Subgroups — C₂×C₇×#1400 #13190771

Order 1792.

306 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×M₆₄	C₂	1	2	yes	maximal, complemented
3	C₇×#1400	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₇×C₈	C₄	1	1	yes	characteristic
5	C₂×C₇×M₃₂	—	2	2	no	—
6	C₂×C₇×M₃₂	—	2	2	no	—
7	C₂×C₇×M₃₂	C₄	1	2	yes	—
8	C₂×C₇×M₃₂	C₂²	1	2	yes	—
9	C₂²×C₇×C₁₆	C₄	1	1	yes	characteristic
10	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
11	C₇×M₆₄	—	2	2	no	—
12	C₇×M₆₄	C₂²	1	4	yes	complemented
13	C₂×C₇×C₃₂	—	2	2	no	—
14	C₂×#1400	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂³×C₄×C₇	C₈	1	1	yes	characteristic
16	C₂²×C₇×C₈	—	2	1	no	—
17	C₂²×C₇×C₈	—	2	1	no	—
18	C₂²×C₇×C₈	—	2	1	no	—
19	C₂²×C₇×C₈	—	2	1	no	—
20	C₂²×C₇×C₈	C₈	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₈	1	2	yes	—
22	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₇×M₃₂	—	2	4	no	—
25	C₇×M₃₂	—	2	4	no	—
26	C₂×C₇×C₁₆	—	2	1	no	—
27	C₂×C₇×C₁₆	—	2	1	no	—
28	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₇×C₁₆	D₄	1	2	yes	—
31	C₂×C₇×C₁₆	D₄	1	2	yes	—
32	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
33	C₇×C₃₂	—	2	4	no	—
34	C₂²×M₃₂	C₂×C₇	1	1	yes	characteristic
35	C₂×M₆₄	C₂×C₇	1	2	yes	complemented
36	#1400	C₂×C₇	1	4	yes	direct product factor
37	C₂⁴×C₇	C₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₇	—	2	1	no	—
39	C₂²×C₄×C₇	—	2	1	no	—
40	C₂²×C₄×C₇	—	2	1	no	—
41	C₂²×C₄×C₇	—	2	1	no	—
42	C₂²×C₄×C₇	C₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₁₆	1	2	yes	—
44	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
46	C₂×C₇×C₈	—	2	1	no	—
47	C₂×C₇×C₈	—	2	1	no	—
48	C₂×C₇×C₈	—	2	2	no	—
49	C₂×C₇×C₈	—	2	2	no	—
50	C₂×C₇×C₈	—	2	2	no	—
51	C₂×C₇×C₈	—	2	2	no	—
52	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
55	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
56	C₂×C₇×C₈	M₁₆	1	1	yes	characteristic
57	C₂×C₇×C₈	M₁₆	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₇×C₁₆	—	2	2	no	—
60	C₇×C₁₆	C₂²⋊C₄	1	2	yes	—
61	C₇×C₁₆	C₂×D₄	1	2	yes	—
62	C₂³×C₈	C₄×C₇	1	1	yes	characteristic
63	C₂×M₃₂	—	2	2	no	—
64	C₂×M₃₂	—	2	2	no	—
65	C₂×M₃₂	C₄×C₇	1	2	yes	—
66	C₂×M₃₂	C₂²×C₇	1	2	yes	—
67	C₂²×C₁₆	C₄×C₇	1	1	yes	characteristic
68	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
69	M₆₄	—	2	2	no	—
70	M₆₄	C₂²×C₇	1	4	yes	complemented
71	C₂×C₃₂	—	2	2	no	—
72	C₂³×C₇	—	2	1	no	—
73	C₂³×C₇	—	2	1	no	—
74	C₂³×C₇	—	4	2	no	—
75	C₂³×C₇	C₂×C₁₆	1	1	yes	characteristic
76	C₂³×C₇	C₂×C₁₆	1	2	yes	—
77	C₂×C₄×C₇	—	2	1	no	—
78	C₂×C₄×C₇	—	2	1	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₇	C₂×C₁₆	1	2	yes	—
86	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇	M₃₂	1	1	yes	characteristic
88	C₂×C₄×C₇	M₃₂	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
90	C₇×C₈	—	2	2	no	—
91	C₇×C₈	C₂²⋊C₈	1	1	yes	characteristic
92	C₇×C₈	C₂²⋊C₈	1	1	yes	characteristic
93	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
95	C₂³×C₄	C₇×C₈	1	1	yes	characteristic
96	C₂²×C₈	—	2	1	no	—
97	C₂²×C₈	—	2	1	no	—
98	C₂²×C₈	—	2	1	no	—
99	C₂²×C₈	—	2	1	no	—
100	C₂²×C₈	C₇×C₈	1	1	yes	characteristic
101	C₂²×C₈	C₇×C₈	1	2	yes	—
102	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
103	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
104	M₃₂	—	2	4	no	—
105	M₃₂	—	2	4	no	—
106	C₂×C₁₆	—	2	1	no	—
107	C₂×C₁₆	—	2	1	no	—
108	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
109	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
110	C₂×C₁₆	C₇×D₄	1	2	yes	—
111	C₂×C₁₆	C₇×D₄	1	2	yes	—
112	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
113	C₃₂	—	2	4	no	—
114	C₂²×C₇	—	2	2	no	—
115	C₂²×C₇	—	2	2	no	—
116	C₂²×C₇	—	4	2	no	—
117	C₂²×C₇	—	4	2	no	—
118	C₂²×C₇	—	4	2	no	—
119	C₂²×C₇	C₂²⋊C₁₆	1	1	yes	characteristic
120	C₂²×C₇	C₂²⋊C₁₆	1	1	yes	characteristic
121	C₂²×C₇	C₂²×C₁₆	1	1	yes	characteristic
122	C₄×C₇	—	2	2	no	—
123	C₄×C₇	C₂²⋊C₁₆	1	1	yes	characteristic
124	C₄×C₇	C₂²⋊C₁₆	1	1	yes	characteristic
125	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
127	C₂⁴	C₇×C₁₆	1	1	yes	characteristic
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₇×C₁₆	1	1	yes	characteristic
133	C₂²×C₄	C₇×C₁₆	1	2	yes	—
134	C₂²×C₄	C₂×C₇×C₈	1	2	yes	—
135	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
145	C₂×C₈	C₂×C₇×C₈	1	1	yes	characteristic
146	C₂×C₈	C₇×M₁₆	1	1	yes	characteristic
147	C₂×C₈	C₇×M₁₆	1	1	yes	characteristic
148	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
149	C₁₆	—	2	2	no	—
150	C₁₆	C₇×C₂²⋊C₄	1	2	yes	—
151	C₁₆	C₂×C₇×D₄	1	2	yes	—
152	C₂×C₇	—	2	1	no	—
153	C₂×C₇	—	2	1	no	—
154	C₂×C₇	—	4	2	no	—
155	C₂×C₇	#1400	1	2	yes	direct product factor
156	C₂×C₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
157	C₂³	—	2	1	no	—
158	C₂³	—	2	1	no	—
159	C₂³	—	4	2	no	—
160	C₂³	C₂×C₇×C₁₆	1	1	yes	characteristic
161	C₂³	C₂×C₇×C₁₆	1	2	yes	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
169	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
170	C₂×C₄	C₂×C₇×C₁₆	1	2	yes	—
171	C₂×C₄	C₂×C₇×C₁₆	1	1	yes	characteristic
172	C₂×C₄	C₇×M₃₂	1	1	yes	characteristic
173	C₂×C₄	C₇×M₃₂	1	1	yes	characteristic
174	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	C₇×C₂²⋊C₈	1	1	yes	characteristic
177	C₈	C₇×C₂²⋊C₈	1	1	yes	characteristic
178	C₈	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
179	C₈	C₂×C₇×M₁₆	1	1	yes	characteristic
180	C₇	C₂×#1400	1	1	yes	characteristic, direct product factor, Sylow
181	C₂²	—	2	2	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₇×C₂²⋊C₁₆	1	1	yes	characteristic
187	C₂²	C₇×C₂²⋊C₁₆	1	1	yes	characteristic
188	C₂²	C₂²×C₇×C₁₆	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	C₇×C₂²⋊C₁₆	1	1	yes	characteristic
191	C₄	C₇×C₂²⋊C₁₆	1	1	yes	characteristic
192	C₄	C₂×C₇×C₂²⋊C₈	1	1	yes	characteristic
193	C₄	C₂×C₇×M₃₂	1	1	yes	characteristic
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	—	4	2	no	—
197	C₂	C₇×#1400	1	2	yes	direct product factor
198	C₂	C₂×C₇×C₂²⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇×C₈	C₂²⋊C₄
commutator	C₂²	C₂²×C₇×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂²×C₇	C₂²⋊C₁₆
prime radical	C₂⁴×C₇	C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×M₆₄	C₂	semidirect
3. C₇×#1400	C₂	direct (N × H)
12. C₇×M₆₄	C₂²	semidirect
14. C₂×#1400	C₇	direct (N × H)
35. C₂×M₆₄	C₂×C₇	semidirect
36. #1400	C₂×C₇	direct (N × H)
70. M₆₄	C₂²×C₇	semidirect
155. C₂×C₇	#1400	direct (N × H)
180. C₇	C₂×#1400	direct (N × H)
197. C₂	C₇×#1400	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×C₇×M₃₂	2
C₂×C₇×M₆₄	2
C₇×#1400	2
C₂×#1400	7

Subgroups — C2×C7×#1400 #13190771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1400 #13190771