Subgroups — C₂×C₇×#1709 #13196813

Order 1792.

982 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#486	C₂	1	4	yes	maximal, complemented
4	C₇×#1709	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×Q₁₆	—	2	2	no	—
7	C₂²×C₇×D₈	—	2	2	no	—
8	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₇×#127	C₂²	1	2	yes	—
10	C₇×#494	—	2	2	no	—
11	C₇×#494	C₂²	1	4	yes	complemented
12	C₇×#493	—	2	2	no	—
13	C₇×#493	C₂²	1	4	yes	—
14	C₇×#486	—	2	8	no	—
15	C₇×#486	C₂²	1	16	yes	complemented
16	C₂×C₇×#102	—	2	8	no	—
17	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₇×C₈²	C₂²	1	4	yes	complemented
20	C₂×#1709	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₇×Q₈	—	2	2	no	—
22	C₂²×C₇×D₄	—	2	4	no	—
23	C₂²×C₇×D₄	—	2	2	no	—
24	C₂×C₇×Q₁₆	—	2	8	no	—
25	C₂×C₇×Q₁₆	—	4	8	no	—
26	C₂×C₇×D₈	—	2	8	no	—
27	C₂×C₇×D₈	—	4	8	no	—
28	C₂²×C₇×C₈	D₄	1	4	yes	—
29	C₂²×C₇×C₈	D₄	1	2	yes	—
30	C₇×C₄⋊Q₈	—	2	4	no	—
31	C₇×C₄⋊Q₈	—	2	2	no	—
32	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
33	C₇×#127	—	2	4	no	—
34	C₇×#127	—	2	2	no	—
35	C₇×#127	C₂³	1	4	yes	—
36	C₂×C₇×C₄⋊C₄	—	2	4	no	—
37	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
38	C₇×#102	—	2	32	no	—
39	C₄×C₇×C₈	D₄	1	4	yes	—
40	C₄×C₇×C₈	D₄	1	2	yes	—
41	C₄×C₇×C₈	C₂³	1	4	yes	—
42	C₄×C₇×C₈	C₂³	1	2	yes	—
43	C₂×#494	C₂×C₇	1	1	yes	characteristic, complemented
44	C₂×#493	C₂×C₇	1	1	yes	characteristic
45	C₂×#486	C₂×C₇	1	4	yes	complemented
46	#1709	C₂×C₇	1	8	yes	direct product factor
47	C₂×C₈²	C₂×C₇	1	1	yes	characteristic, complemented
48	C₂⁴×C₇	—	4	2	no	—
49	C₂×C₇×Q₈	—	2	4	no	—
50	C₂×C₇×Q₈	—	4	2	no	—
51	C₂×C₇×Q₈	—	4	8	no	—
52	C₂×C₇×D₄	—	2	8	no	—
53	C₂×C₇×D₄	—	2	4	no	—
54	C₂×C₇×D₄	—	4	2	no	—
55	C₂×C₇×D₄	—	4	4	no	—
56	C₂×C₇×D₄	—	4	8	no	—
57	C₂×C₇×D₄	—	4	16	no	—
58	C₂²×C₄×C₇	—	4	2	no	—
59	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
61	C₇×Q₁₆	—	4	16	no	—
62	C₇×D₈	—	4	16	no	—
63	C₂×C₇×C₈	—	2	8	no	—
64	C₂×C₇×C₈	D₈	1	8	yes	—
65	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
66	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
67	C₇×C₄⋊C₄	—	2	8	no	—
68	C₇×C₄⋊C₄	—	4	4	no	—
69	C₄²×C₇	C₂×D₄	1	2	yes	—
70	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₇	C₂⁴	1	1	yes	characteristic
72	C₂²×Q₁₆	—	2	2	no	—
73	C₂²×D₈	—	2	2	no	—
74	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
75	C₂×#127	C₂²×C₇	1	2	yes	—
76	#494	—	2	2	no	—
77	#494	C₂²×C₇	1	4	yes	complemented
78	#493	—	2	2	no	—
79	#493	C₂²×C₇	1	4	yes	—
80	#486	—	2	8	no	—
81	#486	C₂²×C₇	1	16	yes	complemented
82	C₂×#102	—	2	8	no	—
83	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
84	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
85	C₈²	C₂²×C₇	1	4	yes	complemented
86	C₂³×C₇	—	4	4	no	—
87	C₂³×C₇	—	8	4	no	—
88	C₂³×C₇	—	8	8	no	—
89	C₂³×C₇	#127	1	1	yes	characteristic
90	C₇×Q₈	—	4	8	no	—
91	C₇×Q₈	—	8	4	no	—
92	C₇×D₄	—	4	8	no	—
93	C₇×D₄	—	4	16	no	—
94	C₇×D₄	—	8	4	no	—
95	C₇×D₄	—	8	8	no	—
96	C₂×C₄×C₇	—	4	4	no	—
97	C₂×C₄×C₇	—	8	4	no	—
98	C₂×C₄×C₇	#127	1	1	yes	characteristic
99	C₂×C₄×C₇	#127	1	2	yes	—
100	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
101	C₂×C₄×C₇	#135	1	8	yes	—
102	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
104	C₇×C₈	—	2	8	no	—
105	C₇×C₈	C₂×D₈	1	8	yes	—
106	C₂²×Q₈	—	2	2	no	—
107	C₂²×D₄	—	2	2	no	—
108	C₂²×D₄	—	2	4	no	—
109	C₂×Q₁₆	—	2	8	no	—
110	C₂×Q₁₆	—	4	8	no	—
111	C₂×D₈	—	2	8	no	—
112	C₂×D₈	—	4	8	no	—
113	C₂²×C₈	C₇×D₄	1	4	yes	—
114	C₂²×C₈	C₇×D₄	1	2	yes	—
115	C₄⋊Q₈	—	2	4	no	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
118	#127	—	2	4	no	—
119	#127	—	2	2	no	—
120	#127	C₂³×C₇	1	4	yes	—
121	C₂×C₄⋊C₄	—	2	4	no	—
122	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
123	#102	—	2	32	no	—
124	C₄×C₈	C₇×D₄	1	4	yes	—
125	C₄×C₈	C₇×D₄	1	2	yes	—
126	C₄×C₈	C₂³×C₇	1	4	yes	—
127	C₄×C₈	C₂³×C₇	1	2	yes	—
128	C₂²×C₇	—	8	4	no	—
129	C₂²×C₇	—	8	8	no	—
130	C₂²×C₇	—	16	8	no	—
131	C₂²×C₇	#493	1	2	yes	—
132	C₂²×C₇	#495	1	4	yes	—
133	C₂²×C₇	C₂×#127	1	1	yes	characteristic
134	C₄×C₇	—	8	4	no	—
135	C₄×C₇	#493	1	2	yes	—
136	C₄×C₇	#495	1	4	yes	—
137	C₄×C₇	C₂²×D₈	1	2	yes	—
138	C₄×C₇	C₂×#135	1	4	yes	—
139	C₂⁴	—	4	2	no	—
140	C₂×Q₈	—	2	4	no	—
141	C₂×Q₈	—	4	2	no	—
142	C₂×Q₈	—	4	8	no	—
143	C₂×D₄	—	2	8	no	—
144	C₂×D₄	—	2	4	no	—
145	C₂×D₄	—	4	2	no	—
146	C₂×D₄	—	4	4	no	—
147	C₂×D₄	—	4	8	no	—
148	C₂×D₄	—	4	16	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
152	Q₁₆	—	4	16	no	—
153	D₈	—	4	16	no	—
154	C₂×C₈	—	2	8	no	—
155	C₂×C₈	C₇×D₈	1	8	yes	—
156	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
157	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
158	C₄⋊C₄	—	2	8	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄²	C₂×C₇×D₄	1	2	yes	—
161	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
162	C₄²	C₂⁴×C₇	1	1	yes	characteristic
163	C₂×C₇	—	16	4	no	—
164	C₂×C₇	#1709	1	4	yes	direct product factor
165	C₂×C₇	C₂×#493	1	1	yes	characteristic
166	C₂×C₇	C₂×#495	1	2	yes	—
167	C₂³	—	4	4	no	—
168	C₂³	—	8	4	no	—
169	C₂³	—	8	8	no	—
170	C₂³	C₇×#127	1	1	yes	characteristic
171	Q₈	—	4	8	no	—
172	Q₈	—	8	4	no	—
173	D₄	—	4	8	no	—
174	D₄	—	4	16	no	—
175	D₄	—	8	4	no	—
176	D₄	—	8	8	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	8	4	no	—
179	C₂×C₄	C₇×#127	1	1	yes	characteristic
180	C₂×C₄	C₇×#127	1	2	yes	—
181	C₂×C₄	C₂×C₇×D₈	1	4	yes	—
182	C₂×C₄	C₇×#135	1	8	yes	—
183	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
185	C₈	—	2	8	no	—
186	C₈	C₂×C₇×D₈	1	8	yes	—
187	C₇	C₂×#1709	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²	—	8	4	no	—
189	C₂²	—	8	8	no	—
190	C₂²	—	16	8	no	—
191	C₂²	C₇×#493	1	2	yes	—
192	C₂²	C₇×#495	1	4	yes	—
193	C₂²	C₂×C₇×#127	1	1	yes	characteristic
194	C₄	—	8	4	no	—
195	C₄	C₇×#493	1	2	yes	—
196	C₄	C₇×#495	1	4	yes	—
197	C₄	C₂²×C₇×D₈	1	2	yes	—
198	C₄	C₂×C₇×#135	1	4	yes	—
199	C₂	—	16	4	no	—
200	C₂	C₇×#1709	1	4	yes	direct product factor
201	C₂	C₂×C₇×#493	1	1	yes	characteristic
202	C₂	C₂×C₇×#495	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂×C₇×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#494	C₂	semidirect
3. C₂×C₇×#486	C₂	semidirect
4. C₇×#1709	C₂	direct (N × H)
5. C₂×C₇×C₈²	C₂	semidirect
11. C₇×#494	C₂²	semidirect
15. C₇×#486	C₂²	semidirect
19. C₇×C₈²	C₂²	semidirect
20. C₂×#1709	C₇	direct (N × H)
43. C₂×#494	C₂×C₇	semidirect
45. C₂×#486	C₂×C₇	semidirect
46. #1709	C₂×C₇	direct (N × H)
47. C₂×C₈²	C₂×C₇	semidirect
77. #494	C₂²×C₇	semidirect
81. #486	C₂²×C₇	semidirect
85. C₈²	C₂²×C₇	semidirect
164. C₂×C₇	#1709	direct (N × H)
187. C₇	C₂×#1709	direct (N × H)
200. C₂	C₇×#1709	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#494	2
C₂×C₇×#493	2
C₂×C₇×#486	2
C₇×#1709	2
C₂×C₇×C₈²	2
C₂×#1709	7

Subgroups — C2×C7×#1709 #13196813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1709 #13196813