Subgroups — C₇×#19636 #13196852

Order 1792.

686 conjugacy classes of subgroups.

The subgroup lattice has 280 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
2	C₇×#3170	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic
4	C₇×#3139	C₂	1	2	yes	maximal
5	C₇×#1718	C₂	1	4	yes	maximal, complemented
6	C₇×#1710	C₂	1	2	yes	maximal, complemented
7	C₇×#1707	C₂	1	2	yes	maximal, complemented
8	C₇×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
9	C₇×#533	C₂²	1	2	yes	—
10	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₇×#500	—	2	2	no	—
12	C₇×#500	C₂²	1	2	yes	—
13	C₇×#500	C₂²	1	1	yes	characteristic
14	C₇×#500	C₂²	1	2	yes	—
15	C₇×#498	C₂²	1	4	yes	—
16	C₇×#494	C₂²	1	1	yes	characteristic
17	C₇×#491	—	2	4	no	—
18	C₇×#487	C₂²	1	4	yes	—
19	C₇×#487	C₂²	1	4	yes	—
20	C₇×#487	C₂²	1	4	yes	—
21	C₇×#487	C₂²	1	2	yes	—
22	C₇×#483	—	2	4	no	—
23	C₇×#475	—	2	4	no	—
24	C₇×#467	—	2	2	no	—
25	C₄×C₇×Q₁₆	—	2	2	no	—
26	C₇×#431	C₂²	1	1	yes	characteristic
27	C₂×C₇×#107	—	2	2	no	—
28	C₇×#419	—	2	4	no	—
29	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
31	C₇×#322	C₂²	1	2	yes	—
32	C₇×C₈²	C₂²	1	2	yes	—
33	#19636	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₇×Q₈	—	2	2	no	—
35	C₂×C₇×Q₁₆	—	2	2	no	—
36	C₂×C₇×M₁₆	D₄	1	2	yes	—
37	C₂²×C₇×C₈	D₄	1	2	yes	—
38	C₇×C₄⋊Q₈	—	2	2	no	—
39	C₇×C₄⋊Q₈	—	2	2	no	—
40	C₇×C₄⋊Q₈	—	2	2	no	—
41	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
42	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
43	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
44	C₇×#125	—	2	2	no	—
45	C₇×C₂²⋊Q₈	—	2	2	no	—
46	C₇×C₂²⋊Q₈	—	2	2	no	—
47	C₄×C₇×Q₈	—	2	2	no	—
48	C₄×C₇×Q₈	—	2	2	no	—
49	C₇×#117	—	2	2	no	—
50	C₂×C₇×C₄⋊C₄	—	2	2	no	—
51	C₂×C₇×C₄⋊C₄	—	2	2	no	—
52	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
53	C₇×#107	—	2	2	no	—
54	C₇×#107	—	2	4	no	—
55	C₇×#107	—	2	4	no	—
56	C₇×#106	—	2	8	no	—
57	C₇×C₄⋊C₈	D₄	1	2	yes	—
58	C₇×#103	—	2	4	no	—
59	C₇×#103	—	2	8	no	—
60	C₇×#103	—	2	8	no	—
61	C₇×#103	—	2	8	no	—
62	C₇×C₂²⋊C₈	D₄	1	2	yes	—
63	C₇×#97	D₄	1	2	yes	—
64	C₄×C₇×C₈	D₄	1	2	yes	—
65	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
66	C₄×C₇×C₈	C₂³	1	2	yes	—
67	C₄×C₇×C₈	C₂³	1	2	yes	—
68	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
69	#3229	C₂×C₇	1	2	yes	—
70	#3170	C₂×C₇	1	1	yes	characteristic
71	C₂×#500	C₂×C₇	1	1	yes	characteristic
72	#3139	C₂×C₇	1	2	yes	—
73	#1718	C₂×C₇	1	4	yes	complemented
74	#1710	C₂×C₇	1	2	yes	complemented
75	#1707	C₂×C₇	1	2	yes	complemented
76	C₈×M₁₆	C₂×C₇	1	1	yes	characteristic
77	C₂×C₇×Q₈	—	2	2	no	—
78	C₂×C₇×Q₈	—	2	4	no	—
79	C₂×C₇×Q₈	—	2	2	no	—
80	C₂×C₇×Q₈	—	4	2	no	—
81	C₂×C₇×Q₈	—	4	8	no	—
82	C₂²×C₄×C₇	—	4	2	no	—
83	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
86	C₇×Q₁₆	—	4	4	no	—
87	C₇×M₁₆	—	2	4	no	—
88	C₂×C₇×C₈	—	2	4	no	—
89	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
90	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
91	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
92	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
93	C₇×C₄⋊C₄	—	2	2	no	—
94	C₇×C₄⋊C₄	—	2	2	no	—
95	C₇×C₄⋊C₄	—	2	4	no	—
96	C₇×C₄⋊C₄	—	2	2	no	—
97	C₇×C₄⋊C₄	—	2	2	no	—
98	C₇×C₄⋊C₄	—	2	4	no	—
99	C₇×C₄⋊C₄	—	4	2	no	—
100	C₇×C₄⋊C₄	—	4	2	no	—
101	C₇×C₄⋊C₄	—	4	2	no	—
102	C₇×C₄⋊C₄	—	4	2	no	—
103	C₇×C₂²⋊C₄	—	4	2	no	—
104	C₄²×C₇	—	4	2	no	—
105	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₇	C₂⁴	1	1	yes	characteristic
109	#533	C₂²×C₇	1	2	yes	—
110	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
111	#500	—	2	2	no	—
112	#500	C₂²×C₇	1	2	yes	—
113	#500	C₂²×C₇	1	2	yes	—
114	#500	C₂²×C₇	1	1	yes	characteristic
115	#498	C₂²×C₇	1	4	yes	—
116	#494	C₂²×C₇	1	1	yes	characteristic
117	#491	—	2	4	no	—
118	#487	C₂²×C₇	1	4	yes	—
119	#487	C₂²×C₇	1	4	yes	—
120	#487	C₂²×C₇	1	2	yes	—
121	#487	C₂²×C₇	1	4	yes	—
122	#483	—	2	4	no	—
123	#475	—	2	4	no	—
124	#467	—	2	2	no	—
125	C₄×Q₁₆	—	2	2	no	—
126	#431	C₂²×C₇	1	1	yes	characteristic
127	C₂×#107	—	2	2	no	—
128	#419	—	2	4	no	—
129	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
131	#322	C₂²×C₇	1	2	yes	—
132	C₈²	C₂²×C₇	1	2	yes	—
133	C₂³×C₇	#127	1	1	yes	characteristic
134	C₇×Q₈	—	4	8	no	—
135	C₇×Q₈	—	4	4	no	—
136	C₇×Q₈	—	4	4	no	—
137	C₇×Q₈	—	8	4	no	—
138	C₂×C₄×C₇	—	2	2	no	—
139	C₂×C₄×C₇	—	2	2	no	—
140	C₂×C₄×C₇	—	4	2	no	—
141	C₂×C₄×C₇	—	4	2	no	—
142	C₂×C₄×C₇	—	4	4	no	—
143	C₂×C₄×C₇	—	8	4	no	—
144	C₂×C₄×C₇	#127	1	1	yes	characteristic
145	C₂×C₄×C₇	#127	1	1	yes	characteristic
146	C₂×C₄×C₇	#127	1	1	yes	characteristic
147	C₂×C₄×C₇	#135	1	4	yes	—
148	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
151	C₇×C₈	—	2	4	no	—
152	C₇×C₈	—	2	2	no	—
153	C₇×C₈	—	2	4	no	—
154	C₇×C₈	#137	1	4	yes	—
155	C₂²×Q₈	—	2	2	no	—
156	C₂×Q₁₆	—	2	2	no	—
157	C₂×M₁₆	C₇×D₄	1	2	yes	—
158	C₂²×C₈	C₇×D₄	1	2	yes	—
159	C₄⋊Q₈	—	2	2	no	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
163	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
164	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
165	#125	—	2	2	no	—
166	C₂²⋊Q₈	—	2	2	no	—
167	C₂²⋊Q₈	—	2	2	no	—
168	C₄×Q₈	—	2	2	no	—
169	C₄×Q₈	—	2	2	no	—
170	#117	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
174	#107	—	2	4	no	—
175	#107	—	2	4	no	—
176	#107	—	2	2	no	—
177	#106	—	2	8	no	—
178	C₄⋊C₈	C₇×D₄	1	2	yes	—
179	#103	—	2	8	no	—
180	#103	—	2	8	no	—
181	#103	—	2	4	no	—
182	#103	—	2	8	no	—
183	C₂²⋊C₈	C₇×D₄	1	2	yes	—
184	#97	C₇×D₄	1	2	yes	—
185	C₄×C₈	C₇×D₄	1	2	yes	—
186	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
187	C₄×C₈	C₂³×C₇	1	2	yes	—
188	C₄×C₈	C₂³×C₇	1	2	yes	—
189	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
190	C₂²×C₇	—	2	1	no	—
191	C₂²×C₇	—	2	1	no	—
192	C₂²×C₇	#495	1	2	yes	—
193	C₂²×C₇	C₂×#127	1	1	yes	characteristic
194	C₄×C₇	—	2	1	no	—
195	C₄×C₇	—	2	1	no	—
196	C₄×C₇	—	8	4	no	—
197	C₄×C₇	—	8	2	no	—
198	C₄×C₇	—	8	2	no	—
199	C₄×C₇	#495	1	2	yes	—
200	C₄×C₇	C₂×#135	1	2	yes	—
201	C₄×C₇	C₂×#137	1	2	yes	—
202	C₄×C₇	#578	1	2	yes	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	2	4	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×Q₈	—	4	8	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
212	Q₁₆	—	4	4	no	—
213	M₁₆	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
216	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
217	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
218	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	4	2	no	—
230	C₄²	—	4	2	no	—
231	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
232	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
233	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
234	C₄²	C₂⁴×C₇	1	1	yes	characteristic
235	C₂×C₇	—	2	2	no	—
236	C₂×C₇	C₂×#495	1	1	yes	characteristic
237	C₂×C₇	#3152	1	1	yes	characteristic
238	C₂×C₇	#3156	1	1	yes	characteristic
239	C₂³	C₇×#127	1	1	yes	characteristic
240	Q₈	—	4	8	no	—
241	Q₈	—	4	4	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	8	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	C₇×#127	1	1	yes	characteristic
251	C₂×C₄	C₇×#127	1	1	yes	characteristic
252	C₂×C₄	C₇×#127	1	1	yes	characteristic
253	C₂×C₄	C₇×#135	1	4	yes	—
254	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
257	C₈	—	2	4	no	—
258	C₈	—	2	2	no	—
259	C₈	—	2	4	no	—
260	C₈	C₇×#137	1	4	yes	—
261	C₇	#19636	1	1	yes	characteristic, direct product factor, Sylow
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	C₇×#495	1	2	yes	—
265	C₂²	C₂×C₇×#127	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	8	2	no	—
269	C₄	—	8	4	no	—
270	C₄	—	8	2	no	—
271	C₄	C₇×#495	1	2	yes	—
272	C₄	C₂×C₇×#135	1	2	yes	—
273	C₄	C₂×C₇×#137	1	2	yes	—
274	C₄	C₇×#578	1	2	yes	—
275	C₂	—	2	2	no	—
276	C₂	C₂×C₇×#495	1	1	yes	characteristic
277	C₂	C₇×#3152	1	1	yes	characteristic
278	C₂	C₇×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂³×C₇	#127
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
5. C₇×#1718	C₂	semidirect
6. C₇×#1710	C₂	semidirect
7. C₇×#1707	C₂	semidirect
33. #19636	C₇	direct (N × H)
73. #1718	C₂×C₇	semidirect
74. #1710	C₂×C₇	semidirect
75. #1707	C₂×C₇	semidirect
261. C₇	#19636	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×#3170	2
C₂×C₇×#500	2
C₇×#3139	2
C₇×#1718	2
C₇×#1710	2
C₇×#1707	2
C₇×C₈×M₁₆	2
#19636	7

Subgroups — C7×#19636 #13196852

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19636 #13196852