Subgroups — C₄²×C₄⋊Dic₇ #13274875

Order 1792.

2478 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄³×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊Dic₇	C₂	1	6	yes	maximal
3	C₂×C₄²×Dic₇	C₂	1	2	yes	maximal
4	C₄×C₇⋊#95	C₂	1	6	yes	maximal
5	C₂²×C₄²×C₇	C₂²	1	3	yes	—
6	C₂²×C₄²×C₇	C₂²	1	3	yes	—
7	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₇	C₂²	1	4	yes	—
9	C₂²×C₄×Dic₇	C₂²	1	12	yes	—
10	C₄³×C₇	C₄	1	8	yes	complemented
11	C₂×C₇⋊#95	C₂²	1	12	yes	—
12	C₄×C₄⋊Dic₇	C₄	1	48	yes	direct product factor
13	C₄²×Dic₇	—	2	8	no	—
14	C₄²×C₄⋊C₄	—	7	1	no	maximal, Sylow
15	C₂³×C₄×C₇	C₂³	1	3	yes	—
16	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
17	C₂³×C₄×C₇	C₂³	1	3	yes	—
18	C₂³×Dic₇	C₂³	1	8	yes	—
19	C₂×C₄²×C₇	—	2	12	no	—
20	C₂×C₄²×C₇	C₂×C₄	1	24	yes	—
21	C₂×C₄²×C₇	C₂×C₄	1	12	yes	—
22	C₂×C₄²×C₇	C₂×C₄	1	12	yes	—
23	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₇	D₄	1	4	yes	—
25	C₂×C₄²×C₇	Q₈	1	4	yes	—
26	C₂×C₄⋊Dic₇	C₂×C₄	1	48	yes	—
27	C₂×C₄×Dic₇	—	2	24	no	—
28	C₂×C₄×Dic₇	—	2	24	no	—
29	C₂×C₄×Dic₇	C₂×C₄	1	48	yes	—
30	C₇⋊#95	C₂×C₄	1	48	yes	—
31	C₂×C₄×C₄⋊C₄	—	7	6	no	—
32	C₂×C₄³	—	7	2	no	—
33	C₂×C₄³	D₇	1	1	yes	characteristic
34	C₄×#95	—	7	6	no	—
35	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
36	C₂²×C₄×C₇	—	2	12	no	—
37	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
38	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
39	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
40	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
42	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
43	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
44	C₂²×C₄×C₇	C₂×Q₈	1	6	yes	—
45	C₂²×C₄×C₇	#41	1	12	yes	—
46	C₂²×Dic₇	—	2	8	no	—
47	C₂²×Dic₇	—	2	24	no	—
48	C₂²×Dic₇	C₂²×C₄	1	48	yes	—
49	C₄²×C₇	—	2	24	no	—
50	C₄²×C₇	C₄²	1	48	yes	complemented
51	C₄²×C₇	C₄⋊C₄	1	16	yes	complemented
52	C₄⋊Dic₇	C₄²	1	64	yes	direct product factor
53	C₄×Dic₇	—	2	96	no	—
54	C₂²×C₄⋊C₄	—	7	4	no	—
55	C₂²×C₄²	—	7	12	no	—
56	C₂²×C₄²	C₂×D₇	1	3	yes	—
57	C₂²×C₄²	C₂×D₇	1	3	yes	—
58	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
59	C₄×C₄⋊C₄	—	7	48	no	—
60	C₂×#95	—	7	12	no	—
61	C₄³	—	14	8	no	—
62	C₄³	Dic₇	1	8	yes	complemented
63	C₂³×C₇	C₂³×C₄	1	6	yes	—
64	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
65	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
66	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
67	C₂³×C₇	C₂×#41	1	6	yes	—
68	C₂×C₄×C₇	—	2	12	no	—
69	C₂×C₄×C₇	—	2	24	no	—
70	C₂×C₄×C₇	C₂×C₄²	1	16	yes	—
71	C₂×C₄×C₇	C₂×C₄²	1	12	yes	—
72	C₂×C₄×C₇	C₂×C₄²	1	12	yes	—
73	C₂×C₄×C₇	C₂×C₄²	1	12	yes	—
74	C₂×C₄×C₇	C₂×C₄⋊C₄	1	12	yes	—
75	C₂×C₄×C₇	#117	1	12	yes	—
76	C₂×C₄×C₇	C₄×D₄	1	24	yes	—
77	C₂×C₄×C₇	C₄×Q₈	1	24	yes	—
78	C₂×Dic₇	—	2	48	no	—
79	C₂×Dic₇	—	2	48	no	—
80	C₂×Dic₇	C₂×C₄²	1	32	yes	—
81	C₂³×C₄	—	7	8	no	—
82	C₂³×C₄	C₂²×D₇	1	3	yes	—
83	C₂³×C₄	C₂²×D₇	1	3	yes	—
84	C₂³×C₄	C₂²×D₇	1	1	yes	characteristic
85	C₂×C₄⋊C₄	—	7	48	no	—
86	C₂×C₄²	—	2	12	no	—
87	C₂×C₄²	—	7	48	no	—
88	C₂×C₄²	—	14	24	no	—
89	C₂×C₄²	—	14	24	no	—
90	C₂×C₄²	Dic₁₄	1	4	yes	—
91	C₂×C₄²	C₄×D₇	1	24	yes	—
92	C₂×C₄²	D₂₈	1	4	yes	—
93	C₂×C₄²	C₂×Dic₇	1	12	yes	—
94	C₂×C₄²	C₂×Dic₇	1	12	yes	—
95	C₂×C₄²	C₂×Dic₇	1	4	yes	—
96	#95	—	7	48	no	—
97	C₂²×C₇	C₂²×C₄²	1	4	yes	—
98	C₂²×C₇	C₂²×C₄²	1	3	yes	—
99	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₇	C₂×#117	1	3	yes	—
101	C₂²×C₇	C₂×C₄×D₄	1	6	yes	—
102	C₂²×C₇	C₂×C₄×Q₈	1	6	yes	—
103	C₂²×C₇	C₄×#41	1	12	yes	—
104	C₄×C₇	—	2	12	no	—
105	C₄×C₇	C₄³	1	8	yes	complemented
106	C₄×C₇	C₄×C₄⋊C₄	1	24	yes	complemented
107	Dic₇	—	2	32	no	—
108	C₂⁴	C₂³×D₇	1	1	yes	characteristic
109	C₂²×C₄	—	2	12	no	—
110	C₂²×C₄	—	7	48	no	—
111	C₂²×C₄	—	14	8	no	—
112	C₂²×C₄	—	14	24	no	—
113	C₂²×C₄	C₂×Dic₁₄	1	6	yes	—
114	C₂²×C₄	C₂×C₄×D₇	1	12	yes	—
115	C₂²×C₄	C₂×C₄×D₇	1	12	yes	—
116	C₂²×C₄	C₂×C₄×D₇	1	12	yes	—
117	C₂²×C₄	C₂×D₂₈	1	6	yes	—
118	C₂²×C₄	#1157	1	12	yes	—
119	C₂²×C₄	C₂²×Dic₇	1	6	yes	—
120	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
121	C₂²×C₄	C₂²×Dic₇	1	6	yes	—
122	C₄⋊C₄	—	7	64	no	—
123	C₄²	—	2	24	no	—
124	C₄²	—	14	96	no	—
125	C₄²	C₄×Dic₇	1	48	yes	complemented
126	C₄²	C₄⋊Dic₇	1	16	yes	direct product factor
127	C₂×C₇	C₂×C₄³	1	1	yes	characteristic
128	C₂×C₇	C₂×C₄×C₄⋊C₄	1	3	yes	—
129	C₂×C₇	C₄×#117	1	3	yes	—
130	C₂×C₇	C₄²×D₄	1	4	yes	—
131	C₂×C₇	C₄²×Q₈	1	4	yes	—
132	C₂³	C₂²×Dic₁₄	1	1	yes	characteristic
133	C₂³	C₂²×C₄×D₇	1	6	yes	—
134	C₂³	C₂²×D₂₈	1	1	yes	characteristic
135	C₂³	C₂×#1157	1	6	yes	—
136	C₂³	C₂³×Dic₇	1	1	yes	characteristic
137	C₂×C₄	—	2	24	no	—
138	C₂×C₄	—	2	12	no	—
139	C₂×C₄	—	7	32	no	—
140	C₂×C₄	—	14	48	no	—
141	C₂×C₄	—	14	48	no	—
142	C₂×C₄	C₄×Dic₁₄	1	24	yes	—
143	C₂×C₄	C₄²×D₇	1	16	yes	—
144	C₂×C₄	C₄×D₂₈	1	24	yes	—
145	C₂×C₄	C₂×C₄×Dic₇	1	12	yes	—
146	C₂×C₄	C₂×C₄×Dic₇	1	12	yes	—
147	C₂×C₄	C₂×C₄×Dic₇	1	12	yes	—
148	C₂×C₄	C₂×C₄⋊Dic₇	1	12	yes	—
149	C₂×C₄	#6489	1	12	yes	—
150	C₇	C₄²×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
151	C₂²	C₂×C₄×Dic₁₄	1	6	yes	—
152	C₂²	C₂×C₄²×D₇	1	4	yes	—
153	C₂²	C₂×C₄×D₂₈	1	6	yes	—
154	C₂²	C₄×#1157	1	12	yes	—
155	C₂²	C₂²×C₄×Dic₇	1	3	yes	—
156	C₂²	C₂²×C₄⋊Dic₇	1	1	yes	characteristic
157	C₂²	C₂×#6489	1	3	yes	—
158	C₄	—	2	12	no	—
159	C₄	—	14	32	no	—
160	C₄	C₄²×Dic₇	1	8	yes	complemented
161	C₄	C₄×C₄⋊Dic₇	1	24	yes	direct product factor
162	C₂	C₄²×D₂₈	1	4	yes	—
163	C₂	C₄²×Dic₁₄	1	4	yes	—
164	C₂	C₂×C₄²×Dic₇	1	1	yes	characteristic
165	C₂	C₂×C₄×C₄⋊Dic₇	1	3	yes	—
166	C₂	C₄×#6489	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₇
commutator	C₂×C₇	C₂×C₄³
Frattini	C₂⁴	C₂³×D₇
Fitting	C₂×C₄³×C₇	C₂
socle, prime radical, cosocle	C₂⁴×C₇	C₂⁴
hypercenter	C₂×C₄³	D₇
hypocenter	C₇	C₄²×C₄⋊C₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
10. C₄³×C₇	C₄	semidirect
12. C₄×C₄⋊Dic₇	C₄	direct (N × H)
50. C₄²×C₇	C₄²	semidirect
51. C₄²×C₇	C₄⋊C₄	semidirect
52. C₄⋊Dic₇	C₄²	direct (N × H)
62. C₄³	Dic₇	semidirect
105. C₄×C₇	C₄³	semidirect
106. C₄×C₇	C₄×C₄⋊C₄	semidirect
125. C₄²	C₄×Dic₇	semidirect
126. C₄²	C₄⋊Dic₇	direct (N × H)
150. C₇	C₄²×C₄⋊C₄	semidirect
160. C₄	C₄²×Dic₇	semidirect
161. C₄	C₄×C₄⋊Dic₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄³×C₇	2
C₂×C₄×C₄⋊Dic₇	2
C₂×C₄²×Dic₇	2
C₄×C₇⋊#95	2
C₄²×C₄⋊C₄	7

Subgroups — C42×C4⋊Dic7 #13274875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₄⋊Dic₇ #13274875