Subgroups — C₄×C₇×#335 #13185488

Order 1792.

470 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₄²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#335	C₂	1	2	yes	maximal
4	C₇×#1277	C₂	1	2	yes	maximal
5	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
6	C₂×C₄×C₇×C₈	—	2	2	no	—
7	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₄³×C₇	C₂²	1	1	yes	characteristic
12	C₇×#336	—	2	2	no	—
13	C₇×#335	C₄	1	8	yes	direct product factor
14	C₄×#335	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂²×C₇×C₈	—	2	4	no	—
16	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
18	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₇	Q₈	1	2	yes	—
23	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
24	C₇×C₄⋊C₈	—	2	4	no	—
25	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
26	C₄×C₇×C₈	—	2	4	no	—
27	C₄×C₇×C₈	—	2	2	no	—
28	C₄×C₇×C₈	—	4	4	no	—
29	C₄×C₇×C₈	C₈	1	8	yes	complemented
30	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
31	C₄×C₄⋊C₈	C₂×C₇	1	2	yes	—
32	C₄²×C₈	C₂×C₇	1	1	yes	characteristic
33	C₂×#335	C₂×C₇	1	2	yes	—
34	#1277	C₂×C₇	1	2	yes	—
35	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
37	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₇×C₈	—	2	2	no	—
41	C₂×C₇×C₈	—	2	2	no	—
42	C₂×C₇×C₈	—	2	2	no	—
43	C₂×C₇×C₈	—	2	8	no	—
44	C₂×C₇×C₈	—	4	4	no	—
45	C₂×C₇×C₈	—	4	4	no	—
46	C₂×C₇×C₈	C₄²	1	2	yes	—
47	C₂×C₇×C₈	C₄²	1	2	yes	—
48	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
49	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
50	C₄²×C₇	—	2	4	no	—
51	C₄²×C₇	—	2	2	no	—
52	C₄²×C₇	C₂×C₈	1	4	yes	—
53	C₄²×C₇	M₁₆	1	4	yes	—
54	C₄²×C₇	D₈	1	2	yes	—
55	C₄²×C₇	Q₁₆	1	2	yes	—
56	C₄²×C₇	C₂²×C₄	1	2	yes	—
57	C₄²×C₇	#41	1	2	yes	—
58	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
59	C₂×C₄×C₈	—	2	2	no	—
60	C₂×C₄×C₈	C₄×C₇	1	2	yes	—
61	C₂×C₄×C₈	C₄×C₇	1	2	yes	—
62	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
64	C₄³	C₂²×C₇	1	1	yes	characteristic
65	#336	—	2	2	no	—
66	#335	C₄×C₇	1	8	yes	direct product factor
67	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇	—	2	2	no	—
69	C₂×C₄×C₇	—	2	2	no	—
70	C₂×C₄×C₇	—	2	2	no	—
71	C₂×C₄×C₇	C₄⋊C₈	1	4	yes	—
72	C₂×C₄×C₇	C₄⋊C₈	1	4	yes	—
73	C₂×C₄×C₇	#107	1	4	yes	—
74	C₂×C₄×C₇	#108	1	4	yes	—
75	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₇	#117	1	1	yes	characteristic
77	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
79	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
80	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
81	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₇	#135	1	2	yes	—
84	C₇×C₈	—	2	2	no	—
85	C₇×C₈	—	2	4	no	—
86	C₇×C₈	—	4	8	no	—
87	C₇×C₈	C₄×C₈	1	4	yes	complemented
88	C₂²×C₈	—	2	4	no	—
89	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
90	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
91	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
92	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
93	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
94	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
95	C₂×C₄²	C₇×Q₈	1	2	yes	—
96	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
97	C₄⋊C₈	—	2	4	no	—
98	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	—
99	C₄×C₈	—	2	4	no	—
100	C₄×C₈	—	2	2	no	—
101	C₄×C₈	—	4	4	no	—
102	C₄×C₈	C₇×C₈	1	8	yes	complemented
103	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
104	C₂²×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₇	C₂×C₄⋊C₈	1	2	yes	—
106	C₂²×C₇	C₂×#107	1	1	yes	characteristic
107	C₂²×C₇	#427	1	1	yes	characteristic
108	C₂²×C₇	C₂×#108	1	2	yes	—
109	C₄×C₇	—	2	4	no	—
110	C₄×C₇	—	2	2	no	—
111	C₄×C₇	#335	1	8	yes	direct product factor
112	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
113	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
114	C₄×C₇	#431	1	2	yes	—
115	C₄×C₇	C₄×D₈	1	2	yes	—
116	C₄×C₇	C₄×Q₁₆	1	2	yes	—
117	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
118	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
119	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
120	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
121	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	8	no	—
126	C₂×C₈	—	4	4	no	—
127	C₂×C₈	—	4	4	no	—
128	C₂×C₈	C₄²×C₇	1	2	yes	—
129	C₂×C₈	C₄²×C₇	1	2	yes	—
130	C₂×C₈	C₂×C₇×C₈	1	4	yes	—
131	C₂×C₈	C₂×C₇×C₈	1	4	yes	—
132	C₄²	—	2	4	no	—
133	C₄²	—	2	2	no	—
134	C₄²	C₂×C₇×C₈	1	4	yes	—
135	C₄²	C₇×M₁₆	1	4	yes	—
136	C₄²	C₇×D₈	1	2	yes	—
137	C₄²	C₇×Q₁₆	1	2	yes	—
138	C₄²	C₂²×C₄×C₇	1	2	yes	—
139	C₄²	C₇×#41	1	2	yes	—
140	C₂×C₇	C₂×#335	1	2	yes	—
141	C₂×C₇	#1564	1	2	yes	—
142	C₂×C₇	C₄×C₄⋊C₈	1	1	yes	characteristic
143	C₂×C₇	C₄×#107	1	1	yes	characteristic
144	C₂×C₇	C₄×#108	1	1	yes	characteristic
145	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	C₇×C₄⋊C₈	1	4	yes	—
150	C₂×C₄	C₇×C₄⋊C₈	1	4	yes	—
151	C₂×C₄	C₇×#107	1	4	yes	—
152	C₂×C₄	C₇×#108	1	4	yes	—
153	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
154	C₂×C₄	C₇×#117	1	1	yes	characteristic
155	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
156	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
157	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
158	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
159	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
160	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
161	C₂×C₄	C₇×#135	1	2	yes	—
162	C₈	—	2	2	no	—
163	C₈	—	2	4	no	—
164	C₈	—	4	8	no	—
165	C₈	C₄×C₇×C₈	1	4	yes	complemented
166	C₇	C₄×#335	1	1	yes	characteristic, direct product factor, Sylow
167	C₂²	C₄×C₇×C₄⋊C₄	1	1	yes	characteristic
168	C₂²	C₂×C₇×C₄⋊C₈	1	2	yes	—
169	C₂²	C₂×C₇×#107	1	1	yes	characteristic
170	C₂²	C₇×#427	1	1	yes	characteristic
171	C₂²	C₂×C₇×#108	1	2	yes	—
172	C₄	—	2	2	no	—
173	C₄	—	2	4	no	—
174	C₄	C₇×#335	1	8	yes	direct product factor
175	C₄	C₂×C₄×C₇×C₈	1	1	yes	characteristic
176	C₄	C₄×C₇×M₁₆	1	1	yes	characteristic
177	C₄	C₇×#431	1	2	yes	—
178	C₄	C₄×C₇×D₈	1	2	yes	—
179	C₄	C₄×C₇×Q₁₆	1	2	yes	—
180	C₂	C₂×C₇×#335	1	2	yes	—
181	C₂	C₇×#1564	1	2	yes	—
182	C₂	C₄×C₇×C₄⋊C₈	1	1	yes	characteristic
183	C₂	C₄×C₇×#107	1	1	yes	characteristic
184	C₂	C₄×C₇×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₇	D₄
commutator	C₄	C₂×C₄×C₇×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle, prime radical	C₂³×C₇	C₂×C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₇ ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₇×#335	C₄	direct (N × H)
14. C₄×#335	C₇	direct (N × H)
29. C₄×C₇×C₈	C₈	semidirect
66. #335	C₄×C₇	direct (N × H)
87. C₇×C₈	C₄×C₈	semidirect
102. C₄×C₈	C₇×C₈	semidirect
111. C₄×C₇	#335	direct (N × H)
165. C₈	C₄×C₇×C₈	semidirect
166. C₇	C₄×#335	direct (N × H)
174. C₄	C₇×#335	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×C₄⋊C₈	2
C₄²×C₇×C₈	2
C₂×C₇×#335	2
C₇×#1277	2
C₄×#335	7

Subgroups — C4×C7×#335 #13185488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₇×#335 #13185488