Subgroups — #29596428

Order 15616.

382 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839583	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839582	C₂	1	2	yes	maximal
3	#21447574	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₆₁×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519717	C₂²	1	2	yes	—
7	C₂×C₄×#92418	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
12	C₄²×Dic₆₁	C₄	1	1	yes	characteristic
13	#16839583	—	2	2	no	—
14	#16839582	—	2	4	no	—
15	#16839580	—	2	4	no	—
16	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#364	—	2	2	no	—
19	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
20	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₆₁	—	2	2	no	—
22	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
24	#14519717	—	2	2	no	—
25	#14519717	—	2	2	no	—
26	C₂×#11758185	—	2	2	no	—
27	C₄×#92418	—	2	2	no	—
28	C₈×Dic₆₁	—	2	2	no	—
29	C₈×Dic₆₁	C₈	1	2	yes	—
30	C₈×Dic₆₁	C₈	1	2	yes	—
31	C₈×Dic₆₁	D₄	1	2	yes	—
32	C₈×Dic₆₁	D₄	1	2	yes	—
33	C₈×Dic₆₁	Q₈	1	2	yes	—
34	C₈×Dic₆₁	Q₈	1	2	yes	—
35	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
38	M₃₂×C₆₁	—	2	2	no	—
39	M₃₂×C₆₁	—	2	2	no	—
40	C₂×C₁₆×C₆₁	—	2	2	no	—
41	C₄×C₈×C₆₁	D₄	1	2	yes	—
42	C₄×C₈×C₆₁	Q₈	1	2	yes	—
43	C₂²×C₄×C₆₁	—	2	1	no	—
44	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₆₁	—	2	1	no	—
46	C₂²×Dic₆₁	M₁₆	1	2	yes	—
47	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
50	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₆₁	—	2	1	no	—
57	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
58	C₄×Dic₆₁	—	2	8	no	—
59	C₄×Dic₆₁	—	2	1	no	—
60	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
61	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
62	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
63	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
64	C₂×#92418	—	2	1	no	—
65	C₂×#92418	—	2	1	no	—
66	C₂×#92418	C₄⋊C₄	1	2	yes	—
67	C₂×#92418	C₄⋊C₄	1	2	yes	—
68	C₂×#92418	C₂×C₈	1	2	yes	—
69	C₂×#92418	M₁₆	1	2	yes	—
70	C₁₆×C₆₁	—	4	2	no	—
71	#11758185	—	4	2	no	—
72	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₆₁	—	2	4	no	—
74	C₂×C₄×C₆₁	—	2	2	no	—
75	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
81	C₂×Dic₆₁	—	2	4	no	—
82	C₂×Dic₆₁	—	2	2	no	—
83	C₂×Dic₆₁	—	2	4	no	—
84	C₂×Dic₆₁	C₄⋊C₈	1	2	yes	—
85	C₂×Dic₆₁	C₄⋊C₈	1	2	yes	—
86	C₈×C₆₁	C₄⋊C₈	1	4	yes	—
87	C₈×C₆₁	#127	1	1	yes	characteristic
88	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
89	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
90	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
91	#92418	—	2	2	no	—
92	#92418	C₄⋊C₈	1	4	yes	—
93	#12900	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	—	2	2	no	—
95	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₂²×C₆₁	#423	1	1	yes	characteristic
97	C₂²×C₆₁	#423	1	1	yes	characteristic
98	C₄×C₆₁	—	2	4	no	—
99	C₄×C₆₁	#384	1	1	yes	characteristic
100	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₄×C₆₁	#423	1	1	yes	characteristic
103	Dic₆₁	—	2	4	no	—
104	Dic₆₁	—	2	2	no	—
105	Dic₆₁	#364	1	4	yes	complemented
106	C₂×#364	—	61	2	no	—
107	C₂×#364	—	61	1	no	—
108	C₂×#364	D₆₁	1	1	yes	characteristic
109	#2150	—	61	2	no	—
110	C₄²×C₈	—	61	1	no	—
111	C₂×C₆₁	—	2	1	no	—
112	C₂×C₆₁	—	2	1	no	—
113	C₂×C₆₁	#1842	1	1	yes	characteristic
114	C₂×C₆₁	C₂×#364	1	2	yes	—
115	C₂×M₃₂	—	61	2	no	—
116	C₂×M₃₂	C₂×D₆₁	1	2	yes	—
117	C₂×C₄×C₈	—	61	1	no	—
118	C₂×C₄×C₈	—	61	1	no	—
119	C₂×C₄×C₈	—	61	1	no	—
120	C₂×C₄×C₈	—	61	1	no	—
121	C₂×C₄×C₈	—	61	1	no	—
122	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
123	C₄³	—	61	1	no	—
124	#364	—	2	2	no	—
125	#364	—	122	4	no	—
126	#364	—	122	2	no	—
127	C₄⋊C₁₆	—	122	4	no	—
128	C₆₁	#12900	1	1	yes	characteristic, complemented, Sylow
129	C₂²×C₈	—	61	1	no	—
130	C₂²×C₈	—	61	1	no	—
131	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
133	C₂×C₄²	—	61	1	no	—
134	C₂×C₄²	—	61	1	no	—
135	C₂×C₄²	—	122	2	no	—
136	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
137	M₃₂	—	2	2	no	—
138	M₃₂	—	2	2	no	—
139	M₃₂	—	122	2	no	—
140	M₃₂	—	122	2	no	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	122	2	no	—
143	C₄×C₈	—	61	2	no	—
144	C₄×C₈	—	61	2	no	—
145	C₄×C₈	—	61	2	no	—
146	C₄×C₈	—	61	2	no	—
147	C₄×C₈	—	61	2	no	—
148	C₄×C₈	—	61	2	no	—
149	C₄×C₈	—	122	2	no	—
150	C₄×C₈	—	122	2	no	—
151	C₄×C₈	Dic₁₂₂	1	2	yes	—
152	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	—	61	2	no	—
155	C₂²×C₄	—	122	1	no	—
156	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
157	C₂×C₈	—	61	2	no	—
158	C₂×C₈	—	61	2	no	—
159	C₂×C₈	—	61	2	no	—
160	C₂×C₈	—	61	2	no	—
161	C₂×C₈	—	122	1	no	—
162	C₂×C₈	—	122	1	no	—
163	C₂×C₈	C₈×D₆₁	1	2	yes	—
164	C₂×C₈	#11758189	1	2	yes	—
165	C₂×C₈	#11758196	1	2	yes	—
166	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
167	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
168	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
169	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
170	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
172	C₄²	—	2	1	no	—
173	C₄²	—	61	2	no	—
174	C₄²	—	61	1	no	—
175	C₄²	—	61	1	no	—
176	C₄²	—	61	2	no	—
177	C₄²	—	122	8	no	—
178	C₄²	—	122	1	no	—
179	C₄²	#11758196	1	2	yes	—
180	C₁₆	—	4	2	no	—
181	C₁₆	—	244	2	no	—
182	C₂³	M₁₆×D₆₁	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	61	2	no	—
186	C₂×C₄	—	61	2	no	—
187	C₂×C₄	—	122	4	no	—
188	C₂×C₄	—	122	4	no	—
189	C₂×C₄	—	122	2	no	—
190	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
191	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
192	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
193	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
194	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
195	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
196	C₈	—	61	4	no	—
197	C₈	—	122	2	no	—
198	C₈	#14519706	1	4	yes	—
199	C₈	#14519773	1	1	yes	characteristic
200	C₈	#14519775	1	1	yes	characteristic
201	C₈	#14519776	1	1	yes	characteristic
202	C₈	#14519776	1	1	yes	characteristic
203	C₂²	—	2	2	no	—
204	C₂²	C₄⋊C₈×D₆₁	1	1	yes	characteristic
205	C₂²	#16839784	1	1	yes	characteristic
206	C₂²	#16839785	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	61	4	no	—
209	C₄	—	122	4	no	—
210	C₄	—	122	2	no	—
211	C₄	#16839631	1	1	yes	characteristic
212	C₄	C₄⋊C₈×D₆₁	1	1	yes	characteristic
213	C₄	C₂×#14519706	1	1	yes	characteristic
214	C₄	#16839784	1	1	yes	characteristic
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	#21445970	1	1	yes	characteristic
218	C₂	#364×D₆₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#364	C₂
socle	C₂²×C₆₁	C₂×C₄⋊C₈
prime radical	C₂³×C₆₁	C₂×M₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×#364	D₆₁
hypocenter	C₆₁	#12900
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
105. Dic₆₁	#364	semidirect
128. C₆₁	#12900	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839583	2
C₂×#16839582	2
#21447574	2
C₄×C₈×Dic₆₁	2
C₂×C₆₁×#364	2
#12900	61