Subgroups — #30624155

Order 15616.

590 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462184	C₂	1	8	yes	maximal, complemented
2	#21462151	C₂	1	2	yes	maximal, complemented
3	#21462046	C₂	1	2	yes	maximal
4	C₄×#14519851	C₂	1	2	yes	maximal, complemented
5	C₂×#16839718	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519851	C₂²	1	4	yes	—
7	C₄⋊#11758187	C₂²	1	8	yes	complemented
8	C₂²⋊#11758187	C₂²	1	8	yes	—
9	#16840650	C₂²	1	4	yes	complemented
10	C₄×#11758187	C₂²	1	4	yes	complemented
11	C₂²×#11758194	C₄	1	1	yes	characteristic
12	C₂³×#92418	C₂²	1	1	yes	characteristic
13	#16839718	C₄	1	4	yes	—
14	#16839718	C₂²	1	4	yes	—
15	C₂×#14519688	C₄	1	1	yes	characteristic
16	C₂×#14519688	C₄	1	1	yes	characteristic
17	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
18	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
19	C₂×#117×C₆₁	C₄	1	1	yes	characteristic
20	#14519851	—	2	8	no	—
21	C₂×#11758187	C₂³	1	8	yes	—
22	C₂×#11758194	—	2	4	no	—
23	C₂×#11758194	C₂×C₄	1	4	yes	—
24	C₂²×#92418	—	2	4	no	—
25	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
26	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
27	C₂²×#92418	C₂×C₄	1	2	yes	—
28	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
29	C₂²×#92418	C₂³	1	2	yes	—
30	C₂²×#92418	C₂³	1	1	yes	characteristic
31	#14519688	C₂×C₄	1	4	yes	—
32	#14519688	C₂×C₄	1	4	yes	—
33	C₄×#92418	C₂×C₄	1	2	yes	—
34	C₄×#92418	C₂×C₄	1	2	yes	—
35	C₄×#92418	C₂³	1	2	yes	—
36	C₄×#92418	C₂³	1	2	yes	—
37	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
38	#117×C₆₁	C₈	1	4	yes	—
39	#117×C₆₁	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊C₄×C₆₁	C₈	1	2	yes	—
41	C₂×C₂²⋊C₄×C₆₁	C₈	1	2	yes	—
42	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	—	2	2	no	—
46	C₂²×C₄×C₆₁	—	2	2	no	—
47	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₄⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
54	C₂²⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
55	C₄²×C₆₁	C₂×C₈	1	2	yes	—
56	C₄²×C₆₁	C₂×C₈	1	2	yes	—
57	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
58	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
59	#11758194	—	2	8	no	—
60	C₂×#92418	—	2	2	no	—
61	C₂×#92418	—	2	2	no	—
62	C₂×#92418	—	2	4	no	—
63	C₂×#92418	—	2	4	no	—
64	C₂×#92418	M₁₆	1	4	yes	—
65	C₂×#92418	C₂²×C₄	1	2	yes	—
66	C₂×#92418	C₂²×C₄	1	2	yes	—
67	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
68	C₂×#92418	C₂²×C₄	1	2	yes	—
69	C₂×#92418	#41	1	4	yes	—
70	C₂×#92418	C₂⁴	1	1	yes	characteristic
71	#11758187	—	2	8	no	—
72	C₂³×C₆₁	—	2	2	no	—
73	C₂³×C₆₁	—	4	2	no	—
74	C₂³×C₆₁	C₂²×C₈	1	2	yes	—
75	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₆₁	—	2	2	no	—
77	C₂×C₄×C₆₁	—	2	2	no	—
78	C₂×C₄×C₆₁	—	2	2	no	—
79	C₂×C₄×C₆₁	—	2	4	no	—
80	C₂×C₄×C₆₁	—	4	2	no	—
81	C₂×C₄×C₆₁	—	4	2	no	—
82	C₂×C₄×C₆₁	#117	1	2	yes	—
83	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
84	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
85	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
89	#92418	—	2	2	no	—
90	#92418	—	2	2	no	—
91	#92418	#117	1	2	yes	—
92	#92418	C₂×M₁₆	1	4	yes	—
93	#92418	C₂×#41	1	2	yes	—
94	#33419	—	61	1	no	maximal, Sylow
95	C₂²×C₆₁	—	2	2	no	—
96	C₂²×C₆₁	—	2	4	no	—
97	C₂²×C₆₁	—	2	2	no	—
98	C₂²×C₆₁	—	4	4	no	—
99	C₂²×C₆₁	#431	1	2	yes	—
100	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
101	C₄×C₆₁	—	2	4	no	—
102	C₄×C₆₁	—	2	4	no	—
103	C₄×C₆₁	—	2	2	no	—
104	C₄×C₆₁	#431	1	2	yes	—
105	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
106	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
107	C₂×#405	—	61	1	no	—
108	#2164	—	61	8	no	—
109	#2150	—	61	2	no	—
110	#2112	—	61	2	no	—
111	C₄×M₃₂	—	61	2	no	—
112	C₂×C₆₁	—	2	2	no	—
113	C₂×C₆₁	—	4	2	no	—
114	C₂×C₆₁	C₂×#431	1	1	yes	characteristic
115	C₂×C₆₁	#3407	1	2	yes	—
116	C₂²×M₁₆	—	61	1	no	—
117	C₂³×C₈	—	61	1	no	—
118	C₂×#117	#6923	1	1	yes	characteristic
119	C₂×M₃₂	—	61	4	no	—
120	#405	—	61	4	no	—
121	#405	—	61	4	no	—
122	C₂×#97	—	61	1	no	—
123	C₂×#97	—	61	1	no	—
124	C₂×C₄×C₈	—	61	1	no	—
125	C₂×C₄×C₈	—	61	1	no	—
126	C₄⋊C₁₆	—	61	8	no	—
127	C₂²⋊C₁₆	—	61	8	no	—
128	#346	—	61	4	no	—
129	C₄×C₁₆	—	61	4	no	—
130	C₆₁	#33419	1	1	yes	characteristic, complemented, Sylow
131	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
132	C₂×M₁₆	—	61	4	no	—
133	C₂×M₁₆	—	122	4	no	—
134	C₂²×C₈	—	61	1	no	—
135	C₂²×C₈	—	61	1	no	—
136	C₂²×C₈	—	61	2	no	—
137	C₂²×C₈	—	61	1	no	—
138	C₂²×C₈	—	61	2	no	—
139	C₂²×C₈	—	61	1	no	—
140	C₂²×C₈	—	122	4	no	—
141	#117	#92420	1	4	yes	—
142	#117	C₂×#6923	1	4	yes	—
143	C₂×C₄⋊C₄	#92420	1	2	yes	—
144	C₂×C₂²⋊C₄	#92420	1	2	yes	—
145	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
146	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
147	M₃₂	—	122	8	no	—
148	C₂×C₁₆	—	61	8	no	—
149	#97	—	61	4	no	—
150	#97	—	61	4	no	—
151	C₄×C₈	—	61	2	no	—
152	C₄×C₈	—	61	2	no	—
153	C₄×C₈	—	61	2	no	—
154	C₄×C₈	—	61	2	no	—
155	C₂⁴	C₂×#92420	1	1	yes	characteristic
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
159	C₂²×C₄	C₂×#92420	1	2	yes	—
160	C₂²×C₄	C₂×#92420	1	2	yes	—
161	C₂²×C₄	C₂×#92420	1	2	yes	—
162	C₂²×C₄	C₂²×#6923	1	2	yes	—
163	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
164	M₁₆	—	122	8	no	—
165	C₂×C₈	—	61	4	no	—
166	C₂×C₈	—	61	4	no	—
167	C₂×C₈	—	61	1	no	—
168	C₂×C₈	—	61	1	no	—
169	C₂×C₈	—	61	2	no	—
170	C₂×C₈	—	61	2	no	—
171	C₂×C₈	—	61	2	no	—
172	C₂×C₈	—	122	4	no	—
173	C₂×C₈	—	122	4	no	—
174	C₂×C₈	—	122	2	no	—
175	C₂×C₈	—	122	2	no	—
176	C₄⋊C₄	C₂×#92420	1	8	yes	—
177	C₂²⋊C₄	C₂×#92420	1	8	yes	—
178	C₄²	C₂×#92420	1	2	yes	—
179	C₄²	C₂×#92420	1	2	yes	—
180	C₄²	C₂²×#6923	1	2	yes	—
181	C₄²	C₂²×#6923	1	2	yes	—
182	C₁₆	—	122	8	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	4	2	no	—
185	C₂³	C₂²×#92420	1	2	yes	—
186	C₂³	C₂²×#92420	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	#14519856	1	2	yes	—
194	C₂×C₄	C₂²×#92420	1	4	yes	—
195	C₂×C₄	C₂²×#92420	1	4	yes	—
196	C₂×C₄	C₂²×#92420	1	2	yes	—
197	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
198	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
199	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
200	C₈	—	61	2	no	—
201	C₈	—	61	2	no	—
202	C₈	—	61	4	no	—
203	C₈	—	122	2	no	—
204	C₈	—	122	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	4	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	4	4	no	—
209	C₂²	#16840778	1	2	yes	—
210	C₂²	C₂³×#92420	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	2	no	—
214	C₄	#16840778	1	2	yes	—
215	C₄	C₂×#14519856	1	1	yes	characteristic
216	C₄	C₂×D₆₁⋊M₁₆	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	4	2	no	—
219	C₂	C₂×#16840778	1	1	yes	characteristic
220	C₂	#21462786	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₂×#117×C₆₁	C₄
prime radical	C₂⁴×C₆₁	C₂×C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	C₂×#117	#6923
hypocenter	C₆₁	#33419
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #21462184	C₂	semidirect
2. #21462151	C₂	semidirect
4. C₄×#14519851	C₂	semidirect
7. C₄⋊#11758187	C₂²	semidirect
9. #16840650	C₂²	semidirect
10. C₄×#11758187	C₂²	semidirect
130. C₆₁	#33419	semidirect

Maximal subgroups

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

subgroup	index
#21462184	2
#21462151	2
#21462046	2
C₄×#14519851	2
C₂×#16839718	2
#33419	61