Subgroups — #29567187

Order 15616.

238 conjugacy classes of subgroups.

The subgroup lattice has 190 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	#21445979	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#364	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519717	—	2	1	no	—
5	C₂×#14519711	—	2	1	no	—
6	C₂×#14519711	—	2	1	no	—
7	#16839632	C₄	1	2	yes	—
8	#16839583	—	2	1	no	—
9	#16839583	C₄	1	2	yes	complemented
10	C₂×M₃₂×C₆₁	—	2	1	no	—
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#364	—	2	1	no	—
13	C₆₁×#364	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₆₁	—	2	2	no	—
15	C₂×C₄⋊Dic₆₁	—	4	2	no	—
16	#14519717	—	2	2	no	—
17	#14519717	—	4	1	no	—
18	C₂×#11758185	—	4	1	no	—
19	#14519711	—	4	1	no	—
20	#14519711	—	4	1	no	—
21	#14519711	—	4	1	no	—
22	#14519711	—	4	1	no	—
23	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₆₁	—	2	2	no	—
27	M₃₂×C₆₁	—	4	1	no	—
28	C₂×C₁₆×C₆₁	—	4	1	no	—
29	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	—	2	1	no	—
34	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×Dic₆₁	—	4	2	no	—
36	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
41	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
42	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
43	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
44	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₆₁	Q₁₆	1	1	yes	characteristic
47	C₂×C₈×C₆₁	Q₁₆	1	1	yes	characteristic
48	C₄²×C₆₁	—	2	1	no	—
49	C₄²×C₆₁	C₄²	1	1	yes	characteristic
50	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
51	C₄⋊Dic₆₁	—	4	2	no	—
52	C₄⋊Dic₆₁	—	4	2	no	—
53	C₄⋊Dic₆₁	—	8	4	no	—
54	C₁₆×C₆₁	—	4	2	no	—
55	#11758185	—	4	2	no	—
56	C₂³×C₆₁	#100	1	1	yes	characteristic
57	C₂×C₄×C₆₁	—	2	2	no	—
58	C₂×C₄×C₆₁	—	2	2	no	—
59	C₂×C₄×C₆₁	—	2	1	no	—
60	C₂×C₄×C₆₁	—	2	1	no	—
61	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
62	C₂×C₄×C₆₁	#101	1	1	yes	characteristic
63	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
64	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
65	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
67	C₂×Dic₆₁	—	8	4	no	—
68	C₂×Dic₆₁	—	8	2	no	—
69	C₈×C₆₁	#102	1	1	yes	characteristic
70	C₈×C₆₁	#102	1	1	yes	characteristic
71	C₈×C₆₁	#103	1	1	yes	characteristic
72	C₈×C₆₁	#103	1	1	yes	characteristic
73	C₈×C₆₁	#106	1	1	yes	characteristic
74	C₈×C₆₁	#106	1	1	yes	characteristic
75	C₈×C₆₁	#107	1	1	yes	characteristic
76	C₈×C₆₁	#107	1	1	yes	characteristic
77	#7269	—	61	1	no	maximal, Sylow
78	C₂²×C₆₁	—	2	1	no	—
79	C₂²×C₆₁	—	2	1	no	—
80	C₂²×C₆₁	#331	1	1	yes	characteristic
81	C₂²×C₆₁	#333	1	1	yes	characteristic
82	C₂²×C₆₁	#343	1	1	yes	characteristic
83	C₄×C₆₁	—	2	2	no	—
84	C₄×C₆₁	—	2	2	no	—
85	C₄×C₆₁	#332	1	1	yes	characteristic
86	C₄×C₆₁	#332	1	1	yes	characteristic
87	C₄×C₆₁	#340	1	1	yes	characteristic
88	C₄×C₆₁	#340	1	1	yes	characteristic
89	Dic₆₁	—	16	2	no	—
90	C₂×#364	—	61	1	no	—
91	C₂×#364	D₆₁	1	1	yes	characteristic
92	#1845	—	61	1	no	—
93	C₂×C₆₁	—	2	1	no	—
94	C₂×C₆₁	—	2	1	no	—
95	C₂×C₆₁	#1296	1	1	yes	characteristic
96	C₂×C₆₁	#1357	1	1	yes	characteristic
97	C₂×C₆₁	#1358	1	1	yes	characteristic
98	C₂×M₃₂	—	2	1	no	—
99	C₂×M₃₂	—	122	1	no	—
100	C₂×#107	—	122	1	no	—
101	C₂×#107	—	122	1	no	—
102	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
103	#384	—	61	2	no	—
104	#364	—	2	1	no	—
105	#364	—	61	2	no	—
106	#364	—	122	1	no	—
107	#364	Dic₆₁	1	2	yes	complemented
108	C₆₁	#7269	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
110	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
111	C₂×C₄⋊C₄	—	122	2	no	—
112	C₂×C₄⋊C₄	—	244	2	no	—
113	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
114	M₃₂	—	2	2	no	—
115	M₃₂	—	4	1	no	—
116	M₃₂	—	122	2	no	—
117	M₃₂	—	244	1	no	—
118	C₂×C₁₆	—	4	1	no	—
119	C₂×C₁₆	—	244	1	no	—
120	#107	—	244	1	no	—
121	#107	—	244	1	no	—
122	#107	—	244	1	no	—
123	#107	—	244	1	no	—
124	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
125	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
126	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
127	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	244	2	no	—
130	C₂²×C₄	#11758198	1	1	yes	characteristic
131	C₂×C₈	#11758190	1	1	yes	characteristic
132	C₂×C₈	#11758190	1	1	yes	characteristic
133	C₂×C₈	D₄₈₈	1	1	yes	characteristic
134	C₂×C₈	Dic₂₄₄	1	1	yes	characteristic
135	C₂×C₈	#11758196	1	1	yes	characteristic
136	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
137	C₂×C₈	#11758198	1	1	yes	characteristic
138	C₂×C₈	#11758199	1	1	yes	characteristic
139	C₂×C₈	#11758200	1	1	yes	characteristic
140	C₂×C₈	#11758201	1	1	yes	characteristic
141	C₂×C₈	#11758202	1	1	yes	characteristic
142	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
143	C₄⋊C₄	—	488	4	no	—
144	C₄⋊C₄	—	244	2	no	—
145	C₄⋊C₄	—	244	2	no	—
146	C₄²	—	2	1	no	—
147	C₄²	C₄×Dic₆₁	1	1	yes	characteristic
148	C₄²	#11758196	1	1	yes	characteristic
149	C₁₆	—	4	2	no	—
150	C₁₆	—	244	2	no	—
151	C₂³	#14519693	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	488	4	no	—
157	C₂×C₄	—	488	2	no	—
158	C₂×C₄	#14519684	1	1	yes	characteristic
159	C₂×C₄	#14519695	1	1	yes	characteristic
160	C₂×C₄	#14519697	1	1	yes	characteristic
161	C₂×C₄	#14519699	1	1	yes	characteristic
162	C₂×C₄	#14519700	1	1	yes	characteristic
163	C₂×C₄	#14519702	1	1	yes	characteristic
164	C₈	#14519698	1	1	yes	characteristic
165	C₈	#14519699	1	1	yes	characteristic
166	C₈	#14519701	1	1	yes	characteristic
167	C₈	#14519702	1	1	yes	characteristic
168	C₈	#14519708	1	1	yes	characteristic
169	C₈	#14519709	1	1	yes	characteristic
170	C₈	#14519710	1	1	yes	characteristic
171	C₈	#14519711	1	1	yes	characteristic
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	#16839505	1	1	yes	characteristic
175	C₂²	#16839510	1	1	yes	characteristic
176	C₂²	#16839530	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	2	2	no	—
179	C₄	—	976	2	no	—
180	C₄	#16839507	1	1	yes	characteristic
181	C₄	#16839509	1	1	yes	characteristic
182	C₄	#16839524	1	1	yes	characteristic
183	C₄	#16839525	1	1	yes	characteristic
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#21443703	1	1	yes	characteristic
187	C₂	#21443855	1	1	yes	characteristic
188	C₂	#21443857	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16839505
commutator	C₄²×C₆₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₁
Fitting	C₂×C₆₁×#364	C₂
socle	C₂²×C₆₁	#331
prime radical	C₂³×C₆₁	#100
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	C₂×#364	D₆₁
hypocenter	C₆₁	#7269
LCS	G ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
9. #16839583	C₄	semidirect
13. C₆₁×#364	C₄	semidirect
107. #364	Dic₆₁	semidirect
108. C₆₁	#7269	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
#21445979	2
C₂×C₆₁×#364	2
#7269	61