Subgroups — C₂²×C₂²⋊#11758187 #30623856

Order 15616.

1470 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11758187	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊#11758187	C₂	1	12	yes	maximal, direct product factor
3	C₂⁴×#92418	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11758187	—	2	8	no	—
5	C₂²×#11758187	C₂²	1	12	yes	complemented
6	C₂²⋊#11758187	C₂²	1	16	yes	direct product factor
7	C₂³×#92418	—	2	8	no	—
8	C₂³×#92418	C₄	1	6	yes	—
9	C₂³×#92418	C₄	1	1	yes	characteristic
10	C₂³×#92418	C₂²	1	6	yes	—
11	C₂³×#92418	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₆₁	C₄	1	1	yes	characteristic
13	C₂×#11758187	—	2	24	no	—
14	C₂×#11758187	C₂³	1	8	yes	complemented
15	C₂²×#92418	—	2	4	no	—
16	C₂²×#92418	—	2	24	no	—
17	C₂²×#92418	—	2	24	no	—
18	C₂²×#92418	C₂×C₄	1	6	yes	—
19	C₂²×#92418	C₂×C₄	1	12	yes	—
20	C₂²×#92418	C₂×C₄	1	3	yes	—
21	C₂²×#92418	D₄	1	8	yes	—
22	C₂²×#92418	C₂³	1	4	yes	—
23	C₂²×#92418	C₂³	1	3	yes	—
24	C₂⁵×C₆₁	C₈	1	1	yes	characteristic
25	C₂³×C₄×C₆₁	—	2	4	no	—
26	C₂³×C₄×C₆₁	—	2	4	no	—
27	C₂³×C₄×C₆₁	C₈	1	1	yes	characteristic
28	C₂³×C₄×C₆₁	C₈	1	6	yes	—
29	C₂³×C₄×C₆₁	C₂×C₄	1	6	yes	—
30	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂⁴×C₆₁	—	2	4	no	—
32	C₂⁴×C₆₁	—	2	4	no	—
33	C₂⁴×C₆₁	C₁₆	1	8	yes	complemented
34	C₂⁴×C₆₁	C₂×C₈	1	6	yes	—
35	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	—	2	4	no	—
37	C₂²×C₄×C₆₁	—	2	24	no	—
38	C₂²×C₄×C₆₁	—	2	12	no	—
39	C₂²×C₄×C₆₁	—	2	12	no	—
40	C₂²×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
41	C₂²×C₄×C₆₁	C₂×C₈	1	6	yes	—
42	C₂²×C₄×C₆₁	C₂×C₈	1	12	yes	—
43	C₂²×C₄×C₆₁	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₆₁	M₁₆	1	4	yes	—
45	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
47	C₂×#92418	—	2	12	no	—
48	C₂×#92418	—	2	24	no	—
49	C₂×#92418	—	2	8	no	—
50	C₂×#92418	C₂²⋊C₄	1	12	yes	—
51	C₂×#92418	C₂²×C₄	1	4	yes	—
52	C₂×#92418	C₂²×C₄	1	3	yes	—
53	C₂×#92418	C₂×D₄	1	12	yes	—
54	C₂×#92418	C₂⁴	1	1	yes	characteristic
55	#11758187	—	2	8	no	—
56	C₂³×C₆₁	—	2	32	no	—
57	C₂³×C₆₁	—	2	12	no	—
58	C₂³×C₆₁	—	2	12	no	—
59	C₂³×C₆₁	C₂²⋊C₈	1	4	yes	—
60	C₂³×C₆₁	C₂×C₁₆	1	4	yes	—
61	C₂³×C₆₁	C₂×C₁₆	1	24	yes	complemented
62	C₂³×C₆₁	M₃₂	1	4	yes	—
63	C₂³×C₆₁	C₂²×C₈	1	4	yes	—
64	C₂³×C₆₁	C₂²×C₈	1	3	yes	—
65	C₂×C₄×C₆₁	—	2	12	no	—
66	C₂×C₄×C₆₁	—	2	4	no	—
67	C₂×C₄×C₆₁	—	2	4	no	—
68	C₂×C₄×C₆₁	—	2	24	no	—
69	C₂×C₄×C₆₁	C₂²⋊C₈	1	12	yes	—
70	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	6	yes	—
71	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
72	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
73	C₂×C₄×C₆₁	C₂×M₁₆	1	6	yes	—
74	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
75	#92418	—	2	4	no	—
76	#92418	C₂×C₂²⋊C₄	1	6	yes	—
77	#92418	C₂²×D₄	1	2	yes	—
78	C₂²×C₂²⋊C₁₆	—	61	1	no	maximal, Sylow
79	C₂²×C₆₁	—	2	4	no	—
80	C₂²×C₆₁	—	2	48	no	—
81	C₂²×C₆₁	—	2	4	no	—
82	C₂²×C₆₁	C₂²⋊C₁₆	1	16	yes	complemented
83	C₂²×C₆₁	C₂×C₂²⋊C₈	1	6	yes	—
84	C₂²×C₆₁	C₂²×C₁₆	1	6	yes	—
85	C₂²×C₆₁	C₂²×C₁₆	1	8	yes	complemented
86	C₂²×C₆₁	C₂×M₃₂	1	6	yes	—
87	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
88	C₄×C₆₁	—	2	4	no	—
89	C₄×C₆₁	C₂×C₂²⋊C₈	1	6	yes	—
90	C₄×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
91	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
92	C₂⁴×C₈	—	61	1	no	—
93	C₂³×C₁₆	—	61	2	no	—
94	C₂×C₂²⋊C₁₆	—	61	12	no	—
95	C₂×C₆₁	—	2	8	no	—
96	C₂×C₆₁	C₂×C₂²⋊C₁₆	1	12	yes	complemented
97	C₂×C₆₁	C₂²×C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₆₁	C₂³×C₁₆	1	1	yes	characteristic
99	C₂×C₆₁	C₂²×M₃₂	1	1	yes	characteristic
100	C₂⁴×C₄	#6923	1	1	yes	characteristic
101	C₂³×C₈	—	61	1	no	—
102	C₂³×C₈	—	61	6	no	—
103	C₂³×C₈	—	61	6	no	—
104	C₂³×C₈	—	61	1	no	—
105	C₂³×C₈	—	122	8	no	—
106	C₂²×C₁₆	—	61	12	no	—
107	C₂²×C₁₆	—	122	8	no	—
108	C₂²⋊C₁₆	—	61	16	no	—
109	C₆₁	C₂²×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
110	C₂⁵	#92420	1	1	yes	characteristic
111	C₂³×C₄	—	2	4	no	—
112	C₂³×C₄	—	2	4	no	—
113	C₂³×C₄	#92420	1	1	yes	characteristic
114	C₂³×C₄	#92420	1	6	yes	—
115	C₂³×C₄	C₂×#6923	1	6	yes	—
116	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
117	C₂²×C₈	—	61	8	no	—
118	C₂²×C₈	—	61	3	no	—
119	C₂²×C₈	—	61	3	no	—
120	C₂²×C₈	—	61	4	no	—
121	C₂²×C₈	—	61	12	no	—
122	C₂²×C₈	—	61	6	no	—
123	C₂²×C₈	—	122	24	no	—
124	C₂²×C₈	—	122	24	no	—
125	C₂²×C₈	—	122	4	no	—
126	C₂×C₁₆	—	61	8	no	—
127	C₂×C₁₆	—	122	24	no	—
128	C₂⁴	—	2	4	no	—
129	C₂⁴	—	2	4	no	—
130	C₂⁴	#11758187	1	8	yes	complemented
131	C₂⁴	C₂×#92420	1	6	yes	—
132	C₂⁴	C₂×#92420	1	1	yes	characteristic
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	—	2	24	no	—
135	C₂²×C₄	—	2	12	no	—
136	C₂²×C₄	—	2	12	no	—
137	C₂²×C₄	C₂×#92420	1	6	yes	—
138	C₂²×C₄	C₂×#92420	1	12	yes	—
139	C₂²×C₄	C₂×#92420	1	3	yes	—
140	C₂²×C₄	#11758217	1	4	yes	—
141	C₂²×C₄	C₂²⋊#6923	1	4	yes	—
142	C₂²×C₄	C₂²×#6923	1	4	yes	—
143	C₂²×C₄	C₂²×#6923	1	3	yes	—
144	C₂×C₈	—	61	12	no	—
145	C₂×C₈	—	61	12	no	—
146	C₂×C₈	—	61	1	no	—
147	C₂×C₈	—	61	3	no	—
148	C₂×C₈	—	61	4	no	—
149	C₂×C₈	—	122	24	no	—
150	C₂×C₈	—	122	8	no	—
151	C₂×C₈	—	122	12	no	—
152	C₁₆	—	122	8	no	—
153	C₂³	—	2	12	no	—
154	C₂³	—	2	12	no	—
155	C₂³	—	2	32	no	—
156	C₂³	C₂²⋊#92420	1	4	yes	—
157	C₂³	C₂×#11758187	1	4	yes	—
158	C₂³	C₂×#11758187	1	24	yes	complemented
159	C₂³	#14519851	1	4	yes	—
160	C₂³	C₂²×#92420	1	4	yes	—
161	C₂³	C₂²×#92420	1	3	yes	—
162	C₂×C₄	—	2	12	no	—
163	C₂×C₄	—	2	24	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	C₂²⋊#92420	1	12	yes	—
167	C₂×C₄	C₂×C₂²⋊#6923	1	6	yes	—
168	C₂×C₄	C₂²×#92420	1	4	yes	—
169	C₂×C₄	C₂²×#92420	1	3	yes	—
170	C₂×C₄	C₂×#11758217	1	6	yes	—
171	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
172	C₈	—	61	6	no	—
173	C₈	—	61	2	no	—
174	C₈	—	122	4	no	—
175	C₂²	—	2	48	no	—
176	C₂²	—	2	4	no	—
177	C₂²	—	2	4	no	—
178	C₂²	C₂²⋊#11758187	1	16	yes	direct product factor
179	C₂²	C₂×C₂²⋊#92420	1	6	yes	—
180	C₂²	C₂²×#11758187	1	6	yes	—
181	C₂²	C₂²×#11758187	1	8	yes	complemented
182	C₂²	C₂×#14519851	1	6	yes	—
183	C₂²	C₂³×#92420	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	C₂×C₂²⋊#92420	1	6	yes	—
186	C₄	C₂²×C₂²⋊#6923	1	1	yes	characteristic
187	C₄	C₂²×#11758217	1	1	yes	characteristic
188	C₂	—	2	8	no	—
189	C₂	C₂×C₂²⋊#11758187	1	12	yes	direct product factor
190	C₂	C₂²×C₂²⋊#92420	1	1	yes	characteristic
191	C₂	C₂³×#11758187	1	1	yes	characteristic
192	C₂	C₂²×#14519851	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×#6923
commutator	C₂×C₆₁	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂⁴×C₄×C₆₁	C₄
socle	C₂⁴×C₆₁	C₂×C₈
prime radical	C₂⁵×C₆₁	C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	C₂⁴×C₄	#6923
hypocenter	C₆₁	C₂²×C₂²⋊C₁₆
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#11758187	C₂	semidirect
2. C₂×C₂²⋊#11758187	C₂	direct (N × H)
5. C₂²×#11758187	C₂²	semidirect
6. C₂²⋊#11758187	C₂²	direct (N × H)
14. C₂×#11758187	C₂³	semidirect
33. C₂⁴×C₆₁	C₁₆	semidirect
61. C₂³×C₆₁	C₂×C₁₆	semidirect
82. C₂²×C₆₁	C₂²⋊C₁₆	semidirect
85. C₂²×C₆₁	C₂²×C₁₆	semidirect
96. C₂×C₆₁	C₂×C₂²⋊C₁₆	semidirect
109. C₆₁	C₂²×C₂²⋊C₁₆	semidirect
130. C₂⁴	#11758187	semidirect
158. C₂³	C₂×#11758187	semidirect
178. C₂²	C₂²⋊#11758187	direct (N × H)
181. C₂²	C₂²×#11758187	semidirect
189. C₂	C₂×C₂²⋊#11758187	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11758187	2
C₂×C₂²⋊#11758187	2
C₂⁴×#92418	2
C₂²×C₂²⋊C₁₆	61

Subgroups — C22×C22⋊#11758187 #30623856

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊#11758187 #30623856