Subgroups — #30606652

Order 15616.

414 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11758187	C₂	1	2	yes	maximal
2	C₂×#16840650	C₂	1	2	yes	maximal
3	#21460857	C₂	1	2	yes	maximal
4	C₄²×#92418	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11758187	C₂²	1	4	yes	—
6	C₄⋊#11758187	C₄	1	8	yes	complemented
7	#16840650	—	2	4	no	—
8	C₂×C₄×#92418	C₄	1	1	yes	characteristic
9	C₂×C₄×#92418	C₄	1	1	yes	characteristic
10	C₂×C₄×#92418	C₄	1	1	yes	characteristic
11	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
12	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
13	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
14	C₄³×C₆₁	C₄	1	1	yes	characteristic
15	C₂×#11758187	—	2	4	no	—
16	C₂×#11758187	—	2	4	no	—
17	C₂×#11758187	C₂×C₄	1	8	yes	—
18	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
19	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
20	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
21	C₂²×#92418	C₂³	1	1	yes	characteristic
22	C₄×#92418	—	2	2	no	—
23	C₄×#92418	—	2	2	no	—
24	C₄×#92418	—	2	2	no	—
25	C₄×#92418	C₈	1	4	yes	—
26	C₄×#92418	C₂×C₄	1	4	yes	—
27	C₄×#92418	D₄	1	2	yes	—
28	C₄×#92418	Q₈	1	2	yes	—
29	C₂×C₄²×C₆₁	C₈	1	2	yes	—
30	C₂×C₄²×C₆₁	C₈	1	2	yes	—
31	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
37	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₆₁	—	2	4	no	—
40	C₄²×C₆₁	—	2	4	no	—
41	C₄²×C₆₁	—	2	1	no	—
42	C₄²×C₆₁	—	2	1	no	—
43	C₄²×C₆₁	C₄²	1	2	yes	—
44	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
45	C₄²×C₆₁	C₂×C₈	1	2	yes	—
46	C₄²×C₆₁	M₁₆	1	2	yes	—
47	C₂×#92418	—	2	1	no	—
48	C₂×#92418	—	2	1	no	—
49	C₂×#92418	—	2	1	no	—
50	C₂×#92418	—	2	1	no	—
51	C₂×#92418	C₄²	1	2	yes	—
52	C₂×#92418	C₄⋊C₄	1	2	yes	—
53	C₂×#92418	C₂×C₈	1	2	yes	—
54	C₂×#92418	C₂×C₈	1	2	yes	—
55	C₂×#92418	M₁₆	1	2	yes	—
56	C₂×#92418	C₂²×C₄	1	2	yes	—
57	C₂×#92418	C₂×D₄	1	1	yes	characteristic
58	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
59	C₂×#92418	#41	1	2	yes	—
60	#11758187	—	2	8	no	—
61	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	—	2	2	no	—
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	—	2	2	no	—
67	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
68	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
70	C₂×C₄×C₆₁	M₃₂	1	4	yes	—
71	C₂×C₄×C₆₁	M₃₂	1	4	yes	—
72	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
78	#92418	—	2	2	no	—
79	#92418	—	2	2	no	—
80	#92418	C₄⋊C₈	1	4	yes	—
81	#92418	C₄×D₄	1	2	yes	—
82	#92418	C₄×Q₈	1	2	yes	—
83	#12653	—	61	1	no	maximal, Sylow
84	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
85	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₂²×C₆₁	#431	1	1	yes	characteristic
87	C₂²×C₆₁	C₂×M₃₂	1	2	yes	—
88	C₂²×C₆₁	C₂×M₃₂	1	2	yes	—
89	C₄×C₆₁	—	2	2	no	—
90	C₄×C₆₁	—	2	4	no	—
91	C₄×C₆₁	—	2	4	no	—
92	C₄×C₆₁	#346	1	4	yes	complemented
93	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
94	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
95	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₄×C₆₁	#431	1	1	yes	characteristic
97	C₂×C₄⋊C₁₆	—	61	2	no	—
98	C₂×#346	—	61	2	no	—
99	C₄²×C₈	—	61	1	no	—
100	#1374	—	61	2	no	—
101	C₂×C₆₁	C₄×C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₆₁	C₂×#346	1	1	yes	characteristic
103	C₂×C₆₁	C₄×M₃₂	1	1	yes	characteristic
104	C₂×C₆₁	#2150	1	2	yes	—
105	C₂×C₆₁	#2163	1	2	yes	—
106	C₂²×C₁₆	—	61	4	no	—
107	C₂×C₄×C₈	—	61	1	no	—
108	C₂×C₄×C₈	—	61	1	no	—
109	C₂×C₄×C₈	—	61	1	no	—
110	C₂×C₄×C₈	—	61	1	no	—
111	C₂×C₄×C₈	—	61	1	no	—
112	C₂×C₄×C₈	—	61	1	no	—
113	C₄³	#6923	1	1	yes	characteristic
114	C₄⋊C₁₆	—	61	8	no	—
115	#346	—	122	4	no	—
116	C₆₁	#12653	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₈	—	61	1	no	—
118	C₂²×C₈	—	61	1	no	—
119	C₂²×C₈	—	61	1	no	—
120	C₂²×C₈	—	61	1	no	—
121	C₂×C₄²	#92420	1	2	yes	—
122	C₂×C₄²	#92420	1	2	yes	—
123	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
124	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
125	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
126	C₂×C₁₆	—	61	8	no	—
127	C₂×C₁₆	—	122	4	no	—
128	C₂×C₁₆	—	122	4	no	—
129	C₄×C₈	—	61	4	no	—
130	C₄×C₈	—	61	4	no	—
131	C₄×C₈	—	61	2	no	—
132	C₄×C₈	—	61	2	no	—
133	C₄×C₈	—	122	2	no	—
134	C₄×C₈	—	122	2	no	—
135	C₄×C₈	—	122	2	no	—
136	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
137	C₂²×C₄	C₂×#92420	1	2	yes	—
138	C₂²×C₄	C₂×#92420	1	2	yes	—
139	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
140	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
141	C₂×C₈	—	61	2	no	—
142	C₂×C₈	—	61	1	no	—
143	C₂×C₈	—	61	2	no	—
144	C₂×C₈	—	61	1	no	—
145	C₂×C₈	—	61	2	no	—
146	C₂×C₈	—	61	2	no	—
147	C₂×C₈	—	61	2	no	—
148	C₂×C₈	—	61	2	no	—
149	C₂×C₈	—	61	2	no	—
150	C₂×C₈	—	122	1	no	—
151	C₂×C₈	—	122	1	no	—
152	C₂×C₈	—	122	1	no	—
153	C₂×C₈	—	122	1	no	—
154	C₄²	—	2	4	no	—
155	C₄²	—	2	4	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	D₆₁⋊C₈	1	2	yes	—
159	C₄²	#11758213	1	2	yes	—
160	C₄²	C₄×#6923	1	2	yes	—
161	C₄²	C₄⋊#6923	1	2	yes	—
162	C₁₆	—	122	8	no	—
163	C₂³	C₂²×#92420	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	C₄×#92420	1	2	yes	—
170	C₂×C₄	C₄×#92420	1	2	yes	—
171	C₂×C₄	C₄⋊#92420	1	4	yes	—
172	C₂×C₄	#14519851	1	4	yes	—
173	C₂×C₄	#14519851	1	4	yes	—
174	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
176	C₂×C₄	#14519856	1	1	yes	characteristic
177	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
178	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
179	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
180	C₈	—	61	4	no	—
181	C₈	—	61	2	no	—
182	C₈	—	61	2	no	—
183	C₈	—	122	2	no	—
184	C₈	—	122	2	no	—
185	C₂²	C₂×C₄×#92420	1	1	yes	characteristic
186	C₂²	C₂×C₄⋊#92420	1	1	yes	characteristic
187	C₂²	#16840778	1	1	yes	characteristic
188	C₂²	C₂×#14519851	1	2	yes	—
189	C₂²	C₂×#14519851	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	4	no	—
193	C₄	#16840650	1	4	yes	complemented
194	C₄	C₄×C₄⋊#6923	1	1	yes	characteristic
195	C₄	C₄×#11758213	1	1	yes	characteristic
196	C₄	D₆₁⋊(C₄⋊C₈)	1	1	yes	characteristic
197	C₄	#16840781	1	1	yes	characteristic
198	C₂	C₄×C₄⋊#92420	1	1	yes	characteristic
199	C₂	C₂×#16840650	1	1	yes	characteristic
200	C₂	C₄×#14519851	1	1	yes	characteristic
201	C₂	#21462151	1	2	yes	—
202	C₂	#21462180	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6923
commutator	C₂²×C₆₁	C₂×C₄×C₈
Fitting	C₄³×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂²×#92418	C₂³
hypercenter	C₄³	#6923
hypocenter	C₆₁	#12653
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
6. C₄⋊#11758187	C₄	semidirect
92. C₄×C₆₁	#346	semidirect
116. C₆₁	#12653	semidirect
193. C₄	#16840650	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11758187	2
C₂×#16840650	2
#21460857	2
C₄²×#92418	2
#12653	61