Subgroups — #29571308

Order 15616.

374 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	C₄×#14519713	C₂	1	1	yes	maximal, characteristic
2	#21444100	C₂	1	2	yes	maximal
3	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
4	#21443655	C₂	1	2	yes	maximal
5	C₆₁×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519713	C₄	1	2	yes	—
7	C₄×#11758194	C₂²	1	2	yes	—
8	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
9	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
10	#16839527	C₄	1	2	yes	—
11	#16839484	—	2	2	no	—
12	C₈×#92418	—	2	2	no	—
13	C₈×#92418	C₄	1	2	yes	—
14	C₈×#92418	C₄	1	2	yes	—
15	C₆₁×#431	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#335	C₄	1	2	yes	—
18	C₆₁×#334	C₄	1	2	yes	—
19	C₂×#11758194	C₂×C₄	1	4	yes	—
20	C₂²×#92418	—	2	1	no	—
21	C₂²×#92418	Q₈	1	2	yes	—
22	#14519713	C₈	1	8	yes	complemented
23	#14519688	—	2	2	no	—
24	C₄×#92418	—	2	2	no	—
25	C₄×#92418	—	2	1	no	—
26	C₄×#92418	C₂×C₄	1	2	yes	—
27	C₄×#92418	C₂×C₄	1	2	yes	—
28	C₄×#92418	D₄	1	1	yes	characteristic
29	C₄×#92418	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
34	C₂²⋊C₈×C₆₁	—	2	2	no	—
35	C₄×C₈×C₆₁	—	2	2	no	—
36	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
43	C₂×C₈×C₆₁	—	2	2	no	—
44	C₂×C₈×C₆₁	—	2	4	no	—
45	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
46	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
50	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₆₁	#41	1	1	yes	characteristic
54	#11758194	—	2	4	no	—
55	#11758194	C₂×C₈	1	8	yes	—
56	C₂×#92418	—	2	4	no	—
57	C₂×#92418	—	2	2	no	—
58	C₂×#92418	—	2	4	no	—
59	C₂×#92418	—	2	4	no	—
60	C₂×#92418	C₄⋊C₄	1	2	yes	—
61	C₂×#92418	C₄⋊C₄	1	2	yes	—
62	C₂³×C₆₁	C₄×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	—	2	2	no	—
65	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₆₁	#131	1	2	yes	—
73	C₈×C₆₁	—	2	1	no	—
74	C₈×C₆₁	—	2	1	no	—
75	C₈×C₆₁	—	4	4	no	—
76	C₈×C₆₁	C₄×C₈	1	2	yes	—
77	C₈×C₆₁	C₄×C₈	1	2	yes	—
78	#92418	—	2	4	no	—
79	#92418	—	2	2	no	—
80	#92418	—	2	8	no	—
81	#92418	#108	1	4	yes	complemented
82	C₈×#108	—	61	1	no	maximal, Sylow
83	C₂²×C₆₁	—	2	1	no	—
84	C₂²×C₆₁	—	2	1	no	—
85	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₆₁	C₈×Q₈	1	2	yes	—
87	C₄×C₆₁	—	2	1	no	—
88	C₄×C₆₁	—	2	1	no	—
89	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
90	C₄×C₆₁	#405	1	1	yes	characteristic
91	C₄×C₆₁	C₂×#108	1	2	yes	—
92	C₄×C₆₁	C₈×D₄	1	2	yes	—
93	C₄×C₆₁	#443	1	2	yes	—
94	C₄×#108	—	61	1	no	—
95	#1564	D₆₁	1	1	yes	characteristic
96	C₈×M₁₆	—	61	2	no	—
97	C₂×C₈²	—	61	1	no	—
98	#1278	—	61	2	no	—
99	C₂×C₆₁	—	2	2	no	—
100	C₂×C₆₁	C₈×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₆₁	#1772	1	1	yes	characteristic
102	C₂×C₆₁	C₄×#108	1	1	yes	characteristic
103	#431	C₂×D₆₁	1	2	yes	—
104	C₂×#108	—	61	2	no	—
105	C₄×M₁₆	—	61	2	no	—
106	C₂×C₄×C₈	—	61	1	no	—
107	C₂×C₄×C₈	—	61	1	no	—
108	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
109	#341	—	61	2	no	—
110	#335	Dic₆₁	1	2	yes	—
111	#334	Dic₆₁	1	2	yes	—
112	#322	—	122	2	no	—
113	C₈²	—	61	2	no	—
114	C₈²	—	61	2	no	—
115	C₈²	—	122	2	no	—
116	C₆₁	C₈×#108	1	1	yes	characteristic, complemented, Sylow
117	C₂×M₁₆	—	61	4	no	—
118	C₂²×C₈	—	61	2	no	—
119	C₂²×C₈	—	122	1	no	—
120	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
121	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
123	#108	—	61	8	no	—
124	C₄⋊C₈	C₂×Dic₆₁	1	4	yes	—
125	C₂²⋊C₈	—	2	2	no	—
126	#97	—	122	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	61	2	no	—
129	C₄×C₈	—	61	2	no	—
130	C₄×C₈	—	61	1	no	—
131	C₄×C₈	—	61	1	no	—
132	C₄×C₈	—	122	2	no	—
133	C₄×C₈	—	122	1	no	—
134	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
135	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
136	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
137	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
139	C₂²×C₄	#11758224	1	1	yes	characteristic
140	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
141	M₁₆	—	61	8	no	—
142	M₁₆	—	122	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	61	2	no	—
146	C₂×C₈	—	61	2	no	—
147	C₂×C₈	—	122	4	no	—
148	C₂×C₈	—	122	4	no	—
149	C₂×C₈	—	122	2	no	—
150	C₂×C₈	—	122	4	no	—
151	C₂×C₈	C₈×D₆₁	1	4	yes	—
152	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
153	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
154	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
155	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
156	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
157	C₄²	D₄×D₆₁	1	1	yes	characteristic
158	C₄²	#11758226	1	1	yes	characteristic
159	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
160	C₂³	#14519742	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
164	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
165	C₂×C₄	#14519735	1	1	yes	characteristic
166	C₂×C₄	#14519739	1	1	yes	characteristic
167	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
168	C₂×C₄	Q₈×Dic₆₁	1	1	yes	characteristic
169	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
170	C₂×C₄	#14519785	1	2	yes	—
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	4	4	no	—
174	C₈	—	61	4	no	—
175	C₈	—	122	4	no	—
176	C₈	—	122	2	no	—
177	C₈	—	122	8	no	—
178	C₈	C₈×Dic₆₁	1	2	yes	—
179	C₈	C₈×Dic₆₁	1	2	yes	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₄⋊C₄×Dic₆₁	1	1	yes	characteristic
183	C₂²	#16839886	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	C₂×C₈×Dic₆₁	1	1	yes	characteristic
187	C₄	#16839720	1	1	yes	characteristic
188	C₄	#108×D₆₁	1	2	yes	—
189	C₄	#16839827	1	2	yes	—
190	C₄	#16839879	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	#21445684	1	1	yes	characteristic
193	C₂	#21445700	1	1	yes	characteristic
194	C₂	#108×Dic₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₆₁
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1564	C₂
socle	C₂²×C₆₁	C₄×C₄⋊C₄
prime radical	C₂³×C₆₁	C₄×Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1564	D₆₁
hypocenter	C₆₁	C₈×#108
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
22. #14519713	C₈	semidirect
81. #92418	#108	semidirect
116. C₆₁	C₈×#108	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519713	2
#21444100	2
C₂×C₈×#92418	2
#21443655	2
C₆₁×#1564	2
C₈×#108	61