Subgroups — #30606657

Order 15616.

382 conjugacy classes of subgroups.

The subgroup lattice has 196 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×#16840650	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11758187	C₂	1	1	yes	maximal, characteristic
4	#21460857	C₂	1	2	yes	maximal
5	#21445506	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758187	C₂²	1	2	yes	—
7	C₂²×#11758187	C₂²	1	2	yes	—
8	C₄⋊#11758187	C₄	1	8	yes	complemented
9	#16840650	—	2	2	no	—
10	C₄×#11758187	—	2	2	no	—
11	C₂×#14519688	C₄	1	1	yes	characteristic
12	C₂×#14519688	C₄	1	1	yes	characteristic
13	C₂×#14519688	C₄	1	1	yes	characteristic
14	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
15	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
16	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
17	C₆₁×#376	C₄	1	1	yes	characteristic
18	C₂×#11758187	—	2	2	no	—
19	C₂×#11758187	—	2	2	no	—
20	C₂×#11758187	—	2	2	no	—
21	C₂×#11758187	—	2	2	no	—
22	C₂×#11758187	C₂×C₄	1	4	yes	—
23	C₂×#11758187	C₂×C₄	1	4	yes	—
24	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
25	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
26	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
27	C₂²×#92418	C₂³	1	1	yes	characteristic
28	#14519688	—	2	2	no	—
29	#14519688	—	2	2	no	—
30	#14519688	C₈	1	4	yes	—
31	C₄×#92418	—	2	2	no	—
32	C₄×#92418	C₂×C₄	1	4	yes	—
33	C₄×#92418	D₄	1	2	yes	—
34	C₄×#92418	Q₈	1	2	yes	—
35	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
38	#95×C₆₁	C₈	1	4	yes	—
39	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₆₁	—	2	1	no	—
44	C₄²×C₆₁	—	2	1	no	—
45	C₄²×C₆₁	C₄²	1	2	yes	—
46	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
47	C₄²×C₆₁	C₂×C₈	1	2	yes	—
48	C₄²×C₆₁	M₁₆	1	2	yes	—
49	C₂×#92418	—	2	1	no	—
50	C₂×#92418	—	2	1	no	—
51	C₂×#92418	—	2	1	no	—
52	C₂×#92418	—	2	1	no	—
53	C₂×#92418	C₄²	1	2	yes	—
54	C₂×#92418	C₄⋊C₄	1	2	yes	—
55	C₂×#92418	C₂×C₈	1	2	yes	—
56	C₂×#92418	C₂×C₈	1	2	yes	—
57	C₂×#92418	M₁₆	1	2	yes	—
58	C₂×#92418	C₂²×C₄	1	2	yes	—
59	C₂×#92418	C₂×D₄	1	1	yes	characteristic
60	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
61	C₂×#92418	#41	1	2	yes	—
62	#11758187	—	2	4	no	—
63	#11758187	—	2	4	no	—
64	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₆₁	—	2	4	no	—
66	C₂×C₄×C₆₁	—	2	4	no	—
67	C₂×C₄×C₆₁	—	2	2	no	—
68	C₂×C₄×C₆₁	—	2	4	no	—
69	C₂×C₄×C₆₁	C₄×C₈	1	4	yes	—
70	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
71	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
77	#92418	—	2	2	no	—
78	#92418	—	2	2	no	—
79	#92418	C₄⋊C₈	1	4	yes	—
80	#92418	C₄×D₄	1	2	yes	—
81	#92418	C₄×Q₈	1	2	yes	—
82	#12654	—	61	1	no	maximal, Sylow
83	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
84	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₂²×C₆₁	#431	1	1	yes	characteristic
86	C₂²×C₆₁	#504	1	4	yes	—
87	C₄×C₆₁	—	2	4	no	—
88	C₄×C₆₁	—	2	4	no	—
89	C₄×C₆₁	—	4	2	no	—
90	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
91	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
92	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
93	C₄×C₆₁	#431	1	1	yes	characteristic
94	C₂×C₄⋊C₁₆	—	61	2	no	—
95	C₂×#346	—	61	1	no	—
96	C₂×C₄×C₁₆	—	61	1	no	—
97	#1726	—	61	1	no	—
98	#1374	—	61	2	no	—
99	C₂×C₆₁	C₄×C₄⋊C₈	1	1	yes	characteristic
100	C₂×C₆₁	#2108	1	2	yes	—
101	C₂×C₆₁	#2151	1	2	yes	—
102	C₂×C₆₁	#2164	1	2	yes	—
103	C₂²×C₁₆	—	61	2	no	—
104	C₂²×C₁₆	—	61	2	no	—
105	C₂×#97	—	61	1	no	—
106	C₂×#97	—	61	1	no	—
107	C₂×#97	—	61	1	no	—
108	C₂×C₄×C₈	—	61	1	no	—
109	C₂×C₄×C₈	—	61	1	no	—
110	C₂×C₄×C₈	—	61	1	no	—
111	#376	#6923	1	1	yes	characteristic
112	C₄⋊C₁₆	—	61	8	no	—
113	#346	—	122	2	no	—
114	C₄×C₁₆	—	122	2	no	—
115	C₆₁	#12654	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₈	—	61	1	no	—
117	C₂²×C₈	—	61	1	no	—
118	C₂²×C₈	—	61	1	no	—
119	C₂²×C₈	—	61	1	no	—
120	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
121	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
122	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
123	C₂×C₁₆	—	61	4	no	—
124	C₂×C₁₆	—	61	4	no	—
125	C₂×C₁₆	—	122	2	no	—
126	C₂×C₁₆	—	122	2	no	—
127	C₂×C₁₆	—	122	2	no	—
128	C₂×C₁₆	—	122	2	no	—
129	#97	—	61	4	no	—
130	#97	—	122	2	no	—
131	#97	—	122	2	no	—
132	C₄×C₈	—	61	4	no	—
133	C₄×C₈	—	61	2	no	—
134	C₄×C₈	—	61	2	no	—
135	C₄×C₈	—	122	2	no	—
136	#95	#92420	1	4	yes	—
137	C₂²×C₄	C₂×#92420	1	2	yes	—
138	C₂²×C₄	C₂×#92420	1	2	yes	—
139	C₂²×C₄	C₂×#92420	1	2	yes	—
140	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
141	C₂×C₈	—	61	2	no	—
142	C₂×C₈	—	61	1	no	—
143	C₂×C₈	—	61	2	no	—
144	C₂×C₈	—	61	1	no	—
145	C₂×C₈	—	61	2	no	—
146	C₂×C₈	—	61	2	no	—
147	C₂×C₈	—	61	2	no	—
148	C₂×C₈	—	61	2	no	—
149	C₂×C₈	—	61	2	no	—
150	C₂×C₈	—	122	1	no	—
151	C₂×C₈	—	122	1	no	—
152	C₂×C₈	—	122	1	no	—
153	C₂×C₈	—	122	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	D₆₁⋊C₈	1	2	yes	—
157	C₄²	#11758213	1	2	yes	—
158	C₄²	C₄×#6923	1	2	yes	—
159	C₄²	C₄⋊#6923	1	2	yes	—
160	C₁₆	—	122	4	no	—
161	C₁₆	—	122	4	no	—
162	C₂³	C₂²×#92420	1	1	yes	characteristic
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	C₄×#92420	1	4	yes	—
168	C₂×C₄	C₄⋊#92420	1	4	yes	—
169	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
171	C₂×C₄	#14519856	1	1	yes	characteristic
172	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
173	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
174	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
175	C₈	—	61	4	no	—
176	C₈	—	61	2	no	—
177	C₈	—	61	2	no	—
178	C₈	—	122	2	no	—
179	C₈	—	122	2	no	—
180	C₂²	C₂×C₄×#92420	1	1	yes	characteristic
181	C₂²	C₂×C₄⋊#92420	1	1	yes	characteristic
182	C₂²	#16840778	1	1	yes	characteristic
183	C₂²	#16840820	1	4	yes	—
184	C₄	—	2	4	no	—
185	C₄	—	2	4	no	—
186	C₄	—	4	2	no	—
187	C₄	C₄×C₄⋊#6923	1	1	yes	characteristic
188	C₄	C₄×#11758213	1	1	yes	characteristic
189	C₄	D₆₁⋊(C₄⋊C₈)	1	1	yes	characteristic
190	C₄	#16840781	1	1	yes	characteristic
191	C₂	C₄×C₄⋊#92420	1	1	yes	characteristic
192	C₂	#21462028	1	2	yes	—
193	C₂	#21462154	1	2	yes	—
194	C₂	#21462184	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6923
commutator	C₂²×C₆₁	C₂×C₄×C₈
Fitting	C₆₁×#376	C₄
socle, prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂²×#92418	C₂³
hypercenter	#376	#6923
hypocenter	C₆₁	#12654
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
8. C₄⋊#11758187	C₄	semidirect
115. C₆₁	#12654	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×#16840650	2
C₂×C₄×#11758187	2
#21460857	2
#21445506	2
#12654	61