Subgroups — C₇×#19662 #13196878

Order 1792.

838 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3226	C₂	1	4	yes	maximal, complemented
2	C₇×#3147	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1722	C₂	1	4	yes	maximal, complemented
5	C₇×#1713	C₂	1	2	yes	maximal, complemented
6	C₇×#1709	C₂	1	2	yes	maximal, complemented
7	C₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×#135	—	2	2	no	—
9	C₂²×C₇×Q₁₆	—	2	2	no	—
10	C₇×#532	C₂²	1	2	yes	—
11	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₇×#500	C₂²	1	8	yes	—
13	C₇×#495	—	2	2	no	—
14	C₇×#494	—	2	2	no	—
15	C₇×#494	C₂²	1	2	yes	—
16	C₇×#494	C₂²	1	1	yes	characteristic
17	C₇×#494	C₂²	1	2	yes	—
18	C₇×#493	C₂²	1	1	yes	characteristic
19	C₇×#492	C₂²	1	2	yes	—
20	C₇×#486	C₂²	1	8	yes	—
21	C₇×#480	—	2	8	no	—
22	C₇×#474	—	2	8	no	—
23	C₇×#431	C₂²	1	2	yes	—
24	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
25	C₇×#322	C₂²	1	2	yes	—
26	C₇×C₈²	C₂²	1	2	yes	—
27	#19662	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₇×#41	—	2	2	no	—
29	C₂²×C₇×Q₈	—	2	2	no	—
30	C₇×#135	—	4	8	no	—
31	C₂×C₇×Q₁₆	—	2	4	no	—
32	C₂×C₇×Q₁₆	—	2	4	no	—
33	C₂×C₇×Q₁₆	—	2	2	no	—
34	C₂×C₇×Q₁₆	—	4	8	no	—
35	C₂×C₇×SD₁₆	—	2	4	no	—
36	C₂×C₇×D₈	—	2	2	no	—
37	C₂²×C₇×C₈	D₄	1	2	yes	—
38	C₇×C₄⋊Q₈	—	2	4	no	—
39	C₇×C₄⋊Q₈	—	2	2	no	—
40	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
41	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
42	C₇×#127	C₂³	1	2	yes	—
43	C₇×#124	—	2	2	no	—
44	C₇×#121	—	2	4	no	—
45	C₄×C₇×D₄	—	2	4	no	—
46	C₂×C₇×C₄⋊C₄	—	2	4	no	—
47	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
48	C₇×#107	—	2	16	no	—
49	C₇×C₄⋊C₈	D₄	1	4	yes	—
50	C₇×#102	—	2	16	no	—
51	C₇×C₂²⋊C₈	D₄	1	4	yes	—
52	C₄×C₇×C₈	D₄	1	2	yes	—
53	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
54	C₄×C₇×C₈	C₂³	1	4	yes	—
55	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
56	#3226	C₂×C₇	1	4	yes	complemented
57	#3147	C₂×C₇	1	1	yes	characteristic
58	C₂×#494	C₂×C₇	1	1	yes	characteristic, complemented
59	#1722	C₂×C₇	1	4	yes	complemented
60	#1713	C₂×C₇	1	2	yes	complemented
61	#1709	C₂×C₇	1	2	yes	complemented
62	#1450	C₂×C₇	1	1	yes	characteristic, complemented
63	C₇×#41	—	4	8	no	—
64	C₂×C₇×Q₈	—	2	2	no	—
65	C₂×C₇×Q₈	—	2	4	no	—
66	C₂×C₇×Q₈	—	4	2	no	—
67	C₂×C₇×Q₈	—	4	8	no	—
68	C₂×C₇×D₄	—	2	4	no	—
69	C₂×C₇×D₄	—	2	2	no	—
70	C₂×C₇×D₄	—	4	2	no	—
71	C₂²×C₄×C₇	—	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	8	no	—
76	C₇×Q₁₆	—	4	8	no	—
77	C₇×Q₁₆	—	4	4	no	—
78	C₇×SD₁₆	—	4	8	no	—
79	C₇×D₈	—	4	4	no	—
80	C₂×C₇×C₈	D₈	1	8	yes	—
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	8	no	—
85	C₇×C₄⋊C₄	—	2	4	no	—
86	C₇×C₄⋊C₄	—	4	4	no	—
87	C₇×C₂²⋊C₄	—	4	4	no	—
88	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₇	C₂×D₄	1	2	yes	—
90	C₄²×C₇	C₂⁴	1	1	yes	characteristic
91	C₂×#135	—	2	2	no	—
92	C₂²×Q₁₆	—	2	2	no	—
93	#532	C₂²×C₇	1	2	yes	—
94	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
95	#500	C₂²×C₇	1	8	yes	—
96	#495	—	2	2	no	—
97	#494	—	2	2	no	—
98	#494	C₂²×C₇	1	2	yes	—
99	#494	C₂²×C₇	1	2	yes	—
100	#494	C₂²×C₇	1	1	yes	characteristic
101	#493	C₂²×C₇	1	1	yes	characteristic
102	#492	C₂²×C₇	1	2	yes	—
103	#486	C₂²×C₇	1	8	yes	—
104	#480	—	2	8	no	—
105	#474	—	2	8	no	—
106	#431	C₂²×C₇	1	2	yes	—
107	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
108	#322	C₂²×C₇	1	2	yes	—
109	C₈²	C₂²×C₇	1	2	yes	—
110	C₂³×C₇	—	4	2	no	—
111	C₂³×C₇	#127	1	1	yes	characteristic
112	C₇×Q₈	—	4	4	no	—
113	C₇×Q₈	—	4	8	no	—
114	C₇×Q₈	—	8	4	no	—
115	C₇×D₄	—	4	4	no	—
116	C₇×D₄	—	4	8	no	—
117	C₇×D₄	—	8	4	no	—
118	C₂×C₄×C₇	—	2	4	no	—
119	C₂×C₄×C₇	—	4	2	no	—
120	C₂×C₄×C₇	—	4	4	no	—
121	C₂×C₄×C₇	—	8	4	no	—
122	C₂×C₄×C₇	—	8	4	no	—
123	C₂×C₄×C₇	#127	1	2	yes	—
124	C₂×C₄×C₇	#127	1	1	yes	characteristic
125	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
126	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
128	C₇×C₈	—	2	8	no	—
129	C₇×C₈	C₂×D₈	1	4	yes	—
130	C₇×C₈	C₂×D₈	1	4	yes	—
131	C₂×#41	—	2	2	no	—
132	C₂²×Q₈	—	2	2	no	—
133	#135	—	4	8	no	—
134	C₂×Q₁₆	—	2	4	no	—
135	C₂×Q₁₆	—	2	4	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	4	8	no	—
138	C₂×SD₁₆	—	2	4	no	—
139	C₂×D₈	—	2	2	no	—
140	C₂²×C₈	C₇×D₄	1	2	yes	—
141	C₄⋊Q₈	—	2	4	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
144	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
145	#127	C₂³×C₇	1	2	yes	—
146	#124	—	2	2	no	—
147	#121	—	2	4	no	—
148	C₄×D₄	—	2	4	no	—
149	C₂×C₄⋊C₄	—	2	4	no	—
150	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
151	#107	—	2	16	no	—
152	C₄⋊C₈	C₇×D₄	1	4	yes	—
153	#102	—	2	16	no	—
154	C₂²⋊C₈	C₇×D₄	1	4	yes	—
155	C₄×C₈	C₇×D₄	1	2	yes	—
156	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
157	C₄×C₈	C₂³×C₇	1	4	yes	—
158	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
159	C₂²×C₇	—	2	2	no	—
160	C₂²×C₇	—	8	2	no	—
161	C₂²×C₇	—	8	4	no	—
162	C₂²×C₇	#493	1	2	yes	—
163	C₂²×C₇	C₂×#127	1	1	yes	characteristic
164	C₄×C₇	—	2	2	no	—
165	C₄×C₇	—	8	2	no	—
166	C₄×C₇	—	8	4	no	—
167	C₄×C₇	#493	1	2	yes	—
168	C₄×C₇	C₂²×D₈	1	2	yes	—
169	C₄×C₇	#578	1	4	yes	—
170	#41	—	4	8	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	8	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
181	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
182	Q₁₆	—	4	8	no	—
183	Q₁₆	—	4	8	no	—
184	Q₁₆	—	4	4	no	—
185	SD₁₆	—	4	8	no	—
186	D₈	—	4	4	no	—
187	C₂×C₈	C₇×D₈	1	8	yes	—
188	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
189	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
190	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₂²⋊C₄	—	4	4	no	—
195	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
196	C₄²	C₂×C₇×D₄	1	2	yes	—
197	C₄²	C₂⁴×C₇	1	1	yes	characteristic
198	C₂×C₇	—	2	2	no	—
199	C₂×C₇	—	16	2	no	—
200	C₂×C₇	C₂×#493	1	1	yes	characteristic
201	C₂×C₇	#3156	1	2	yes	—
202	C₂³	—	4	2	no	—
203	C₂³	C₇×#127	1	1	yes	characteristic
204	Q₈	—	4	4	no	—
205	Q₈	—	4	8	no	—
206	Q₈	—	8	4	no	—
207	D₄	—	4	4	no	—
208	D₄	—	4	8	no	—
209	D₄	—	8	4	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	8	4	no	—
214	C₂×C₄	—	8	4	no	—
215	C₂×C₄	C₇×#127	1	2	yes	—
216	C₂×C₄	C₇×#127	1	1	yes	characteristic
217	C₂×C₄	C₂×C₇×D₈	1	4	yes	—
218	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
219	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
220	C₈	—	2	8	no	—
221	C₈	C₂×C₇×D₈	1	4	yes	—
222	C₈	C₂×C₇×D₈	1	4	yes	—
223	C₇	#19662	1	1	yes	characteristic, direct product factor, Sylow
224	C₂²	—	2	2	no	—
225	C₂²	—	8	4	no	—
226	C₂²	—	8	2	no	—
227	C₂²	C₇×#493	1	2	yes	—
228	C₂²	C₂×C₇×#127	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	8	2	no	—
231	C₄	—	8	4	no	—
232	C₄	C₇×#493	1	2	yes	—
233	C₄	C₂²×C₇×D₈	1	2	yes	—
234	C₄	C₇×#578	1	4	yes	—
235	C₂	—	2	2	no	—
236	C₂	—	16	2	no	—
237	C₂	C₂×C₇×#493	1	1	yes	characteristic
238	C₂	C₇×#3156	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₇×#3226	C₂	semidirect
3. C₂×C₇×#494	C₂	semidirect
4. C₇×#1722	C₂	semidirect
5. C₇×#1713	C₂	semidirect
6. C₇×#1709	C₂	semidirect
7. C₇×#1450	C₂	semidirect
27. #19662	C₇	direct (N × H)
56. #3226	C₂×C₇	semidirect
58. C₂×#494	C₂×C₇	semidirect
59. #1722	C₂×C₇	semidirect
60. #1713	C₂×C₇	semidirect
61. #1709	C₂×C₇	semidirect
62. #1450	C₂×C₇	semidirect
223. C₇	#19662	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×#3226	2
C₇×#3147	2
C₂×C₇×#494	2
C₇×#1722	2
C₇×#1713	2
C₇×#1709	2
C₇×#1450	2
#19662	7

Subgroups — C7×#19662 #13196878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19662 #13196878