Subgroups — C₂×C₇×#830 #13056241

Order 1344.

404 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	C₂³×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₇×#830	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₇×C₂²⋊C₈	—	3	1	no	maximal, Hall
5	C₂⁴×C₃×C₇	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₄×C₇	—	2	4	no	—
7	C₂²×C₃×C₄×C₇	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₇	C₄	1	2	yes	—
9	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
10	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₃⋊C₈	—	2	4	no	—
12	C₂×C₇×C₃⋊C₈	C₂²	1	4	yes	complemented
13	C₂³×C₄×C₇	D₃	1	1	yes	characteristic
14	C₂²×C₇×C₈	—	3	2	no	—
15	C₇×C₂²⋊C₈	—	3	4	no	—
16	C₂×#830	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂³×C₃×C₇	—	2	2	no	—
18	C₂³×C₃×C₇	—	2	2	no	—
19	C₂³×C₃×C₇	C₈	1	4	yes	complemented
20	C₂³×C₃×C₇	C₂×C₄	1	2	yes	—
21	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₄×C₇	—	2	2	no	—
23	C₂×C₃×C₄×C₇	—	2	4	no	—
24	C₂×C₃×C₄×C₇	—	2	4	no	—
25	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₇	D₄	1	4	yes	—
28	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₇×C₃⋊C₈	—	2	4	no	—
30	C₂⁴×C₇	Dic₃	1	1	yes	characteristic
31	C₂²×C₄×C₇	—	2	4	no	—
32	C₂²×C₄×C₇	Dic₃	1	1	yes	characteristic
33	C₂²×C₄×C₇	Dic₃	1	2	yes	—
34	C₂²×C₄×C₇	C₂×D₃	1	2	yes	—
35	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
36	C₂×C₇×C₈	—	3	4	no	—
37	C₂×C₇×C₈	—	6	4	no	—
38	C₂³×C₃×C₄	C₂×C₇	1	1	yes	characteristic
39	C₂²×C₃⋊C₈	C₂×C₇	1	2	yes	complemented
40	#830	C₂×C₇	1	4	yes	direct product factor
41	C₂²×C₃×C₇	—	2	8	no	—
42	C₂²×C₃×C₇	—	2	2	no	—
43	C₂²×C₃×C₇	—	2	2	no	—
44	C₂²×C₃×C₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₃×C₇	C₂×C₈	1	2	yes	—
46	C₂²×C₃×C₇	C₂×C₈	1	4	yes	complemented
47	C₂²×C₃×C₇	M₁₆	1	2	yes	—
48	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
49	C₃×C₄×C₇	—	2	2	no	—
50	C₃×C₄×C₇	C₂²⋊C₄	1	2	yes	—
51	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
52	C₂×C₂²⋊C₈	—	3	1	no	Sylow
53	C₂³×C₇	—	2	2	no	—
54	C₂³×C₇	—	2	2	no	—
55	C₂³×C₇	C₃⋊C₈	1	4	yes	complemented
56	C₂³×C₇	C₂×Dic₃	1	2	yes	—
57	C₂³×C₇	C₂×Dic₃	1	1	yes	characteristic
58	C₂×C₄×C₇	—	2	2	no	—
59	C₂×C₄×C₇	—	2	4	no	—
60	C₂×C₄×C₇	—	2	4	no	—
61	C₂×C₄×C₇	C₂×Dic₃	1	2	yes	—
62	C₂×C₄×C₇	C₂×Dic₃	1	1	yes	characteristic
63	C₂×C₄×C₇	#67	1	4	yes	—
64	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
65	C₇×C₈	—	6	4	no	—
66	C₂⁴×C₃	C₄×C₇	1	1	yes	characteristic
67	C₂²×C₃×C₄	—	2	4	no	—
68	C₂²×C₃×C₄	C₄×C₇	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₄×C₇	1	2	yes	—
70	C₂²×C₃×C₄	C₂²×C₇	1	2	yes	—
71	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
72	C₂×C₃⋊C₈	—	2	4	no	—
73	C₂×C₃⋊C₈	C₂²×C₇	1	4	yes	complemented
74	C₂×C₃×C₇	—	2	4	no	—
75	C₂×C₃×C₇	C₂²⋊C₈	1	4	yes	complemented
76	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₇	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₃×C₇	C₂×M₁₆	1	1	yes	characteristic
79	C₂³×C₄	D₃×C₇	1	1	yes	characteristic
80	C₂²×C₈	—	3	2	no	—
81	C₂²⋊C₈	—	3	4	no	—
82	C₂²×C₇	—	2	8	no	—
83	C₂²×C₇	—	2	2	no	—
84	C₂²×C₇	—	2	2	no	—
85	C₂²×C₇	C₂×C₃⋊C₈	1	2	yes	—
86	C₂²×C₇	C₂×C₃⋊C₈	1	4	yes	complemented
87	C₂²×C₇	#208	1	2	yes	—
88	C₂²×C₇	#217	1	2	yes	—
89	C₂²×C₇	C₂²×Dic₃	1	1	yes	characteristic
90	C₄×C₇	—	2	2	no	—
91	C₄×C₇	#217	1	2	yes	—
92	C₄×C₇	C₂×#67	1	2	yes	—
93	C₂³×C₃	—	2	2	no	—
94	C₂³×C₃	—	2	2	no	—
95	C₂³×C₃	C₇×C₈	1	4	yes	complemented
96	C₂³×C₃	C₂×C₄×C₇	1	2	yes	—
97	C₂³×C₃	C₂×C₄×C₇	1	1	yes	characteristic
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	4	no	—
100	C₂×C₃×C₄	—	2	4	no	—
101	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
102	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
103	C₂×C₃×C₄	C₇×D₄	1	4	yes	—
104	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
105	C₃⋊C₈	—	2	4	no	—
106	C₃×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Hall
107	C₂⁴	C₇×Dic₃	1	1	yes	characteristic
108	C₂²×C₄	—	2	4	no	—
109	C₂²×C₄	C₇×Dic₃	1	1	yes	characteristic
110	C₂²×C₄	C₇×Dic₃	1	2	yes	—
111	C₂²×C₄	C₂×D₃×C₇	1	2	yes	—
112	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
113	C₂×C₈	—	3	4	no	—
114	C₂×C₈	—	6	4	no	—
115	C₂×C₇	—	2	4	no	—
116	C₂×C₇	#830	1	4	yes	direct product factor
117	C₂×C₇	C₂²×C₃⋊C₈	1	1	yes	characteristic
118	C₂×C₇	C₂×#208	1	1	yes	characteristic
119	C₂×C₇	C₂×#217	1	1	yes	characteristic
120	C₂²×C₃	—	2	8	no	—
121	C₂²×C₃	—	2	2	no	—
122	C₂²×C₃	—	2	2	no	—
123	C₂²×C₃	C₇×C₂²⋊C₄	1	2	yes	—
124	C₂²×C₃	C₂×C₇×C₈	1	2	yes	—
125	C₂²×C₃	C₂×C₇×C₈	1	4	yes	complemented
126	C₂²×C₃	C₇×M₁₆	1	2	yes	—
127	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
128	C₃×C₄	—	2	2	no	—
129	C₃×C₄	C₇×C₂²⋊C₄	1	2	yes	—
130	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
131	C₂³	—	2	2	no	—
132	C₂³	—	2	2	no	—
133	C₂³	C₇×C₃⋊C₈	1	4	yes	complemented
134	C₂³	C₂×C₇×Dic₃	1	2	yes	—
135	C₂³	C₂×C₇×Dic₃	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	C₂×C₇×Dic₃	1	2	yes	—
140	C₂×C₄	C₂×C₇×Dic₃	1	1	yes	characteristic
141	C₂×C₄	C₇×#67	1	4	yes	—
142	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
143	C₈	—	6	4	no	—
144	C₇	C₂×#830	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×C₃	—	2	4	no	—
146	C₂×C₃	C₇×C₂²⋊C₈	1	4	yes	complemented
147	C₂×C₃	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₃	C₂²×C₇×C₈	1	1	yes	characteristic
149	C₂×C₃	C₂×C₇×M₁₆	1	1	yes	characteristic
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	2	8	no	—
153	C₂²	C₂×C₇×C₃⋊C₈	1	2	yes	—
154	C₂²	C₂×C₇×C₃⋊C₈	1	4	yes	complemented
155	C₂²	C₇×#208	1	2	yes	—
156	C₂²	C₇×#217	1	2	yes	—
157	C₂²	C₂²×C₇×Dic₃	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	C₇×#217	1	2	yes	—
160	C₄	C₂×C₇×#67	1	2	yes	—
161	C₃	C₂×C₇×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
162	C₂	—	2	4	no	—
163	C₂	C₇×#830	1	4	yes	direct product factor
164	C₂	C₂²×C₇×C₃⋊C₈	1	1	yes	characteristic
165	C₂	C₂×C₇×#208	1	1	yes	characteristic
166	C₂	C₂×C₇×#217	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₃
commutator	C₂×C₃	C₂²×C₇×C₈
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₂³×C₃×C₄×C₇	C₂
socle	C₂³×C₃×C₇	C₂×C₄
prime radical	C₂⁴×C₃×C₇	C₄
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₂³×C₄×C₇	D₃
hypocenter	C₃	C₂×C₇×C₂²⋊C₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂³×C₄×C₇

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×C₃⋊C₈	C₂	semidirect
3. C₇×#830	C₂	direct (N × H)
12. C₂×C₇×C₃⋊C₈	C₂²	semidirect
16. C₂×#830	C₇	direct (N × H)
19. C₂³×C₃×C₇	C₈	semidirect
39. C₂²×C₃⋊C₈	C₂×C₇	semidirect
40. #830	C₂×C₇	direct (N × H)
46. C₂²×C₃×C₇	C₂×C₈	semidirect
55. C₂³×C₇	C₃⋊C₈	semidirect
73. C₂×C₃⋊C₈	C₂²×C₇	semidirect
75. C₂×C₃×C₇	C₂²⋊C₈	semidirect
86. C₂²×C₇	C₂×C₃⋊C₈	semidirect
95. C₂³×C₃	C₇×C₈	semidirect
106. C₃×C₇	C₂×C₂²⋊C₈	semidirect
116. C₂×C₇	#830	direct (N × H)
125. C₂²×C₃	C₂×C₇×C₈	semidirect
133. C₂³	C₇×C₃⋊C₈	semidirect
144. C₇	C₂×#830	direct (N × H)
146. C₂×C₃	C₇×C₂²⋊C₈	semidirect
154. C₂²	C₂×C₇×C₃⋊C₈	semidirect
161. C₃	C₂×C₇×C₂²⋊C₈	semidirect
163. C₂	C₇×#830	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₇	2
C₂²×C₇×C₃⋊C₈	2
C₇×#830	2
C₂×C₇×C₂²⋊C₈	3
C₂×#830	7

Subgroups — C2×C7×#830 #13056241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#830 #13056241