Subgroups — #30594222

Order 15616.

262 conjugacy classes of subgroups.

The subgroup lattice has 166 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	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11758187	—	2	2	no	—
4	C₄⋊#11758187	—	2	2	no	—
5	C₄⋊#11758187	C₄	1	4	yes	—
6	C₂×C₄×#92418	C₄	1	1	yes	characteristic
7	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
8	C₈×#92418	—	2	2	no	—
9	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
10	C₂×#11758187	—	2	4	no	—
11	C₂×#11758187	—	4	2	no	—
12	C₂×#11758187	—	4	2	no	—
13	C₂²×#92418	C₈	1	2	yes	—
14	C₂²×#92418	D₄	1	1	yes	characteristic
15	C₂²×#92418	Q₈	1	1	yes	characteristic
16	C₄×#92418	—	2	1	no	—
17	C₄×#92418	—	2	1	no	—
18	C₄×#92418	C₂×C₄	1	2	yes	—
19	C₄×#92418	D₄	1	2	yes	—
20	C₂²×C₈×C₆₁	—	2	1	no	—
21	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₆₁	—	2	1	no	—
23	C₄×C₈×C₆₁	C₈	1	2	yes	—
24	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₆₁	—	2	4	no	—
28	C₂×C₈×C₆₁	—	2	2	no	—
29	C₄²×C₆₁	C₄²	1	1	yes	characteristic
30	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
31	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
32	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
33	C₂×#92418	—	2	2	no	—
34	C₂×#92418	—	2	1	no	—
35	C₂×#92418	—	2	1	no	—
36	C₂×#92418	—	2	2	no	—
37	C₂×#92418	C₁₆	1	4	yes	complemented
38	C₂×#92418	C₂²⋊C₄	1	2	yes	—
39	C₂×#92418	C₄⋊C₄	1	2	yes	—
40	C₂×#92418	D₈	1	1	yes	characteristic
41	C₂×#92418	SD₁₆	1	2	yes	—
42	C₂×#92418	Q₁₆	1	1	yes	characteristic
43	#11758187	—	4	4	no	—
44	C₂³×C₆₁	C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₆₁	—	2	2	no	—
46	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
54	C₂×C₄×C₆₁	#108	1	1	yes	characteristic
55	C₂×C₄×C₆₁	#108	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₆₁	M₃₂	1	2	yes	—
58	C₈×C₆₁	—	2	4	no	—
59	#92418	—	2	2	no	—
60	#92418	—	2	4	no	—
61	#92418	#102	1	2	yes	—
62	#92418	#103	1	2	yes	—
63	#7151	—	61	1	no	maximal, Sylow
64	C₂²×C₆₁	#334	1	1	yes	characteristic
65	C₂²×C₆₁	#335	1	1	yes	characteristic
66	C₂²×C₆₁	#336	1	1	yes	characteristic
67	C₂²×C₆₁	C₄⋊C₁₆	1	2	yes	—
68	C₂²×C₆₁	#364	1	2	yes	—
69	C₄×C₆₁	—	2	1	no	—
70	C₄×C₆₁	—	2	1	no	—
71	C₄×C₆₁	#334	1	1	yes	characteristic
72	C₄×C₆₁	#335	1	1	yes	characteristic
73	C₄×C₆₁	#340	1	1	yes	characteristic
74	C₄×C₆₁	#341	1	1	yes	characteristic
75	C₄×C₆₁	C₄×C₁₆	1	1	yes	characteristic
76	C₄×C₆₁	#346	1	1	yes	characteristic
77	C₄×C₆₁	C₂²⋊C₁₆	1	2	yes	—
78	C₂×C₄⋊C₁₆	—	61	2	no	—
79	C₂×C₈²	—	61	1	no	—
80	C₂×C₆₁	#1277	1	1	yes	characteristic
81	C₂×C₆₁	#1367	1	2	yes	—
82	C₂×C₆₁	#1370	1	2	yes	—
83	C₂×C₆₁	#1374	1	1	yes	characteristic
84	C₂×C₆₁	#1376	1	1	yes	characteristic
85	C₂²×C₁₆	—	122	2	no	—
86	C₂×C₄×C₈	—	61	1	no	—
87	C₂×C₄×C₈	—	61	1	no	—
88	C₂×C₄×C₈	#6923	1	1	yes	characteristic
89	C₄⋊C₁₆	—	61	4	no	—
90	C₄⋊C₁₆	—	122	2	no	—
91	C₈²	—	122	2	no	—
92	C₆₁	#7151	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₈	—	2	1	no	—
94	C₂²×C₈	—	61	1	no	—
95	C₂²×C₈	—	61	2	no	—
96	C₂²×C₈	—	61	1	no	—
97	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
98	C₂×C₁₆	—	122	4	no	—
99	C₂×C₁₆	—	244	2	no	—
100	C₂×C₁₆	—	244	2	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	61	2	no	—
103	C₄×C₈	—	61	2	no	—
104	C₄×C₈	—	122	1	no	—
105	C₄×C₈	—	122	1	no	—
106	C₄×C₈	#92420	1	2	yes	—
107	C₂²×C₄	D₆₁⋊C₈	1	1	yes	characteristic
108	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
109	C₂²×C₄	#11758217	1	1	yes	characteristic
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	61	2	no	—
113	C₂×C₈	—	61	2	no	—
114	C₂×C₈	—	61	1	no	—
115	C₂×C₈	—	61	1	no	—
116	C₂×C₈	—	61	2	no	—
117	C₂×C₈	—	61	4	no	—
118	C₂×C₈	—	122	2	no	—
119	C₂×C₈	—	122	1	no	—
120	C₂×C₈	—	122	1	no	—
121	C₂×C₈	—	122	2	no	—
122	C₄²	#11758213	1	1	yes	characteristic
123	C₄²	C₄×#6923	1	1	yes	characteristic
124	C₄²	C₂×#92420	1	1	yes	characteristic
125	C₄²	C₂²⋊#6923	1	1	yes	characteristic
126	C₁₆	—	244	4	no	—
127	C₂³	#14519844	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
130	C₂×C₄	C₄×#92420	1	1	yes	characteristic
131	C₂×C₄	#14519832	1	1	yes	characteristic
132	C₂×C₄	C₂²⋊#92420	1	1	yes	characteristic
133	C₂×C₄	#14519834	1	1	yes	characteristic
134	C₂×C₄	C₄⋊#92420	1	1	yes	characteristic
135	C₂×C₄	#14519845	1	1	yes	characteristic
136	C₂×C₄	#14519846	1	1	yes	characteristic
137	C₂×C₄	#14519847	1	1	yes	characteristic
138	C₂×C₄	#14519848	1	1	yes	characteristic
139	C₂×C₄	D₆₁⋊C₁₆	1	2	yes	—
140	C₂×C₄	#14519852	1	2	yes	—
141	C₈	—	2	4	no	—
142	C₈	—	61	2	no	—
143	C₈	—	61	2	no	—
144	C₈	—	122	2	no	—
145	C₈	—	122	4	no	—
146	C₂²	#16840624	1	1	yes	characteristic
147	C₂²	#16840626	1	1	yes	characteristic
148	C₂²	#16840628	1	1	yes	characteristic
149	C₂²	#16840683	1	2	yes	—
150	C₂²	#16840685	1	2	yes	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	#16840623	1	1	yes	characteristic
154	C₄	#16840625	1	1	yes	characteristic
155	C₄	#16840636	1	1	yes	characteristic
156	C₄	#16840638	1	1	yes	characteristic
157	C₄	#16840648	1	1	yes	characteristic
158	C₄	#16840651	1	1	yes	characteristic
159	C₄	#16840657	1	2	yes	—
160	C₂	#21460649	1	1	yes	characteristic
161	C₂	#21460842	1	2	yes	—
162	C₂	#21460849	1	2	yes	—
163	C₂	#21460858	1	1	yes	characteristic
164	C₂	#21460863	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄⋊#6923
commutator	C₄×C₆₁	C₄×C₁₆
Frattini	C₂×C₄²	C₂×#6923
Fitting	C₂×C₄×C₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₄⋊C₈
cosocle	C₂×C₄×#92418	C₂²
hypercenter	C₂×C₄×C₈	#6923
hypocenter	C₆₁	#7151
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
37. C₂×#92418	C₁₆	semidirect
92. C₆₁	#7151	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
C₂×C₈×#92418	2
#7151	61