Subgroups — C₂²×C₇×#358 #13210591

Order 1792.

1214 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇×#107	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#358	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₇×Q₈	—	2	1	no	—
6	C₂²×C₇×Q₁₆	—	2	3	no	—
7	C₂²×C₇×Q₁₆	—	2	4	no	—
8	C₂²×C₇×Q₁₆	C₄	1	8	yes	complemented
9	C₂²×C₇×Q₁₆	C₂²	1	6	yes	—
10	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
12	C₂²×C₇×C₁₆	—	2	4	no	—
13	C₂²×C₇×C₁₆	C₂²	1	6	yes	—
14	C₂×C₇×#107	—	2	3	no	—
15	C₂×C₇×#107	C₂²	1	6	yes	—
16	C₇×#358	C₂²	1	16	yes	direct product factor
17	C₂²×#358	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₇×Q₈	—	2	8	no	—
19	C₂²×C₇×Q₈	—	2	6	no	—
20	C₂²×C₇×Q₈	—	4	3	no	—
21	C₂²×C₇×Q₈	—	4	4	no	—
22	C₂³×C₄×C₇	—	4	1	no	—
23	C₂³×C₄×C₇	—	4	1	no	—
24	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
25	C₂×C₇×Q₁₆	—	2	24	no	—
26	C₂×C₇×Q₁₆	—	2	6	no	—
27	C₂×C₇×Q₁₆	—	2	12	no	—
28	C₂×C₇×Q₁₆	C₂×C₄	1	24	yes	complemented
29	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
30	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
31	C₂²×C₇×C₈	D₄	1	3	yes	—
32	C₂²×C₇×C₈	D₄	1	4	yes	—
33	C₂²×C₇×C₈	C₂³	1	3	yes	—
34	C₂×C₇×C₄⋊C₄	—	2	6	no	—
35	C₂×C₇×C₄⋊C₄	—	4	3	no	—
36	C₂×C₇×C₁₆	—	2	12	no	—
37	C₂×C₇×C₁₆	C₂³	1	4	yes	—
38	C₇×#107	—	2	6	no	—
39	C₇×#107	C₂³	1	4	yes	—
40	C₂³×Q₁₆	C₂×C₇	1	1	yes	characteristic
41	C₂³×C₁₆	C₂×C₇	1	1	yes	characteristic
42	C₂²×#107	C₂×C₇	1	1	yes	characteristic
43	C₂×#358	C₂×C₇	1	12	yes	direct product factor
44	C₂⁴×C₇	D₈	1	1	yes	characteristic
45	C₂×C₇×Q₈	—	2	24	no	—
46	C₂×C₇×Q₈	—	2	4	no	—
47	C₂×C₇×Q₈	—	4	24	no	—
48	C₂×C₇×Q₈	—	4	6	no	—
49	C₂×C₇×Q₈	—	4	12	no	—
50	C₂²×C₄×C₇	—	4	8	no	—
51	C₂²×C₄×C₇	—	4	6	no	—
52	C₂²×C₄×C₇	—	4	6	no	—
53	C₂²×C₄×C₇	—	8	4	no	—
54	C₂²×C₄×C₇	C₂²⋊C₄	1	4	yes	—
55	C₂²×C₄×C₇	D₈	1	3	yes	—
56	C₂²×C₄×C₇	SD₁₆	1	4	yes	—
57	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
58	C₇×Q₁₆	—	2	24	no	—
59	C₇×Q₁₆	—	2	4	no	—
60	C₇×Q₁₆	C₂²×C₄	1	8	yes	complemented
61	C₂×C₇×C₈	C₂²⋊C₄	1	12	yes	—
62	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
63	C₂×C₇×C₈	C₂×D₄	1	3	yes	—
64	C₂×C₇×C₈	C₂×D₄	1	6	yes	—
65	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
66	C₇×C₄⋊C₄	—	2	4	no	—
67	C₇×C₄⋊C₄	—	4	6	no	—
68	C₇×C₁₆	—	2	4	no	—
69	C₂³×Q₈	—	2	1	no	—
70	C₂²×Q₁₆	—	2	3	no	—
71	C₂²×Q₁₆	—	2	4	no	—
72	C₂²×Q₁₆	C₄×C₇	1	8	yes	complemented
73	C₂²×Q₁₆	C₂²×C₇	1	6	yes	—
74	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
75	C₂²×C₄⋊C₄	—	2	1	no	—
76	C₂²×C₁₆	—	2	4	no	—
77	C₂²×C₁₆	C₂²×C₇	1	6	yes	—
78	C₂×#107	—	2	3	no	—
79	C₂×#107	C₂²×C₇	1	6	yes	—
80	#358	C₂²×C₇	1	16	yes	direct product factor
81	C₂³×C₇	#102	1	4	yes	—
82	C₂³×C₇	SD₃₂	1	4	yes	—
83	C₂³×C₇	Q₃₂	1	4	yes	—
84	C₂³×C₇	C₂×D₈	1	3	yes	—
85	C₇×Q₈	—	2	8	no	—
86	C₇×Q₈	—	4	4	no	—
87	C₇×Q₈	—	4	24	no	—
88	C₂×C₄×C₇	—	4	4	no	—
89	C₂×C₄×C₇	—	4	24	no	—
90	C₂×C₄×C₇	—	4	4	no	—
91	C₂×C₄×C₇	—	8	12	no	—
92	C₂×C₄×C₇	#102	1	12	yes	—
93	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	6	yes	—
94	C₂×C₄×C₇	C₂×D₈	1	3	yes	—
95	C₂×C₄×C₇	C₂×SD₁₆	1	6	yes	—
96	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
97	C₇×C₈	C₂×C₂²⋊C₄	1	6	yes	—
98	C₇×C₈	C₂³×C₄	1	1	yes	characteristic
99	C₇×C₈	C₂²×D₄	1	1	yes	characteristic
100	C₂²×Q₈	—	2	8	no	—
101	C₂²×Q₈	—	2	6	no	—
102	C₂²×Q₈	—	4	3	no	—
103	C₂²×Q₈	—	4	4	no	—
104	C₂³×C₄	—	4	1	no	—
105	C₂³×C₄	—	4	1	no	—
106	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
107	C₂×Q₁₆	—	2	24	no	—
108	C₂×Q₁₆	—	2	6	no	—
109	C₂×Q₁₆	—	2	12	no	—
110	C₂×Q₁₆	C₂×C₄×C₇	1	24	yes	complemented
111	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
112	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
113	C₂²×C₈	C₇×D₄	1	3	yes	—
114	C₂²×C₈	C₇×D₄	1	4	yes	—
115	C₂²×C₈	C₂³×C₇	1	3	yes	—
116	C₂×C₄⋊C₄	—	2	6	no	—
117	C₂×C₄⋊C₄	—	4	3	no	—
118	C₂×C₁₆	—	2	12	no	—
119	C₂×C₁₆	C₂³×C₇	1	4	yes	—
120	#107	—	2	6	no	—
121	#107	C₂³×C₇	1	4	yes	—
122	C₂²×C₇	#358	1	16	yes	direct product factor
123	C₂²×C₇	C₂×#102	1	6	yes	—
124	C₂²×C₇	C₂×SD₃₂	1	6	yes	—
125	C₂²×C₇	C₂×Q₃₂	1	6	yes	—
126	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
127	C₄×C₇	—	4	8	no	—
128	C₄×C₇	—	8	4	no	—
129	C₄×C₇	C₂×#102	1	6	yes	—
130	C₄×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
131	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
132	C₂⁴	C₇×D₈	1	1	yes	characteristic
133	C₂×Q₈	—	2	24	no	—
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	—	4	24	no	—
136	C₂×Q₈	—	4	6	no	—
137	C₂×Q₈	—	4	12	no	—
138	C₂²×C₄	—	4	8	no	—
139	C₂²×C₄	—	4	6	no	—
140	C₂²×C₄	—	4	6	no	—
141	C₂²×C₄	—	8	4	no	—
142	C₂²×C₄	C₇×C₂²⋊C₄	1	4	yes	—
143	C₂²×C₄	C₇×D₈	1	3	yes	—
144	C₂²×C₄	C₇×SD₁₆	1	4	yes	—
145	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
146	Q₁₆	—	2	24	no	—
147	Q₁₆	—	2	4	no	—
148	Q₁₆	C₂²×C₄×C₇	1	8	yes	complemented
149	C₂×C₈	C₇×C₂²⋊C₄	1	12	yes	—
150	C₂×C₈	C₂²×C₄×C₇	1	6	yes	—
151	C₂×C₈	C₂×C₇×D₄	1	3	yes	—
152	C₂×C₈	C₂×C₇×D₄	1	6	yes	—
153	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	6	no	—
156	C₁₆	—	2	4	no	—
157	C₂×C₇	C₂×#358	1	12	yes	direct product factor
158	C₂×C₇	C₂²×#102	1	1	yes	characteristic
159	C₂×C₇	C₂²×SD₃₂	1	1	yes	characteristic
160	C₂×C₇	C₂²×Q₃₂	1	1	yes	characteristic
161	C₂³	C₇×#102	1	4	yes	—
162	C₂³	C₇×SD₃₂	1	4	yes	—
163	C₂³	C₇×Q₃₂	1	4	yes	—
164	C₂³	C₂×C₇×D₈	1	3	yes	—
165	Q₈	—	2	8	no	—
166	Q₈	—	4	24	no	—
167	Q₈	—	4	4	no	—
168	C₂×C₄	—	4	24	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	8	12	no	—
172	C₂×C₄	C₇×#102	1	12	yes	—
173	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	6	yes	—
174	C₂×C₄	C₂×C₇×D₈	1	3	yes	—
175	C₂×C₄	C₂×C₇×SD₁₆	1	6	yes	—
176	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
177	C₈	C₂×C₇×C₂²⋊C₄	1	6	yes	—
178	C₈	C₂³×C₄×C₇	1	1	yes	characteristic
179	C₈	C₂²×C₇×D₄	1	1	yes	characteristic
180	C₇	C₂²×#358	1	1	yes	characteristic, direct product factor, Sylow
181	C₂²	C₇×#358	1	16	yes	direct product factor
182	C₂²	C₂×C₇×#102	1	6	yes	—
183	C₂²	C₂×C₇×SD₃₂	1	6	yes	—
184	C₂²	C₂×C₇×Q₃₂	1	6	yes	—
185	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
186	C₄	—	4	8	no	—
187	C₄	—	8	4	no	—
188	C₄	C₂×C₇×#102	1	6	yes	—
189	C₄	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
190	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
191	C₂	C₂×C₇×#358	1	12	yes	direct product factor
192	C₂	C₂²×C₇×#102	1	1	yes	characteristic
193	C₂	C₂²×C₇×SD₃₂	1	1	yes	characteristic
194	C₂	C₂²×C₇×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₇	D₈
commutator	C₈	C₂³×C₄×C₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₇
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂³×C₄×C₇ ◁ C₂³×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₇×#358	C₂	direct (N × H)
8. C₂²×C₇×Q₁₆	C₄	semidirect
16. C₇×#358	C₂²	direct (N × H)
17. C₂²×#358	C₇	direct (N × H)
28. C₂×C₇×Q₁₆	C₂×C₄	semidirect
43. C₂×#358	C₂×C₇	direct (N × H)
60. C₇×Q₁₆	C₂²×C₄	semidirect
72. C₂²×Q₁₆	C₄×C₇	semidirect
80. #358	C₂²×C₇	direct (N × H)
110. C₂×Q₁₆	C₂×C₄×C₇	semidirect
122. C₂²×C₇	#358	direct (N × H)
148. Q₁₆	C₂²×C₄×C₇	semidirect
157. C₂×C₇	C₂×#358	direct (N × H)
180. C₇	C₂²×#358	direct (N × H)
181. C₂²	C₇×#358	direct (N × H)
191. C₂	C₂×C₇×#358	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×C₇×Q₁₆	2
C₂³×C₇×C₁₆	2
C₂²×C₇×#107	2
C₂×C₇×#358	2
C₂²×#358	7

Subgroups — C22×C7×#358 #13210591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#358 #13210591