Subgroups — #29596452

Order 15616.

342 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839589	C₂	1	2	yes	maximal
2	C₂×#16839587	C₂	1	2	yes	maximal
3	C₂×C₄×#11758185	C₂	1	1	yes	maximal, characteristic
4	#21445983	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1846	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758185	C₂²	1	1	yes	characteristic
7	C₂²×#11758185	C₂²	1	1	yes	characteristic
8	C₂×#14519711	C₂²	1	2	yes	—
9	C₂×#14519709	—	2	1	no	—
10	#16839628	—	2	1	no	—
11	#16839589	—	2	4	no	—
12	#16839587	—	2	4	no	—
13	C₄×#11758185	C₄	1	4	yes	complemented
14	C₂×#107×C₆₁	C₂²	1	2	yes	—
15	C₂×#106×C₆₁	—	2	1	no	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#382	—	2	1	no	—
18	C₂×C₄⋊Dic₆₁	—	2	2	no	—
19	C₂×#11758185	C₂×C₄	1	2	yes	—
20	C₂×#11758185	C₂×C₄	1	2	yes	—
21	C₂×#11758185	D₄	1	2	yes	—
22	C₂×#11758185	D₄	1	2	yes	—
23	C₂×#11758185	Q₈	1	2	yes	—
24	C₂×#11758185	Q₈	1	2	yes	—
25	#14519711	—	2	2	no	—
26	#14519711	—	2	2	no	—
27	#14519709	—	4	2	no	—
28	#14519684	—	4	2	no	—
29	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
32	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
33	#107×C₆₁	—	2	2	no	—
34	#107×C₆₁	—	2	2	no	—
35	#106×C₆₁	—	4	2	no	—
36	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₆₁	Q₈	1	2	yes	—
38	#95×C₆₁	—	4	2	no	—
39	C₂²×C₄×C₆₁	—	4	2	no	—
40	C₂²×C₄×C₆₁	D₈	1	2	yes	—
41	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₆₁	—	4	2	no	—
43	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₆₁	D₈	1	2	yes	—
45	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
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×D₄	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
52	C₄⋊C₄×C₆₁	—	4	2	no	—
53	C₄⋊C₄×C₆₁	—	4	2	no	—
54	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
55	C₄²×C₆₁	Q₁₆	1	2	yes	—
56	C₄⋊Dic₆₁	—	4	2	no	—
57	C₄⋊Dic₆₁	—	4	2	no	—
58	#11758185	C₄⋊C₄	1	4	yes	—
59	#11758185	C₄⋊C₄	1	4	yes	—
60	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₆₁	—	2	4	no	—
62	C₂×C₄×C₆₁	—	8	2	no	—
63	C₂×C₄×C₆₁	—	8	4	no	—
64	C₂×C₄×C₆₁	#107	1	4	yes	—
65	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×Dic₆₁	—	8	2	no	—
72	C₂×Dic₆₁	—	8	4	no	—
73	C₈×C₆₁	#107	1	4	yes	—
74	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
77	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
78	#12907	—	61	1	no	maximal, Sylow
79	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
80	C₂²×C₆₁	#493	1	1	yes	characteristic
81	C₂²×C₆₁	#500	1	1	yes	characteristic
82	C₂²×C₆₁	#508	1	4	yes	—
83	C₄×C₆₁	—	2	4	no	—
84	C₄×C₆₁	—	16	2	no	—
85	C₄×C₆₁	#384	1	1	yes	characteristic
86	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
87	C₄×C₆₁	#494	1	1	yes	characteristic
88	C₄×C₆₁	#500	1	1	yes	characteristic
89	Dic₆₁	—	16	2	no	—
90	C₂×#367	—	61	2	no	—
91	C₂×#366	—	61	2	no	—
92	C₂×C₄×C₁₆	—	61	1	no	—
93	#1846	—	61	1	no	—
94	#1846	D₆₁	1	1	yes	characteristic
95	C₂×C₆₁	#1845	1	1	yes	characteristic
96	C₂×C₆₁	#2158	1	2	yes	—
97	C₂×C₆₁	#2249	1	2	yes	—
98	C₂×C₆₁	#2253	1	2	yes	—
99	C₂²×C₁₆	—	61	1	no	—
100	C₂²×C₁₆	—	61	1	no	—
101	C₂×#107	—	61	2	no	—
102	C₂×#107	C₂×D₆₁	1	2	yes	—
103	C₂×#106	—	2	1	no	—
104	C₂×#106	—	122	1	no	—
105	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
106	#382	—	2	1	no	—
107	#382	—	122	1	no	—
108	#367	—	122	4	no	—
109	#366	—	122	4	no	—
110	C₄×C₁₆	—	61	4	no	—
111	C₆₁	#12907	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	122	2	no	—
116	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
117	C₂×C₁₆	—	61	2	no	—
118	C₂×C₁₆	—	61	2	no	—
119	C₂×C₁₆	—	61	2	no	—
120	C₂×C₁₆	—	61	2	no	—
121	C₂×C₁₆	—	61	2	no	—
122	C₂×C₁₆	—	61	2	no	—
123	#107	—	2	2	no	—
124	#107	—	2	2	no	—
125	#107	—	122	2	no	—
126	#107	—	122	2	no	—
127	#106	—	4	2	no	—
128	#106	—	244	2	no	—
129	C₄×C₈	Dic₁₂₂	1	2	yes	—
130	C₄×C₈	C₄×D₆₁	1	2	yes	—
131	#95	—	4	2	no	—
132	#95	—	244	2	no	—
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	—	244	2	no	—
135	C₂²×C₄	#11758199	1	2	yes	—
136	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
137	C₂×C₈	#11758196	1	2	yes	—
138	C₂×C₈	#11758199	1	2	yes	—
139	C₂×C₈	#11758202	1	2	yes	—
140	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
141	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
142	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
143	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
145	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	244	2	no	—
149	C₄⋊C₄	—	244	2	no	—
150	C₄²	#11758196	1	2	yes	—
151	C₄²	#11758202	1	2	yes	—
152	C₁₆	—	61	4	no	—
153	C₁₆	—	61	4	no	—
154	C₂³	C₂×#11758199	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	8	4	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	—	488	4	no	—
159	C₂×C₄	—	488	2	no	—
160	C₂×C₄	#14519710	1	4	yes	—
161	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
162	C₂×C₄	#14519773	1	1	yes	characteristic
163	C₂×C₄	#14519775	1	1	yes	characteristic
164	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
165	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
166	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
167	C₈	#14519710	1	4	yes	—
168	C₈	C₄⋊C₄×D₆₁	1	1	yes	characteristic
169	C₈	C₄⋊C₄×D₆₁	1	1	yes	characteristic
170	C₈	#14519776	1	1	yes	characteristic
171	C₈	#14519776	1	1	yes	characteristic
172	C₂²	C₂×#14519710	1	1	yes	characteristic
173	C₂²	#16840138	1	1	yes	characteristic
174	C₂²	#16840165	1	1	yes	characteristic
175	C₂²	#16840192	1	4	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	16	2	no	—
178	C₄	—	976	2	no	—
179	C₄	#16839631	1	1	yes	characteristic
180	C₄	C₂×#14519710	1	1	yes	characteristic
181	C₄	#16840141	1	1	yes	characteristic
182	C₄	#16840165	1	1	yes	characteristic
183	C₂	#21445978	1	1	yes	characteristic
184	C₂	#21447601	1	2	yes	—
185	C₂	#21448078	1	2	yes	—
186	C₂	#21448095	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11758199
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1846	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1846	D₆₁
hypocenter	C₆₁	#12907
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

Complements

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

N	H ≅ G/N	type
13. C₄×#11758185	C₄	semidirect
111. C₆₁	#12907	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839589	2
C₂×#16839587	2
C₂×C₄×#11758185	2
#21445983	2
C₆₁×#1846	2
#12907	61