C₂	1	1	yes	maximal, characteristic
7	C₂×#14519851	C₂²	1	2	yes	—
8	C₂²×#11758187	C₂²	1	1	yes	characteristic
9	C₂²×#11758187	C₂²	1	1	yes	characteristic
10	#16840683	C₂²	1	8	yes	—
11	C₄⋊#11758187	C₂²	1	4	yes	—
12	#16840651	C₂²	1	4	yes	—
13	#16840650	C₂²	1	4	yes	—
14	#16840648	C₂²	1	4	yes	—
15	#16840319	C₄	1	1	yes	characteristic
16	#16839889	C₄	1	4	yes	—
17	#16839886	C₂²	1	4	yes	—
18	#16839790	C₄	1	2	yes	—
19	#16839720	C₂²	1	2	yes	—
20	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
21	C₆₁×#423	C₄	1	1	yes	characteristic
22	#14519851	—	2	4	no	—
23	C₂×#11758187	—	2	2	no	—
24	C₂×#11758187	Q₈	1	4	yes	complemented
25	C₂×#11758187	C₂³	1	2	yes	—
26	C₂×#11758187	C₂³	1	4	yes	—
27	C₂×#11758187	C₂³	1	2	yes	—
28	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
29	C₂²×#92418	C₂³	1	1	yes	characteristic
30	#14519777	C₈	1	2	yes	—
31	#14519769	C₈	1	2	yes	—
32	#14519756	C₈	1	4	yes	—
33	C₄×Dic₁₂₂	C₂×C₄	1	4	yes	—
34	#14519736	C₂×C₄	1	2	yes	—
35	#14519706	C₂×C₄	1	8	yes	—
36	#14519687	C₂×C₄	1	4	yes	—
37	C₄×#92418	C₂×C₄	1	2	yes	—
38	C₄×#92418	C₂³	1	2	yes	—
39	C₈×Dic₆₁	C₂³	1	4	yes	—
40	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
43	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₁₂₂	C₂×C₈	1	4	yes	—
46	M₁₆×C₆₁	—	2	4	no	—
47	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
48	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
51	C₂²⋊Dic₆₁	C₂×C₈	1	4	yes	—
52	C₄⋊Dic₆₁	C₂×C₈	1	4	yes	—
53	C₄⋊Dic₆₁	C₂×C₈	1	4	yes	—
54	#11758196	C₂×C₈	1	8	yes	—
55	C₄×Dic₆₁	C₂²×C₄	1	4	yes	—
56	C₂×#92418	—	2	2	no	—
57	C₂×#92418	C₂²×C₄	1	2	yes	—
58	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
59	C₂×#92418	C₂×Q₈	1	2	yes	—
60	C₂×#92418	#41	1	2	yes	—
61	C₂×#92418	C₂⁴	1	1	yes	characteristic
62	#11758187	—	2	2	no	—
63	#11758187	—	2	4	no	—
64	#11758187	C₂×Q₈	1	4	yes	—
65	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₆₁	—	4	2	no	—
67	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
68	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
69	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₆₁	#131	1	2	yes	—
72	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
73	C₂×Dic₆₁	C₂²×C₈	1	8	yes	—
74	Dic₁₂₂	—	4	4	no	—
75	C₈×C₆₁	—	2	4	no	—
76	#92418	—	2	2	no	—
77	#92418	C₄×Q₈	1	2	yes	—
78	#92418	C₂²×Q₈	1	1	yes	characteristic
79	#92418	C₂×#41	1	1	yes	characteristic
80	#33475	—	61	1	no	maximal, Sylow
81	C₂²×C₆₁	—	2	2	no	—
82	C₂²×C₆₁	C₈×Q₈	1	2	yes	—
83	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
84	C₄×C₆₁	—	2	2	no	—
85	C₄×C₆₁	—	2	2	no	—
86	C₄×C₆₁	C₈×Q₈	1	2	yes	—
87	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
89	Dic₆₁	—	2	8	no	—
90	#2960	—	61	1	no	—
91	#2183	—	61	8	no	—
92	#2151	—	61	2	no	—
93	#2150	—	61	1	no	—
94	#2108	—	61	2	no	—
95	C₂×#346	—	61	1	no	—
96	C₂×C₆₁	—	2	2	no	—
97	C₂×C₆₁	C₂×C₈×Q₈	1	1	yes	characteristic
98	C₂×C₆₁	#3407	1	2	yes	—
99	#533	—	61	1	no	—
100	C₂×M₃₂	—	61	2	no	—
101	C₂²×C₁₆	—	61	1	no	—
102	C₂²×C₁₆	—	61	1	no	—
103	#446	—	61	4	no	—
104	C₈×Q₈	—	61	4	no	—
105	#424	—	61	2	no	—
106	#423	#6923	1	1	yes	characteristic
107	#405	—	61	2	no	—
108	C₂×C₄×C₈	—	61	1	no	—
109	C₄⋊C₁₆	—	61	4	no	—
110	C₄⋊C₁₆	—	61	8	no	—
111	#346	—	61	4	no	—
112	#346	—	61	4	no	—
113	C₄×C₁₆	—	61	4	no	—
114	C₆₁	#33475	1	1	yes	characteristic, complemented, Sylow
115	C₂×M₁₆	C₂×#6923	1	2	yes	—
116	C₂²×C₈	—	61	1	no	—
117	C₂²×C₈	—	61	1	no	—
118	C₄⋊Q₈	—	61	2	no	—
119	#125	—	61	2	no	—
120	C₂²⋊Q₈	—	61	4	no	—
121	C₄×Q₈	—	61	4	no	—
122	#117	—	61	2	no	—
123	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
124	M₃₂	—	122	4	no	—
125	C₂×C₁₆	—	61	4	no	—
126	C₂×C₁₆	—	61	2	no	—
127	C₂×C₁₆	—	61	4	no	—
128	C₂×C₁₆	—	61	2	no	—
129	C₂×C₁₆	—	122	2	no	—
130	C₄⋊C₈	—	61	8	no	—
131	C₄⋊C₈	C₂×#6923	1	4	yes	—
132	#97	—	61	4	no	—
133	C₄×C₈	—	61	4	no	—
134	C₄×C₈	—	61	2	no	—
135	C₄×C₈	—	61	2	no	—
136	C₂×Q₈	—	61	4	no	—
137	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
138	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
139	M₁₆	—	2	4	no	—
140	C₂×C₈	—	61	2	no	—
141	C₂×C₈	—	61	2	no	—
142	C₂×C₈	—	61	1	no	—
143	C₂×C₈	—	61	1	no	—
144	C₂×C₈	—	61	2	no	—
145	C₂×C₈	—	122	2	no	—
146	C₂×C₈	C₂²×#6923	1	4	yes	—
147	C₄⋊C₄	—	61	8	no	—
148	C₄⋊C₄	—	61	4	no	—
149	C₄⋊C₄	—	61	4	no	—
150	C₂²⋊C₄	—	61	4	no	—
151	C₄²	—	61	4	no	—
152	C₄²	D₆₁⋊C₈	1	1	yes	characteristic
153	C₄²	D₆₁⋊C₈	1	1	yes	characteristic
154	C₄²	C₂²×#6923	1	2	yes	—
155	C₁₆	—	61	4	no	—
156	C₁₆	—	122	2	no	—
157	C₁₆	—	122	4	no	—
158	C₂³	C₂×D₆₁⋊C₈	1	1	yes	characteristic
159	Q₈	—	244	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	61	8	no	—
162	C₂×C₄	Q₈×#6923	1	2	yes	—
163	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
164	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
165	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
166	C₂×C₄	#14519864	1	2	yes	—
167	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
168	C₈	—	2	4	no	—
169	C₈	—	61	2	no	—
170	C₈	—	61	1	no	—
171	C₈	—	61	1	no	—
172	C₈	—	122	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	#16840811	1	2	yes	—
175	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	122	8	no	—
179	C₄	#16840811	1	2	yes	—
180	C₄	C₂×Q₈×#6923	1	1	yes	characteristic
181	C₄	C₂×#14519864	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×#16840811	1	1	yes	characteristic
184	C₂	#21462788	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₆₁×#423	C₄
prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	#423	#6923
hypocenter	C₆₁	#33475
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
1. #21462236	C₂	semidirect
24. C₂×#11758187	Q₈	semidirect
114. C₆₁	#33475	semidirect

Maximal subgroups

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

subgroup	index
#21462236	2
#21462156	2
#21462151	2
#21462030	2
C₂×#16840650	2
#21454749	2
#33475	61