Subgroups — #97×#92420 #30595955

Order 15616.

606 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461533	C₂	1	1	yes	maximal, characteristic
2	C₄²×#92420	C₂	1	1	yes	maximal, characteristic
3	C₂×#16840603	C₂	1	2	yes	maximal
4	C₂×C₈×#92420	C₂	1	2	yes	maximal
5	#97×Dic₆₁	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	#16840603	C₄	1	8	yes	complemented
12	C₈×#92420	C₄	1	8	yes	complemented
13	C₂×#14519688	C₄	1	1	yes	characteristic
14	C₂×#14519687	C₄	1	2	yes	—
15	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
16	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
17	C₂×#97×C₆₁	C₄	1	1	yes	characteristic
18	C₂²×#92420	C₂×C₄	1	2	yes	—
19	C₂²×#92420	C₂×C₄	1	4	yes	—
20	C₂²×#92420	C₂×C₄	1	2	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	2	yes	—
28	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
31	#14519688	C₈	1	4	yes	complemented
32	#14519687	C₈	1	8	yes	complemented
33	C₈×Dic₆₁	C₂×C₄	1	4	yes	—
34	C₈×Dic₆₁	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
37	#97×C₆₁	C₈	1	4	yes	complemented
38	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
39	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₆₁	C₄²	1	2	yes	—
41	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×#92420	—	2	4	no	—
44	C₂×#92420	—	2	8	no	—
45	C₂×#92420	C₄²	1	4	yes	—
46	C₂×#92420	C₄²	1	8	yes	—
47	C₂×#92420	C₄²	1	4	yes	—
48	C₂×#92420	M₁₆	1	8	yes	—
49	C₂×C₈×C₆₁	C₄²	1	4	yes	—
50	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
51	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
52	C₄²×C₆₁	C₂×C₈	1	2	yes	—
53	C₄²×C₆₁	C₂×C₈	1	2	yes	—
54	C₄×Dic₆₁	—	2	4	no	—
55	C₄×Dic₆₁	—	2	4	no	—
56	C₄×Dic₆₁	C₂×C₈	1	4	yes	—
57	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
58	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
59	C₂×#92418	C₄²	1	4	yes	—
60	C₂×#92418	C₂×C₈	1	4	yes	—
61	C₂×#92418	C₂×C₈	1	4	yes	—
62	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	—	2	2	no	—
65	C₂×C₄×C₆₁	C₄×C₈	1	4	yes	—
66	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
67	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
70	C₂×Dic₆₁	—	2	2	no	—
71	C₂×Dic₆₁	—	2	2	no	—
72	C₂×Dic₆₁	C₄×C₈	1	4	yes	—
73	C₂×Dic₆₁	C₂×C₄²	1	1	yes	characteristic
74	C₂×Dic₆₁	C₂×C₄²	1	1	yes	characteristic
75	C₂×Dic₆₁	C₂×C₄²	1	2	yes	—
76	C₂×Dic₆₁	C₂×M₁₆	1	4	yes	—
77	C₂×Dic₆₁	#131	1	4	yes	—
78	#92420	—	2	4	no	—
79	#92420	—	2	8	no	—
80	#92420	#97	1	8	yes	direct product factor
81	C₈×C₆₁	C₄×C₈	1	8	yes	complemented
82	#92418	C₄×C₈	1	8	yes	complemented
83	C₈×#97	—	61	1	no	maximal, Sylow
84	C₂²×C₆₁	C₄³	1	1	yes	characteristic
85	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
86	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
87	C₂²×C₆₁	C₄×M₁₆	1	2	yes	—
88	C₂²×C₆₁	#405	1	2	yes	—
89	C₄×C₆₁	—	2	4	no	—
90	C₄×C₆₁	C₂×C₄×C₈	1	2	yes	—
91	C₄×C₆₁	C₂×C₄×C₈	1	2	yes	—
92	Dic₆₁	—	2	4	no	—
93	Dic₆₁	C₂×#97	1	2	yes	—
94	Dic₆₁	C₄×M₁₆	1	2	yes	—
95	Dic₆₁	#405	1	4	yes	—
96	#1744	—	61	1	no	—
97	C₄×#97	—	61	1	no	—
98	C₄²×C₈	—	61	1	no	—
99	C₂×#322	—	61	2	no	—
100	C₂×C₈²	—	61	2	no	—
101	C₂×C₆₁	C₈×M₁₆	1	2	yes	—
102	C₂×C₆₁	C₈×M₁₆	1	2	yes	—
103	C₂×C₆₁	C₄²×C₈	1	1	yes	characteristic
104	C₂×C₆₁	C₄×#97	1	1	yes	characteristic
105	C₂×C₆₁	#1726	1	1	yes	characteristic
106	C₂×#97	—	61	2	no	—
107	C₂×#97	—	61	1	no	—
108	C₂×#97	#6923	1	1	yes	characteristic
109	C₂×C₄×C₈	—	61	1	no	—
110	C₂×C₄×C₈	—	61	2	no	—
111	C₂×C₄×C₈	—	61	1	no	—
112	C₂×C₄×C₈	—	61	2	no	—
113	C₂×C₄×C₈	—	61	4	no	—
114	C₄³	—	61	1	no	—
115	#336	—	61	4	no	—
116	#322	—	61	8	no	—
117	C₈²	—	61	8	no	—
118	C₆₁	C₈×#97	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₈	—	61	2	no	—
120	C₂²×C₈	—	61	2	no	—
121	C₂²×C₈	—	61	4	no	—
122	C₂²×C₈	—	61	2	no	—
123	C₂²×C₈	C₂×#6923	1	2	yes	—
124	C₂×C₄²	—	61	1	no	—
125	C₂×C₄²	—	61	1	no	—
126	C₂×C₄²	—	61	2	no	—
127	C₂×C₄²	—	61	2	no	—
128	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
129	#97	—	61	8	no	—
130	#97	—	61	4	no	—
131	#97	#92420	1	4	yes	direct product factor
132	C₄×C₈	—	61	4	no	—
133	C₄×C₈	—	61	4	no	—
134	C₄×C₈	—	61	4	no	—
135	C₄×C₈	—	61	4	no	—
136	C₄×C₈	—	61	4	no	—
137	C₄×C₈	—	61	4	no	—
138	C₄×C₈	—	122	8	no	—
139	C₂²×C₄	—	61	1	no	—
140	C₂²×C₄	—	61	1	no	—
141	C₂²×C₄	—	61	2	no	—
142	C₂²×C₄	C₄×#6923	1	2	yes	—
143	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
144	C₂×C₈	—	61	4	no	—
145	C₂×C₈	—	61	8	no	—
146	C₂×C₈	—	61	4	no	—
147	C₂×C₈	—	61	4	no	—
148	C₂×C₈	—	61	4	no	—
149	C₂×C₈	—	61	8	no	—
150	C₂×C₈	—	61	4	no	—
151	C₂×C₈	—	122	4	no	—
152	C₂×C₈	—	122	8	no	—
153	C₂×C₈	D₆₁⋊C₈	1	4	yes	—
154	C₂×C₈	C₄×#6923	1	4	yes	—
155	C₂×C₈	C₂×#92420	1	4	yes	—
156	C₄²	—	61	2	no	—
157	C₄²	—	61	2	no	—
158	C₄²	—	61	4	no	—
159	C₄²	—	122	4	no	—
160	C₄²	—	122	4	no	—
161	C₄²	D₆₁⋊C₈	1	2	yes	—
162	C₄²	C₂×#92420	1	2	yes	—
163	C₂³	C₂×C₄×#6923	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	61	4	no	—
167	C₂×C₄	—	61	4	no	—
168	C₂×C₄	—	61	1	no	—
169	C₂×C₄	—	61	1	no	—
170	C₂×C₄	—	61	2	no	—
171	C₂×C₄	—	61	4	no	—
172	C₂×C₄	—	122	2	no	—
173	C₂×C₄	—	122	2	no	—
174	C₂×C₄	C₄×#92420	1	4	yes	—
175	C₂×C₄	C₂×C₄×#6923	1	2	yes	—
176	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
177	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
178	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
179	C₈	—	61	8	no	—
180	C₈	—	61	8	no	—
181	C₈	—	122	4	no	—
182	C₈	—	122	8	no	—
183	C₈	C₄×#92420	1	8	yes	complemented
184	C₂²	C₄²×#6923	1	1	yes	characteristic
185	C₂²	C₂×C₄×#92420	1	1	yes	characteristic
186	C₂²	C₄×D₆₁⋊C₈	1	1	yes	characteristic
187	C₂²	M₁₆×#6923	1	2	yes	—
188	C₂²	#16840728	1	2	yes	—
189	C₄	—	2	4	no	—
190	C₄	—	61	2	no	—
191	C₄	—	61	2	no	—
192	C₄	—	61	4	no	—
193	C₄	—	122	4	no	—
194	C₄	C₂×C₄×#92420	1	2	yes	—
195	C₄	C₄×D₆₁⋊C₈	1	2	yes	—
196	C₂	M₁₆×#92420	1	2	yes	—
197	C₂	#21461017	1	2	yes	—
198	C₂	C₄²×#92420	1	1	yes	characteristic
199	C₂	#97×#6923	1	1	yes	characteristic
200	C₂	#21461477	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6923
commutator	C₂×C₆₁	C₄²×C₈
Fitting	C₂×#97×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₄²
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×#97	#6923
hypocenter	C₆₁	C₈×#97
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
11. #16840603	C₄	semidirect
12. C₈×#92420	C₄	semidirect
31. #14519688	C₈	semidirect
32. #14519687	C₈	semidirect
37. #97×C₆₁	C₈	semidirect
80. #92420	#97	direct (N × H)
81. C₈×C₆₁	C₄×C₈	semidirect
82. #92418	C₄×C₈	semidirect
118. C₆₁	C₈×#97	semidirect
131. #97	#92420	direct (N × H)
183. C₈	C₄×#92420	semidirect

Maximal subgroups

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

subgroup	index
#21461533	2
C₄²×#92420	2
C₂×#16840603	2
C₂×C₈×#92420	2
#97×Dic₆₁	2
C₈×#97	61