Subgroups — C₇×#19703 #13196919

Order 1792.

704 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3227	C₂	1	2	yes	maximal
2	C₇×#3209	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3131	C₂	1	1	yes	maximal, characteristic
5	C₇×#1720	C₂	1	4	yes	maximal, complemented
6	C₇×#1716	C₂	1	4	yes	maximal, complemented
7	C₇×#1452	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#533	C₂²	1	1	yes	characteristic
9	C₇×#532	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₇×#498	—	2	2	no	—
12	C₇×#498	C₂²	1	4	yes	—
13	C₇×#498	C₂²	1	4	yes	—
14	C₇×#494	C₂²	1	4	yes	—
15	C₇×#488	—	2	2	no	—
16	C₇×#487	C₂²	1	2	yes	—
17	C₇×#487	C₂²	1	4	yes	—
18	C₇×#486	C₂²	1	4	yes	—
19	C₇×#486	C₂²	1	2	yes	—
20	C₇×#481	—	2	4	no	—
21	C₇×#476	—	2	4	no	—
22	C₇×#462	—	2	4	no	—
23	C₇×#451	—	2	4	no	—
24	C₇×#431	C₂²	1	2	yes	—
25	C₂×C₇×#106	—	2	2	no	—
26	C₇×#416	—	2	2	no	—
27	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
28	C₇×#322	C₂²	1	4	yes	—
29	#19703	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	C₂²×C₇×C₈	D₄	1	2	yes	—
34	C₇×C₄⋊Q₈	—	2	2	no	—
35	C₇×C₄⋊Q₈	—	2	2	no	—
36	C₇×C₄⋊Q₈	—	2	2	no	—
37	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
38	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
39	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
40	C₇×#127	C₂³	1	1	yes	characteristic
41	C₇×#125	—	2	1	no	—
42	C₇×#124	—	2	1	no	—
43	C₇×C₂²⋊Q₈	—	2	2	no	—
44	C₇×#121	—	2	2	no	—
45	C₄×C₇×Q₈	—	2	2	no	—
46	C₄×C₇×D₄	—	2	2	no	—
47	C₇×#117	—	2	1	no	—
48	C₂×C₇×C₄⋊C₄	—	2	2	no	—
49	C₂×C₇×C₄⋊C₄	—	2	2	no	—
50	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
51	C₇×#106	—	2	8	no	—
52	C₇×#106	—	2	8	no	—
53	C₇×C₄⋊C₈	D₄	1	4	yes	—
54	C₇×#103	—	2	4	no	—
55	C₇×#103	—	2	8	no	—
56	C₇×#102	—	2	8	no	—
57	C₇×#102	—	2	4	no	—
58	C₇×C₂²⋊C₈	D₄	1	4	yes	—
59	C₄×C₇×C₈	D₄	1	2	yes	—
60	C₄×C₇×C₈	C₂³	1	4	yes	—
61	C₄×C₇×C₈	C₂³	1	2	yes	—
62	#3227	C₂×C₇	1	2	yes	—
63	#3209	C₂×C₇	1	2	yes	complemented
64	C₂×#498	C₂×C₇	1	1	yes	characteristic, complemented
65	#3131	C₂×C₇	1	1	yes	characteristic
66	#1720	C₂×C₇	1	4	yes	complemented
67	#1716	C₂×C₇	1	4	yes	complemented
68	#1452	C₂×C₇	1	1	yes	characteristic, complemented
69	C₇×#41	—	4	4	no	—
70	C₂×C₇×Q₈	—	2	1	no	—
71	C₂×C₇×Q₈	—	2	4	no	—
72	C₂×C₇×Q₈	—	2	2	no	—
73	C₂×C₇×Q₈	—	4	2	no	—
74	C₂×C₇×Q₈	—	8	4	no	—
75	C₂×C₇×D₄	—	2	2	no	—
76	C₂×C₇×D₄	—	2	1	no	—
77	C₂×C₇×D₄	—	4	1	no	—
78	C₂²×C₄×C₇	—	4	2	no	—
79	C₂²×C₄×C₇	—	4	1	no	—
80	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
82	C₇×Q₁₆	—	8	8	no	—
83	C₂×C₇×C₈	—	2	4	no	—
84	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
85	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
86	C₇×C₄⋊C₄	—	2	2	no	—
87	C₇×C₄⋊C₄	—	2	4	no	—
88	C₇×C₄⋊C₄	—	2	2	no	—
89	C₇×C₄⋊C₄	—	2	4	no	—
90	C₇×C₄⋊C₄	—	2	2	no	—
91	C₇×C₄⋊C₄	—	4	2	no	—
92	C₇×C₄⋊C₄	—	4	2	no	—
93	C₇×C₄⋊C₄	—	4	2	no	—
94	C₇×C₂²⋊C₄	—	4	1	no	—
95	C₇×C₂²⋊C₄	—	4	2	no	—
96	C₄²×C₇	—	4	1	no	—
97	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₇	C₂×D₄	1	2	yes	—
99	C₄²×C₇	C₂⁴	1	1	yes	characteristic
100	#533	C₂²×C₇	1	1	yes	characteristic
101	#532	C₂²×C₇	1	1	yes	characteristic
102	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
103	#498	—	2	2	no	—
104	#498	C₂²×C₇	1	4	yes	—
105	#498	C₂²×C₇	1	4	yes	—
106	#494	C₂²×C₇	1	4	yes	—
107	#488	—	2	2	no	—
108	#487	C₂²×C₇	1	2	yes	—
109	#487	C₂²×C₇	1	4	yes	—
110	#486	C₂²×C₇	1	2	yes	—
111	#486	C₂²×C₇	1	4	yes	—
112	#481	—	2	4	no	—
113	#476	—	2	4	no	—
114	#462	—	2	4	no	—
115	#451	—	2	4	no	—
116	#431	C₂²×C₇	1	2	yes	—
117	C₂×#106	—	2	2	no	—
118	#416	—	2	2	no	—
119	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
120	#322	C₂²×C₇	1	4	yes	—
121	C₂³×C₇	—	4	1	no	—
122	C₂³×C₇	#127	1	1	yes	characteristic
123	C₇×Q₈	—	4	2	no	—
124	C₇×Q₈	—	4	4	no	—
125	C₇×Q₈	—	4	8	no	—
126	C₇×Q₈	—	8	4	no	—
127	C₇×D₄	—	4	2	no	—
128	C₇×D₄	—	4	4	no	—
129	C₇×D₄	—	8	2	no	—
130	C₂×C₄×C₇	—	2	4	no	—
131	C₂×C₄×C₇	—	4	2	no	—
132	C₂×C₄×C₇	—	4	1	no	—
133	C₂×C₄×C₇	—	4	4	no	—
134	C₂×C₄×C₇	—	8	4	no	—
135	C₂×C₄×C₇	—	8	2	no	—
136	C₂×C₄×C₇	#127	1	2	yes	—
137	C₂×C₄×C₇	#127	1	1	yes	characteristic
138	C₂×C₄×C₇	#137	1	4	yes	—
139	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
141	C₇×C₈	—	2	4	no	—
142	C₇×C₈	—	2	8	no	—
143	C₂×#41	—	2	1	no	—
144	C₂²×Q₈	—	2	2	no	—
145	C₂×Q₁₆	—	2	8	no	—
146	C₂²×C₈	C₇×D₄	1	2	yes	—
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
151	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
152	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
153	#127	C₂³×C₇	1	1	yes	characteristic
154	#125	—	2	1	no	—
155	#124	—	2	1	no	—
156	C₂²⋊Q₈	—	2	2	no	—
157	#121	—	2	2	no	—
158	C₄×Q₈	—	2	2	no	—
159	C₄×D₄	—	2	2	no	—
160	#117	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
164	#106	—	2	8	no	—
165	#106	—	2	8	no	—
166	C₄⋊C₈	C₇×D₄	1	4	yes	—
167	#103	—	2	4	no	—
168	#103	—	2	8	no	—
169	#102	—	2	4	no	—
170	#102	—	2	8	no	—
171	C₂²⋊C₈	C₇×D₄	1	4	yes	—
172	C₄×C₈	C₇×D₄	1	2	yes	—
173	C₄×C₈	C₂³×C₇	1	2	yes	—
174	C₄×C₈	C₂³×C₇	1	4	yes	—
175	C₂²×C₇	—	2	2	no	—
176	C₂²×C₇	—	8	1	no	—
177	C₂²×C₇	—	8	2	no	—
178	C₂²×C₇	#497	1	2	yes	—
179	C₂²×C₇	C₂×#127	1	1	yes	characteristic
180	C₄×C₇	—	2	2	no	—
181	C₄×C₇	—	8	1	no	—
182	C₄×C₇	—	8	4	no	—
183	C₄×C₇	—	8	2	no	—
184	C₄×C₇	#497	1	2	yes	—
185	C₄×C₇	C₂×#137	1	2	yes	—
186	C₄×C₇	#577	1	4	yes	—
187	#41	—	4	4	no	—
188	C₂×Q₈	—	2	1	no	—
189	C₂×Q₈	—	2	4	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	4	2	no	—
192	C₂×Q₈	—	8	4	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	2	1	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
199	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
200	Q₁₆	—	8	8	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
203	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₂²⋊C₄	—	4	1	no	—
213	C₂²⋊C₄	—	4	2	no	—
214	C₄²	—	4	1	no	—
215	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
216	C₄²	C₂×C₇×D₄	1	2	yes	—
217	C₄²	C₂⁴×C₇	1	1	yes	characteristic
218	C₂×C₇	—	2	2	no	—
219	C₂×C₇	—	16	1	no	—
220	C₂×C₇	C₂×#497	1	1	yes	characteristic
221	C₂×C₇	#3154	1	2	yes	—
222	C₂³	—	4	1	no	—
223	C₂³	C₇×#127	1	1	yes	characteristic
224	Q₈	—	4	2	no	—
225	Q₈	—	4	4	no	—
226	Q₈	—	4	8	no	—
227	Q₈	—	8	4	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	4	no	—
230	D₄	—	8	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	C₇×#127	1	2	yes	—
238	C₂×C₄	C₇×#127	1	1	yes	characteristic
239	C₂×C₄	C₇×#137	1	4	yes	—
240	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
242	C₈	—	2	4	no	—
243	C₈	—	2	8	no	—
244	C₇	#19703	1	1	yes	characteristic, direct product factor, Sylow
245	C₂²	—	2	2	no	—
246	C₂²	—	8	1	no	—
247	C₂²	—	8	2	no	—
248	C₂²	C₇×#497	1	2	yes	—
249	C₂²	C₂×C₇×#127	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	8	1	no	—
252	C₄	—	8	4	no	—
253	C₄	—	8	2	no	—
254	C₄	C₇×#497	1	2	yes	—
255	C₄	C₂×C₇×#137	1	2	yes	—
256	C₄	C₇×#577	1	4	yes	—
257	C₂	—	2	2	no	—
258	C₂	—	16	1	no	—
259	C₂	C₂×C₇×#497	1	1	yes	characteristic
260	C₂	C₇×#3154	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
2. C₇×#3209	C₂	semidirect
3. C₂×C₇×#498	C₂	semidirect
5. C₇×#1720	C₂	semidirect
6. C₇×#1716	C₂	semidirect
7. C₇×#1452	C₂	semidirect
29. #19703	C₇	direct (N × H)
63. #3209	C₂×C₇	semidirect
64. C₂×#498	C₂×C₇	semidirect
66. #1720	C₂×C₇	semidirect
67. #1716	C₂×C₇	semidirect
68. #1452	C₂×C₇	semidirect
244. C₇	#19703	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×#3227	2
C₇×#3209	2
C₂×C₇×#498	2
C₇×#3131	2
C₇×#1720	2
C₇×#1716	2
C₇×#1452	2
#19703	7

Subgroups — C7×#19703 #13196919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19703 #13196919