Subgroups — C₇×#17828 #13195044

Order 1792.

702 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₈×#41	C₂	1	1	yes	maximal, characteristic
2	C₇×#2906	C₂	1	2	yes	maximal
3	C₇×#1571	C₂	1	6	yes	maximal, complemented
4	C₇×#1567	C₂	1	3	yes	maximal, complemented
5	C₂×C₇×#335	C₂	1	3	yes	maximal, complemented
6	C₂×C₇×#131	—	2	2	no	—
7	C₄×C₇×#41	C₂²	1	1	yes	characteristic
8	C₇×C₈×Q₈	C₄	1	1	yes	characteristic
9	C₇×C₈×Q₈	C₄	1	1	yes	characteristic
10	C₇×C₈×D₄	C₄	1	3	yes	—
11	C₇×C₈×D₄	C₄	1	3	yes	—
12	C₇×#431	C₂²	1	3	yes	—
13	C₇×#424	—	2	6	no	—
14	C₇×#423	C₂²	1	6	yes	—
15	C₂×C₇×C₄⋊C₈	C₂²	1	6	yes	—
16	C₂×C₄×C₇×C₈	C₂²	1	3	yes	—
17	C₇×#335	C₂²	1	1	yes	characteristic
18	C₇×#335	C₂²	1	3	yes	—
19	C₇×#335	C₂²	1	6	yes	—
20	C₇×#335	C₂²	1	6	yes	—
21	#17828	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₇×#41	D₄	1	1	yes	characteristic
23	C₇×#131	—	4	8	no	—
24	C₂×C₇×M₁₆	—	2	6	no	—
25	C₂²×C₇×C₈	—	2	6	no	—
26	C₂²×C₇×C₈	C₂×C₄	1	3	yes	—
27	C₂²×C₇×C₈	C₂×C₄	1	3	yes	—
28	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₇×Q₈	Q₈	1	1	yes	characteristic
30	C₄×C₇×D₄	C₂×C₄	1	3	yes	—
31	C₄×C₇×D₄	Q₈	1	3	yes	—
32	C₇×#117	D₄	1	3	yes	—
33	C₂×C₄²×C₇	C₂³	1	3	yes	—
34	C₇×C₄⋊C₈	—	2	6	no	—
35	C₇×C₄⋊C₈	—	2	6	no	—
36	C₇×C₄⋊C₈	C₂×C₄	1	3	yes	—
37	C₇×C₄⋊C₈	C₂×C₄	1	3	yes	—
38	C₇×C₄⋊C₈	C₂³	1	2	yes	—
39	C₇×C₄⋊C₈	C₂³	1	6	yes	—
40	C₇×C₂²⋊C₈	C₂×C₄	1	6	yes	—
41	C₄×C₇×C₈	C₂×C₄	1	3	yes	—
42	C₄×C₇×C₈	C₂×C₄	1	3	yes	—
43	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
44	C₄×C₇×C₈	C₂³	1	3	yes	—
45	C₈×#41	C₂×C₇	1	1	yes	characteristic
46	#2906	C₂×C₇	1	2	yes	—
47	#1571	C₂×C₇	1	6	yes	complemented
48	#1567	C₂×C₇	1	3	yes	complemented
49	C₂×#335	C₂×C₇	1	3	yes	complemented
50	C₇×#41	D₈	1	4	yes	—
51	C₇×#41	Q₁₆	1	4	yes	—
52	C₂×C₇×Q₈	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₇×D₄	C₄⋊C₄	1	3	yes	—
54	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
55	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₇	C₂×Q₈	1	3	yes	—
57	C₇×M₁₆	—	4	12	no	—
58	C₂×C₇×C₈	—	2	6	no	—
59	C₂×C₇×C₈	—	2	2	no	—
60	C₂×C₇×C₈	—	2	6	no	—
61	C₂×C₇×C₈	—	4	3	no	—
62	C₂×C₇×C₈	—	4	12	no	—
63	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
65	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
66	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₇×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
68	C₇×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
69	C₇×C₂²⋊C₄	C₄⋊C₄	1	6	yes	—
70	C₄²×C₇	C₂²×C₄	1	3	yes	—
71	C₄²×C₇	C₂×D₄	1	3	yes	—
72	C₄²×C₇	C₂×Q₈	1	3	yes	—
73	C₄²×C₇	C₂⁴	1	1	yes	characteristic
74	C₂×#131	—	2	2	no	—
75	C₄×#41	C₂²×C₇	1	1	yes	characteristic
76	C₈×Q₈	C₄×C₇	1	1	yes	characteristic
77	C₈×Q₈	C₄×C₇	1	1	yes	characteristic
78	C₈×D₄	C₄×C₇	1	3	yes	—
79	C₈×D₄	C₄×C₇	1	3	yes	—
80	#431	C₂²×C₇	1	3	yes	—
81	#424	—	2	6	no	—
82	#423	C₂²×C₇	1	6	yes	—
83	C₂×C₄⋊C₈	C₂²×C₇	1	6	yes	—
84	C₂×C₄×C₈	C₂²×C₇	1	3	yes	—
85	#335	C₂²×C₇	1	1	yes	characteristic
86	#335	C₂²×C₇	1	3	yes	—
87	#335	C₂²×C₇	1	6	yes	—
88	#335	C₂²×C₇	1	6	yes	—
89	C₂³×C₇	C₂×C₄⋊C₄	1	3	yes	—
90	C₇×Q₈	#107	1	4	yes	—
91	C₇×D₄	#107	1	12	yes	—
92	C₂×C₄×C₇	—	2	3	no	—
93	C₂×C₄×C₇	—	2	3	no	—
94	C₂×C₄×C₇	—	4	3	no	—
95	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
96	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
97	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
98	C₂×C₄×C₇	C₂×D₈	1	6	yes	—
99	C₂×C₄×C₇	C₂×Q₁₆	1	6	yes	—
100	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
103	C₇×C₈	—	2	3	no	—
104	C₇×C₈	—	2	1	no	—
105	C₇×C₈	—	2	3	no	—
106	C₇×C₈	—	4	2	no	—
107	C₇×C₈	—	4	6	no	—
108	C₇×C₈	#131	1	2	yes	complemented
109	C₂×#41	C₇×D₄	1	1	yes	characteristic
110	#131	—	4	8	no	—
111	C₂×M₁₆	—	2	6	no	—
112	C₂²×C₈	—	2	6	no	—
113	C₂²×C₈	C₂×C₄×C₇	1	3	yes	—
114	C₂²×C₈	C₂×C₄×C₇	1	3	yes	—
115	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
116	C₄×Q₈	C₇×Q₈	1	1	yes	characteristic
117	C₄×D₄	C₂×C₄×C₇	1	3	yes	—
118	C₄×D₄	C₇×Q₈	1	3	yes	—
119	#117	C₇×D₄	1	3	yes	—
120	C₂×C₄²	C₂³×C₇	1	3	yes	—
121	C₄⋊C₈	—	2	6	no	—
122	C₄⋊C₈	—	2	6	no	—
123	C₄⋊C₈	C₂×C₄×C₇	1	3	yes	—
124	C₄⋊C₈	C₂×C₄×C₇	1	3	yes	—
125	C₄⋊C₈	C₂³×C₇	1	2	yes	—
126	C₄⋊C₈	C₂³×C₇	1	6	yes	—
127	C₂²⋊C₈	C₂×C₄×C₇	1	6	yes	—
128	C₄×C₈	C₂×C₄×C₇	1	3	yes	—
129	C₄×C₈	C₂×C₄×C₇	1	3	yes	—
130	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
131	C₄×C₈	C₂³×C₇	1	3	yes	—
132	C₂²×C₇	—	2	3	no	—
133	C₂²×C₇	—	2	3	no	—
134	C₂²×C₇	C₂×#107	1	6	yes	—
135	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₄×C₇	—	2	3	no	—
137	C₄×C₇	—	2	1	no	—
138	C₄×C₇	—	2	3	no	—
139	C₄×C₇	C₂×#107	1	6	yes	—
140	C₄×C₇	C₂×#131	1	1	yes	characteristic
141	C₄×C₇	#568	1	1	yes	characteristic
142	C₄×C₇	C₂²×D₈	1	1	yes	characteristic
143	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
144	#41	C₇×D₈	1	4	yes	—
145	#41	C₇×Q₁₆	1	4	yes	—
146	C₂×Q₈	C₇×C₄⋊C₄	1	1	yes	characteristic
147	C₂×D₄	C₇×C₄⋊C₄	1	3	yes	—
148	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
149	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
150	C₂²×C₄	C₂×C₇×Q₈	1	3	yes	—
151	M₁₆	—	4	12	no	—
152	C₂×C₈	—	2	6	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	6	no	—
155	C₂×C₈	—	4	3	no	—
156	C₂×C₈	—	4	12	no	—
157	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
158	C₂×C₈	C₂²×C₄×C₇	1	3	yes	—
159	C₂×C₈	C₂²×C₄×C₇	1	3	yes	—
160	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
161	C₄⋊C₄	C₇×C₄⋊C₄	1	3	yes	—
162	C₄⋊C₄	C₇×C₄⋊C₄	1	3	yes	—
163	C₂²⋊C₄	C₇×C₄⋊C₄	1	6	yes	—
164	C₄²	C₂²×C₄×C₇	1	3	yes	—
165	C₄²	C₂×C₇×D₄	1	3	yes	—
166	C₄²	C₂×C₇×Q₈	1	3	yes	—
167	C₄²	C₂⁴×C₇	1	1	yes	characteristic
168	C₂×C₇	—	2	6	no	—
169	C₂×C₇	#2906	1	1	yes	characteristic
170	C₂×C₇	C₂²×#107	1	1	yes	characteristic
171	C₂×C₇	#2916	1	1	yes	characteristic
172	C₂³	C₂×C₇×C₄⋊C₄	1	3	yes	—
173	Q₈	C₇×#107	1	4	yes	—
174	D₄	C₇×#107	1	12	yes	—
175	C₂×C₄	—	2	3	no	—
176	C₂×C₄	—	2	3	no	—
177	C₂×C₄	—	4	3	no	—
178	C₂×C₄	C₂×C₇×C₄⋊C₄	1	3	yes	—
179	C₂×C₄	C₂×C₇×C₄⋊C₄	1	3	yes	—
180	C₂×C₄	C₂×C₇×C₄⋊C₄	1	3	yes	—
181	C₂×C₄	C₂×C₇×D₈	1	6	yes	—
182	C₂×C₄	C₂×C₇×Q₁₆	1	6	yes	—
183	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
185	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
186	C₈	—	2	3	no	—
187	C₈	—	2	1	no	—
188	C₈	—	2	3	no	—
189	C₈	—	4	2	no	—
190	C₈	—	4	6	no	—
191	C₈	C₇×#131	1	2	yes	complemented
192	C₇	#17828	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²	—	2	3	no	—
194	C₂²	—	2	3	no	—
195	C₂²	C₂×C₇×#107	1	6	yes	—
196	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
197	C₄	—	2	3	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	3	no	—
200	C₄	C₂×C₇×#107	1	6	yes	—
201	C₄	C₂×C₇×#131	1	1	yes	characteristic
202	C₄	C₇×#568	1	1	yes	characteristic
203	C₄	C₂²×C₇×D₈	1	1	yes	characteristic
204	C₄	C₂²×C₇×Q₁₆	1	1	yes	characteristic
205	C₂	—	2	6	no	—
206	C₂	C₇×#2906	1	1	yes	characteristic
207	C₂	C₂²×C₇×#107	1	1	yes	characteristic
208	C₂	C₇×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂×C₇×#41	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₄×C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1571	C₂	semidirect
4. C₇×#1567	C₂	semidirect
5. C₂×C₇×#335	C₂	semidirect
21. #17828	C₇	direct (N × H)
47. #1571	C₂×C₇	semidirect
48. #1567	C₂×C₇	semidirect
49. C₂×#335	C₂×C₇	semidirect
108. C₇×C₈	#131	semidirect
191. C₈	C₇×#131	semidirect
192. C₇	#17828	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×C₈×#41	2
C₇×#2906	2
C₇×#1571	2
C₇×#1567	2
C₂×C₇×#335	2
#17828	7

Subgroups — C7×#17828 #13195044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#17828 #13195044