Subgroups — C₂×C₇×#2864 #13213850

Order 1792.

2078 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	C₇×#2864	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₇×#389	C₂	1	4	yes	maximal
3	C₂×C₇×#387	C₂	1	8	yes	maximal
4	C₂×C₇×#382	C₂	1	2	yes	maximal
5	C₂×C₄³×C₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×C₄⋊C₄	C₂²	1	4	yes	—
7	C₂²×C₇×C₄⋊C₄	C₂²	1	8	yes	—
8	C₂²×C₄²×C₇	C₂²	1	4	yes	—
9	C₂²×C₄²×C₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₇×#389	C₂²	1	32	yes	—
12	C₇×#387	C₂²	1	64	yes	—
13	C₇×#382	C₂²	1	16	yes	—
14	C₂×C₇×#95	C₂²	1	8	yes	—
15	C₂×C₇×#95	C₂²	1	8	yes	—
16	C₄³×C₇	C₂²	1	8	yes	—
17	C₂×#2864	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₇	C₂³	1	8	yes	—
19	C₂³×C₄×C₇	C₂³	1	2	yes	—
20	C₂³×C₄×C₇	C₂³	1	4	yes	—
21	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂×C₇×C₄⋊C₄	—	2	32	no	—
23	C₂×C₇×C₄⋊C₄	—	2	16	no	—
24	C₂×C₇×C₄⋊C₄	C₂³	1	16	yes	—
25	C₂×C₇×C₄⋊C₄	C₂³	1	32	yes	—
26	C₂×C₄²×C₇	—	2	16	no	—
27	C₂×C₄²×C₇	—	2	8	no	—
28	C₂×C₄²×C₇	Q₈	1	8	yes	—
29	C₂×C₄²×C₇	C₂³	1	16	yes	—
30	C₂×C₄²×C₇	C₂³	1	8	yes	—
31	C₂×C₄²×C₇	C₂³	1	4	yes	—
32	C₇×#95	C₂³	1	32	yes	—
33	C₇×#95	C₂³	1	32	yes	—
34	#2864	C₂×C₇	1	16	yes	direct product factor
35	C₂×#389	C₂×C₇	1	4	yes	—
36	C₂×#387	C₂×C₇	1	8	yes	—
37	C₂×#382	C₂×C₇	1	2	yes	—
38	C₂×C₄³	C₂×C₇	1	1	yes	characteristic
39	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
40	C₂²×C₄×C₇	—	2	8	no	—
41	C₂²×C₄×C₇	—	2	8	no	—
42	C₂²×C₄×C₇	—	2	8	no	—
43	C₂²×C₄×C₇	—	2	16	no	—
44	C₂²×C₄×C₇	—	2	32	no	—
45	C₂²×C₄×C₇	C₂×Q₈	1	8	yes	—
46	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
47	C₂²×C₄×C₇	#41	1	16	yes	—
48	C₂²×C₄×C₇	#41	1	8	yes	—
49	C₂²×C₄×C₇	C₂⁴	1	16	yes	—
50	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
51	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
52	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
53	C₇×C₄⋊C₄	—	2	64	no	—
54	C₇×C₄⋊C₄	—	2	32	no	—
55	C₄²×C₇	—	2	32	no	—
56	C₄²×C₇	—	2	16	no	—
57	C₄²×C₇	C₂×Q₈	1	16	yes	—
58	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	—
59	C₂²×C₄⋊C₄	C₂²×C₇	1	8	yes	—
60	C₂²×C₄²	C₂²×C₇	1	4	yes	—
61	C₂²×C₄²	C₂²×C₇	1	2	yes	—
62	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
63	#389	C₂²×C₇	1	32	yes	—
64	#387	C₂²×C₇	1	64	yes	—
65	#382	C₂²×C₇	1	16	yes	—
66	C₂×#95	C₂²×C₇	1	8	yes	—
67	C₂×#95	C₂²×C₇	1	8	yes	—
68	C₄³	C₂²×C₇	1	8	yes	—
69	C₂³×C₇	C₂²×Q₈	1	2	yes	—
70	C₂³×C₇	C₂×#41	1	4	yes	—
71	C₂³×C₇	C₂×#41	1	8	yes	—
72	C₂³×C₇	C₂⁵	1	1	yes	characteristic
73	C₂×C₄×C₇	—	2	32	no	—
74	C₂×C₄×C₇	—	2	16	no	—
75	C₂×C₄×C₇	—	2	8	no	—
76	C₂×C₄×C₇	—	2	8	no	—
77	C₂×C₄×C₇	—	2	8	no	—
78	C₂×C₄×C₇	—	2	16	no	—
79	C₂×C₄×C₇	—	2	32	no	—
80	C₂×C₄×C₇	—	4	32	no	—
81	C₂×C₄×C₇	#125	1	16	yes	—
82	C₂×C₄×C₇	C₂²×Q₈	1	8	yes	—
83	C₂×C₄×C₇	C₂²×Q₈	1	4	yes	—
84	C₂×C₄×C₇	C₂×#41	1	16	yes	—
85	C₂×C₄×C₇	C₂×#41	1	8	yes	—
86	C₂³×C₄	C₂³×C₇	1	8	yes	—
87	C₂³×C₄	C₂³×C₇	1	2	yes	—
88	C₂³×C₄	C₂³×C₇	1	4	yes	—
89	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	2	32	no	—
91	C₂×C₄⋊C₄	—	2	16	no	—
92	C₂×C₄⋊C₄	C₂³×C₇	1	16	yes	—
93	C₂×C₄⋊C₄	C₂³×C₇	1	32	yes	—
94	C₂×C₄²	—	2	16	no	—
95	C₂×C₄²	—	2	8	no	—
96	C₂×C₄²	C₇×Q₈	1	8	yes	—
97	C₂×C₄²	C₂³×C₇	1	16	yes	—
98	C₂×C₄²	C₂³×C₇	1	8	yes	—
99	C₂×C₄²	C₂³×C₇	1	4	yes	—
100	#95	C₂³×C₇	1	32	yes	—
101	#95	C₂³×C₇	1	32	yes	—
102	C₂²×C₇	C₂×#125	1	4	yes	—
103	C₂²×C₇	#529	1	16	yes	—
104	C₂²×C₇	#533	1	8	yes	—
105	C₂²×C₇	C₂³×Q₈	1	1	yes	characteristic
106	C₂²×C₇	C₂²×#41	1	2	yes	—
107	C₂²×C₇	C₂²×#41	1	4	yes	—
108	C₄×C₇	—	2	16	no	—
109	C₄×C₇	—	2	8	no	—
110	C₄×C₇	—	4	16	no	—
111	C₄×C₇	C₂×#125	1	8	yes	—
112	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	—	2	8	no	—
115	C₂²×C₄	—	2	8	no	—
116	C₂²×C₄	—	2	16	no	—
117	C₂²×C₄	—	2	32	no	—
118	C₂²×C₄	C₂×C₇×Q₈	1	8	yes	—
119	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
120	C₂²×C₄	C₇×#41	1	16	yes	—
121	C₂²×C₄	C₇×#41	1	8	yes	—
122	C₂²×C₄	C₂⁴×C₇	1	16	yes	—
123	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
124	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
125	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
126	C₄⋊C₄	—	2	64	no	—
127	C₄⋊C₄	—	2	32	no	—
128	C₄²	—	2	32	no	—
129	C₄²	—	2	16	no	—
130	C₄²	C₂×C₇×Q₈	1	16	yes	—
131	C₂×C₇	#2864	1	8	yes	direct product factor
132	C₂×C₇	C₂²×#125	1	1	yes	characteristic
133	C₂×C₇	C₂×#529	1	4	yes	—
134	C₂×C₇	C₂×#533	1	2	yes	—
135	C₂³	C₂²×C₇×Q₈	1	2	yes	—
136	C₂³	C₂×C₇×#41	1	4	yes	—
137	C₂³	C₂×C₇×#41	1	8	yes	—
138	C₂³	C₂⁵×C₇	1	1	yes	characteristic
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	8	no	—
141	C₂×C₄	—	2	8	no	—
142	C₂×C₄	—	2	32	no	—
143	C₂×C₄	—	2	16	no	—
144	C₂×C₄	—	2	16	no	—
145	C₂×C₄	—	2	32	no	—
146	C₂×C₄	—	4	32	no	—
147	C₂×C₄	C₇×#125	1	16	yes	—
148	C₂×C₄	C₂²×C₇×Q₈	1	8	yes	—
149	C₂×C₄	C₂²×C₇×Q₈	1	4	yes	—
150	C₂×C₄	C₂×C₇×#41	1	16	yes	—
151	C₂×C₄	C₂×C₇×#41	1	8	yes	—
152	C₇	C₂×#2864	1	1	yes	characteristic, direct product factor, Sylow
153	C₂²	C₂×C₇×#125	1	4	yes	—
154	C₂²	C₇×#529	1	16	yes	—
155	C₂²	C₇×#533	1	8	yes	—
156	C₂²	C₂³×C₇×Q₈	1	1	yes	characteristic
157	C₂²	C₂²×C₇×#41	1	2	yes	—
158	C₂²	C₂²×C₇×#41	1	4	yes	—
159	C₄	—	2	16	no	—
160	C₄	—	2	8	no	—
161	C₄	—	4	16	no	—
162	C₄	C₂×C₇×#125	1	8	yes	—
163	C₂	C₇×#2864	1	8	yes	direct product factor
164	C₂	C₂²×C₇×#125	1	1	yes	characteristic
165	C₂	C₂×C₇×#529	1	4	yes	—
166	C₂	C₂×C₇×#533	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₇×#2864	C₂	direct (N × H)
17. C₂×#2864	C₇	direct (N × H)
34. #2864	C₂×C₇	direct (N × H)
131. C₂×C₇	#2864	direct (N × H)
152. C₇	C₂×#2864	direct (N × H)
163. C₂	C₇×#2864	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×#2864	2
C₂×C₇×#389	2
C₂×C₇×#387	2
C₂×C₇×#382	2
C₂×C₄³×C₇	2
C₂×#2864	7

Subgroups — C2×C7×#2864 #13213850

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2864 #13213850