Subgroups — #29598622

Order 15616.

374 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840164	C₂	1	1	yes	maximal, characteristic
2	#21445979	C₂	1	1	yes	maximal, characteristic
3	#21443911	C₂	1	4	yes	maximal
4	C₂×C₆₁×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519769	—	2	1	no	—
6	#16840164	—	2	2	no	—
7	#16840164	C₄	1	4	yes	complemented
8	C₂×#14519711	—	2	1	no	—
9	C₂×#14519711	C₂²	1	2	yes	—
10	C₂×#14519711	C₂²	1	2	yes	—
11	C₂×#14519709	—	2	1	no	—
12	#16839632	—	2	1	no	—
13	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#346	—	2	2	no	—
16	#14519769	—	2	4	no	—
17	#14519769	—	4	2	no	—
18	C₂×C₄⋊Dic₆₁	—	2	2	no	—
19	C₂×C₄⋊Dic₆₁	—	2	2	no	—
20	C₂×C₄⋊Dic₆₁	—	4	2	no	—
21	C₂×#11758196	—	4	2	no	—
22	#14519711	—	2	2	no	—
23	#14519711	—	2	4	no	—
24	#14519711	—	4	2	no	—
25	#14519711	C₂×C₄	1	4	yes	—
26	#14519709	—	2	4	no	—
27	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₆₁	—	2	2	no	—
31	C₂×C₁₆×C₆₁	—	2	2	no	—
32	C₂×C₁₆×C₆₁	Q₈	1	4	yes	—
33	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₆₁	D₄	1	2	yes	—
35	C₂²×C₄×C₆₁	D₈	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₆₁	—	4	2	no	—
38	C₂²×Dic₆₁	—	4	2	no	—
39	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
41	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
42	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₆₁	#41	1	2	yes	—
46	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
47	C₄²×C₆₁	SD₁₆	1	2	yes	—
48	C₄⋊Dic₆₁	—	2	4	no	—
49	C₄⋊Dic₆₁	—	4	2	no	—
50	C₄⋊Dic₆₁	—	4	2	no	—
51	C₄⋊Dic₆₁	—	4	2	no	—
52	C₄⋊Dic₆₁	—	8	4	no	—
53	#11758196	—	4	8	no	—
54	C₁₆×C₆₁	—	2	4	no	—
55	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₆₁	—	2	2	no	—
57	C₂×C₄×C₆₁	—	2	2	no	—
58	C₂×C₄×C₆₁	#102	1	4	yes	—
59	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₄⋊Q₈	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×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×Dic₆₁	—	4	4	no	—
66	C₂×Dic₆₁	—	8	2	no	—
67	C₂×Dic₆₁	—	8	2	no	—
68	C₂×Dic₆₁	—	8	2	no	—
69	C₂×Dic₆₁	—	8	4	no	—
70	C₈×C₆₁	#103	1	4	yes	—
71	C₈×C₆₁	C₄×Q₈	1	2	yes	—
72	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
73	#13284	—	61	1	no	maximal, Sylow
74	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
75	C₂²×C₆₁	#486	1	1	yes	characteristic
76	C₂²×C₆₁	#500	1	1	yes	characteristic
77	C₂²×C₆₁	#509	1	2	yes	—
78	C₂²×C₆₁	#510	1	2	yes	—
79	C₄×C₆₁	—	2	4	no	—
80	C₄×C₆₁	#391	1	1	yes	characteristic
81	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
82	C₄×C₆₁	#487	1	1	yes	characteristic
83	C₄×C₆₁	#498	1	1	yes	characteristic
84	Dic₆₁	—	8	4	no	—
85	Dic₆₁	—	16	2	no	—
86	C₂×#499	—	61	1	no	—
87	C₂×#346	D₆₁	1	1	yes	characteristic
88	#1845	—	61	1	no	—
89	#1380	—	61	4	no	—
90	C₂×C₆₁	#1990	1	1	yes	characteristic
91	C₂×C₆₁	#2139	1	2	yes	—
92	C₂×C₆₁	#2244	1	1	yes	characteristic
93	C₂×C₆₁	#2245	1	1	yes	characteristic
94	C₂×C₆₁	#2255	1	2	yes	—
95	C₂×#125	—	122	1	no	—
96	C₂²×C₁₆	C₂×D₆₁	1	2	yes	—
97	#499	—	61	4	no	—
98	#499	—	122	2	no	—
99	C₂×#107	—	61	2	no	—
100	C₂×#107	—	61	2	no	—
101	C₂×#107	—	122	1	no	—
102	C₂×#106	—	122	1	no	—
103	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
104	#384	—	122	1	no	—
105	#346	—	2	2	no	—
106	C₆₁	#13284	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
109	#125	—	122	4	no	—
110	#125	—	244	2	no	—
111	C₂×C₄⋊C₄	—	122	2	no	—
112	C₂×C₄⋊C₄	—	122	2	no	—
113	C₂×C₄⋊C₄	—	244	2	no	—
114	C₂×C₄⋊C₄	—	244	2	no	—
115	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	Dic₁₂₂	1	4	yes	—
119	#107	—	61	4	no	—
120	#107	—	122	2	no	—
121	#107	—	122	4	no	—
122	#107	—	244	2	no	—
123	#106	—	122	4	no	—
124	C₄×C₈	C₄×D₆₁	1	2	yes	—
125	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
126	C₂²×C₄	—	244	2	no	—
127	C₂²×C₄	—	244	2	no	—
128	C₂²×C₄	D₄₈₈	1	2	yes	—
129	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
130	C₂×C₈	#11758190	1	2	yes	—
131	C₂×C₈	Dic₂₄₄	1	2	yes	—
132	C₂×C₈	#11758198	1	2	yes	—
133	C₂×C₈	C₂×Dic₁₂₂	1	2	yes	—
134	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
135	C₂×C₈	#11758222	1	2	yes	—
136	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
137	C₄⋊C₄	—	122	4	no	—
138	C₄⋊C₄	—	488	4	no	—
139	C₄⋊C₄	—	244	2	no	—
140	C₄⋊C₄	—	244	2	no	—
141	C₄⋊C₄	—	244	8	no	—
142	C₄⋊C₄	—	244	2	no	—
143	C₄²	#11758190	1	2	yes	—
144	C₄²	#11758198	1	2	yes	—
145	C₁₆	—	2	4	no	—
146	C₂³	C₂×D₄₈₈	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	488	4	no	—
150	C₂×C₄	—	488	2	no	—
151	C₂×C₄	—	488	2	no	—
152	C₂×C₄	—	488	2	no	—
153	C₂×C₄	—	244	4	no	—
154	C₂×C₄	#14519699	1	4	yes	—
155	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
156	C₂×C₄	#14519765	1	1	yes	characteristic
157	C₂×C₄	#14519777	1	1	yes	characteristic
158	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
159	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
160	C₂×C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
161	C₈	#14519702	1	4	yes	—
162	C₈	C₄×Dic₁₂₂	1	2	yes	—
163	C₈	#14519756	1	2	yes	—
164	C₂²	C₂×#14519699	1	1	yes	characteristic
165	C₂²	#16840109	1	1	yes	characteristic
166	C₂²	#16840167	1	1	yes	characteristic
167	C₂²	#16840200	1	2	yes	—
168	C₂²	#16840205	1	2	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	976	2	no	—
171	C₄	—	488	4	no	—
172	C₄	#16839668	1	1	yes	characteristic
173	C₄	C₂×#14519702	1	1	yes	characteristic
174	C₄	#16840113	1	1	yes	characteristic
175	C₄	#16840160	1	1	yes	characteristic
176	C₂	#21446780	1	1	yes	characteristic
177	C₂	#21447532	1	2	yes	—
178	C₂	#21448061	1	1	yes	characteristic
179	C₂	#21448065	1	1	yes	characteristic
180	C₂	#21448105	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₈₈
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#346	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×#346	D₆₁
hypocenter	C₆₁	#13284
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
7. #16840164	C₄	semidirect
106. C₆₁	#13284	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840164	2
#21445979	2
#21443911	2
C₂×C₆₁×#346	2
#13284	61