Subgroups — #29595358

Order 15616.

398 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11758185	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₆₁	C₂	1	2	yes	maximal
3	C₈×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	#21443896	C₂	1	2	yes	maximal
5	C₂×C₆₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758185	C₂²	1	1	yes	characteristic
7	C₂²×#11758185	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#92418	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
10	C₄×C₄⋊Dic₆₁	C₄	1	1	yes	characteristic
11	C₄×#11758185	C₄	1	4	yes	complemented
12	C₁₆×Dic₆₁	—	2	4	no	—
13	#16839516	C₄	1	2	yes	—
14	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×C₄⋊C₁₆	C₄	1	4	yes	complemented
17	C₂²×#92418	C₂×C₄	1	2	yes	—
18	C₂×C₄⋊Dic₆₁	C₈	1	2	yes	—
19	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
20	C₂×#11758185	—	2	1	no	—
21	C₂×#11758185	—	2	1	no	—
22	C₂×#11758185	C₂×C₄	1	2	yes	—
23	C₂×#11758185	C₂×C₄	1	2	yes	—
24	C₂×#11758185	D₄	1	2	yes	—
25	C₂×#11758185	Q₈	1	2	yes	—
26	C₄⋊#92418	C₈	1	4	yes	—
27	C₈×Dic₆₁	—	2	2	no	—
28	C₈×Dic₆₁	—	2	2	no	—
29	#14519684	C₈	1	2	yes	—
30	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₆₁	—	2	2	no	—
34	C₂×C₁₆×C₆₁	—	2	2	no	—
35	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₆₁	C₂×C₈	1	4	yes	—
42	C₂×C₈×C₆₁	—	2	1	no	—
43	C₂×C₈×C₆₁	—	2	1	no	—
44	C₂×C₈×C₆₁	C₄²	1	2	yes	—
45	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
50	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
51	C₄²×C₆₁	C₄²	1	2	yes	—
52	C₄²×C₆₁	C₂×C₈	1	2	yes	—
53	C₄⋊Dic₆₁	C₁₆	1	8	yes	complemented
54	C₄×Dic₆₁	—	2	2	no	—
55	C₄×Dic₆₁	—	2	2	no	—
56	C₂×#92418	—	2	2	no	—
57	C₂×#92418	—	2	2	no	—
58	C₂×#92418	C₂×C₈	1	4	yes	—
59	C₁₆×C₆₁	—	2	4	no	—
60	#11758185	—	2	2	no	—
61	#11758185	C₄⋊C₄	1	4	yes	complemented
62	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
65	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
66	C₂×C₄×C₆₁	C₂×C₁₆	1	4	yes	—
67	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
73	C₂×Dic₆₁	—	2	4	no	—
74	C₂×Dic₆₁	—	2	4	no	—
75	C₂×Dic₆₁	C₂×C₁₆	1	8	yes	—
76	C₈×C₆₁	—	2	2	no	—
77	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₈×C₆₁	#117	1	1	yes	characteristic
79	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
80	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
81	#92418	—	2	4	no	—
82	C₁₆×C₄⋊C₄	—	61	1	no	maximal, Sylow
83	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
84	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
85	C₂²×C₆₁	C₈×Q₈	1	1	yes	characteristic
86	C₂²×C₆₁	C₂²×C₁₆	1	2	yes	—
87	C₂²×C₆₁	#504	1	2	yes	—
88	C₄×C₆₁	—	2	2	no	—
89	C₄×C₆₁	C₄×C₁₆	1	4	yes	complemented
90	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
91	C₄×C₆₁	#405	1	1	yes	characteristic
92	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
93	C₄×C₆₁	C₈×Q₈	1	1	yes	characteristic
94	Dic₆₁	—	2	8	no	—
95	C₂×C₄⋊C₁₆	D₆₁	1	1	yes	characteristic
96	C₂×C₄×C₁₆	—	61	1	no	—
97	C₂×C₄×C₁₆	—	61	2	no	—
98	C₈×C₄⋊C₄	—	61	1	no	—
99	#1374	—	61	2	no	—
100	C₂×C₆₁	C₈×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₆₁	C₂×C₄×C₁₆	1	1	yes	characteristic
102	C₂×C₆₁	#2108	1	1	yes	characteristic
103	C₂×C₆₁	D₄×C₁₆	1	2	yes	—
104	C₂×C₆₁	Q₈×C₁₆	1	2	yes	—
105	C₂²×C₁₆	—	61	1	no	—
106	C₂²×C₁₆	—	61	1	no	—
107	C₂²×C₁₆	C₂×D₆₁	1	2	yes	—
108	C₂×C₄⋊C₈	—	61	1	no	—
109	C₂×C₄×C₈	—	61	2	no	—
110	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
111	C₄×C₄⋊C₄	—	61	1	no	—
112	C₄⋊C₁₆	Dic₆₁	1	4	yes	complemented
113	C₄×C₁₆	—	61	4	no	—
114	C₄×C₁₆	—	122	4	no	—
115	#336	—	61	2	no	—
116	C₆₁	C₁₆×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₈	—	61	2	no	—
118	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	61	2	no	—
121	C₂×C₄²	—	61	2	no	—
122	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	61	2	no	—
126	C₂×C₁₆	—	61	2	no	—
127	C₂×C₁₆	—	61	2	no	—
128	C₂×C₁₆	—	61	2	no	—
129	C₂×C₁₆	—	122	1	no	—
130	C₂×C₁₆	—	122	1	no	—
131	C₂×C₁₆	C₂×Dic₆₁	1	4	yes	—
132	C₄⋊C₈	—	61	4	no	—
133	C₄×C₈	—	122	2	no	—
134	C₄×C₈	—	122	2	no	—
135	C₄×C₈	C₄×D₆₁	1	2	yes	—
136	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
137	#95	—	61	2	no	—
138	C₂²×C₄	—	61	4	no	—
139	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
140	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	61	4	no	—
145	C₂×C₈	—	122	2	no	—
146	C₂×C₈	—	122	2	no	—
147	C₂×C₈	C₄×Dic₆₁	1	2	yes	—
148	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
149	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
150	C₂×C₈	#11758224	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
152	C₂×C₈	#11758226	1	1	yes	characteristic
153	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
154	C₄⋊C₄	—	61	8	no	—
155	C₄²	—	122	2	no	—
156	C₄²	—	122	2	no	—
157	C₄²	C₈×D₆₁	1	2	yes	—
158	C₄²	C₄×Dic₆₁	1	2	yes	—
159	C₁₆	—	2	4	no	—
160	C₁₆	—	61	4	no	—
161	C₁₆	—	122	2	no	—
162	C₂³	C₂×C₈×D₆₁	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	61	8	no	—
165	C₂×C₄	—	122	4	no	—
166	C₂×C₄	—	122	4	no	—
167	C₂×C₄	C₈×Dic₆₁	1	2	yes	—
168	C₂×C₄	C₈×Dic₆₁	1	2	yes	—
169	C₂×C₄	C₁₆×D₆₁	1	4	yes	—
170	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
171	C₂×C₄	#14519739	1	1	yes	characteristic
172	C₂×C₄	#14519742	1	1	yes	characteristic
173	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
174	C₂×C₄	#14519785	1	1	yes	characteristic
175	C₂×C₄	#14519785	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	122	4	no	—
178	C₈	C₄⋊C₄×D₆₁	1	1	yes	characteristic
179	C₈	#14519735	1	1	yes	characteristic
180	C₈	D₄×Dic₆₁	1	1	yes	characteristic
181	C₈	Q₈×Dic₆₁	1	1	yes	characteristic
182	C₂²	C₂×C₈×Dic₆₁	1	1	yes	characteristic
183	C₂²	#16839827	1	1	yes	characteristic
184	C₂²	#16839886	1	1	yes	characteristic
185	C₂²	C₂×C₁₆×D₆₁	1	2	yes	—
186	C₂²	#16840180	1	2	yes	—
187	C₄	—	2	2	no	—
188	C₄	—	122	8	no	—
189	C₄	C₁₆×Dic₆₁	1	4	yes	complemented
190	C₄	C₄⋊C₄×Dic₆₁	1	1	yes	characteristic
191	C₄	#16839720	1	1	yes	characteristic
192	C₄	#16839827	1	1	yes	characteristic
193	C₄	#16839886	1	1	yes	characteristic
194	C₂	#21445684	1	1	yes	characteristic
195	C₂	C₂×C₁₆×Dic₆₁	1	1	yes	characteristic
196	C₂	#21447418	1	1	yes	characteristic
197	C₂	#21447638	1	2	yes	—
198	C₂	#21447721	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₁
commutator	C₂×C₆₁	C₂×C₄×C₁₆
Fitting	C₂×C₆₁×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₆₁
hypocenter	C₆₁	C₁₆×C₄⋊C₄
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
11. C₄×#11758185	C₄	semidirect
16. C₆₁×C₄⋊C₁₆	C₄	semidirect
53. C₄⋊Dic₆₁	C₁₆	semidirect
61. #11758185	C₄⋊C₄	semidirect
89. C₄×C₆₁	C₄×C₁₆	semidirect
112. C₄⋊C₁₆	Dic₆₁	semidirect
116. C₆₁	C₁₆×C₄⋊C₄	semidirect
189. C₄	C₁₆×Dic₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11758185	2
C₂×C₁₆×Dic₆₁	2
C₈×C₄⋊Dic₆₁	2
#21443896	2
C₂×C₆₁×C₄⋊C₁₆	2
C₁₆×C₄⋊C₄	61