Subgroups — #29833201

Order 15616.

690 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840180	C₂	1	2	yes	maximal, complemented
2	#21454532	C₂	1	1	yes	maximal, characteristic
3	#21447650	C₂	1	4	yes	maximal, complemented
4	#21447638	C₂	1	4	yes	maximal, complemented
5	#21447434	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×#11758185	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14519783	C₂²	1	2	yes	—
9	#16840312	C₄	1	1	yes	characteristic
10	#16840180	—	2	8	no	—
11	C₂×#14519716	C₂²	1	4	yes	—
12	C₂²×#11758185	C₂²	1	1	yes	characteristic
13	C₂²×#11758185	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₆₁	C₂²	1	4	yes	—
15	#16839841	C₄	1	4	yes	—
16	C₈×D₂₄₄	C₂²	1	4	yes	—
17	#16839786	C₄	1	1	yes	characteristic
18	#16839728	C₄	1	2	yes	—
19	#16839541	C₂²	1	8	yes	—
20	C₄×#11758185	C₂²	1	2	yes	—
21	C₄×#11758185	C₂²	1	2	yes	—
22	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₆₁×C₄⋊C₁₆	C₂²	1	4	yes	—
25	C₂×#11758222	C₂×C₄	1	2	yes	—
26	#14519783	—	2	4	no	—
27	#14519783	—	2	4	no	—
28	C₂×#11758194	C₂×C₄	1	2	yes	—
29	C₂×#11758189	C₂×C₄	1	4	yes	—
30	C₂×C₈×D₆₁	C₂³	1	4	yes	—
31	#14519777	C₈	1	1	yes	characteristic
32	#14519774	C₈	1	1	yes	characteristic
33	#14519765	C₈	1	2	yes	—
34	#14519751	C₈	1	4	yes	—
35	C₄×D₂₄₄	C₂×C₄	1	4	yes	—
36	#14519716	—	2	8	no	—
37	C₂×#11758185	—	2	2	no	—
38	C₂×#11758185	D₄	1	4	yes	complemented
39	C₂×#11758185	C₂³	1	2	yes	—
40	C₂×#11758185	C₂³	1	1	yes	characteristic
41	C₂×#11758185	C₂³	1	1	yes	characteristic
42	C₁₆×D₆₁	—	2	8	no	—
43	C₄⋊#92418	C₂×C₄	1	4	yes	—
44	#14519689	C₂×C₄	1	8	yes	—
45	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
48	M₃₂×C₆₁	—	2	4	no	—
49	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
50	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₆₁	C₂³	1	2	yes	—
52	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₆₁	C₂×C₈	1	4	yes	—
55	#11758222	—	2	4	no	—
56	#11758222	—	2	4	no	—
57	C₂×D₂₄₄	C₂×C₈	1	2	yes	—
58	C₂×D₂₄₄	C₂×C₈	1	4	yes	—
59	C₂×C₄×D₆₁	C₂²×C₄	1	4	yes	—
60	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
61	C₂×C₈×C₆₁	—	2	2	no	—
62	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₆₁	#41	1	2	yes	—
66	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
67	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
68	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
69	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
70	#11758198	C₂×C₈	1	8	yes	—
71	C₄⋊Dic₆₁	C₂×C₈	1	4	yes	—
72	#11758194	—	2	4	no	—
73	C₂×#92418	C₂²×C₄	1	4	yes	—
74	#11758189	—	2	8	no	—
75	C₈×D₆₁	—	2	4	no	—
76	C₈×D₆₁	—	2	4	no	—
77	C₁₆×C₆₁	—	2	4	no	—
78	#11758185	—	2	2	no	—
79	#11758185	C₂×D₄	1	2	yes	—
80	#11758185	C₂×D₄	1	2	yes	—
81	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
82	C₂²×D₆₁	C₂²×C₈	1	4	yes	—
83	C₂×C₄×C₆₁	—	2	4	no	—
84	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₆₁	#131	1	2	yes	—
89	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
90	C₂²⋊D₆₁	—	2	8	no	—
91	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
92	D₂₄₄	—	2	4	no	—
93	D₂₄₄	—	2	8	no	—
94	C₄×D₆₁	—	2	4	no	—
95	C₄×D₆₁	—	2	4	no	—
96	Dic₁₂₂	—	2	4	no	—
97	C₈×C₆₁	—	2	2	no	—
98	C₈×C₆₁	C₄×D₄	1	2	yes	—
99	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
101	#92418	—	2	4	no	—
102	#33434	—	61	1	no	maximal, Sylow
103	C₂²×C₆₁	—	2	2	no	—
104	C₂²×C₆₁	C₈×D₄	1	2	yes	—
105	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
106	C₂×D₆₁	—	2	8	no	—
107	C₂×D₆₁	—	2	4	no	—
108	C₄×C₆₁	—	2	2	no	—
109	C₄×C₆₁	C₈×D₄	1	2	yes	—
110	C₄×C₆₁	#504	1	4	yes	complemented
111	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
112	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
113	Dic₆₁	—	2	4	no	—
114	C₂×#504	—	61	2	no	—
115	#2931	—	61	1	no	—
116	#2169	—	61	4	no	—
117	D₄×C₁₆	—	61	4	no	—
118	#2150	D₆₁	1	1	yes	characteristic, complemented
119	#2113	—	61	2	no	—
120	C₂×C₄×C₁₆	—	61	1	no	—
121	C₂×C₆₁	—	2	2	no	—
122	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
123	C₂×C₆₁	C₂×#504	1	2	yes	—
124	D₆₁	—	4	4	no	—
125	C₂×#131	—	61	2	no	—
126	#532	—	61	1	no	—
127	#504	—	122	8	no	—
128	C₂×M₃₂	—	61	4	no	—
129	C₂×M₃₂	C₂×D₆₁	1	2	yes	—
130	C₂²×C₁₆	—	61	1	no	—
131	C₂²×C₁₆	—	61	1	no	—
132	C₂²×C₁₆	—	61	4	no	—
133	#436	—	61	4	no	—
134	C₈×D₄	—	61	4	no	—
135	#423	—	61	1	no	—
136	#408	—	61	2	no	—
137	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
138	C₄⋊C₁₆	C₂×D₆₁	1	4	yes	—
139	C₂²⋊C₁₆	—	61	8	no	—
140	C₄×C₁₆	—	61	2	no	—
141	C₄×C₁₆	—	61	2	no	—
142	C₆₁	#33434	1	1	yes	characteristic, complemented, Sylow
143	C₂×#41	—	61	2	no	—
144	#131	—	122	4	no	—
145	#131	—	122	4	no	—
146	C₂×M₁₆	—	61	2	no	—
147	C₂×M₁₆	—	61	4	no	—
148	C₂²×C₈	—	61	4	no	—
149	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
151	C₄⋊Q₈	—	61	1	no	—
152	#127	—	61	1	no	—
153	#124	—	61	2	no	—
154	#121	—	61	4	no	—
155	C₄×D₄	—	61	4	no	—
156	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
157	M₃₂	—	2	4	no	—
158	M₃₂	—	122	8	no	—
159	C₂×C₁₆	—	61	4	no	—
160	C₂×C₁₆	—	61	1	no	—
161	C₂×C₁₆	—	61	1	no	—
162	C₂×C₁₆	—	61	2	no	—
163	C₂×C₁₆	—	122	8	no	—
164	C₂×C₁₆	—	122	2	no	—
165	C₂×C₁₆	C₂²×D₆₁	1	4	yes	—
166	C₄⋊C₈	—	61	4	no	—
167	C₂²⋊C₈	—	61	8	no	—
168	C₄×C₈	C₄×D₆₁	1	2	yes	—
169	C₄×C₈	C₂²×D₆₁	1	2	yes	—
170	#41	—	122	4	no	—
171	#41	—	122	4	no	—
172	C₂×Q₈	—	61	2	no	—
173	C₂×D₄	—	61	4	no	—
174	C₂×D₄	—	61	4	no	—
175	C₂×D₄	—	61	2	no	—
176	C₂²×C₄	—	61	4	no	—
177	C₂²×C₄	C₈×D₆₁	1	2	yes	—
178	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
179	M₁₆	—	122	4	no	—
180	M₁₆	—	122	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	61	4	no	—
183	C₂×C₈	—	122	4	no	—
184	C₂×C₈	—	122	4	no	—
185	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
186	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
187	C₂×C₈	D₄×D₆₁	1	2	yes	—
188	C₂×C₈	#11758226	1	2	yes	—
189	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
190	C₄⋊C₄	—	61	4	no	—
191	C₂²⋊C₄	—	61	8	no	—
192	C₄²	C₈×D₆₁	1	1	yes	characteristic
193	C₄²	C₈×D₆₁	1	1	yes	characteristic
194	C₄²	C₂×C₄×D₆₁	1	2	yes	—
195	C₁₆	—	2	4	no	—
196	C₁₆	—	61	2	no	—
197	C₁₆	—	61	2	no	—
198	C₁₆	—	122	2	no	—
199	C₂³	—	61	4	no	—
200	C₂³	C₂×C₈×D₆₁	1	1	yes	characteristic
201	Q₈	—	122	4	no	—
202	D₄	—	122	8	no	—
203	D₄	—	122	4	no	—
204	D₄	—	122	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	61	4	no	—
207	C₂×C₄	—	122	4	no	—
208	C₂×C₄	—	122	4	no	—
209	C₂×C₄	#14519739	1	2	yes	—
210	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
211	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
212	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
213	C₂×C₄	#14519785	1	2	yes	—
214	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	122	4	no	—
217	C₈	#14519739	1	2	yes	—
218	C₈	C₂×D₄×D₆₁	1	1	yes	characteristic
219	C₈	C₂×#11758226	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	122	4	no	—
222	C₂²	—	122	8	no	—
223	C₂²	#16839827	1	2	yes	—
224	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	122	4	no	—
227	C₄	#16839827	1	2	yes	—
228	C₄	#16840180	1	4	yes	complemented
229	C₄	C₂×#14519739	1	1	yes	characteristic
230	C₄	C₂×#14519785	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	244	4	no	—
233	C₂	C₂×#16839827	1	1	yes	characteristic
234	C₂	C₂×#16840180	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₆₁×#2150	C₂
prime radical	#16840312	C₄
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#2150	D₆₁
hypocenter	C₆₁	#33434
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
1. C₂×#16840180	C₂	semidirect
3. #21447650	C₂	semidirect
4. #21447638	C₂	semidirect
5. #21447434	C₂	semidirect
6. C₂×C₄×#11758185	C₂	semidirect
7. C₆₁×#2150	C₂	semidirect
38. C₂×#11758185	D₄	semidirect
110. C₄×C₆₁	#504	semidirect
118. #2150	D₆₁	semidirect
142. C₆₁	#33434	semidirect
228. C₄	#16840180	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840180	2
#21454532	2
#21447650	2
#21447638	2
#21447434	2
C₂×C₄×#11758185	2
C₆₁×#2150	2
#33434	61