Subgroups — C₂×C₇×#1371 #13189222

Order 1792.

262 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#335	C₂	1	2	yes	maximal
3	C₇×#1371	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₇×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₇×C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₇×C₄⋊C₈	—	2	2	no	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₄×C₇×C₁₆	C₄	1	2	yes	—
9	C₄×C₇×C₁₆	C₂²	1	2	yes	—
10	C₇×#335	—	2	2	no	—
11	C₇×#335	C₂²	1	4	yes	—
12	C₂×#1371	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₇×C₈	—	4	2	no	—
14	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
16	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
17	C₂×C₇×C₁₆	—	2	2	no	—
18	C₂×C₇×C₁₆	C₈	1	4	yes	complemented
19	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
21	C₇×C₄⋊C₈	—	2	4	no	—
22	C₇×C₄⋊C₈	—	4	2	no	—
23	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₇×C₈	D₄	1	1	yes	characteristic
25	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
26	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₁₆	C₂×C₇	1	1	yes	characteristic
28	C₂×#335	C₂×C₇	1	2	yes	—
29	#1371	C₂×C₇	1	4	yes	direct product factor
30	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₇	Q₁₆	1	1	yes	characteristic
33	C₂×C₇×C₈	—	2	2	no	—
34	C₂×C₇×C₈	—	4	4	no	—
35	C₂×C₇×C₈	—	8	4	no	—
36	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
39	C₂×C₇×C₈	M₁₆	1	2	yes	—
40	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₇	D₈	1	1	yes	characteristic
44	C₄²×C₇	Q₁₆	1	1	yes	characteristic
45	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
46	C₇×C₁₆	—	2	2	no	—
47	C₇×C₁₆	C₂×C₈	1	4	yes	complemented
48	C₂²×C₁₆	C₄×C₇	1	1	yes	characteristic
49	C₂²×C₁₆	C₄×C₇	1	1	yes	characteristic
50	C₂×C₄⋊C₈	—	2	2	no	—
51	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
52	C₄×C₁₆	C₄×C₇	1	2	yes	—
53	C₄×C₁₆	C₂²×C₇	1	2	yes	—
54	#335	—	2	2	no	—
55	#335	C₂²×C₇	1	4	yes	—
56	C₂³×C₇	#107	1	1	yes	characteristic
57	C₂×C₄×C₇	—	2	2	no	—
58	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₇	#107	1	1	yes	characteristic
60	C₂×C₄×C₇	#107	1	1	yes	characteristic
61	C₂×C₄×C₇	#107	1	1	yes	characteristic
62	C₂×C₄×C₇	#108	1	2	yes	—
63	C₂×C₄×C₇	D₁₆	1	2	yes	—
64	C₂×C₄×C₇	Q₃₂	1	2	yes	—
65	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
68	C₇×C₈	—	2	1	no	—
69	C₇×C₈	—	2	1	no	—
70	C₇×C₈	—	8	4	no	—
71	C₇×C₈	C₄⋊C₈	1	2	yes	—
72	C₇×C₈	C₂²×C₈	1	1	yes	characteristic
73	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
74	C₂²×C₈	—	4	2	no	—
75	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
76	C₂²×C₈	C₇×Q₈	1	1	yes	characteristic
77	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	C₇×C₈	1	4	yes	complemented
80	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
81	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
82	C₄⋊C₈	—	2	4	no	—
83	C₄⋊C₈	—	4	2	no	—
84	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
85	C₄×C₈	C₇×D₄	1	1	yes	characteristic
86	C₄×C₈	C₇×Q₈	1	1	yes	characteristic
87	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
88	C₂²×C₇	#335	1	2	yes	—
89	C₂²×C₇	#366	1	2	yes	—
90	C₂²×C₇	#368	1	2	yes	—
91	C₂²×C₇	C₂×#107	1	1	yes	characteristic
92	C₄×C₇	—	2	1	no	—
93	C₄×C₇	—	2	1	no	—
94	C₄×C₇	#335	1	2	yes	—
95	C₄×C₇	#366	1	2	yes	—
96	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₄×C₇	C₂×#108	1	1	yes	characteristic
98	C₄×C₇	C₂×D₁₆	1	1	yes	characteristic
99	C₄×C₇	C₂×Q₃₂	1	1	yes	characteristic
100	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
102	C₂²×C₄	C₇×Q₁₆	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	4	4	no	—
105	C₂×C₈	—	8	4	no	—
106	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
109	C₂×C₈	C₇×M₁₆	1	2	yes	—
110	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
111	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
112	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
113	C₄²	C₇×D₈	1	1	yes	characteristic
114	C₄²	C₇×Q₁₆	1	1	yes	characteristic
115	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
116	C₁₆	—	2	2	no	—
117	C₁₆	C₂×C₇×C₈	1	4	yes	complemented
118	C₂×C₇	#1371	1	4	yes	direct product factor
119	C₂×C₇	C₂×#335	1	1	yes	characteristic
120	C₂×C₇	C₂×#366	1	1	yes	characteristic
121	C₂×C₇	C₂×#368	1	1	yes	characteristic
122	C₂³	C₇×#107	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	C₇×C₄⋊C₈	1	2	yes	—
125	C₂×C₄	C₇×#107	1	1	yes	characteristic
126	C₂×C₄	C₇×#107	1	1	yes	characteristic
127	C₂×C₄	C₇×#107	1	1	yes	characteristic
128	C₂×C₄	C₇×#108	1	2	yes	—
129	C₂×C₄	C₇×D₁₆	1	2	yes	—
130	C₂×C₄	C₇×Q₃₂	1	2	yes	—
131	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
133	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	8	4	no	—
137	C₈	C₇×C₄⋊C₈	1	2	yes	—
138	C₈	C₂²×C₇×C₈	1	1	yes	characteristic
139	C₈	C₂×C₇×M₁₆	1	1	yes	characteristic
140	C₇	C₂×#1371	1	1	yes	characteristic, direct product factor, Sylow
141	C₂²	C₇×#335	1	2	yes	—
142	C₂²	C₇×#366	1	2	yes	—
143	C₂²	C₇×#368	1	2	yes	—
144	C₂²	C₂×C₇×#107	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	C₇×#335	1	2	yes	—
148	C₄	C₇×#366	1	2	yes	—
149	C₄	C₂×C₇×C₄⋊C₈	1	1	yes	characteristic
150	C₄	C₂×C₇×#108	1	1	yes	characteristic
151	C₄	C₂×C₇×D₁₆	1	1	yes	characteristic
152	C₄	C₂×C₇×Q₃₂	1	1	yes	characteristic
153	C₂	C₇×#1371	1	4	yes	direct product factor
154	C₂	C₂×C₇×#335	1	1	yes	characteristic
155	C₂	C₂×C₇×#366	1	1	yes	characteristic
156	C₂	C₂×C₇×#368	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	D₈
commutator	C₈	C₂²×C₇×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle, prime radical	C₂³×C₇	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1371	C₂	direct (N × H)
12. C₂×#1371	C₇	direct (N × H)
18. C₂×C₇×C₁₆	C₈	semidirect
29. #1371	C₂×C₇	direct (N × H)
47. C₇×C₁₆	C₂×C₈	semidirect
79. C₂×C₁₆	C₇×C₈	semidirect
117. C₁₆	C₂×C₇×C₈	semidirect
118. C₂×C₇	#1371	direct (N × H)
140. C₇	C₂×#1371	direct (N × H)
153. C₂	C₇×#1371	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×C₄×C₇×C₁₆	2
C₂×C₇×#335	2
C₇×#1371	2
C₂×#1371	7

Subgroups — C2×C7×#1371 #13189222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1371 #13189222