Subgroups — #29595339

Order 15616.

334 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	#21447575	C₂	1	2	yes	maximal
2	C₂×#16839540	C₂	1	2	yes	maximal
3	#21443899	C₂	1	2	yes	maximal
4	C₆₁×#1726	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519717	C₂²	1	4	yes	—
6	C₄⋊#11758185	C₄	1	8	yes	complemented
7	#16839540	—	2	4	no	—
8	C₂×#97×C₆₁	C₄	1	1	yes	characteristic
9	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#376	C₄	1	1	yes	characteristic
15	#14519717	—	2	4	no	—
16	#14519717	—	2	4	no	—
17	C₂×#11758185	C₂×C₄	1	8	yes	—
18	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
25	#97×C₆₁	—	2	2	no	—
26	#97×C₆₁	—	2	2	no	—
27	#97×C₆₁	D₄	1	2	yes	—
28	#97×C₆₁	Q₈	1	2	yes	—
29	C₄×C₈×C₆₁	—	2	2	no	—
30	C₄×C₈×C₆₁	—	2	2	no	—
31	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
32	#95×C₆₁	—	2	2	no	—
33	C₂²×C₄×C₆₁	—	2	1	no	—
34	C₂²×C₄×C₆₁	—	2	1	no	—
35	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₆₁	—	2	1	no	—
38	C₂×C₈×C₆₁	—	2	1	no	—
39	C₂×C₈×C₆₁	—	2	1	no	—
40	C₂×C₈×C₆₁	—	2	1	no	—
41	C₂×C₈×C₆₁	—	2	1	no	—
42	C₂×C₈×C₆₁	—	2	1	no	—
43	C₂×C₈×C₆₁	C₄²	1	2	yes	—
44	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
46	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₆₁	#41	1	2	yes	—
50	C₄²×C₆₁	—	2	1	no	—
51	C₄²×C₆₁	—	2	1	no	—
52	C₄²×C₆₁	C₄²	1	2	yes	—
53	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
54	C₄²×C₆₁	M₁₆	1	2	yes	—
55	C₄²×C₆₁	M₁₆	1	2	yes	—
56	#11758185	—	2	8	no	—
57	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₆₁	—	2	2	no	—
59	C₂×C₄×C₆₁	—	2	2	no	—
60	C₂×C₄×C₆₁	—	4	2	no	—
61	C₂×C₄×C₆₁	—	4	2	no	—
62	C₂×C₄×C₆₁	—	4	2	no	—
63	C₂×C₄×C₆₁	#97	1	4	yes	—
64	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₆₁	—	2	2	no	—
71	C₈×C₆₁	—	2	2	no	—
72	C₈×C₆₁	—	2	2	no	—
73	C₈×C₆₁	C₄×D₄	1	2	yes	—
74	C₈×C₆₁	C₄×Q₈	1	2	yes	—
75	#12657	—	61	1	no	maximal, Sylow
76	C₂²×C₆₁	—	2	2	no	—
77	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
78	C₂²×C₆₁	#423	1	1	yes	characteristic
79	C₂²×C₆₁	#432	1	1	yes	characteristic
80	C₄×C₆₁	—	2	4	no	—
81	C₄×C₆₁	—	4	2	no	—
82	C₄×C₆₁	—	4	2	no	—
83	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
85	C₄×C₆₁	#423	1	1	yes	characteristic
86	C₄×C₆₁	#432	1	1	yes	characteristic
87	#2150	—	61	2	no	—
88	C₂×#347	—	61	2	no	—
89	#1726	D₆₁	1	1	yes	characteristic
90	#1375	—	61	2	no	—
91	C₂×C₆₁	—	2	1	no	—
92	C₂×C₆₁	—	2	1	no	—
93	C₂×C₆₁	#1767	1	1	yes	characteristic
94	C₂×C₆₁	#2110	1	2	yes	—
95	C₂×M₃₂	—	61	4	no	—
96	C₂×#97	Dic₆₁	1	1	yes	characteristic
97	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
98	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
99	C₂×C₄×C₈	Dic₆₁	1	1	yes	characteristic
100	C₂×C₄×C₈	Dic₆₁	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
102	#376	Dic₆₁	1	1	yes	characteristic
103	C₄⋊C₁₆	—	61	8	no	—
104	#347	—	122	4	no	—
105	C₆₁	#12657	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
108	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
109	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
110	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
111	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
112	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
113	M₃₂	—	122	4	no	—
114	M₃₂	—	122	4	no	—
115	C₂×C₁₆	—	61	8	no	—
116	#97	—	2	2	no	—
117	#97	—	2	2	no	—
118	#97	Dic₁₂₂	1	2	yes	—
119	#97	D₂₄₄	1	2	yes	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	C₄×D₆₁	1	4	yes	—
123	#95	—	2	2	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	#11758194	1	2	yes	—
127	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	#11758194	1	2	yes	—
135	C₂×C₈	C₄×Dic₆₁	1	2	yes	—
136	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
137	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
138	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
139	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
140	C₂×C₈	#11758222	1	2	yes	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	#11758194	1	2	yes	—
144	C₄²	#11758194	1	2	yes	—
145	C₄²	C₄×Dic₆₁	1	2	yes	—
146	C₄²	C₄⋊Dic₆₁	1	2	yes	—
147	C₁₆	—	122	8	no	—
148	C₂³	C₂×#11758194	1	1	yes	characteristic
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	#14519688	1	4	yes	—
155	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
157	C₂×C₄	#14519736	1	1	yes	characteristic
158	C₂×C₄	C₂×#11758194	1	1	yes	characteristic
159	C₂×C₄	C₂×#11758194	1	1	yes	characteristic
160	C₂×C₄	C₂×#11758194	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	—	2	2	no	—
164	C₈	C₄×D₂₄₄	1	2	yes	—
165	C₈	C₄×Dic₁₂₂	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	C₂×#14519688	1	1	yes	characteristic
168	C₂²	#16839786	1	1	yes	characteristic
169	C₂²	#16839819	1	1	yes	characteristic
170	C₄	—	2	4	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	2	no	—
173	C₄	C₄×C₄⋊Dic₆₁	1	1	yes	characteristic
174	C₄	C₄×#11758194	1	1	yes	characteristic
175	C₄	#16839786	1	1	yes	characteristic
176	C₄	#16839819	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#21445672	1	1	yes	characteristic
180	C₂	#21447424	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₆₁
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1726	C₂
socle	C₂²×C₆₁	C₂×#97
prime radical	C₂³×C₆₁	C₂×M₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1726	D₆₁
hypocenter	C₆₁	#12657
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #1726

Complements

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

N	H ≅ G/N	type
6. C₄⋊#11758185	C₄	semidirect
105. C₆₁	#12657	semidirect

Maximal subgroups

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

subgroup	index
#21447575	2
C₂×#16839540	2
#21443899	2
C₆₁×#1726	2
#12657	61