Subgroups — C₇×#19608 #13196824

Order 1792.

798 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3167	C₂	1	4	yes	maximal, complemented
2	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3147	C₂	1	1	yes	maximal, characteristic
4	C₇×#1716	C₂	1	2	yes	maximal, complemented
5	C₇×#1715	C₂	1	2	yes	maximal, complemented
6	C₇×#1714	C₂	1	4	yes	maximal, complemented
7	C₂×C₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×#135	—	2	2	no	—
9	C₇×#533	C₂²	1	2	yes	—
10	C₇×#532	C₂²	1	2	yes	—
11	C₇×#500	C₂²	1	2	yes	—
12	C₇×#499	—	2	2	no	—
13	C₇×#498	C₂²	1	2	yes	—
14	C₇×#498	C₂²	1	4	yes	—
15	C₇×#495	—	2	2	no	—
16	C₇×#494	C₂²	1	1	yes	characteristic
17	C₇×#493	C₂²	1	1	yes	characteristic
18	C₇×#492	C₂²	1	2	yes	—
19	C₇×#492	C₂²	1	4	yes	—
20	C₇×#487	C₂²	1	4	yes	—
21	C₇×#486	C₂²	1	4	yes	—
22	C₇×#465	—	2	8	no	—
23	C₄×C₇×SD₁₆	—	2	8	no	—
24	C₇×#427	—	2	2	no	—
25	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
27	C₇×#322	C₂²	1	2	yes	—
28	C₇×#322	C₂²	1	2	yes	—
29	#19608	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₇×#41	—	2	2	no	—
31	C₇×#135	—	4	8	no	—
32	C₂×C₇×Q₁₆	—	2	2	no	—
33	C₂×C₇×SD₁₆	—	2	4	no	—
34	C₂×C₇×SD₁₆	—	2	8	no	—
35	C₂×C₇×D₈	—	2	2	no	—
36	C₂²×C₇×C₈	D₄	1	4	yes	—
37	C₂²×C₇×C₈	D₄	1	2	yes	—
38	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
39	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
40	C₇×#127	C₂³	1	2	yes	—
41	C₇×#125	—	2	2	no	—
42	C₇×#124	—	2	2	no	—
43	C₇×C₂²⋊Q₈	—	2	4	no	—
44	C₇×#121	—	2	4	no	—
45	C₄×C₇×Q₈	—	2	4	no	—
46	C₄×C₇×D₄	—	2	4	no	—
47	C₇×#117	—	2	2	no	—
48	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
49	C₇×#107	—	2	4	no	—
50	C₇×#106	—	2	4	no	—
51	C₇×#106	—	2	8	no	—
52	C₇×#103	—	2	8	no	—
53	C₇×#102	—	2	8	no	—
54	C₄×C₇×C₈	D₄	1	2	yes	—
55	C₄×C₇×C₈	D₄	1	4	yes	—
56	C₄×C₇×C₈	C₂³	1	4	yes	—
57	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
58	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
59	#3167	C₂×C₇	1	4	yes	complemented
60	#3160	C₂×C₇	1	1	yes	characteristic, complemented
61	#3147	C₂×C₇	1	1	yes	characteristic
62	#1716	C₂×C₇	1	2	yes	complemented
63	#1715	C₂×C₇	1	2	yes	complemented
64	#1714	C₂×C₇	1	4	yes	complemented
65	C₂×#322	C₂×C₇	1	1	yes	characteristic, complemented
66	C₇×#41	—	4	8	no	—
67	C₂×C₇×Q₈	—	2	4	no	—
68	C₂×C₇×Q₈	—	2	2	no	—
69	C₂×C₇×D₄	—	2	4	no	—
70	C₂×C₇×D₄	—	2	2	no	—
71	C₂×C₇×D₄	—	4	2	no	—
72	C₂²×C₄×C₇	—	4	2	no	—
73	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
75	C₇×Q₁₆	—	4	4	no	—
76	C₇×SD₁₆	—	4	16	no	—
77	C₇×SD₁₆	—	4	8	no	—
78	C₇×D₈	—	4	4	no	—
79	C₂×C₇×C₈	—	4	4	no	—
80	C₂×C₇×C₈	SD₁₆	1	8	yes	complemented
81	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
82	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
83	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
84	C₇×C₄⋊C₄	—	2	4	no	—
85	C₇×C₄⋊C₄	—	2	4	no	—
86	C₇×C₄⋊C₄	—	2	4	no	—
87	C₇×C₄⋊C₄	—	4	4	no	—
88	C₇×C₂²⋊C₄	—	4	2	no	—
89	C₇×C₂²⋊C₄	—	4	4	no	—
90	C₄²×C₇	—	4	2	no	—
91	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₇	C₂×D₄	1	2	yes	—
93	C₄²×C₇	C₂⁴	1	1	yes	characteristic
94	C₂×#135	—	2	2	no	—
95	#533	C₂²×C₇	1	2	yes	—
96	#532	C₂²×C₇	1	2	yes	—
97	#500	C₂²×C₇	1	2	yes	—
98	#499	—	2	2	no	—
99	#498	C₂²×C₇	1	2	yes	—
100	#498	C₂²×C₇	1	4	yes	—
101	#495	—	2	2	no	—
102	#494	C₂²×C₇	1	1	yes	characteristic
103	#493	C₂²×C₇	1	1	yes	characteristic
104	#492	C₂²×C₇	1	2	yes	—
105	#492	C₂²×C₇	1	4	yes	—
106	#487	C₂²×C₇	1	4	yes	—
107	#486	C₂²×C₇	1	4	yes	—
108	#465	—	2	8	no	—
109	C₄×SD₁₆	—	2	8	no	—
110	#427	—	2	2	no	—
111	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
113	#322	C₂²×C₇	1	2	yes	—
114	#322	C₂²×C₇	1	2	yes	—
115	C₂³×C₇	—	4	2	no	—
116	C₂³×C₇	#127	1	1	yes	characteristic
117	C₇×Q₈	—	4	4	no	—
118	C₇×Q₈	—	4	8	no	—
119	C₇×D₄	—	4	4	no	—
120	C₇×D₄	—	4	8	no	—
121	C₇×D₄	—	8	4	no	—
122	C₂×C₄×C₇	—	2	4	no	—
123	C₂×C₄×C₇	—	4	4	no	—
124	C₂×C₄×C₇	—	4	2	no	—
125	C₂×C₄×C₇	—	8	4	no	—
126	C₂×C₄×C₇	#127	1	2	yes	—
127	C₂×C₄×C₇	#127	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
129	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
131	C₇×C₈	—	2	8	no	—
132	C₇×C₈	C₂×SD₁₆	1	4	yes	—
133	C₇×C₈	C₂×SD₁₆	1	4	yes	—
134	C₂×#41	—	2	2	no	—
135	#135	—	4	8	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×SD₁₆	—	2	4	no	—
138	C₂×SD₁₆	—	2	8	no	—
139	C₂×D₈	—	2	2	no	—
140	C₂²×C₈	C₇×D₄	1	4	yes	—
141	C₂²×C₈	C₇×D₄	1	2	yes	—
142	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
143	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
144	#127	C₂³×C₇	1	2	yes	—
145	#125	—	2	2	no	—
146	#124	—	2	2	no	—
147	C₂²⋊Q₈	—	2	4	no	—
148	#121	—	2	4	no	—
149	C₄×Q₈	—	2	4	no	—
150	C₄×D₄	—	2	4	no	—
151	#117	—	2	2	no	—
152	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
153	#107	—	2	4	no	—
154	#106	—	2	4	no	—
155	#106	—	2	8	no	—
156	#103	—	2	8	no	—
157	#102	—	2	8	no	—
158	C₄×C₈	C₇×D₄	1	4	yes	—
159	C₄×C₈	C₇×D₄	1	2	yes	—
160	C₄×C₈	C₂³×C₇	1	4	yes	—
161	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
162	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
163	C₂²×C₇	—	2	2	no	—
164	C₂²×C₇	—	8	2	no	—
165	C₂²×C₇	—	8	4	no	—
166	C₂²×C₇	#492	1	2	yes	—
167	C₂²×C₇	C₂×#127	1	1	yes	characteristic
168	C₄×C₇	—	2	2	no	—
169	C₄×C₇	—	8	2	no	—
170	C₄×C₇	—	8	4	no	—
171	C₄×C₇	#492	1	2	yes	—
172	C₄×C₇	C₂²×SD₁₆	1	2	yes	—
173	C₄×C₇	#575	1	4	yes	—
174	#41	—	4	8	no	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×D₄	—	2	4	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
182	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
183	Q₁₆	—	4	4	no	—
184	SD₁₆	—	4	16	no	—
185	SD₁₆	—	4	8	no	—
186	D₈	—	4	4	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	C₇×SD₁₆	1	8	yes	complemented
189	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
190	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
191	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	4	4	no	—
198	C₄²	—	4	2	no	—
199	C₄²	C₂×C₇×D₄	1	2	yes	—
200	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
201	C₄²	C₂⁴×C₇	1	1	yes	characteristic
202	C₂×C₇	—	2	2	no	—
203	C₂×C₇	—	16	2	no	—
204	C₂×C₇	C₂×#492	1	1	yes	characteristic
205	C₂×C₇	#3150	1	2	yes	—
206	C₂³	—	4	2	no	—
207	C₂³	C₇×#127	1	1	yes	characteristic
208	Q₈	—	4	4	no	—
209	Q₈	—	4	8	no	—
210	D₄	—	4	4	no	—
211	D₄	—	4	8	no	—
212	D₄	—	8	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	8	4	no	—
217	C₂×C₄	C₇×#127	1	2	yes	—
218	C₂×C₄	C₇×#127	1	1	yes	characteristic
219	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
220	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
222	C₈	—	2	8	no	—
223	C₈	C₂×C₇×SD₁₆	1	4	yes	—
224	C₈	C₂×C₇×SD₁₆	1	4	yes	—
225	C₇	#19608	1	1	yes	characteristic, direct product factor, Sylow
226	C₂²	—	2	2	no	—
227	C₂²	—	8	2	no	—
228	C₂²	—	8	4	no	—
229	C₂²	C₇×#492	1	2	yes	—
230	C₂²	C₂×C₇×#127	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	8	2	no	—
233	C₄	—	8	4	no	—
234	C₄	C₇×#492	1	2	yes	—
235	C₄	C₂²×C₇×SD₁₆	1	2	yes	—
236	C₄	C₇×#575	1	4	yes	—
237	C₂	—	2	2	no	—
238	C₂	—	16	2	no	—
239	C₂	C₂×C₇×#492	1	1	yes	characteristic
240	C₂	C₇×#3150	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂²×C₇	C₂×#127
prime radical	C₇×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3167	C₂	semidirect
2. C₇×#3160	C₂	semidirect
4. C₇×#1716	C₂	semidirect
5. C₇×#1715	C₂	semidirect
6. C₇×#1714	C₂	semidirect
7. C₂×C₇×#322	C₂	semidirect
29. #19608	C₇	direct (N × H)
59. #3167	C₂×C₇	semidirect
60. #3160	C₂×C₇	semidirect
62. #1716	C₂×C₇	semidirect
63. #1715	C₂×C₇	semidirect
64. #1714	C₂×C₇	semidirect
65. C₂×#322	C₂×C₇	semidirect
80. C₂×C₇×C₈	SD₁₆	semidirect
188. C₂×C₈	C₇×SD₁₆	semidirect
225. C₇	#19608	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×#3167	2
C₇×#3160	2
C₇×#3147	2
C₇×#1716	2
C₇×#1715	2
C₇×#1714	2
C₂×C₇×#322	2
#19608	7

Subgroups — C7×#19608 #13196824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19608 #13196824