Subgroups — C₂²×C₂²⋊(C₇⋊C₁₆) #13506461

Order 1792.

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	C₂⁴×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₇⋊C₁₆)	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₄×C₇	C₄	1	1	yes	characteristic
5	C₂³×C₇×C₈	—	2	8	no	—
6	C₂³×C₇×C₈	C₄	1	1	yes	characteristic
7	C₂³×C₇×C₈	C₄	1	6	yes	—
8	C₂³×C₇×C₈	C₂²	1	6	yes	—
9	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₇⋊C₁₆	—	2	8	no	—
11	C₂²×C₇⋊C₁₆	C₂²	1	12	yes	complemented
12	C₂²⋊(C₇⋊C₁₆)	C₂²	1	16	yes	direct product factor
13	C₂²×C₂²⋊C₁₆	—	7	1	no	maximal, Sylow
14	C₂⁵×C₇	C₈	1	1	yes	characteristic
15	C₂³×C₄×C₇	—	2	4	no	—
16	C₂³×C₄×C₇	—	2	4	no	—
17	C₂³×C₄×C₇	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₇	C₈	1	6	yes	—
19	C₂³×C₄×C₇	C₂×C₄	1	6	yes	—
20	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	—	2	4	no	—
22	C₂²×C₇×C₈	—	2	24	no	—
23	C₂²×C₇×C₈	—	2	24	no	—
24	C₂²×C₇×C₈	C₂×C₄	1	6	yes	—
25	C₂²×C₇×C₈	C₂×C₄	1	12	yes	—
26	C₂²×C₇×C₈	C₂×C₄	1	3	yes	—
27	C₂²×C₇×C₈	D₄	1	8	yes	—
28	C₂²×C₇×C₈	C₂³	1	4	yes	—
29	C₂²×C₇×C₈	C₂³	1	3	yes	—
30	C₂×C₇⋊C₁₆	—	2	24	no	—
31	C₂×C₇⋊C₁₆	C₂³	1	8	yes	complemented
32	C₂⁴×C₈	D₇	1	1	yes	characteristic
33	C₂³×C₁₆	—	7	2	no	—
34	C₂×C₂²⋊C₁₆	—	7	12	no	—
35	C₂⁴×C₇	—	2	4	no	—
36	C₂⁴×C₇	—	2	4	no	—
37	C₂⁴×C₇	C₁₆	1	8	yes	complemented
38	C₂⁴×C₇	C₂×C₈	1	6	yes	—
39	C₂⁴×C₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₇	—	2	4	no	—
41	C₂²×C₄×C₇	—	2	24	no	—
42	C₂²×C₄×C₇	—	2	12	no	—
43	C₂²×C₄×C₇	—	2	12	no	—
44	C₂²×C₄×C₇	C₂²⋊C₄	1	4	yes	—
45	C₂²×C₄×C₇	C₂×C₈	1	6	yes	—
46	C₂²×C₄×C₇	C₂×C₈	1	12	yes	—
47	C₂²×C₄×C₇	C₂×C₈	1	3	yes	—
48	C₂²×C₄×C₇	M₁₆	1	4	yes	—
49	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
51	C₂×C₇×C₈	—	2	12	no	—
52	C₂×C₇×C₈	—	2	24	no	—
53	C₂×C₇×C₈	—	2	8	no	—
54	C₂×C₇×C₈	C₂²⋊C₄	1	12	yes	—
55	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
56	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
57	C₂×C₇×C₈	C₂×D₄	1	12	yes	—
58	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
59	C₇⋊C₁₆	—	2	8	no	—
60	C₂⁴×C₄	Dic₇	1	1	yes	characteristic
61	C₂³×C₈	—	2	8	no	—
62	C₂³×C₈	Dic₇	1	1	yes	characteristic
63	C₂³×C₈	Dic₇	1	6	yes	—
64	C₂³×C₈	C₂×D₇	1	6	yes	—
65	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
66	C₂²×C₁₆	—	7	12	no	—
67	C₂²×C₁₆	—	14	8	no	—
68	C₂²⋊C₁₆	—	7	16	no	—
69	C₂³×C₇	—	2	32	no	—
70	C₂³×C₇	—	2	12	no	—
71	C₂³×C₇	—	2	12	no	—
72	C₂³×C₇	C₂²⋊C₈	1	4	yes	—
73	C₂³×C₇	C₂×C₁₆	1	4	yes	—
74	C₂³×C₇	C₂×C₁₆	1	24	yes	complemented
75	C₂³×C₇	M₃₂	1	4	yes	—
76	C₂³×C₇	C₂²×C₈	1	4	yes	—
77	C₂³×C₇	C₂²×C₈	1	3	yes	—
78	C₂×C₄×C₇	—	2	12	no	—
79	C₂×C₄×C₇	—	2	4	no	—
80	C₂×C₄×C₇	—	2	4	no	—
81	C₂×C₄×C₇	—	2	24	no	—
82	C₂×C₄×C₇	C₂²⋊C₈	1	12	yes	—
83	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	6	yes	—
84	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
85	C₂×C₄×C₇	C₂²×C₈	1	3	yes	—
86	C₂×C₄×C₇	C₂×M₁₆	1	6	yes	—
87	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
88	C₇×C₈	—	2	4	no	—
89	C₇×C₈	C₂×C₂²⋊C₄	1	6	yes	—
90	C₇×C₈	C₂²×D₄	1	2	yes	—
91	C₂⁵	C₇⋊C₈	1	1	yes	characteristic
92	C₂³×C₄	—	2	4	no	—
93	C₂³×C₄	—	2	4	no	—
94	C₂³×C₄	C₇⋊C₈	1	1	yes	characteristic
95	C₂³×C₄	C₇⋊C₈	1	6	yes	—
96	C₂³×C₄	C₂×Dic₇	1	6	yes	—
97	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
98	C₂²×C₈	—	2	4	no	—
99	C₂²×C₈	—	2	24	no	—
100	C₂²×C₈	—	2	24	no	—
101	C₂²×C₈	C₂×Dic₇	1	6	yes	—
102	C₂²×C₈	C₂×Dic₇	1	12	yes	—
103	C₂²×C₈	C₂×Dic₇	1	3	yes	—
104	C₂²×C₈	C₂²⋊D₇	1	8	yes	—
105	C₂²×C₈	C₂²×D₇	1	4	yes	—
106	C₂²×C₈	C₂²×D₇	1	3	yes	—
107	C₂×C₁₆	—	7	8	no	—
108	C₂×C₁₆	—	14	24	no	—
109	C₂²×C₇	—	2	4	no	—
110	C₂²×C₇	—	2	48	no	—
111	C₂²×C₇	—	2	4	no	—
112	C₂²×C₇	C₂²⋊C₁₆	1	16	yes	complemented
113	C₂²×C₇	C₂×C₂²⋊C₈	1	6	yes	—
114	C₂²×C₇	C₂²×C₁₆	1	6	yes	—
115	C₂²×C₇	C₂²×C₁₆	1	8	yes	complemented
116	C₂²×C₇	C₂×M₃₂	1	6	yes	—
117	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
118	C₄×C₇	—	2	4	no	—
119	C₄×C₇	C₂×C₂²⋊C₈	1	6	yes	—
120	C₄×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
121	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
122	C₂⁴	—	2	4	no	—
123	C₂⁴	—	2	4	no	—
124	C₂⁴	C₇⋊C₁₆	1	8	yes	complemented
125	C₂⁴	C₂×C₇⋊C₈	1	6	yes	—
126	C₂⁴	C₂×C₇⋊C₈	1	1	yes	characteristic
127	C₂²×C₄	—	2	4	no	—
128	C₂²×C₄	—	2	24	no	—
129	C₂²×C₄	—	2	12	no	—
130	C₂²×C₄	—	2	12	no	—
131	C₂²×C₄	C₂×C₇⋊C₈	1	6	yes	—
132	C₂²×C₄	C₂×C₇⋊C₈	1	12	yes	—
133	C₂²×C₄	C₂×C₇⋊C₈	1	3	yes	—
134	C₂²×C₄	#1136	1	4	yes	—
135	C₂²×C₄	C₂²⋊Dic₇	1	4	yes	—
136	C₂²×C₄	C₂²×Dic₇	1	4	yes	—
137	C₂²×C₄	C₂²×Dic₇	1	3	yes	—
138	C₂×C₈	—	2	12	no	—
139	C₂×C₈	—	2	24	no	—
140	C₂×C₈	—	2	8	no	—
141	C₂×C₈	C₂²⋊Dic₇	1	12	yes	—
142	C₂×C₈	C₂²×Dic₇	1	4	yes	—
143	C₂×C₈	C₂²×Dic₇	1	3	yes	—
144	C₂×C₈	C₂×C₂²⋊D₇	1	12	yes	—
145	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
146	C₁₆	—	14	8	no	—
147	C₂×C₇	—	2	8	no	—
148	C₂×C₇	C₂×C₂²⋊C₁₆	1	12	yes	complemented
149	C₂×C₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
150	C₂×C₇	C₂³×C₁₆	1	1	yes	characteristic
151	C₂×C₇	C₂²×M₃₂	1	1	yes	characteristic
152	C₂³	—	2	32	no	—
153	C₂³	—	2	12	no	—
154	C₂³	—	2	12	no	—
155	C₂³	C₂×C₇⋊C₁₆	1	4	yes	—
156	C₂³	C₂×C₇⋊C₁₆	1	24	yes	complemented
157	C₂³	#6386	1	4	yes	—
158	C₂³	C₂²⋊(C₇⋊C₈)	1	4	yes	—
159	C₂³	C₂²×C₇⋊C₈	1	4	yes	—
160	C₂³	C₂²×C₇⋊C₈	1	3	yes	—
161	C₂×C₄	—	2	12	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	24	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	12	yes	—
166	C₂×C₄	C₂²×C₇⋊C₈	1	4	yes	—
167	C₂×C₄	C₂²×C₇⋊C₈	1	3	yes	—
168	C₂×C₄	C₂×#1136	1	6	yes	—
169	C₂×C₄	C₂×C₂²⋊Dic₇	1	6	yes	—
170	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
171	C₈	—	2	4	no	—
172	C₈	C₂×C₂²⋊Dic₇	1	6	yes	—
173	C₈	C₂²×C₂²⋊D₇	1	2	yes	—
174	C₇	C₂²×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
175	C₂²	—	2	4	no	—
176	C₂²	—	2	48	no	—
177	C₂²	—	2	4	no	—
178	C₂²	C₂²⋊(C₇⋊C₁₆)	1	16	yes	direct product factor
179	C₂²	C₂²×C₇⋊C₁₆	1	6	yes	—
180	C₂²	C₂²×C₇⋊C₁₆	1	8	yes	complemented
181	C₂²	C₂×#6386	1	6	yes	—
182	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	6	yes	—
183	C₂²	C₂³×C₇⋊C₈	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	C₂×C₂²⋊(C₇⋊C₈)	1	6	yes	—
186	C₄	C₂²×#1136	1	1	yes	characteristic
187	C₄	C₂²×C₂²⋊Dic₇	1	1	yes	characteristic
188	C₂	—	2	8	no	—
189	C₂	C₂×C₂²⋊(C₇⋊C₁₆)	1	12	yes	direct product factor
190	C₂	C₂³×C₇⋊C₁₆	1	1	yes	characteristic
191	C₂	C₂²×#6386	1	1	yes	characteristic
192	C₂	C₂²×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₇
commutator	C₂×C₇	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂⁴×C₇×C₈	C₂
socle	C₂⁴×C₇	C₂×C₈
prime radical	C₂⁵×C₇	C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂⁴×C₈	D₇
hypocenter	C₇	C₂²×C₂²⋊C₁₆
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₈ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂³×C₇⋊C₁₆	C₂	semidirect
3. C₂×C₂²⋊(C₇⋊C₁₆)	C₂	direct (N × H)
11. C₂²×C₇⋊C₁₆	C₂²	semidirect
12. C₂²⋊(C₇⋊C₁₆)	C₂²	direct (N × H)
31. C₂×C₇⋊C₁₆	C₂³	semidirect
37. C₂⁴×C₇	C₁₆	semidirect
74. C₂³×C₇	C₂×C₁₆	semidirect
112. C₂²×C₇	C₂²⋊C₁₆	semidirect
115. C₂²×C₇	C₂²×C₁₆	semidirect
124. C₂⁴	C₇⋊C₁₆	semidirect
148. C₂×C₇	C₂×C₂²⋊C₁₆	semidirect
156. C₂³	C₂×C₇⋊C₁₆	semidirect
174. C₇	C₂²×C₂²⋊C₁₆	semidirect
178. C₂²	C₂²⋊(C₇⋊C₁₆)	direct (N × H)
180. C₂²	C₂²×C₇⋊C₁₆	semidirect
189. C₂	C₂×C₂²⋊(C₇⋊C₁₆)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₇×C₈	2
C₂³×C₇⋊C₁₆	2
C₂×C₂²⋊(C₇⋊C₁₆)	2
C₂²×C₂²⋊C₁₆	7

Subgroups — C22×C22⋊(C7⋊C16) #13506461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊(C₇⋊C₁₆) #13506461