Subgroups — C₄×#89307 #13268623

Order 1792.

418 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#89307	C₂	1	4	yes	maximal
3	C₄×#6389	C₂	1	1	yes	maximal, characteristic
4	C₄×#6386	C₂	1	1	yes	maximal, characteristic
5	C₂×C₇×M₃₂	C₂²	1	2	yes	—
6	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂×#6389	C₄	1	2	yes	—
8	C₂×#6389	C₂²	1	2	yes	—
9	C₂×#6386	C₂²	1	2	yes	—
10	C₄²×Dic₇	C₄	1	1	yes	characteristic
11	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
12	C₇×#346	C₄	1	2	yes	complemented
13	C₄×C₇×C₁₆	C₄	1	2	yes	complemented
14	#89307	C₄	1	16	yes	direct product factor
15	#89256	C₄	1	2	yes	complemented
16	C₄×C₇⋊C₁₆	C₄	1	2	yes	complemented
17	C₄×#347	—	7	1	no	maximal, Sylow
18	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇	—	2	2	no	—
22	C₂×C₄×Dic₇	C₂×C₄	1	2	yes	—
23	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₇×M₃₂	C₂×C₄	1	8	yes	—
25	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	—
26	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
30	#6389	—	2	4	no	—
31	#6389	C₂×C₄	1	8	yes	—
32	#6386	C₂×C₄	1	8	yes	—
33	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
34	C₄×C₇⋊C₈	—	2	2	no	—
35	C₂×#347	—	7	4	no	—
36	C₄×M₃₂	—	7	1	no	—
37	C₄×M₃₂	D₇	1	1	yes	characteristic
38	C₄×#97	—	7	1	no	—
39	C₂²×C₄×C₇	M₁₆	1	2	yes	—
40	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₇	—	2	2	no	—
42	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
43	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
44	C₂×C₇×C₈	M₁₆	1	4	yes	—
45	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
47	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₇	C₄²	1	1	yes	characteristic
49	C₄²×C₇	C₄²	1	1	yes	characteristic
50	C₄²×C₇	M₁₆	1	2	yes	—
51	C₄×Dic₇	—	2	2	no	—
52	C₄×Dic₇	—	4	4	no	—
53	C₄×Dic₇	C₄²	1	4	yes	—
54	C₂×C₇⋊C₈	—	2	2	no	—
55	C₂×C₇⋊C₈	—	2	2	no	—
56	C₂×C₇⋊C₈	C₄²	1	4	yes	—
57	C₇×C₁₆	C₄²	1	8	yes	complemented
58	C₇⋊C₁₆	C₄²	1	8	yes	complemented
59	C₂×M₃₂	—	7	2	no	—
60	C₂×M₃₂	C₂×D₇	1	2	yes	—
61	C₂×#97	—	7	2	no	—
62	C₂×#97	—	7	2	no	—
63	C₂×C₄×C₈	—	7	1	no	—
64	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
65	C₄³	—	7	1	no	—
66	#347	—	7	16	no	—
67	#346	—	7	2	no	—
68	#346	Dic₇	1	2	yes	complemented
69	C₄×C₁₆	—	7	2	no	—
70	C₄×C₁₆	Dic₇	1	2	yes	complemented
71	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇	—	2	4	no	—
73	C₂×C₄×C₇	#97	1	2	yes	—
74	C₂×C₄×C₇	#97	1	2	yes	—
75	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
76	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
79	C₂×Dic₇	—	2	4	no	—
80	C₂×Dic₇	—	4	4	no	—
81	C₇×C₈	#97	1	2	yes	—
82	C₇×C₈	#97	1	2	yes	—
83	C₇×C₈	C₂×C₄²	1	2	yes	—
84	C₇×C₈	C₂×C₄²	1	2	yes	—
85	C₇⋊C₈	—	2	4	no	—
86	C₂²×C₈	—	7	2	no	—
87	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
88	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
89	C₂×C₄²	—	7	2	no	—
90	C₂×C₄²	—	14	2	no	—
91	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
92	M₃₂	—	7	8	no	—
93	M₃₂	C₄×D₇	1	8	yes	—
94	C₂×C₁₆	—	7	4	no	—
95	C₂×C₁₆	C₂×Dic₇	1	4	yes	—
96	#97	—	7	8	no	—
97	#97	—	14	4	no	—
98	C₄×C₈	—	14	2	no	—
99	C₄×C₈	C₄×D₇	1	1	yes	characteristic
100	C₄×C₈	C₄×D₇	1	1	yes	characteristic
101	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
102	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
103	C₂²×C₇	—	2	2	no	—
104	C₂²×C₇	C₂×#97	1	1	yes	characteristic
105	C₂²×C₇	C₄×M₁₆	1	2	yes	—
106	C₄×C₇	—	2	2	no	—
107	C₄×C₇	#347	1	4	yes	complemented
108	C₄×C₇	C₄³	1	1	yes	characteristic
109	C₄×C₇	C₂×#97	1	1	yes	characteristic
110	C₄×C₇	C₄×M₁₆	1	2	yes	—
111	Dic₇	—	4	4	no	—
112	C₂²×C₄	—	14	2	no	—
113	C₂²×C₄	#1131	1	2	yes	—
114	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
115	C₂×C₈	—	7	4	no	—
116	C₂×C₈	—	14	2	no	—
117	C₂×C₈	—	14	2	no	—
118	C₂×C₈	#1131	1	4	yes	—
119	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
120	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
121	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
122	C₂×C₈	C₂×C₄×D₇	1	4	yes	—
123	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
124	C₄²	—	7	4	no	—
125	C₄²	—	14	2	no	—
126	C₄²	—	28	4	no	—
127	C₄²	#1131	1	2	yes	—
128	C₄²	C₄×Dic₇	1	1	yes	characteristic
129	C₄²	C₄×Dic₇	1	1	yes	characteristic
130	C₁₆	—	7	8	no	—
131	C₁₆	C₄×Dic₇	1	8	yes	complemented
132	C₂×C₇	—	2	2	no	—
133	C₂×C₇	C₄×#97	1	1	yes	characteristic
134	C₂×C₇	C₂×#347	1	2	yes	—
135	C₂³	C₂×#1131	1	1	yes	characteristic
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	14	4	no	—
138	C₂×C₄	—	28	4	no	—
139	C₂×C₄	#6389	1	2	yes	—
140	C₂×C₄	#6389	1	2	yes	—
141	C₂×C₄	C₄²×D₇	1	2	yes	—
142	C₂×C₄	C₂×#1131	1	2	yes	—
143	C₂×C₄	C₂×#1131	1	1	yes	characteristic
144	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
145	C₈	—	14	4	no	—
146	C₈	#6389	1	2	yes	—
147	C₈	#6389	1	2	yes	—
148	C₈	C₄²×D₇	1	2	yes	—
149	C₈	C₂×C₄×Dic₇	1	2	yes	—
150	C₇	C₄×#347	1	1	yes	characteristic, complemented, Sylow
151	C₂²	—	2	2	no	—
152	C₂²	C₄×#1131	1	2	yes	—
153	C₂²	C₂×#6389	1	1	yes	characteristic
154	C₄	—	2	2	no	—
155	C₄	—	28	4	no	—
156	C₄	#89307	1	4	yes	direct product factor
157	C₄	C₄×#1131	1	2	yes	—
158	C₄	C₄²×Dic₇	1	1	yes	characteristic
159	C₄	C₂×#6389	1	1	yes	characteristic
160	C₂	—	2	2	no	—
161	C₂	C₄×#6389	1	1	yes	characteristic
162	C₂	C₂×#89307	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₇
commutator	C₄×C₇	C₄³
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₄×C₇×M₃₂	C₂
socle	C₂²×C₇	C₂×#97
prime radical	C₂³×C₇	C₂×M₁₆
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₄×M₃₂	D₇
hypocenter	C₇	C₄×#347
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₄² ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
12. C₇×#346	C₄	semidirect
13. C₄×C₇×C₁₆	C₄	semidirect
14. #89307	C₄	direct (N × H)
15. #89256	C₄	semidirect
16. C₄×C₇⋊C₁₆	C₄	semidirect
57. C₇×C₁₆	C₄²	semidirect
58. C₇⋊C₁₆	C₄²	semidirect
68. #346	Dic₇	semidirect
70. C₄×C₁₆	Dic₇	semidirect
107. C₄×C₇	#347	semidirect
131. C₁₆	C₄×Dic₇	semidirect
150. C₇	C₄×#347	semidirect
156. C₄	#89307	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×M₃₂	2
C₂×#89307	2
C₄×#6389	2
C₄×#6386	2
C₄×#347	7

Subgroups — C4×#89307 #13268623

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#89307 #13268623