Subgroups — #29571223

Order 15616.

394 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	C₄×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
2	#21445565	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839512	C₂	1	2	yes	maximal
4	#21443653	C₂	1	2	yes	maximal
5	C₆₁×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×C₄×#92418	—	2	1	no	—
8	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
9	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
10	#16839517	—	2	1	no	—
11	#16839517	C₄	1	4	yes	—
12	#16839512	C₄	1	8	yes	complemented
13	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
14	C₂×#97×C₆₁	—	2	1	no	—
15	C₄³×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#336	—	2	1	no	—
17	C₂²×#92418	—	2	2	no	—
18	C₂²×#92418	C₂×C₄	1	2	yes	—
19	C₂²×#92418	C₂×C₄	1	2	yes	—
20	C₄⋊#92418	—	2	2	no	—
21	C₄⋊#92418	C₂×C₄	1	4	yes	—
22	C₄×#92418	—	2	2	no	—
23	C₄×#92418	—	4	2	no	—
24	C₄×#92418	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₆₁	—	2	2	no	—
26	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
30	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₆₁	—	2	2	no	—
32	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
33	#97×C₆₁	—	4	2	no	—
34	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₆₁	—	2	4	no	—
40	C₂×C₈×C₆₁	—	4	2	no	—
41	C₂×C₈×C₆₁	—	4	2	no	—
42	C₄²×C₆₁	—	2	4	no	—
43	C₄²×C₆₁	—	2	2	no	—
44	C₄²×C₆₁	—	4	2	no	—
45	C₄²×C₆₁	SD₁₆	1	2	yes	—
46	C₄²×C₆₁	SD₁₆	1	2	yes	—
47	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
48	C₄²×C₆₁	#41	1	2	yes	—
49	C₂×#92418	—	2	4	no	—
50	C₂×#92418	—	2	2	no	—
51	C₂×#92418	—	2	4	no	—
52	C₂×#92418	—	2	4	no	—
53	C₂×#92418	—	4	2	no	—
54	C₂×#92418	—	4	2	no	—
55	C₂×#92418	C₄²	1	4	yes	—
56	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₆₁	—	2	2	no	—
58	C₂×C₄×C₆₁	—	2	2	no	—
59	C₂×C₄×C₆₁	—	2	2	no	—
60	C₂×C₄×C₆₁	—	2	2	no	—
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	#106	1	4	yes	—
64	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
68	C₂×C₄×C₆₁	#131	1	4	yes	—
69	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₆₁	#135	1	2	yes	—
72	C₈×C₆₁	—	4	4	no	—
73	#92418	—	2	4	no	—
74	#92418	—	4	2	no	—
75	#92418	—	4	4	no	—
76	#8277	—	61	1	no	maximal, Sylow
77	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₆₁	#424	1	2	yes	—
79	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
80	C₂²×C₆₁	#427	1	1	yes	characteristic
81	C₂²×C₆₁	#430	1	2	yes	—
82	C₄×C₆₁	—	2	4	no	—
83	C₄×C₆₁	—	2	2	no	—
84	C₄×C₆₁	—	4	2	no	—
85	C₄×C₆₁	#405	1	2	yes	—
86	C₄×C₆₁	#433	1	2	yes	—
87	C₄×C₆₁	C₄×SD₁₆	1	2	yes	—
88	C₄×C₆₁	C₄×SD₁₆	1	2	yes	—
89	#1767	D₆₁	1	1	yes	characteristic
90	C₄×C₄⋊C₈	—	61	1	no	—
91	#1744	—	61	1	no	—
92	C₂×#334	—	61	2	no	—
93	#1277	—	61	2	no	—
94	C₂×C₆₁	#1570	1	2	yes	—
95	C₂×C₆₁	#1572	1	2	yes	—
96	C₂×C₆₁	#1768	1	1	yes	characteristic
97	C₂×C₆₁	C₄×#106	1	1	yes	characteristic
98	C₂×C₆₁	#1778	1	1	yes	characteristic
99	C₂×C₄⋊C₈	—	61	2	no	—
100	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
101	C₂×#97	—	2	1	no	—
102	C₂×C₄×C₈	—	61	1	no	—
103	C₂×C₄×C₈	—	61	1	no	—
104	C₂×C₄×C₈	—	122	1	no	—
105	C₄³	C₂×D₆₁	1	1	yes	characteristic
106	#336	—	2	1	no	—
107	#336	—	61	4	no	—
108	#336	—	122	1	no	—
109	#334	—	61	8	no	—
110	C₆₁	#8277	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	61	2	no	—
113	C₂²×C₈	—	61	2	no	—
114	C₂²×C₈	—	122	2	no	—
115	C₂×C₄²	Dic₁₂₂	1	2	yes	—
116	C₂×C₄²	C₄×D₆₁	1	2	yes	—
117	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
118	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
119	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	61	4	no	—
122	C₄⋊C₈	—	122	2	no	—
123	C₄⋊C₈	C₄×D₆₁	1	4	yes	—
124	#97	—	4	2	no	—
125	C₄×C₈	—	61	4	no	—
126	C₄×C₈	—	122	2	no	—
127	C₄×C₈	—	244	2	no	—
128	C₂²×C₄	#11758196	1	2	yes	—
129	C₂²×C₄	#11758196	1	2	yes	—
130	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	61	4	no	—
137	C₂×C₈	—	122	4	no	—
138	C₂×C₈	—	122	2	no	—
139	C₂×C₈	—	122	4	no	—
140	C₂×C₈	—	122	4	no	—
141	C₂×C₈	—	244	2	no	—
142	C₂×C₈	—	244	2	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	#11758200	1	2	yes	—
147	C₄²	#11758201	1	2	yes	—
148	C₄²	C₂×C₄×D₆₁	1	2	yes	—
149	C₄²	#11758222	1	2	yes	—
150	C₂³	C₂×#11758196	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	#14519708	1	4	yes	—
158	C₂×C₄	C₄²×D₆₁	1	1	yes	characteristic
159	C₂×C₄	#14519734	1	1	yes	characteristic
160	C₂×C₄	C₄×C₂²⋊D₆₁	1	2	yes	—
161	C₂×C₄	C₄×Dic₁₂₂	1	2	yes	—
162	C₂×C₄	#14519785	1	4	yes	—
163	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
164	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
165	C₂×C₄	#14519797	1	2	yes	—
166	C₈	—	4	4	no	—
167	C₈	—	122	4	no	—
168	C₈	—	244	2	no	—
169	C₈	—	244	4	no	—
170	C₂²	C₄×#11758196	1	1	yes	characteristic
171	C₂²	#16839790	1	2	yes	—
172	C₂²	C₂×#14519708	1	1	yes	characteristic
173	C₂²	#16839798	1	1	yes	characteristic
174	C₂²	#16839810	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	4	no	—
177	C₄	—	4	2	no	—
178	C₄	#16839719	1	2	yes	—
179	C₄	#16839824	1	2	yes	—
180	C₄	C₄×#11758200	1	2	yes	—
181	C₄	C₄×#11758201	1	2	yes	—
182	C₂	#21444591	1	2	yes	—
183	C₂	#21444600	1	2	yes	—
184	C₂	#21445679	1	1	yes	characteristic
185	C₂	C₄×#14519708	1	1	yes	characteristic
186	C₂	#21445726	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1767	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1767	D₆₁
hypocenter	C₆₁	#8277
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
12. #16839512	C₄	semidirect
110. C₆₁	#8277	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#92418	2
#21445565	2
C₂×#16839512	2
#21443653	2
C₆₁×#1767	2
#8277	61