Subgroups — C₂×C₇×#1372 #13189223

Order 1792.

262 conjugacy classes of subgroups.

The subgroup lattice has 154 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×#1372	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×#1372	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	#1372	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	SD₃₂	1	4	yes	—
64	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₇×C₈	—	2	1	no	—
68	C₇×C₈	—	2	1	no	—
69	C₇×C₈	—	8	4	no	—
70	C₇×C₈	C₄⋊C₈	1	2	yes	—
71	C₇×C₈	C₂²×C₈	1	1	yes	characteristic
72	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
73	C₂²×C₈	—	4	2	no	—
74	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
75	C₂²×C₈	C₇×Q₈	1	1	yes	characteristic
76	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	C₇×C₈	1	4	yes	complemented
79	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
80	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
81	C₄⋊C₈	—	2	4	no	—
82	C₄⋊C₈	—	4	2	no	—
83	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
84	C₄×C₈	C₇×D₄	1	1	yes	characteristic
85	C₄×C₈	C₇×Q₈	1	1	yes	characteristic
86	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
87	C₂²×C₇	#335	1	2	yes	—
88	C₂²×C₇	#367	1	2	yes	—
89	C₂²×C₇	#368	1	2	yes	—
90	C₂²×C₇	C₂×#107	1	1	yes	characteristic
91	C₄×C₇	—	2	1	no	—
92	C₄×C₇	—	2	1	no	—
93	C₄×C₇	#335	1	2	yes	—
94	C₄×C₇	#367	1	2	yes	—
95	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₄×C₇	C₂×#108	1	1	yes	characteristic
97	C₄×C₇	C₂×SD₃₂	1	2	yes	—
98	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
100	C₂²×C₄	C₇×Q₁₆	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	4	4	no	—
103	C₂×C₈	—	8	4	no	—
104	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
107	C₂×C₈	C₇×M₁₆	1	2	yes	—
108	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
109	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
110	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
111	C₄²	C₇×D₈	1	1	yes	characteristic
112	C₄²	C₇×Q₁₆	1	1	yes	characteristic
113	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
114	C₁₆	—	2	2	no	—
115	C₁₆	C₂×C₇×C₈	1	4	yes	complemented
116	C₂×C₇	#1372	1	4	yes	direct product factor
117	C₂×C₇	C₂×#335	1	1	yes	characteristic
118	C₂×C₇	C₂×#367	1	1	yes	characteristic
119	C₂×C₇	C₂×#368	1	1	yes	characteristic
120	C₂³	C₇×#107	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	C₇×C₄⋊C₈	1	2	yes	—
123	C₂×C₄	C₇×#107	1	1	yes	characteristic
124	C₂×C₄	C₇×#107	1	1	yes	characteristic
125	C₂×C₄	C₇×#107	1	1	yes	characteristic
126	C₂×C₄	C₇×#108	1	2	yes	—
127	C₂×C₄	C₇×SD₃₂	1	4	yes	—
128	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
130	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
131	C₈	—	2	1	no	—
132	C₈	—	2	1	no	—
133	C₈	—	8	4	no	—
134	C₈	C₇×C₄⋊C₈	1	2	yes	—
135	C₈	C₂²×C₇×C₈	1	1	yes	characteristic
136	C₈	C₂×C₇×M₁₆	1	1	yes	characteristic
137	C₇	C₂×#1372	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²	C₇×#335	1	2	yes	—
139	C₂²	C₇×#367	1	2	yes	—
140	C₂²	C₇×#368	1	2	yes	—
141	C₂²	C₂×C₇×#107	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	C₇×#335	1	2	yes	—
145	C₄	C₇×#367	1	2	yes	—
146	C₄	C₂×C₇×C₄⋊C₈	1	1	yes	characteristic
147	C₄	C₂×C₇×#108	1	1	yes	characteristic
148	C₄	C₂×C₇×SD₃₂	1	2	yes	—
149	C₂	C₇×#1372	1	4	yes	direct product factor
150	C₂	C₂×C₇×#335	1	1	yes	characteristic
151	C₂	C₂×C₇×#367	1	1	yes	characteristic
152	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×#1372	C₂	direct (N × H)
12. C₂×#1372	C₇	direct (N × H)
18. C₂×C₇×C₁₆	C₈	semidirect
29. #1372	C₂×C₇	direct (N × H)
47. C₇×C₁₆	C₂×C₈	semidirect
78. C₂×C₁₆	C₇×C₈	semidirect
115. C₁₆	C₂×C₇×C₈	semidirect
116. C₂×C₇	#1372	direct (N × H)
137. C₇	C₂×#1372	direct (N × H)
149. C₂	C₇×#1372	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×#1372	2
C₂×#1372	7

Subgroups — C2×C7×#1372 #13189223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1372 #13189223