Subgroups — #30595970

Order 15616.

526 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	#21461534	C₂	1	1	yes	maximal, characteristic
2	C₄²×#92420	C₂	1	1	yes	maximal, characteristic
3	C₂×#92420⋊C₈	C₂	1	2	yes	maximal
4	C₂×#16840603	C₂	1	2	yes	maximal
5	#21445563	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#92420	C₄	1	4	yes	—
7	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
8	C₂×C₄×#92420	C₂²	1	2	yes	—
9	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
10	#16840628	C₄	1	4	yes	—
11	#92420⋊C₈	C₄	1	8	yes	complemented
12	#16840603	C₄	1	8	yes	complemented
13	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
14	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
15	#16839517	C₄	1	1	yes	characteristic
16	#16839516	C₄	1	2	yes	—
17	C₆₁×#336	C₄	1	1	yes	characteristic
18	C₂²×#92420	C₂×C₄	1	2	yes	—
19	C₂²×#92420	C₂×C₄	1	2	yes	—
20	C₂²×#92420	C₂×C₄	1	4	yes	—
21	C₄×#92420	—	2	8	no	—
22	C₄×#92420	C₂×C₄	1	4	yes	—
23	C₄×#92420	C₂×C₄	1	4	yes	—
24	C₄×#92420	C₂×C₄	1	4	yes	—
25	C₄×#92420	C₂×C₄	1	4	yes	—
26	C₂²×#92418	C₂×C₄	1	2	yes	—
27	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
31	C₈×Dic₆₁	C₂×C₄	1	4	yes	—
32	C₈×Dic₆₁	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₂²×Dic₆₁	C₄²	1	2	yes	—
38	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×#92420	—	2	4	no	—
41	C₂×#92420	—	2	4	no	—
42	C₂×#92420	—	2	8	no	—
43	C₂×#92420	C₄²	1	4	yes	—
44	C₂×#92420	C₄²	1	4	yes	—
45	C₂×#92420	C₄²	1	8	yes	—
46	C₂×C₈×C₆₁	—	2	2	no	—
47	C₂×C₈×C₆₁	—	2	2	no	—
48	C₂×C₈×C₆₁	C₄²	1	4	yes	—
49	C₄²×C₆₁	—	2	2	no	—
50	C₄×Dic₆₁	—	2	1	no	—
51	C₄×Dic₆₁	—	2	1	no	—
52	C₄×Dic₆₁	—	2	4	no	—
53	C₄×Dic₆₁	M₁₆	1	8	yes	—
54	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
55	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
56	C₂×#92418	—	2	2	no	—
57	C₂×#92418	—	2	2	no	—
58	C₂×#92418	C₄²	1	4	yes	—
59	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₆₁	—	2	2	no	—
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	#97	1	4	yes	—
63	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
64	C₂×C₄×C₆₁	#131	1	2	yes	—
65	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
67	C₂×Dic₆₁	—	2	2	no	—
68	C₂×Dic₆₁	—	2	2	no	—
69	C₂×Dic₆₁	#97	1	4	yes	—
70	C₂×Dic₆₁	C₂×C₄²	1	1	yes	characteristic
71	C₂×Dic₆₁	C₂×C₄²	1	1	yes	characteristic
72	C₂×Dic₆₁	C₂×C₄²	1	2	yes	—
73	C₂×Dic₆₁	C₂×M₁₆	1	4	yes	—
74	C₂×Dic₆₁	#131	1	4	yes	—
75	#92420	—	2	4	no	—
76	#92420	—	2	4	no	—
77	#92420	—	2	8	no	—
78	C₈×C₆₁	—	2	4	no	—
79	#92418	—	2	4	no	—
80	#7843	—	61	1	no	maximal, Sylow
81	C₂²×C₆₁	C₄³	1	1	yes	characteristic
82	C₂²×C₆₁	C₂×#97	1	2	yes	—
83	C₂²×C₆₁	#405	1	1	yes	characteristic
84	C₂²×C₆₁	#405	1	1	yes	characteristic
85	C₂²×C₆₁	#405	1	2	yes	—
86	C₄×C₆₁	—	2	4	no	—
87	C₄×C₆₁	#405	1	2	yes	—
88	C₄×C₆₁	#405	1	2	yes	—
89	Dic₆₁	—	2	4	no	—
90	Dic₆₁	C₄×M₁₆	1	2	yes	—
91	Dic₆₁	C₄×M₁₆	1	2	yes	—
92	Dic₆₁	#405	1	4	yes	—
93	#1744	—	61	1	no	—
94	#1744	—	61	1	no	—
95	C₄²×C₈	—	61	1	no	—
96	C₂×#322	—	61	2	no	—
97	C₂×#322	—	61	2	no	—
98	C₂×C₆₁	#1452	1	2	yes	—
99	C₂×C₆₁	#1452	1	2	yes	—
100	C₂×C₆₁	C₄×#97	1	1	yes	characteristic
101	C₂×C₆₁	#1726	1	1	yes	characteristic
102	C₂×C₆₁	#1726	1	1	yes	characteristic
103	C₂×C₄×C₈	—	61	1	no	—
104	C₂×C₄×C₈	—	61	1	no	—
105	C₂×C₄×C₈	—	61	2	no	—
106	C₂×C₄×C₈	—	61	2	no	—
107	C₂×C₄×C₈	—	61	4	no	—
108	C₄³	—	61	1	no	—
109	#336	—	61	4	no	—
110	#336	—	61	2	no	—
111	#336	—	61	1	no	—
112	#336	#6923	1	1	yes	characteristic
113	#322	—	61	8	no	—
114	#322	—	61	8	no	—
115	C₆₁	#7843	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₈	—	61	4	no	—
117	C₂²×C₈	—	61	2	no	—
118	C₂²×C₈	—	61	2	no	—
119	C₂²×C₈	—	61	2	no	—
120	C₂²×C₈	C₂×#6923	1	2	yes	—
121	C₂×C₄²	—	61	2	no	—
122	C₂×C₄²	—	61	1	no	—
123	C₂×C₄²	—	61	2	no	—
124	C₂×C₄²	—	61	1	no	—
125	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
126	C₄×C₈	—	61	4	no	—
127	C₄×C₈	—	61	4	no	—
128	C₄×C₈	—	61	4	no	—
129	C₄×C₈	—	61	4	no	—
130	C₄×C₈	—	61	4	no	—
131	C₄×C₈	—	61	4	no	—
132	C₄×C₈	—	122	8	no	—
133	C₂²×C₄	—	61	1	no	—
134	C₂²×C₄	—	61	1	no	—
135	C₂²×C₄	—	61	2	no	—
136	C₂²×C₄	C₄×#6923	1	2	yes	—
137	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	61	8	no	—
141	C₂×C₈	—	61	4	no	—
142	C₂×C₈	—	61	4	no	—
143	C₂×C₈	—	61	4	no	—
144	C₂×C₈	—	122	4	no	—
145	C₂×C₈	—	122	4	no	—
146	C₂×C₈	—	122	8	no	—
147	C₂×C₈	—	122	2	no	—
148	C₂×C₈	—	122	2	no	—
149	C₂×C₈	C₄×#6923	1	4	yes	—
150	C₄²	—	2	2	no	—
151	C₄²	—	61	8	no	—
152	C₄²	—	61	2	no	—
153	C₄²	—	61	2	no	—
154	C₄²	—	122	4	no	—
155	C₄²	—	122	1	no	—
156	C₄²	—	122	1	no	—
157	C₂³	C₂×C₄×#6923	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	61	4	no	—
161	C₂×C₄	—	61	4	no	—
162	C₂×C₄	—	61	1	no	—
163	C₂×C₄	—	61	1	no	—
164	C₂×C₄	—	61	2	no	—
165	C₂×C₄	—	61	4	no	—
166	C₂×C₄	—	122	2	no	—
167	C₂×C₄	—	122	2	no	—
168	C₂×C₄	#14519831	1	4	yes	—
169	C₂×C₄	C₂×C₄×#6923	1	2	yes	—
170	C₂×C₄	#14519864	1	1	yes	characteristic
171	C₂×C₄	#14519864	1	2	yes	—
172	C₂×C₄	#14519865	1	1	yes	characteristic
173	C₈	—	2	4	no	—
174	C₈	—	122	4	no	—
175	C₈	—	122	4	no	—
176	C₈	—	122	8	no	—
177	C₈	—	122	4	no	—
178	C₂²	C₄²×#6923	1	1	yes	characteristic
179	C₂²	C₂×#14519831	1	2	yes	—
180	C₂²	#16840726	1	1	yes	characteristic
181	C₂²	#16840727	1	1	yes	characteristic
182	C₂²	#16840728	1	2	yes	—
183	C₄	—	2	4	no	—
184	C₄	—	61	2	no	—
185	C₄	—	61	2	no	—
186	C₄	—	61	4	no	—
187	C₄	—	122	4	no	—
188	C₄	#16840726	1	2	yes	—
189	C₄	#16840727	1	2	yes	—
190	C₂	#21461034	1	2	yes	—
191	C₂	#21461035	1	2	yes	—
192	C₂	C₄×#14519831	1	1	yes	characteristic
193	C₂	#21461476	1	1	yes	characteristic
194	C₂	#21461477	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6923
commutator	C₂²×C₆₁	C₄³
Fitting	C₆₁×#336	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₄²
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	#336	#6923
hypocenter	C₆₁	#7843
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
11. #92420⋊C₈	C₄	semidirect
12. #16840603	C₄	semidirect
115. C₆₁	#7843	semidirect

Maximal subgroups

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

subgroup	index
#21461534	2
C₄²×#92420	2
C₂×#92420⋊C₈	2
C₂×#16840603	2
#21445563	2
#7843	61