Subgroups — #29596444

Order 15616.

318 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839586	C₂	1	4	yes	maximal
2	C₄×#14519717	C₂	1	1	yes	maximal, characteristic
3	#21445983	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519717	C₂²	1	2	yes	—
6	C₂×#14519711	—	2	1	no	—
7	C₂×#14519709	C₂²	1	2	yes	—
8	#16839628	—	2	1	no	—
9	#16839586	—	2	8	no	—
10	#16839538	C₄	1	2	yes	complemented
11	C₄×#11758185	C₄	1	2	yes	complemented
12	C₂×#108×C₆₁	—	2	1	no	—
13	C₂×#108×C₆₁	C₂²	1	2	yes	—
14	C₆₁×#423	—	2	1	no	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₆₁	—	2	2	no	—
17	#14519717	D₄	1	4	yes	—
18	#14519717	Q₈	1	4	yes	—
19	C₂×#11758185	C₂×C₄	1	4	yes	—
20	#14519711	—	4	2	no	—
21	#14519709	—	2	4	no	—
22	#14519684	—	4	2	no	—
23	C₂×M₁₆×C₆₁	—	2	2	no	—
24	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
27	#108×C₆₁	—	2	4	no	—
28	#108×C₆₁	—	4	2	no	—
29	C₄⋊C₈×C₆₁	—	4	2	no	—
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	SD₁₆	1	2	yes	—
35	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₆₁	—	4	2	no	—
37	M₁₆×C₆₁	—	4	2	no	—
38	M₁₆×C₆₁	—	4	2	no	—
39	C₂×C₈×C₆₁	—	4	2	no	—
40	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
43	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
44	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₆₁	SD₁₆	1	2	yes	—
51	C₄⋊Dic₆₁	—	4	2	no	—
52	C₄⋊Dic₆₁	—	4	2	no	—
53	#11758185	C₄⋊C₄	1	8	yes	—
54	C₂³×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×C₄×C₆₁	—	4	2	no	—
56	C₂×C₄×C₆₁	#106	1	2	yes	—
57	C₂×C₄×C₆₁	#106	1	2	yes	—
58	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×Dic₆₁	—	8	4	no	—
65	C₂×Dic₆₁	—	8	2	no	—
66	C₈×C₆₁	—	8	2	no	—
67	C₈×C₆₁	#106	1	2	yes	—
68	C₈×C₆₁	#106	1	2	yes	—
69	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
70	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
71	#12904	—	61	1	no	maximal, Sylow
72	C₂²×C₆₁	—	2	2	no	—
73	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
74	C₂²×C₆₁	#492	1	1	yes	characteristic
75	C₂²×C₆₁	#498	1	1	yes	characteristic
76	C₄×C₆₁	—	2	2	no	—
77	C₄×C₆₁	—	2	2	no	—
78	C₄×C₆₁	#384	1	1	yes	characteristic
79	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
80	C₄×C₆₁	#492	1	1	yes	characteristic
81	C₄×C₆₁	#498	1	1	yes	characteristic
82	Dic₆₁	—	16	2	no	—
83	C₂×#365	—	61	4	no	—
84	C₄×M₃₂	—	61	1	no	—
85	#1848	D₆₁	1	1	yes	characteristic
86	#1846	—	61	1	no	—
87	C₂×C₆₁	—	2	1	no	—
88	C₂×C₆₁	—	2	1	no	—
89	C₂×C₆₁	#1844	1	1	yes	characteristic
90	C₂×C₆₁	#2155	1	2	yes	—
91	C₂×M₃₂	—	61	2	no	—
92	C₂×#108	—	2	1	no	—
93	C₂×#108	C₂×D₆₁	1	2	yes	—
94	C₂×#107	—	122	1	no	—
95	C₂×#106	—	61	2	no	—
96	#423	—	2	1	no	—
97	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
98	#382	—	122	1	no	—
99	#365	—	122	8	no	—
100	#346	—	61	2	no	—
101	C₄×C₁₆	—	61	2	no	—
102	C₆₁	#12904	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₁₆	—	2	2	no	—
104	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
105	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	122	2	no	—
107	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
108	M₃₂	—	61	4	no	—
109	M₃₂	—	61	4	no	—
110	C₂×C₁₆	—	61	4	no	—
111	#108	—	2	4	no	—
112	#108	—	4	2	no	—
113	#107	—	244	2	no	—
114	#106	—	122	4	no	—
115	C₄⋊C₈	—	4	2	no	—
116	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
117	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
118	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
119	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
120	#95	—	244	2	no	—
121	C₂²×C₄	—	244	2	no	—
122	C₂²×C₄	#11758201	1	2	yes	—
123	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
124	M₁₆	—	4	2	no	—
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	#11758196	1	1	yes	characteristic
128	C₂×C₈	#11758196	1	1	yes	characteristic
129	C₂×C₈	#11758200	1	2	yes	—
130	C₂×C₈	#11758201	1	2	yes	—
131	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
133	C₂×C₈	D₄×D₆₁	1	2	yes	—
134	C₂×C₈	Q₈×D₆₁	1	2	yes	—
135	C₄⋊C₄	—	244	2	no	—
136	C₄⋊C₄	—	244	2	no	—
137	C₄²	#11758196	1	1	yes	characteristic
138	C₄²	#11758196	1	1	yes	characteristic
139	C₄²	#11758200	1	2	yes	—
140	C₁₆	—	61	8	no	—
141	C₂³	C₂×#11758201	1	1	yes	characteristic
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	488	4	no	—
144	C₂×C₄	—	488	2	no	—
145	C₂×C₄	#14519708	1	2	yes	—
146	C₂×C₄	#14519708	1	2	yes	—
147	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
148	C₂×C₄	#14519773	1	1	yes	characteristic
149	C₂×C₄	#14519775	1	1	yes	characteristic
150	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
151	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
152	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
153	C₈	—	8	2	no	—
154	C₈	#14519708	1	2	yes	—
155	C₈	#14519708	1	2	yes	—
156	C₈	C₄⋊C₄×D₆₁	1	2	yes	—
157	C₈	#14519776	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	C₂×#14519708	1	1	yes	characteristic
160	C₂²	#16840136	1	1	yes	characteristic
161	C₂²	#16840158	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	976	2	no	—
165	C₄	#16839631	1	1	yes	characteristic
166	C₄	C₂×#14519708	1	1	yes	characteristic
167	C₄	#16840134	1	1	yes	characteristic
168	C₄	#16840158	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#21445975	1	1	yes	characteristic
172	C₂	#21447593	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14519708
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1848	C₂
socle	C₂²×C₆₁	C₂×#106
prime radical	C₂³×C₆₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1848	D₆₁
hypocenter	C₆₁	#12904
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
10. #16839538	C₄	semidirect
11. C₄×#11758185	C₄	semidirect
102. C₆₁	#12904	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839586	2
C₄×#14519717	2
#21445983	2
C₆₁×#1848	2
#12904	61