Subgroups — C₂×C₃×C₇×#103 #13056926

Order 1344.

324 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×C₇×#103	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₇×#103	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₇×Q₈	—	2	2	no	—
7	C₂×C₃×C₇×Q₈	—	2	1	no	—
8	C₂×C₃×C₇×Q₈	C₄	1	4	yes	complemented
9	C₂×C₃×C₇×Q₈	C₂²	1	2	yes	—
10	C₂²×C₃×C₄×C₇	—	2	1	no	—
11	C₂²×C₃×C₄×C₇	—	2	1	no	—
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₇×C₈	—	2	2	no	—
14	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
15	C₃×C₇×C₄⋊C₄	—	2	1	no	—
16	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂²×C₇×Q₈	C₂×C₃	1	1	yes	characteristic
18	C₂²×C₇×C₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
20	C₇×#103	C₂×C₃	1	4	yes	direct product factor
21	C₂×C₃×#103	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂³×C₃×C₇	D₄	1	1	yes	characteristic
23	C₃×C₇×Q₈	—	2	4	no	—
24	C₃×C₇×Q₈	—	2	2	no	—
25	C₃×C₇×Q₈	C₂×C₄	1	4	yes	complemented
26	C₂×C₃×C₄×C₇	—	2	4	no	—
27	C₂×C₃×C₄×C₇	—	2	2	no	—
28	C₂×C₃×C₄×C₇	—	2	2	no	—
29	C₂×C₃×C₄×C₇	—	4	2	no	—
30	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
33	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
34	C₃×C₇×C₈	—	2	2	no	—
35	C₂×C₇×Q₈	—	2	2	no	—
36	C₂×C₇×Q₈	—	2	1	no	—
37	C₂×C₇×Q₈	C₃×C₄	1	4	yes	complemented
38	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
39	C₂²×C₄×C₇	—	2	1	no	—
40	C₂²×C₄×C₇	—	2	1	no	—
41	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₇×C₈	—	2	2	no	—
43	C₂×C₇×C₈	C₂²×C₃	1	2	yes	—
44	C₇×C₄⋊C₄	—	2	1	no	—
45	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
46	C₂²×C₃×Q₈	C₂×C₇	1	1	yes	characteristic
47	C₂²×C₃×C₈	C₂×C₇	1	1	yes	characteristic
48	C₂×C₃×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
49	C₃×#103	C₂×C₇	1	4	yes	direct product factor
50	C₂²×C₃×C₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₃×C₇	SD₁₆	1	2	yes	—
52	C₂²×C₃×C₇	Q₁₆	1	2	yes	—
53	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
54	C₃×C₄×C₇	—	2	4	no	—
55	C₃×C₄×C₇	—	4	2	no	—
56	C₃×C₄×C₇	C₂²⋊C₄	1	2	yes	—
57	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
58	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
59	C₂×#103	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
60	C₂³×C₇	C₃×D₄	1	1	yes	characteristic
61	C₇×Q₈	—	2	4	no	—
62	C₇×Q₈	—	2	2	no	—
63	C₇×Q₈	C₂×C₃×C₄	1	4	yes	complemented
64	C₂×C₄×C₇	—	2	4	no	—
65	C₂×C₄×C₇	—	2	2	no	—
66	C₂×C₄×C₇	—	2	2	no	—
67	C₂×C₄×C₇	—	4	2	no	—
68	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
69	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
71	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
72	C₇×C₈	—	2	2	no	—
73	C₂×C₃×Q₈	—	2	2	no	—
74	C₂×C₃×Q₈	—	2	1	no	—
75	C₂×C₃×Q₈	C₄×C₇	1	4	yes	complemented
76	C₂×C₃×Q₈	C₂²×C₇	1	2	yes	—
77	C₂²×C₃×C₄	—	2	1	no	—
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	C₂²×C₇	1	2	yes	—
82	C₃×C₄⋊C₄	—	2	1	no	—
83	C₃×C₄⋊C₄	C₂²×C₇	1	2	yes	—
84	C₂×C₃×C₇	#103	1	4	yes	direct product factor
85	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃×C₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₃×C₇	C₂×Q₁₆	1	1	yes	characteristic
88	C₂²×Q₈	C₂×C₃×C₇	1	1	yes	characteristic
89	C₂²×C₈	C₂×C₃×C₇	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂×C₃×C₇	1	1	yes	characteristic
91	#103	C₂×C₃×C₇	1	4	yes	direct product factor
92	C₂²×C₇	C₃×C₂²⋊C₄	1	2	yes	—
93	C₂²×C₇	C₃×SD₁₆	1	2	yes	—
94	C₂²×C₇	C₃×Q₁₆	1	2	yes	—
95	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
96	C₄×C₇	—	2	4	no	—
97	C₄×C₇	—	4	2	no	—
98	C₄×C₇	C₃×C₂²⋊C₄	1	2	yes	—
99	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
100	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
101	C₂³×C₃	C₇×D₄	1	1	yes	characteristic
102	C₃×Q₈	—	2	4	no	—
103	C₃×Q₈	—	2	2	no	—
104	C₃×Q₈	C₂×C₄×C₇	1	4	yes	complemented
105	C₂×C₃×C₄	—	2	4	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	4	2	no	—
109	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
110	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
111	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
112	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
113	C₃×C₈	—	2	2	no	—
114	C₃×C₇	C₂×#103	1	1	yes	characteristic, direct product factor, Hall
115	C₂×Q₈	—	2	2	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	C₃×C₄×C₇	1	4	yes	complemented
118	C₂×Q₈	C₂²×C₃×C₇	1	2	yes	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	C₂²×C₃×C₇	1	2	yes	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	C₂²×C₃×C₇	1	2	yes	—
126	C₂×C₇	C₃×#103	1	4	yes	direct product factor
127	C₂×C₇	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₇	C₂×C₃×SD₁₆	1	1	yes	characteristic
129	C₂×C₇	C₂×C₃×Q₁₆	1	1	yes	characteristic
130	C₂²×C₃	C₇×C₂²⋊C₄	1	2	yes	—
131	C₂²×C₃	C₇×SD₁₆	1	2	yes	—
132	C₂²×C₃	C₇×Q₁₆	1	2	yes	—
133	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
134	C₃×C₄	—	2	4	no	—
135	C₃×C₄	—	4	2	no	—
136	C₃×C₄	C₇×C₂²⋊C₄	1	2	yes	—
137	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
138	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
139	C₂³	C₃×C₇×D₄	1	1	yes	characteristic
140	Q₈	—	2	4	no	—
141	Q₈	—	2	2	no	—
142	Q₈	C₂×C₃×C₄×C₇	1	4	yes	complemented
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	C₂×C₃×C₄×C₇	1	2	yes	—
148	C₂×C₄	C₃×C₇×D₄	1	1	yes	characteristic
149	C₂×C₄	C₃×C₇×D₄	1	2	yes	—
150	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₇	C₂×C₃×#103	1	1	yes	characteristic, direct product factor, Sylow
153	C₂×C₃	C₇×#103	1	4	yes	direct product factor
154	C₂×C₃	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₃	C₂×C₇×SD₁₆	1	1	yes	characteristic
156	C₂×C₃	C₂×C₇×Q₁₆	1	1	yes	characteristic
157	C₂²	C₃×C₇×C₂²⋊C₄	1	2	yes	—
158	C₂²	C₃×C₇×SD₁₆	1	2	yes	—
159	C₂²	C₃×C₇×Q₁₆	1	2	yes	—
160	C₂²	C₂×C₃×C₇×D₄	1	1	yes	characteristic
161	C₄	—	2	4	no	—
162	C₄	—	4	2	no	—
163	C₄	C₃×C₇×C₂²⋊C₄	1	2	yes	—
164	C₄	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
165	C₄	C₂×C₃×C₇×D₄	1	1	yes	characteristic
166	C₃	C₂×C₇×#103	1	1	yes	characteristic, direct product factor, Sylow
167	C₂	C₃×C₇×#103	1	4	yes	direct product factor
168	C₂	C₂×C₃×C₇×C₂²⋊C₄	1	1	yes	characteristic
169	C₂	C₂×C₃×C₇×SD₁₆	1	1	yes	characteristic
170	C₂	C₂×C₃×C₇×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₇	D₄
commutator	C₄	C₂²×C₃×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₇ ◁ C₂²×C₃×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×C₇×#103	C₂	direct (N × H)
5. C₂×C₇×#103	C₃	direct (N × H)
8. C₂×C₃×C₇×Q₈	C₄	semidirect
20. C₇×#103	C₂×C₃	direct (N × H)
21. C₂×C₃×#103	C₇	direct (N × H)
25. C₃×C₇×Q₈	C₂×C₄	semidirect
37. C₂×C₇×Q₈	C₃×C₄	semidirect
49. C₃×#103	C₂×C₇	direct (N × H)
59. C₂×#103	C₃×C₇	direct (N × H)
63. C₇×Q₈	C₂×C₃×C₄	semidirect
75. C₂×C₃×Q₈	C₄×C₇	semidirect
84. C₂×C₃×C₇	#103	direct (N × H)
91. #103	C₂×C₃×C₇	direct (N × H)
104. C₃×Q₈	C₂×C₄×C₇	semidirect
114. C₃×C₇	C₂×#103	direct (N × H)
117. C₂×Q₈	C₃×C₄×C₇	semidirect
126. C₂×C₇	C₃×#103	direct (N × H)
142. Q₈	C₂×C₃×C₄×C₇	semidirect
152. C₇	C₂×C₃×#103	direct (N × H)
153. C₂×C₃	C₇×#103	direct (N × H)
166. C₃	C₂×C₇×#103	direct (N × H)
167. C₂	C₃×C₇×#103	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₇×Q₈	2
C₂²×C₃×C₇×C₈	2
C₂×C₃×C₇×C₄⋊C₄	2
C₃×C₇×#103	2
C₂×C₇×#103	3
C₂×C₃×#103	7

Subgroups — C2×C3×C7×#103 #13056926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₇×#103 #13056926