Subgroups — C₂³×C₇×#100 #13237287

Order 1792.

7134 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₇×M₁₆	C₂	1	2	yes	maximal, complemented
3	C₂²×C₇×#100	C₂	1	28	yes	maximal, direct product factor
4	C₂⁶×C₇	C₄	1	2	yes	—
5	C₂³×C₇×D₄	—	2	14	no	—
6	C₂³×C₇×D₄	—	2	2	no	—
7	C₂³×C₇×D₄	C₄	1	14	yes	—
8	C₂³×C₇×D₄	C₂²	1	14	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	complemented
12	C₂³×C₇×C₈	—	2	2	no	—
13	C₂×C₇×#100	C₂²	1	112	yes	direct product factor
14	C₂³×#100	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂⁵×C₇	—	2	14	no	—
16	C₂⁵×C₇	—	2	2	no	—
17	C₂⁵×C₇	—	4	16	no	—
18	C₂⁵×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂⁵×C₇	C₂×C₄	1	28	yes	—
20	C₂²×C₇×D₄	—	2	84	no	—
21	C₂²×C₇×D₄	—	2	28	no	—
22	C₂²×C₇×D₄	—	2	28	no	—
23	C₂²×C₇×D₄	—	2	84	no	—
24	C₂²×C₇×D₄	C₂×C₄	1	84	yes	—
25	C₂²×C₇×D₄	C₂³	1	28	yes	—
26	C₂³×C₄×C₇	C₂×C₄	1	7	yes	—
27	C₂³×C₄×C₇	D₄	1	14	yes	—
28	C₂³×C₄×C₇	D₄	1	2	yes	—
29	C₂³×C₄×C₇	C₂³	1	7	yes	—
30	C₂×C₇×M₁₆	—	2	84	no	—
31	C₂×C₇×M₁₆	C₂³	1	56	yes	complemented
32	C₂²×C₇×C₈	—	2	28	no	—
33	C₇×#100	C₂³	1	64	yes	direct product factor
34	C₂⁴×D₄	C₂×C₇	1	1	yes	characteristic
35	C₂³×M₁₆	C₂×C₇	1	2	yes	complemented
36	C₂²×#100	C₂×C₇	1	28	yes	direct product factor
37	C₂⁴×C₇	—	2	8	no	—
38	C₂⁴×C₇	—	2	84	no	—
39	C₂⁴×C₇	—	2	28	no	—
40	C₂⁴×C₇	—	4	112	no	—
41	C₂⁴×C₇	—	4	112	no	—
42	C₂⁴×C₇	—	4	16	no	—
43	C₂⁴×C₇	C₂²⋊C₄	1	7	yes	—
44	C₂⁴×C₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂⁴×C₇	C₂²×C₄	1	7	yes	—
46	C₂⁴×C₇	C₂²×C₄	1	56	yes	—
47	C₂×C₇×D₄	—	2	168	no	—
48	C₂×C₇×D₄	—	2	168	no	—
49	C₂×C₇×D₄	—	2	56	no	—
50	C₂×C₇×D₄	—	2	56	no	—
51	C₂×C₇×D₄	C₂²×C₄	1	56	yes	—
52	C₂×C₇×D₄	C₂⁴	1	8	yes	—
53	C₂²×C₄×C₇	C₂²⋊C₄	1	42	yes	—
54	C₂²×C₄×C₇	C₂²⋊C₄	1	14	yes	—
55	C₂²×C₄×C₇	C₂²×C₄	1	21	yes	—
56	C₂²×C₄×C₇	C₂×D₄	1	14	yes	—
57	C₂²×C₄×C₇	C₂×D₄	1	42	yes	—
58	C₂²×C₄×C₇	C₂⁴	1	7	yes	—
59	C₇×M₁₆	—	2	56	no	—
60	C₇×M₁₆	C₂⁴	1	16	yes	complemented
61	C₂×C₇×C₈	—	2	56	no	—
62	C₂⁶	C₄×C₇	1	2	yes	—
63	C₂³×D₄	—	2	2	no	—
64	C₂³×D₄	—	2	14	no	—
65	C₂³×D₄	C₄×C₇	1	14	yes	—
66	C₂³×D₄	C₂²×C₇	1	14	yes	—
67	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
68	C₂²×M₁₆	—	2	14	no	—
69	C₂²×M₁₆	C₂²×C₇	1	28	yes	complemented
70	C₂³×C₈	—	2	2	no	—
71	C₂×#100	C₂²×C₇	1	112	yes	direct product factor
72	C₂³×C₇	—	2	28	no	—
73	C₂³×C₇	—	2	28	no	—
74	C₂³×C₇	—	2	56	no	—
75	C₂³×C₇	—	2	56	no	—
76	C₂³×C₇	—	4	112	no	—
77	C₂³×C₇	—	4	336	no	—
78	C₂³×C₇	—	4	112	no	—
79	C₂³×C₇	#100	1	8	yes	direct product factor
80	C₂³×C₇	C₂×C₂²⋊C₄	1	21	yes	—
81	C₂³×C₇	C₂×C₂²⋊C₄	1	7	yes	—
82	C₂³×C₇	C₂³×C₄	1	7	yes	—
83	C₂³×C₇	C₂³×C₄	1	16	yes	—
84	C₇×D₄	—	2	112	no	—
85	C₇×D₄	—	2	16	no	—
86	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	42	yes	—
87	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	42	yes	—
88	C₂×C₄×C₇	C₂³×C₄	1	7	yes	—
89	C₂×C₄×C₇	C₂²×D₄	1	14	yes	—
90	C₂×C₄×C₇	C₂²×D₄	1	14	yes	—
91	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
92	C₇×C₈	—	2	16	no	—
93	C₂⁵	—	2	14	no	—
94	C₂⁵	—	2	2	no	—
95	C₂⁵	—	4	16	no	—
96	C₂⁵	C₂×C₄×C₇	1	1	yes	characteristic
97	C₂⁵	C₂×C₄×C₇	1	28	yes	—
98	C₂²×D₄	—	2	84	no	—
99	C₂²×D₄	—	2	28	no	—
100	C₂²×D₄	—	2	28	no	—
101	C₂²×D₄	—	2	84	no	—
102	C₂²×D₄	C₂×C₄×C₇	1	84	yes	—
103	C₂²×D₄	C₂³×C₇	1	28	yes	—
104	C₂³×C₄	C₂×C₄×C₇	1	7	yes	—
105	C₂³×C₄	C₇×D₄	1	2	yes	—
106	C₂³×C₄	C₇×D₄	1	14	yes	—
107	C₂³×C₄	C₂³×C₇	1	7	yes	—
108	C₂×M₁₆	—	2	84	no	—
109	C₂×M₁₆	C₂³×C₇	1	56	yes	complemented
110	C₂²×C₈	—	2	28	no	—
111	#100	C₂³×C₇	1	64	yes	direct product factor
112	C₂²×C₇	—	2	42	no	—
113	C₂²×C₇	—	2	14	no	—
114	C₂²×C₇	—	2	16	no	—
115	C₂²×C₇	—	4	16	no	—
116	C₂²×C₇	—	4	112	no	—
117	C₂²×C₇	—	4	112	no	—
118	C₂²×C₇	C₂×#100	1	28	yes	direct product factor
119	C₂²×C₇	C₂²×C₂²⋊C₄	1	7	yes	—
120	C₂²×C₇	C₂²×C₂²⋊C₄	1	7	yes	—
121	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
122	C₄×C₇	C₂²×C₂²⋊C₄	1	14	yes	—
123	C₄×C₇	C₂³×D₄	1	2	yes	—
124	C₂⁴	—	2	8	no	—
125	C₂⁴	—	2	84	no	—
126	C₂⁴	—	2	28	no	—
127	C₂⁴	—	4	112	no	—
128	C₂⁴	—	4	112	no	—
129	C₂⁴	—	4	16	no	—
130	C₂⁴	C₇×C₂²⋊C₄	1	7	yes	—
131	C₂⁴	C₇×C₂²⋊C₄	1	1	yes	characteristic
132	C₂⁴	C₂²×C₄×C₇	1	7	yes	—
133	C₂⁴	C₂²×C₄×C₇	1	56	yes	—
134	C₂×D₄	—	2	168	no	—
135	C₂×D₄	—	2	168	no	—
136	C₂×D₄	—	2	56	no	—
137	C₂×D₄	—	2	56	no	—
138	C₂×D₄	C₂²×C₄×C₇	1	56	yes	—
139	C₂×D₄	C₂⁴×C₇	1	8	yes	—
140	C₂²×C₄	C₇×C₂²⋊C₄	1	42	yes	—
141	C₂²×C₄	C₇×C₂²⋊C₄	1	14	yes	—
142	C₂²×C₄	C₂²×C₄×C₇	1	21	yes	—
143	C₂²×C₄	C₂×C₇×D₄	1	14	yes	—
144	C₂²×C₄	C₂×C₇×D₄	1	42	yes	—
145	C₂²×C₄	C₂⁴×C₇	1	7	yes	—
146	M₁₆	—	2	56	no	—
147	M₁₆	C₂⁴×C₇	1	16	yes	complemented
148	C₂×C₈	—	2	56	no	—
149	C₂×C₇	—	2	7	no	—
150	C₂×C₇	—	2	1	no	—
151	C₂×C₇	—	4	16	no	—
152	C₂×C₇	C₂²×#100	1	14	yes	direct product factor
153	C₂×C₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
154	C₂³	—	2	28	no	—
155	C₂³	—	2	28	no	—
156	C₂³	—	2	56	no	—
157	C₂³	—	2	56	no	—
158	C₂³	—	4	112	no	—
159	C₂³	—	4	336	no	—
160	C₂³	—	4	112	no	—
161	C₂³	C₇×#100	1	8	yes	direct product factor
162	C₂³	C₂×C₇×C₂²⋊C₄	1	21	yes	—
163	C₂³	C₂×C₇×C₂²⋊C₄	1	7	yes	—
164	C₂³	C₂³×C₄×C₇	1	7	yes	—
165	C₂³	C₂³×C₄×C₇	1	16	yes	—
166	D₄	—	2	16	no	—
167	D₄	—	2	112	no	—
168	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	42	yes	—
169	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	42	yes	—
170	C₂×C₄	C₂³×C₄×C₇	1	7	yes	—
171	C₂×C₄	C₂²×C₇×D₄	1	14	yes	—
172	C₂×C₄	C₂²×C₇×D₄	1	14	yes	—
173	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
174	C₈	—	2	16	no	—
175	C₇	C₂³×#100	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²	—	2	42	no	—
177	C₂²	—	2	14	no	—
178	C₂²	—	2	16	no	—
179	C₂²	—	4	16	no	—
180	C₂²	—	4	112	no	—
181	C₂²	—	4	112	no	—
182	C₂²	C₂×C₇×#100	1	28	yes	direct product factor
183	C₂²	C₂²×C₇×C₂²⋊C₄	1	7	yes	—
184	C₂²	C₂²×C₇×C₂²⋊C₄	1	7	yes	—
185	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
186	C₄	C₂²×C₇×C₂²⋊C₄	1	14	yes	—
187	C₄	C₂³×C₇×D₄	1	2	yes	—
188	C₂	—	2	7	no	—
189	C₂	—	2	1	no	—
190	C₂	—	4	16	no	—
191	C₂	C₂²×C₇×#100	1	14	yes	direct product factor
192	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	C₂⁴×C₇×D₄	C₂
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
2. C₂³×C₇×M₁₆	C₂	semidirect
3. C₂²×C₇×#100	C₂	direct (N × H)
11. C₂²×C₇×M₁₆	C₂²	semidirect
13. C₂×C₇×#100	C₂²	direct (N × H)
14. C₂³×#100	C₇	direct (N × H)
31. C₂×C₇×M₁₆	C₂³	semidirect
33. C₇×#100	C₂³	direct (N × H)
35. C₂³×M₁₆	C₂×C₇	semidirect
36. C₂²×#100	C₂×C₇	direct (N × H)
60. C₇×M₁₆	C₂⁴	semidirect
69. C₂²×M₁₆	C₂²×C₇	semidirect
71. C₂×#100	C₂²×C₇	direct (N × H)
79. C₂³×C₇	#100	direct (N × H)
109. C₂×M₁₆	C₂³×C₇	semidirect
111. #100	C₂³×C₇	direct (N × H)
118. C₂²×C₇	C₂×#100	direct (N × H)
147. M₁₆	C₂⁴×C₇	semidirect
152. C₂×C₇	C₂²×#100	direct (N × H)
161. C₂³	C₇×#100	direct (N × H)
175. C₇	C₂³×#100	direct (N × H)
182. C₂²	C₂×C₇×#100	direct (N × H)
191. C₂	C₂²×C₇×#100	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×D₄	2
C₂³×C₇×M₁₆	2
C₂²×C₇×#100	2
C₂³×#100	7

Subgroups — C23×C7×#100 #13237287

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×#100 #13237287