Subgroups — C₇×#19699 #13196915

Order 1792.

744 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3229	C₂	1	2	yes	maximal, complemented
2	C₇×#3226	C₂	1	2	yes	maximal
3	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3131	C₂	1	1	yes	maximal, characteristic
5	C₇×#1723	C₂	1	4	yes	maximal, complemented
6	C₇×#1720	C₂	1	4	yes	maximal, complemented
7	C₇×#1452	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₇×Q₁₆	—	2	2	no	—
9	C₇×#533	C₂²	1	1	yes	characteristic
10	C₇×#532	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₇×#500	C₂²	1	4	yes	—
13	C₇×#498	C₂²	1	4	yes	—
14	C₇×#494	—	2	2	no	—
15	C₇×#494	C₂²	1	4	yes	—
16	C₇×#488	—	2	2	no	—
17	C₇×#487	C₂²	1	4	yes	—
18	C₇×#487	C₂²	1	2	yes	—
19	C₇×#486	C₂²	1	4	yes	—
20	C₇×#486	C₂²	1	2	yes	—
21	C₇×#483	—	2	4	no	—
22	C₇×#480	—	2	4	no	—
23	C₇×#475	—	2	4	no	—
24	C₇×#474	—	2	4	no	—
25	C₇×#431	C₂²	1	2	yes	—
26	C₇×#416	—	2	2	no	—
27	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
28	C₇×#322	C₂²	1	4	yes	—
29	#19699	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₇×#41	—	2	1	no	—
31	C₂²×C₇×Q₈	—	2	2	no	—
32	C₂×C₇×Q₁₆	—	2	8	no	—
33	C₂×C₇×Q₁₆	—	4	8	no	—
34	C₂²×C₇×C₈	D₄	1	2	yes	—
35	C₇×C₄⋊Q₈	—	2	2	no	—
36	C₇×C₄⋊Q₈	—	2	2	no	—
37	C₇×C₄⋊Q₈	—	2	2	no	—
38	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
39	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
40	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
41	C₇×#127	C₂³	1	1	yes	characteristic
42	C₇×#125	—	2	1	no	—
43	C₇×#124	—	2	1	no	—
44	C₇×C₂²⋊Q₈	—	2	2	no	—
45	C₇×#121	—	2	2	no	—
46	C₄×C₇×Q₈	—	2	2	no	—
47	C₄×C₇×D₄	—	2	2	no	—
48	C₇×#117	—	2	1	no	—
49	C₂×C₇×C₄⋊C₄	—	2	2	no	—
50	C₂×C₇×C₄⋊C₄	—	2	2	no	—
51	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
52	C₇×#107	—	2	8	no	—
53	C₇×#106	—	2	8	no	—
54	C₇×C₄⋊C₈	D₄	1	4	yes	—
55	C₇×#103	—	2	8	no	—
56	C₇×#103	—	2	4	no	—
57	C₇×#102	—	2	8	no	—
58	C₇×#102	—	2	4	no	—
59	C₇×C₂²⋊C₈	D₄	1	4	yes	—
60	C₄×C₇×C₈	D₄	1	2	yes	—
61	C₄×C₇×C₈	C₂³	1	2	yes	—
62	C₄×C₇×C₈	C₂³	1	4	yes	—
63	#3229	C₂×C₇	1	2	yes	complemented
64	#3226	C₂×C₇	1	2	yes	—
65	C₂×#494	C₂×C₇	1	1	yes	characteristic, complemented
66	#3131	C₂×C₇	1	1	yes	characteristic
67	#1723	C₂×C₇	1	4	yes	complemented
68	#1720	C₂×C₇	1	4	yes	complemented
69	#1452	C₂×C₇	1	1	yes	characteristic, complemented
70	C₇×#41	—	4	4	no	—
71	C₂×C₇×Q₈	—	2	2	no	—
72	C₂×C₇×Q₈	—	2	1	no	—
73	C₂×C₇×Q₈	—	2	4	no	—
74	C₂×C₇×Q₈	—	4	2	no	—
75	C₂×C₇×Q₈	—	4	8	no	—
76	C₂×C₇×D₄	—	2	2	no	—
77	C₂×C₇×D₄	—	2	1	no	—
78	C₂×C₇×D₄	—	4	1	no	—
79	C₂²×C₄×C₇	—	4	2	no	—
80	C₂²×C₄×C₇	—	4	1	no	—
81	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
83	C₇×Q₁₆	—	4	16	no	—
84	C₂×C₇×C₈	—	2	4	no	—
85	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
86	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
87	C₇×C₄⋊C₄	—	2	2	no	—
88	C₇×C₄⋊C₄	—	2	4	no	—
89	C₇×C₄⋊C₄	—	2	4	no	—
90	C₇×C₄⋊C₄	—	2	2	no	—
91	C₇×C₄⋊C₄	—	2	2	no	—
92	C₇×C₄⋊C₄	—	4	2	no	—
93	C₇×C₄⋊C₄	—	4	2	no	—
94	C₇×C₄⋊C₄	—	4	2	no	—
95	C₇×C₂²⋊C₄	—	4	1	no	—
96	C₇×C₂²⋊C₄	—	4	2	no	—
97	C₄²×C₇	—	4	1	no	—
98	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₇	C₂×D₄	1	2	yes	—
100	C₄²×C₇	C₂⁴	1	1	yes	characteristic
101	C₂²×Q₁₆	—	2	2	no	—
102	#533	C₂²×C₇	1	1	yes	characteristic
103	#532	C₂²×C₇	1	1	yes	characteristic
104	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
105	#500	C₂²×C₇	1	4	yes	—
106	#498	C₂²×C₇	1	4	yes	—
107	#494	—	2	2	no	—
108	#494	C₂²×C₇	1	4	yes	—
109	#488	—	2	2	no	—
110	#487	C₂²×C₇	1	2	yes	—
111	#487	C₂²×C₇	1	4	yes	—
112	#486	C₂²×C₇	1	2	yes	—
113	#486	C₂²×C₇	1	4	yes	—
114	#483	—	2	4	no	—
115	#480	—	2	4	no	—
116	#475	—	2	4	no	—
117	#474	—	2	4	no	—
118	#431	C₂²×C₇	1	2	yes	—
119	#416	—	2	2	no	—
120	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
121	#322	C₂²×C₇	1	4	yes	—
122	C₂³×C₇	—	4	1	no	—
123	C₂³×C₇	#127	1	1	yes	characteristic
124	C₇×Q₈	—	4	8	no	—
125	C₇×Q₈	—	4	2	no	—
126	C₇×Q₈	—	4	4	no	—
127	C₇×Q₈	—	8	4	no	—
128	C₇×D₄	—	4	2	no	—
129	C₇×D₄	—	4	4	no	—
130	C₇×D₄	—	8	2	no	—
131	C₂×C₄×C₇	—	2	4	no	—
132	C₂×C₄×C₇	—	4	2	no	—
133	C₂×C₄×C₇	—	4	1	no	—
134	C₂×C₄×C₇	—	4	4	no	—
135	C₂×C₄×C₇	—	8	4	no	—
136	C₂×C₄×C₇	—	8	2	no	—
137	C₂×C₄×C₇	#127	1	2	yes	—
138	C₂×C₄×C₇	#127	1	1	yes	characteristic
139	C₂×C₄×C₇	AGL(1,ℤ/8)	1	4	yes	—
140	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
142	C₇×C₈	—	2	4	no	—
143	C₇×C₈	—	2	8	no	—
144	C₂×#41	—	2	1	no	—
145	C₂²×Q₈	—	2	2	no	—
146	C₂×Q₁₆	—	2	8	no	—
147	C₂×Q₁₆	—	4	8	no	—
148	C₂²×C₈	C₇×D₄	1	2	yes	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
153	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
154	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
155	#127	C₂³×C₇	1	1	yes	characteristic
156	#125	—	2	1	no	—
157	#124	—	2	1	no	—
158	C₂²⋊Q₈	—	2	2	no	—
159	#121	—	2	2	no	—
160	C₄×Q₈	—	2	2	no	—
161	C₄×D₄	—	2	2	no	—
162	#117	—	2	1	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
166	#107	—	2	8	no	—
167	#106	—	2	8	no	—
168	C₄⋊C₈	C₇×D₄	1	4	yes	—
169	#103	—	2	4	no	—
170	#103	—	2	8	no	—
171	#102	—	2	4	no	—
172	#102	—	2	8	no	—
173	C₂²⋊C₈	C₇×D₄	1	4	yes	—
174	C₄×C₈	C₇×D₄	1	2	yes	—
175	C₄×C₈	C₂³×C₇	1	2	yes	—
176	C₄×C₈	C₂³×C₇	1	4	yes	—
177	C₂²×C₇	—	2	2	no	—
178	C₂²×C₇	—	8	1	no	—
179	C₂²×C₇	—	8	2	no	—
180	C₂²×C₇	#496	1	2	yes	—
181	C₂²×C₇	C₂×#127	1	1	yes	characteristic
182	C₄×C₇	—	2	2	no	—
183	C₄×C₇	—	8	1	no	—
184	C₄×C₇	—	8	4	no	—
185	C₄×C₇	—	8	2	no	—
186	C₄×C₇	#496	1	2	yes	—
187	C₄×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
188	C₄×C₇	#578	1	4	yes	—
189	#41	—	4	4	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×Q₈	—	4	8	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	1	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
201	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
202	Q₁₆	—	4	16	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
205	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₂²⋊C₄	—	4	1	no	—
215	C₂²⋊C₄	—	4	2	no	—
216	C₄²	—	4	1	no	—
217	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
218	C₄²	C₂×C₇×D₄	1	2	yes	—
219	C₄²	C₂⁴×C₇	1	1	yes	characteristic
220	C₂×C₇	—	2	2	no	—
221	C₂×C₇	—	16	1	no	—
222	C₂×C₇	C₂×#496	1	1	yes	characteristic
223	C₂×C₇	#3156	1	2	yes	—
224	C₂³	—	4	1	no	—
225	C₂³	C₇×#127	1	1	yes	characteristic
226	Q₈	—	4	8	no	—
227	Q₈	—	4	2	no	—
228	Q₈	—	4	4	no	—
229	Q₈	—	8	4	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	4	no	—
232	D₄	—	8	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	C₇×#127	1	2	yes	—
240	C₂×C₄	C₇×#127	1	1	yes	characteristic
241	C₂×C₄	C₇×AGL(1,ℤ/8)	1	4	yes	—
242	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
244	C₈	—	2	4	no	—
245	C₈	—	2	8	no	—
246	C₇	#19699	1	1	yes	characteristic, direct product factor, Sylow
247	C₂²	—	2	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	1	no	—
250	C₂²	C₇×#496	1	2	yes	—
251	C₂²	C₂×C₇×#127	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	8	2	no	—
254	C₄	—	8	1	no	—
255	C₄	—	8	4	no	—
256	C₄	C₇×#496	1	2	yes	—
257	C₄	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
258	C₄	C₇×#578	1	4	yes	—
259	C₂	—	2	2	no	—
260	C₂	—	16	1	no	—
261	C₂	C₂×C₇×#496	1	1	yes	characteristic
262	C₂	C₇×#3156	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#532	C₂²
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×#3229	C₂	semidirect
3. C₂×C₇×#494	C₂	semidirect
5. C₇×#1723	C₂	semidirect
6. C₇×#1720	C₂	semidirect
7. C₇×#1452	C₂	semidirect
29. #19699	C₇	direct (N × H)
63. #3229	C₂×C₇	semidirect
65. C₂×#494	C₂×C₇	semidirect
67. #1723	C₂×C₇	semidirect
68. #1720	C₂×C₇	semidirect
69. #1452	C₂×C₇	semidirect
246. C₇	#19699	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×#3229	2
C₇×#3226	2
C₂×C₇×#494	2
C₇×#3131	2
C₇×#1723	2
C₇×#1720	2
C₇×#1452	2
#19699	7

Subgroups — C7×#19699 #13196915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19699 #13196915