Subgroups — C₂×C₇×#3556 #13239976

Order 1792.

6338 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	C₇×#3556	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₇×#564	C₂	1	4	yes	maximal, complemented
3	C₂×C₇×#557	C₂	1	12	yes	maximal, complemented
4	C₂×C₇×#541	C₂	1	12	yes	maximal
5	C₂×C₇×#533	C₂	1	3	yes	maximal
6	C₇×#564	C₂²	1	64	yes	complemented
7	C₇×#557	C₂²	1	192	yes	complemented
8	C₇×#541	C₂²	1	192	yes	—
9	C₇×#533	C₂²	1	48	yes	—
10	C₂×C₇×C₄⋊Q₈	C₂²	1	6	yes	—
11	C₂×C₇×C₄⋊Q₈	C₂²	1	12	yes	—
12	C₂×C₇×#126	C₂²	1	16	yes	—
13	C₂×C₇×#125	C₂²	1	6	yes	—
14	C₂×C₇×#125	C₂²	1	12	yes	—
15	C₂×C₇×#125	C₂²	1	48	yes	—
16	C₂×C₇×C₂²⋊Q₈	C₂²	1	12	yes	—
17	C₂×C₄×C₇×Q₈	C₂²	1	12	yes	—
18	C₂×C₄×C₇×Q₈	C₂²	1	24	yes	—
19	C₂×C₇×#117	C₂²	1	6	yes	—
20	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
21	C₂×#3556	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂²×C₇×Q₈	C₂³	1	12	yes	—
23	C₂³×C₄×C₇	C₂³	1	3	yes	—
24	C₇×C₄⋊Q₈	C₂³	1	48	yes	—
25	C₇×C₄⋊Q₈	C₂³	1	96	yes	—
26	C₇×#126	C₂³	1	128	yes	—
27	C₇×#125	C₂³	1	48	yes	—
28	C₇×#125	C₂³	1	96	yes	—
29	C₇×#125	C₂³	1	384	yes	—
30	C₇×C₂²⋊Q₈	C₂³	1	96	yes	—
31	C₄×C₇×Q₈	C₂³	1	96	yes	—
32	C₄×C₇×Q₈	C₂³	1	192	yes	—
33	C₇×#117	C₂³	1	48	yes	—
34	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
35	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
36	C₂×C₇×C₄⋊C₄	C₂³	1	24	yes	—
37	C₂×C₇×C₄⋊C₄	C₂³	1	48	yes	—
38	C₂×C₇×C₂²⋊C₄	C₂³	1	12	yes	—
39	C₂×C₄²×C₇	C₂³	1	3	yes	—
40	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₇	C₂³	1	8	yes	—
42	C₂×C₄²×C₇	C₂³	1	12	yes	—
43	C₂×C₄²×C₇	C₂³	1	16	yes	—
44	#3556	C₂×C₇	1	32	yes	direct product factor
45	C₂×#564	C₂×C₇	1	4	yes	complemented
46	C₂×#557	C₂×C₇	1	12	yes	complemented
47	C₂×#541	C₂×C₇	1	12	yes	—
48	C₂×#533	C₂×C₇	1	3	yes	—
49	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
50	C₂×C₇×Q₈	—	2	48	no	—
51	C₂×C₇×Q₈	C₂⁴	1	48	yes	—
52	C₂²×C₄×C₇	—	2	12	no	—
53	C₂²×C₄×C₇	C₂⁴	1	12	yes	—
54	C₂²×C₄×C₇	C₂⁴	1	24	yes	—
55	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
56	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
57	C₇×C₄⋊C₄	C₂⁴	1	48	yes	—
58	C₇×C₄⋊C₄	C₂⁴	1	48	yes	—
59	C₇×C₄⋊C₄	C₂⁴	1	96	yes	—
60	C₇×C₄⋊C₄	C₂⁴	1	192	yes	—
61	C₇×C₂²⋊C₄	C₂⁴	1	48	yes	—
62	C₄²×C₇	C₂⁴	1	12	yes	—
63	C₄²×C₇	C₂⁴	1	4	yes	—
64	C₄²×C₇	C₂⁴	1	48	yes	—
65	C₄²×C₇	C₂⁴	1	64	yes	—
66	#564	C₂²×C₇	1	64	yes	complemented
67	#557	C₂²×C₇	1	192	yes	complemented
68	#541	C₂²×C₇	1	192	yes	—
69	#533	C₂²×C₇	1	48	yes	—
70	C₂×C₄⋊Q₈	C₂²×C₇	1	6	yes	—
71	C₂×C₄⋊Q₈	C₂²×C₇	1	12	yes	—
72	C₂×#126	C₂²×C₇	1	16	yes	—
73	C₂×#125	C₂²×C₇	1	6	yes	—
74	C₂×#125	C₂²×C₇	1	12	yes	—
75	C₂×#125	C₂²×C₇	1	48	yes	—
76	C₂×C₂²⋊Q₈	C₂²×C₇	1	12	yes	—
77	C₂×C₄×Q₈	C₂²×C₇	1	12	yes	—
78	C₂×C₄×Q₈	C₂²×C₇	1	24	yes	—
79	C₂×#117	C₂²×C₇	1	6	yes	—
80	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
81	C₂³×C₇	—	2	6	no	—
82	C₂³×C₇	C₂⁵	1	2	yes	—
83	C₂³×C₇	C₂⁵	1	1	yes	characteristic
84	C₇×Q₈	—	2	96	no	—
85	C₂×C₄×C₇	—	2	6	no	—
86	C₂×C₄×C₇	—	2	48	no	—
87	C₂×C₄×C₇	—	2	24	no	—
88	C₂×C₄×C₇	2₋¹⁺⁴	1	12	yes	—
89	C₂×C₄×C₇	C₂⁵	1	48	yes	—
90	C₂×C₄×C₇	C₂⁵	1	6	yes	—
91	C₂×C₄×C₇	C₂⁵	1	6	yes	—
92	C₂²×Q₈	C₂³×C₇	1	12	yes	—
93	C₂³×C₄	C₂³×C₇	1	3	yes	—
94	C₄⋊Q₈	C₂³×C₇	1	48	yes	—
95	C₄⋊Q₈	C₂³×C₇	1	96	yes	—
96	#126	C₂³×C₇	1	128	yes	—
97	#125	C₂³×C₇	1	48	yes	—
98	#125	C₂³×C₇	1	96	yes	—
99	#125	C₂³×C₇	1	384	yes	—
100	C₂²⋊Q₈	C₂³×C₇	1	96	yes	—
101	C₄×Q₈	C₂³×C₇	1	96	yes	—
102	C₄×Q₈	C₂³×C₇	1	192	yes	—
103	#117	C₂³×C₇	1	48	yes	—
104	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
105	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
106	C₂×C₄⋊C₄	C₂³×C₇	1	24	yes	—
107	C₂×C₄⋊C₄	C₂³×C₇	1	48	yes	—
108	C₂×C₂²⋊C₄	C₂³×C₇	1	12	yes	—
109	C₂×C₄²	C₂³×C₇	1	12	yes	—
110	C₂×C₄²	C₂³×C₇	1	3	yes	—
111	C₂×C₄²	C₂³×C₇	1	16	yes	—
112	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
113	C₂×C₄²	C₂³×C₇	1	8	yes	—
114	C₂²×C₇	—	2	6	no	—
115	C₂²×C₇	—	4	4	no	—
116	C₂²×C₇	C₂×2₋¹⁺⁴	1	6	yes	—
117	C₂²×C₇	C₂⁶	1	1	yes	characteristic
118	C₄×C₇	—	2	6	no	—
119	C₄×C₇	—	2	48	no	—
120	C₄×C₇	C₂×2₋¹⁺⁴	1	12	yes	—
121	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
122	C₂×Q₈	—	2	48	no	—
123	C₂×Q₈	C₂⁴×C₇	1	48	yes	—
124	C₂²×C₄	—	2	12	no	—
125	C₂²×C₄	C₂⁴×C₇	1	24	yes	—
126	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
127	C₂²×C₄	C₂⁴×C₇	1	12	yes	—
128	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
129	C₄⋊C₄	C₂⁴×C₇	1	48	yes	—
130	C₄⋊C₄	C₂⁴×C₇	1	48	yes	—
131	C₄⋊C₄	C₂⁴×C₇	1	96	yes	—
132	C₄⋊C₄	C₂⁴×C₇	1	192	yes	—
133	C₂²⋊C₄	C₂⁴×C₇	1	48	yes	—
134	C₄²	C₂⁴×C₇	1	12	yes	—
135	C₄²	C₂⁴×C₇	1	4	yes	—
136	C₄²	C₂⁴×C₇	1	48	yes	—
137	C₄²	C₂⁴×C₇	1	64	yes	—
138	C₂×C₇	—	4	2	no	—
139	C₂×C₇	#3556	1	4	yes	direct product factor
140	C₂×C₇	C₂²×2₋¹⁺⁴	1	3	yes	—
141	C₂³	—	2	6	no	—
142	C₂³	C₂⁵×C₇	1	2	yes	—
143	C₂³	C₂⁵×C₇	1	1	yes	characteristic
144	Q₈	—	2	96	no	—
145	C₂×C₄	—	2	6	no	—
146	C₂×C₄	—	2	48	no	—
147	C₂×C₄	—	2	24	no	—
148	C₂×C₄	C₇×2₋¹⁺⁴	1	12	yes	—
149	C₂×C₄	C₂⁵×C₇	1	48	yes	—
150	C₂×C₄	C₂⁵×C₇	1	6	yes	—
151	C₂×C₄	C₂⁵×C₇	1	6	yes	—
152	C₇	C₂×#3556	1	1	yes	characteristic, direct product factor, Sylow
153	C₂²	—	2	6	no	—
154	C₂²	—	4	4	no	—
155	C₂²	C₂×C₇×2₋¹⁺⁴	1	6	yes	—
156	C₂²	C₂⁶×C₇	1	1	yes	characteristic
157	C₄	—	2	6	no	—
158	C₄	—	2	48	no	—
159	C₄	C₂×C₇×2₋¹⁺⁴	1	12	yes	—
160	C₂	—	4	2	no	—
161	C₂	C₇×#3556	1	4	yes	direct product factor
162	C₂	C₂²×C₇×2₋¹⁺⁴	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₇
prime radical	C₂⁴×C₇	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3556	C₂	direct (N × H)
2. C₂×C₇×#564	C₂	semidirect
3. C₂×C₇×#557	C₂	semidirect
6. C₇×#564	C₂²	semidirect
7. C₇×#557	C₂²	semidirect
21. C₂×#3556	C₇	direct (N × H)
44. #3556	C₂×C₇	direct (N × H)
45. C₂×#564	C₂×C₇	semidirect
46. C₂×#557	C₂×C₇	semidirect
66. #564	C₂²×C₇	semidirect
67. #557	C₂²×C₇	semidirect
139. C₂×C₇	#3556	direct (N × H)
152. C₇	C₂×#3556	direct (N × H)
161. C₂	C₇×#3556	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#3556	2
C₂×C₇×#564	2
C₂×C₇×#557	2
C₂×C₇×#541	2
C₂×C₇×#533	2
C₂×#3556	7

Subgroups — C2×C7×#3556 #13239976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3556 #13239976