Subgroups — #30544406

Order 15616.

1810 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	#21454760	C₂	1	12	yes	maximal, complemented
2	#21454754	C₂	1	12	yes	maximal, complemented
3	#21454383	C₂	1	3	yes	maximal
4	C₂×#16839718	C₂	1	3	yes	maximal
5	C₆₁×#3458	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758194	C₂²	1	3	yes	—
7	C₂³×#92418	C₂²	1	1	yes	characteristic
8	#16839893	C₂²	1	48	yes	complemented
9	Q₈×#92418	C₂²	1	16	yes	complemented
10	#16839789	C₂²	1	24	yes	—
11	#16839786	C₂²	1	12	yes	—
12	#16839718	C₂²	1	12	yes	—
13	#16839718	C₂²	1	12	yes	—
14	C₂×#14519688	C₂²	1	6	yes	—
15	C₂×C₄×#92418	C₂²	1	6	yes	—
16	C₆₁×#544	C₄	1	12	yes	—
17	C₆₁×#543	C₄	1	4	yes	—
18	C₆₁×#533	C₂²	1	12	yes	—
19	C₂×#117×C₆₁	C₂²	1	3	yes	—
20	C₂×#11758194	—	2	12	no	—
21	C₂×#11758194	C₂³	1	12	yes	—
22	C₂²×#92418	Q₈	1	8	yes	complemented
23	C₂²×#92418	C₂³	1	6	yes	—
24	C₂²×#92418	C₂³	1	3	yes	—
25	C₂²×#92418	C₂³	1	3	yes	—
26	C₄⋊#92418	C₂³	1	48	yes	—
27	#14519688	C₂³	1	24	yes	—
28	C₄×#92418	C₂³	1	12	yes	—
29	C₄×#92418	C₂³	1	12	yes	—
30	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₆₁	C₂×C₄	1	24	yes	—
32	#125×C₆₁	C₂×C₄	1	24	yes	—
33	C₂²⋊Q₈×C₆₁	C₂×C₄	1	48	yes	—
34	C₄×Q₈×C₆₁	C₂³	1	16	yes	—
35	#117×C₆₁	C₂×C₄	1	12	yes	—
36	#117×C₆₁	C₂³	1	12	yes	—
37	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	6	yes	—
38	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	6	yes	—
39	C₂×C₄²×C₆₁	C₂³	1	6	yes	—
40	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×Q₈×C₆₁	C₂²×C₄	1	16	yes	—
42	C₂²×C₄×C₆₁	C₂²×C₄	1	12	yes	—
43	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
44	C₂²×C₄×C₆₁	C₂×Q₈	1	4	yes	—
45	C₂²×C₄×C₆₁	#41	1	4	yes	—
46	C₂²×C₄×C₆₁	C₂⁴	1	3	yes	—
47	C₄⋊C₄×C₆₁	C₂²×C₄	1	24	yes	—
48	C₄⋊C₄×C₆₁	C₂²×C₄	1	48	yes	—
49	C₂²⋊C₄×C₆₁	C₂²×C₄	1	24	yes	—
50	C₄²×C₆₁	C₂²×C₄	1	12	yes	—
51	C₄²×C₆₁	C₂⁴	1	12	yes	—
52	#11758194	—	2	24	no	—
53	C₂×#92418	—	2	12	no	—
54	C₂×#92418	C₂×Q₈	1	12	yes	—
55	C₂×#92418	C₂×Q₈	1	12	yes	—
56	C₂×#92418	C₂⁴	1	12	yes	—
57	C₂×#92418	C₂⁴	1	1	yes	characteristic
58	C₂×#92418	C₂⁴	1	3	yes	—
59	C₂³×C₆₁	—	4	2	no	—
60	C₂³×C₆₁	C₄×Q₈	1	4	yes	—
61	C₂³×C₆₁	C₂³×C₄	1	3	yes	—
62	Q₈×C₆₁	—	4	16	no	—
63	C₂×C₄×C₆₁	—	4	12	no	—
64	C₂×C₄×C₆₁	C₄×Q₈	1	12	yes	—
65	C₂×C₄×C₆₁	C₂³×C₄	1	24	yes	—
66	C₂×C₄×C₆₁	C₂³×C₄	1	3	yes	—
67	C₂×C₄×C₆₁	C₂²×Q₈	1	6	yes	—
68	C₂×C₄×C₆₁	C₂×#41	1	6	yes	—
69	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
70	#92418	—	2	12	no	—
71	#92418	C₂²×Q₈	1	2	yes	—
72	#92418	C₂²×Q₈	1	6	yes	—
73	#60191	—	61	1	no	maximal, Sylow
74	C₂²×C₆₁	—	2	6	no	—
75	C₂²×C₆₁	—	4	4	no	—
76	C₂²×C₆₁	C₂×C₄×Q₈	1	6	yes	—
77	C₂²×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
78	C₄×C₆₁	—	2	24	no	—
79	C₄×C₆₁	C₂×C₄×Q₈	1	6	yes	—
80	C₄×C₆₁	C₂³×Q₈	1	1	yes	characteristic
81	C₄×C₆₁	C₂²×#41	1	1	yes	characteristic
82	#3458	D₆₁	1	1	yes	characteristic
83	#2961	—	61	12	no	—
84	#2960	—	61	12	no	—
85	#2905	—	61	3	no	—
86	C₂×#405	—	61	3	no	—
87	C₂×C₆₁	—	2	6	no	—
88	C₂×C₆₁	C₂²×C₄×Q₈	1	1	yes	characteristic
89	C₂×C₆₁	#3573	1	2	yes	—
90	C₂²×M₁₆	—	61	3	no	—
91	C₂³×C₈	—	61	1	no	—
92	#544	Dic₆₁	1	12	yes	—
93	#543	Dic₆₁	1	4	yes	—
94	#533	C₂×D₆₁	1	12	yes	—
95	C₂×#117	C₂×D₆₁	1	3	yes	—
96	#446	—	61	48	no	—
97	C₈×Q₈	—	61	16	no	—
98	#424	—	61	24	no	—
99	#423	—	61	12	no	—
100	#405	—	61	12	no	—
101	#405	—	61	12	no	—
102	C₂×#97	—	61	6	no	—
103	C₂×C₄×C₈	—	61	6	no	—
104	C₆₁	#60191	1	1	yes	characteristic, complemented, Sylow
105	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
106	C₂×M₁₆	—	61	12	no	—
107	C₂×M₁₆	—	122	12	no	—
108	C₂²×C₈	—	61	8	no	—
109	C₂²×C₈	—	61	3	no	—
110	C₂²×C₈	—	61	3	no	—
111	C₂²×C₈	—	61	6	no	—
112	C₄⋊Q₈	C₂×Dic₆₁	1	24	yes	—
113	#125	C₂×Dic₆₁	1	24	yes	—
114	C₂²⋊Q₈	C₂×Dic₆₁	1	48	yes	—
115	C₄×Q₈	C₂²×D₆₁	1	16	yes	—
116	#117	C₂×Dic₆₁	1	12	yes	—
117	#117	C₂²×D₆₁	1	12	yes	—
118	C₂×C₄⋊C₄	C₂×Dic₆₁	1	6	yes	—
119	C₂×C₂²⋊C₄	C₂×Dic₆₁	1	6	yes	—
120	C₂×C₄²	C₂²×D₆₁	1	6	yes	—
121	C₄⋊C₈	—	61	48	no	—
122	#97	—	61	24	no	—
123	C₄×C₈	—	61	12	no	—
124	C₄×C₈	—	61	12	no	—
125	C₂⁴	C₂²×Dic₆₁	1	1	yes	characteristic
126	C₂×Q₈	C₂²×Dic₆₁	1	16	yes	—
127	C₂²×C₄	Q₈×D₆₁	1	4	yes	—
128	C₂²×C₄	#11758226	1	4	yes	—
129	C₂²×C₄	C₂²×Dic₆₁	1	12	yes	—
130	C₂²×C₄	C₂²×Dic₆₁	1	3	yes	—
131	C₂²×C₄	C₂³×D₆₁	1	3	yes	—
132	M₁₆	—	122	24	no	—
133	C₂×C₈	—	61	12	no	—
134	C₂×C₈	—	61	12	no	—
135	C₂×C₈	—	61	1	no	—
136	C₂×C₈	—	61	3	no	—
137	C₂×C₈	—	61	12	no	—
138	C₂×C₈	—	122	12	no	—
139	C₄⋊C₄	C₂²×Dic₆₁	1	24	yes	—
140	C₄⋊C₄	C₂²×Dic₆₁	1	48	yes	—
141	C₂²⋊C₄	C₂²×Dic₆₁	1	24	yes	—
142	C₄²	C₂²×Dic₆₁	1	12	yes	—
143	C₄²	C₂³×D₆₁	1	12	yes	—
144	C₂³	—	4	2	no	—
145	C₂³	Q₈×Dic₆₁	1	4	yes	—
146	C₂³	C₂³×Dic₆₁	1	3	yes	—
147	Q₈	—	4	16	no	—
148	C₂×C₄	—	4	12	no	—
149	C₂×C₄	Q₈×Dic₆₁	1	12	yes	—
150	C₂×C₄	C₂³×Dic₆₁	1	24	yes	—
151	C₂×C₄	C₂³×Dic₆₁	1	3	yes	—
152	C₂×C₄	C₂×Q₈×D₆₁	1	6	yes	—
153	C₂×C₄	C₂×#11758226	1	6	yes	—
154	C₂×C₄	C₂⁴×D₆₁	1	1	yes	characteristic
155	C₈	—	61	2	no	—
156	C₈	—	61	6	no	—
157	C₈	—	122	12	no	—
158	C₂²	—	2	6	no	—
159	C₂²	—	4	4	no	—
160	C₂²	C₂×Q₈×Dic₆₁	1	6	yes	—
161	C₂²	C₂⁴×Dic₆₁	1	1	yes	characteristic
162	C₄	—	2	24	no	—
163	C₄	C₂×Q₈×Dic₆₁	1	6	yes	—
164	C₄	C₂²×Q₈×D₆₁	1	1	yes	characteristic
165	C₄	C₂²×#11758226	1	1	yes	characteristic
166	C₂	—	2	6	no	—
167	C₂	C₂²×Q₈×Dic₆₁	1	1	yes	characteristic
168	C₂	#21460531	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₆₁
commutator, socle	C₂²×C₆₁	C₂⁴×C₄
Fitting	C₆₁×#3458	C₂
prime radical	C₂⁴×C₆₁	C₂²×C₄
cosocle	C₂×C₄×C₆₁	C₂⁵
hypercenter	#3458	D₆₁
hypocenter	C₆₁	#60191
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #3458

Complements

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

N	H ≅ G/N	type
1. #21454760	C₂	semidirect
2. #21454754	C₂	semidirect
8. #16839893	C₂²	semidirect
9. Q₈×#92418	C₂²	semidirect
22. C₂²×#92418	Q₈	semidirect
104. C₆₁	#60191	semidirect

Maximal subgroups

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

subgroup	index
#21454760	2
#21454754	2
#21454383	2
C₂×#16839718	2
C₆₁×#3458	2
#60191	61