Subgroups — C₂³×C₇×#101 #13237288

Order 1792.

4990 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	C₂⁴×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₇×M₁₆	C₂	1	2	yes	maximal
3	C₂²×C₇×#101	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₇×Q₈	—	2	14	no	—
5	C₂³×C₇×Q₈	—	2	2	no	—
6	C₂³×C₇×Q₈	C₄	1	14	yes	—
7	C₂³×C₇×Q₈	C₂²	1	14	yes	—
8	C₂⁴×C₄×C₇	C₄	1	2	yes	—
9	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂²×C₇×M₁₆	—	2	14	no	—
11	C₂²×C₇×M₁₆	C₂²	1	28	yes	—
12	C₂³×C₇×C₈	—	2	2	no	—
13	C₂×C₇×#101	C₂²	1	112	yes	direct product factor
14	C₂³×#101	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂⁵×C₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₇×Q₈	—	2	84	no	—
17	C₂²×C₇×Q₈	—	2	28	no	—
18	C₂²×C₇×Q₈	—	2	28	no	—
19	C₂²×C₇×Q₈	—	2	84	no	—
20	C₂²×C₇×Q₈	C₂×C₄	1	84	yes	—
21	C₂²×C₇×Q₈	C₂³	1	28	yes	—
22	C₂³×C₄×C₇	—	2	14	no	—
23	C₂³×C₄×C₇	—	2	2	no	—
24	C₂³×C₄×C₇	C₂×C₄	1	28	yes	—
25	C₂³×C₄×C₇	C₂×C₄	1	7	yes	—
26	C₂³×C₄×C₇	D₄	1	14	yes	—
27	C₂³×C₄×C₇	D₄	1	2	yes	—
28	C₂³×C₄×C₇	C₂³	1	7	yes	—
29	C₂×C₇×M₁₆	—	2	84	no	—
30	C₂×C₇×M₁₆	C₂³	1	56	yes	—
31	C₂²×C₇×C₈	—	2	28	no	—
32	C₇×#101	C₂³	1	64	yes	direct product factor
33	C₂⁴×Q₈	C₂×C₇	1	1	yes	characteristic
34	C₂³×M₁₆	C₂×C₇	1	2	yes	—
35	C₂²×#101	C₂×C₇	1	28	yes	direct product factor
36	C₂⁴×C₇	—	2	8	no	—
37	C₂⁴×C₇	C₂²⋊C₄	1	7	yes	—
38	C₂⁴×C₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂⁴×C₇	C₂²×C₄	1	7	yes	—
40	C₂×C₇×Q₈	—	2	168	no	—
41	C₂×C₇×Q₈	—	2	168	no	—
42	C₂×C₇×Q₈	—	2	56	no	—
43	C₂×C₇×Q₈	—	2	56	no	—
44	C₂×C₇×Q₈	C₂²×C₄	1	56	yes	—
45	C₂×C₇×Q₈	C₂⁴	1	8	yes	—
46	C₂²×C₄×C₇	—	2	84	no	—
47	C₂²×C₄×C₇	—	2	28	no	—
48	C₂²×C₄×C₇	C₂²⋊C₄	1	42	yes	—
49	C₂²×C₄×C₇	C₂²⋊C₄	1	14	yes	—
50	C₂²×C₄×C₇	C₂²×C₄	1	56	yes	—
51	C₂²×C₄×C₇	C₂²×C₄	1	21	yes	—
52	C₂²×C₄×C₇	C₂×D₄	1	14	yes	—
53	C₂²×C₄×C₇	C₂×D₄	1	42	yes	—
54	C₂²×C₄×C₇	C₂⁴	1	7	yes	—
55	C₇×M₁₆	—	2	56	no	—
56	C₇×M₁₆	C₂⁴	1	16	yes	—
57	C₂×C₇×C₈	—	2	56	no	—
58	C₂³×Q₈	—	2	2	no	—
59	C₂³×Q₈	—	2	14	no	—
60	C₂³×Q₈	C₄×C₇	1	14	yes	—
61	C₂³×Q₈	C₂²×C₇	1	14	yes	—
62	C₂⁴×C₄	C₄×C₇	1	2	yes	—
63	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
64	C₂²×M₁₆	—	2	14	no	—
65	C₂²×M₁₆	C₂²×C₇	1	28	yes	—
66	C₂³×C₈	—	2	2	no	—
67	C₂×#101	C₂²×C₇	1	112	yes	direct product factor
68	C₂³×C₇	—	2	28	no	—
69	C₂³×C₇	—	2	28	no	—
70	C₂³×C₇	#101	1	8	yes	direct product factor
71	C₂³×C₇	C₂×C₂²⋊C₄	1	21	yes	—
72	C₂³×C₇	C₂×C₂²⋊C₄	1	7	yes	—
73	C₂³×C₇	C₂³×C₄	1	7	yes	—
74	C₇×Q₈	—	2	112	no	—
75	C₇×Q₈	—	2	16	no	—
76	C₂×C₄×C₇	—	2	56	no	—
77	C₂×C₄×C₇	—	2	56	no	—
78	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	42	yes	—
79	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	42	yes	—
80	C₂×C₄×C₇	C₂³×C₄	1	16	yes	—
81	C₂×C₄×C₇	C₂³×C₄	1	7	yes	—
82	C₂×C₄×C₇	C₂²×D₄	1	14	yes	—
83	C₂×C₄×C₇	C₂²×D₄	1	14	yes	—
84	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
85	C₇×C₈	—	2	16	no	—
86	C₂⁵	C₂×C₄×C₇	1	1	yes	characteristic
87	C₂²×Q₈	—	2	84	no	—
88	C₂²×Q₈	—	2	28	no	—
89	C₂²×Q₈	—	2	28	no	—
90	C₂²×Q₈	—	2	84	no	—
91	C₂²×Q₈	C₂×C₄×C₇	1	84	yes	—
92	C₂²×Q₈	C₂³×C₇	1	28	yes	—
93	C₂³×C₄	—	2	14	no	—
94	C₂³×C₄	—	2	2	no	—
95	C₂³×C₄	C₂×C₄×C₇	1	28	yes	—
96	C₂³×C₄	C₂×C₄×C₇	1	7	yes	—
97	C₂³×C₄	C₇×D₄	1	2	yes	—
98	C₂³×C₄	C₇×D₄	1	14	yes	—
99	C₂³×C₄	C₂³×C₇	1	7	yes	—
100	C₂×M₁₆	—	2	84	no	—
101	C₂×M₁₆	C₂³×C₇	1	56	yes	—
102	C₂²×C₈	—	2	28	no	—
103	#101	C₂³×C₇	1	64	yes	direct product factor
104	C₂²×C₇	—	2	42	no	—
105	C₂²×C₇	—	2	14	no	—
106	C₂²×C₇	C₂×#101	1	28	yes	direct product factor
107	C₂²×C₇	C₂²×C₂²⋊C₄	1	7	yes	—
108	C₂²×C₇	C₂²×C₂²⋊C₄	1	7	yes	—
109	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
110	C₄×C₇	—	2	16	no	—
111	C₄×C₇	C₂²×C₂²⋊C₄	1	14	yes	—
112	C₄×C₇	C₂³×D₄	1	2	yes	—
113	C₂⁴	—	2	8	no	—
114	C₂⁴	C₇×C₂²⋊C₄	1	7	yes	—
115	C₂⁴	C₇×C₂²⋊C₄	1	1	yes	characteristic
116	C₂⁴	C₂²×C₄×C₇	1	7	yes	—
117	C₂×Q₈	—	2	168	no	—
118	C₂×Q₈	—	2	168	no	—
119	C₂×Q₈	—	2	56	no	—
120	C₂×Q₈	—	2	56	no	—
121	C₂×Q₈	C₂²×C₄×C₇	1	56	yes	—
122	C₂×Q₈	C₂⁴×C₇	1	8	yes	—
123	C₂²×C₄	—	2	84	no	—
124	C₂²×C₄	—	2	28	no	—
125	C₂²×C₄	C₇×C₂²⋊C₄	1	42	yes	—
126	C₂²×C₄	C₇×C₂²⋊C₄	1	14	yes	—
127	C₂²×C₄	C₂²×C₄×C₇	1	56	yes	—
128	C₂²×C₄	C₂²×C₄×C₇	1	21	yes	—
129	C₂²×C₄	C₂×C₇×D₄	1	14	yes	—
130	C₂²×C₄	C₂×C₇×D₄	1	42	yes	—
131	C₂²×C₄	C₂⁴×C₇	1	7	yes	—
132	M₁₆	—	2	56	no	—
133	M₁₆	C₂⁴×C₇	1	16	yes	—
134	C₂×C₈	—	2	56	no	—
135	C₂×C₇	—	2	7	no	—
136	C₂×C₇	—	2	1	no	—
137	C₂×C₇	C₂²×#101	1	14	yes	direct product factor
138	C₂×C₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
139	C₂³	—	2	28	no	—
140	C₂³	—	2	28	no	—
141	C₂³	C₇×#101	1	8	yes	direct product factor
142	C₂³	C₂×C₇×C₂²⋊C₄	1	21	yes	—
143	C₂³	C₂×C₇×C₂²⋊C₄	1	7	yes	—
144	C₂³	C₂³×C₄×C₇	1	7	yes	—
145	Q₈	—	2	16	no	—
146	Q₈	—	2	112	no	—
147	C₂×C₄	—	2	56	no	—
148	C₂×C₄	—	2	56	no	—
149	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	42	yes	—
150	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	42	yes	—
151	C₂×C₄	C₂³×C₄×C₇	1	16	yes	—
152	C₂×C₄	C₂³×C₄×C₇	1	7	yes	—
153	C₂×C₄	C₂²×C₇×D₄	1	14	yes	—
154	C₂×C₄	C₂²×C₇×D₄	1	14	yes	—
155	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
156	C₈	—	2	16	no	—
157	C₇	C₂³×#101	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²	—	2	42	no	—
159	C₂²	—	2	14	no	—
160	C₂²	C₂×C₇×#101	1	28	yes	direct product factor
161	C₂²	C₂²×C₇×C₂²⋊C₄	1	7	yes	—
162	C₂²	C₂²×C₇×C₂²⋊C₄	1	7	yes	—
163	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
164	C₄	—	2	16	no	—
165	C₄	C₂²×C₇×C₂²⋊C₄	1	14	yes	—
166	C₄	C₂³×C₇×D₄	1	2	yes	—
167	C₂	—	2	7	no	—
168	C₂	—	2	1	no	—
169	C₂	C₂²×C₇×#101	1	14	yes	direct product factor
170	C₂	C₂³×C₇×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂²⋊C₄
commutator	C₂²	C₂⁴×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂⁵×C₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂⁴×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₇×#101	C₂	direct (N × H)
13. C₂×C₇×#101	C₂²	direct (N × H)
14. C₂³×#101	C₇	direct (N × H)
32. C₇×#101	C₂³	direct (N × H)
35. C₂²×#101	C₂×C₇	direct (N × H)
67. C₂×#101	C₂²×C₇	direct (N × H)
70. C₂³×C₇	#101	direct (N × H)
103. #101	C₂³×C₇	direct (N × H)
106. C₂²×C₇	C₂×#101	direct (N × H)
137. C₂×C₇	C₂²×#101	direct (N × H)
141. C₂³	C₇×#101	direct (N × H)
157. C₇	C₂³×#101	direct (N × H)
160. C₂²	C₂×C₇×#101	direct (N × H)
169. C₂	C₂²×C₇×#101	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₇×Q₈	2
C₂³×C₇×M₁₆	2
C₂²×C₇×#101	2
C₂³×#101	7

Subgroups — C23×C7×#101 #13237288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×#101 #13237288