Subgroups — #29595336

Order 15616.

358 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839582	C₂	1	2	yes	maximal
2	C₄×#14519717	C₂	1	1	yes	maximal, characteristic
3	#21445505	C₂	1	1	yes	maximal, characteristic
4	#21443900	C₂	1	2	yes	maximal
5	C₄×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519717	C₂²	1	2	yes	—
7	C₂×#14519688	C₄	1	1	yes	characteristic
8	C₂×#14519687	C₄	1	1	yes	characteristic
9	C₂×#14519687	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
12	#16839608	C₄	1	1	yes	characteristic
13	#16839582	C₄	1	8	yes	complemented
14	#16839538	—	2	1	no	—
15	C₄×#11758185	—	2	1	no	—
16	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#346	—	2	1	no	—
19	C₄×C₁₆×C₆₁	—	2	1	no	—
20	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
21	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
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	C₂×C₄	1	4	yes	—
26	C₂×#11758185	—	2	2	no	—
27	#14519688	C₈	1	4	yes	—
28	#14519687	—	2	2	no	—
29	#14519687	C₈	1	4	yes	—
30	C₈×Dic₆₁	—	2	2	no	—
31	C₈×Dic₆₁	C₂×C₄	1	4	yes	—
32	#14519684	—	2	2	no	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₆₁	—	2	2	no	—
37	M₃₂×C₆₁	C₂×C₄	1	4	yes	—
38	C₂×C₁₆×C₆₁	—	2	2	no	—
39	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
40	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
42	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
44	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₆₁	—	2	1	no	—
46	C₂²×Dic₆₁	—	2	1	no	—
47	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
51	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₆₁	#41	1	2	yes	—
55	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
56	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
57	C₄²×C₆₁	C₂×C₈	1	2	yes	—
58	C₄×Dic₆₁	—	2	1	no	—
59	C₄×Dic₆₁	—	2	1	no	—
60	C₄×Dic₆₁	C₄²	1	2	yes	—
61	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
62	C₂×#92418	—	2	1	no	—
63	C₂×#92418	—	2	1	no	—
64	C₂×#92418	C₄²	1	2	yes	—
65	C₂×#92418	C₂×C₈	1	2	yes	—
66	C₂×#92418	C₂×C₈	1	2	yes	—
67	C₂×#92418	C₂×C₈	1	2	yes	—
68	C₁₆×C₆₁	—	2	4	no	—
69	#11758185	—	2	4	no	—
70	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₆₁	—	4	2	no	—
72	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
73	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂²×C₈	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×Dic₆₁	—	2	2	no	—
81	C₂×Dic₆₁	—	2	2	no	—
82	C₂×Dic₆₁	—	4	2	no	—
83	C₂×Dic₆₁	—	4	2	no	—
84	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
85	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
86	C₈×C₆₁	C₄×D₄	1	2	yes	—
87	C₈×C₆₁	C₄×Q₈	1	2	yes	—
88	#92418	—	2	2	no	—
89	#92418	C₄×C₈	1	4	yes	—
90	#12656	—	61	1	no	maximal, Sylow
91	C₂²×C₆₁	—	2	2	no	—
92	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
93	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₂²×C₆₁	#431	1	1	yes	characteristic
95	C₄×C₆₁	—	2	2	no	—
96	C₄×C₆₁	—	2	2	no	—
97	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
99	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₄×C₆₁	#431	1	1	yes	characteristic
101	Dic₆₁	—	4	2	no	—
102	Dic₆₁	—	4	2	no	—
103	C₂×#364	—	61	2	no	—
104	C₄×M₃₂	—	61	1	no	—
105	C₄×M₃₂	D₆₁	1	1	yes	characteristic
106	#1726	—	61	1	no	—
107	#1376	—	61	2	no	—
108	C₂×C₆₁	—	2	1	no	—
109	C₂×C₆₁	—	2	1	no	—
110	C₂×C₆₁	C₄×C₄⋊C₈	1	1	yes	characteristic
111	C₂×C₆₁	#2153	1	2	yes	—
112	C₂×M₃₂	—	61	2	no	—
113	C₂×M₃₂	C₂×D₆₁	1	2	yes	—
114	C₂×#97	—	61	1	no	—
115	C₂×#97	—	61	1	no	—
116	C₂×#97	—	61	1	no	—
117	C₂×C₄×C₈	—	61	1	no	—
118	C₂×C₄×C₈	—	61	1	no	—
119	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
120	#376	—	61	1	no	—
121	#364	—	61	8	no	—
122	#346	—	2	1	no	—
123	#346	—	122	1	no	—
124	C₄×C₁₆	—	2	1	no	—
125	C₄×C₁₆	—	122	1	no	—
126	C₆₁	#12656	1	1	yes	characteristic, complemented, Sylow
127	C₂²×C₈	—	61	1	no	—
128	C₂²×C₈	—	61	1	no	—
129	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
131	C₂×C₄²	—	61	1	no	—
132	C₂×C₄²	—	61	1	no	—
133	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
134	M₃₂	—	2	2	no	—
135	M₃₂	—	61	4	no	—
136	M₃₂	—	122	2	no	—
137	M₃₂	C₄×D₆₁	1	4	yes	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	—	122	2	no	—
140	#97	—	61	4	no	—
141	#97	—	61	4	no	—
142	#97	—	122	2	no	—
143	C₄×C₈	—	61	4	no	—
144	C₄×C₈	—	122	2	no	—
145	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
146	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
149	#95	—	122	2	no	—
150	C₂²×C₄	—	122	1	no	—
151	C₂²×C₄	—	122	1	no	—
152	C₂²×C₄	#11758189	1	2	yes	—
153	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
154	C₂×C₈	—	61	2	no	—
155	C₂×C₈	—	61	2	no	—
156	C₂×C₈	—	61	2	no	—
157	C₂×C₈	—	61	2	no	—
158	C₂×C₈	—	122	1	no	—
159	C₂×C₈	—	122	1	no	—
160	C₂×C₈	C₈×D₆₁	1	2	yes	—
161	C₂×C₈	#11758189	1	2	yes	—
162	C₂×C₈	#11758196	1	1	yes	characteristic
163	C₂×C₈	#11758196	1	1	yes	characteristic
164	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
165	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
166	C₂×C₈	#11758222	1	2	yes	—
167	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
168	C₄²	—	61	2	no	—
169	C₄²	—	61	2	no	—
170	C₄²	—	122	1	no	—
171	C₄²	—	122	1	no	—
172	C₄²	C₈×D₆₁	1	2	yes	—
173	C₄²	#11758196	1	1	yes	characteristic
174	C₄²	#11758196	1	1	yes	characteristic
175	C₁₆	—	2	4	no	—
176	C₁₆	—	122	4	no	—
177	C₂³	C₂×#11758189	1	1	yes	characteristic
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	122	2	no	—
180	C₂×C₄	—	122	2	no	—
181	C₂×C₄	—	244	2	no	—
182	C₂×C₄	—	244	2	no	—
183	C₂×C₄	#14519706	1	2	yes	—
184	C₂×C₄	#14519706	1	2	yes	—
185	C₂×C₄	C₄²×D₆₁	1	1	yes	characteristic
186	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
187	C₂×C₄	#14519734	1	1	yes	characteristic
188	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
189	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
190	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
191	C₈	—	61	4	no	—
192	C₈	—	122	2	no	—
193	C₈	#14519706	1	2	yes	—
194	C₈	#14519706	1	2	yes	—
195	C₈	C₄×C₂²⋊D₆₁	1	2	yes	—
196	C₈	C₄×Dic₁₂₂	1	2	yes	—
197	C₂²	—	2	2	no	—
198	C₂²	C₄×#11758189	1	1	yes	characteristic
199	C₂²	C₂×#14519706	1	1	yes	characteristic
200	C₂²	#16839813	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	244	2	no	—
204	C₄	—	244	2	no	—
205	C₄	C₄×#11758196	1	1	yes	characteristic
206	C₄	C₄×C₈×D₆₁	1	1	yes	characteristic
207	C₄	C₂×#14519706	1	1	yes	characteristic
208	C₄	#16839813	1	1	yes	characteristic
209	C₂	—	2	1	no	—
210	C₂	—	2	1	no	—
211	C₂	C₄×#14519706	1	1	yes	characteristic
212	C₂	#21447584	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₁
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₄×M₃₂×C₆₁	C₂
socle	C₂²×C₆₁	C₂×C₄⋊C₈
prime radical	C₂³×C₆₁	C₂×M₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₄×M₃₂	D₆₁
hypocenter	C₆₁	#12656
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
13. #16839582	C₄	semidirect
126. C₆₁	#12656	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839582	2
C₄×#14519717	2
#21445505	2
#21443900	2
C₄×M₃₂×C₆₁	2
#12656	61