Subgroups — C₇×#17972 #13195188

Order 1792.

560 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3170	C₂	1	1	yes	maximal, characteristic
2	C₇×#2985	C₂	1	2	yes	maximal
3	C₇×#1593	C₂	1	2	yes	maximal, complemented
4	C₇×#1591	C₂	1	2	yes	maximal, complemented
5	C₇×#1579	C₂	1	2	yes	maximal, complemented
6	C₇×C₈×Q₁₆	C₂	1	2	yes	maximal, complemented
7	C₇×#1568	C₂	1	1	yes	maximal, characteristic
8	C₇×#1497	C₂	1	2	yes	maximal
9	C₇×#1450	C₂	1	1	yes	maximal, characteristic
10	C₇×#533	C₂²	1	2	yes	—
11	C₇×#500	C₄	1	1	yes	characteristic
12	C₇×#494	C₄	1	1	yes	characteristic
13	C₇×#487	C₄	1	2	yes	—
14	C₇×#467	C₄	1	2	yes	—
15	C₇×#467	C₄	1	2	yes	—
16	C₇×#446	—	2	4	no	—
17	C₇×#446	C₂²	1	4	yes	—
18	C₇×C₈×Q₈	C₂²	1	4	yes	—
19	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
20	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
21	C₇×#433	—	2	4	no	—
22	C₇×#432	C₂²	1	2	yes	—
23	C₇×#431	C₂²	1	1	yes	characteristic
24	C₇×#431	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
26	C₇×#335	C₂²	1	2	yes	—
27	C₇×#334	C₂²	1	2	yes	—
28	C₇×Q₈⋊C₈	C₂²	1	4	yes	—
29	C₇×Q₈⋊C₈	C₂²	1	4	yes	—
30	C₇×#322	C₂²	1	1	yes	characteristic
31	C₇×#322	C₂²	1	1	yes	characteristic
32	C₇×C₈²	C₂²	1	1	yes	characteristic
33	C₇×C₈²	C₂²	1	1	yes	characteristic
34	#17972	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
37	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
39	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
41	C₇×#125	—	2	2	no	—
42	C₇×C₂²⋊Q₈	—	2	2	no	—
43	C₇×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
44	C₄×C₇×Q₈	—	2	2	no	—
45	C₄×C₇×Q₈	C₂³	1	4	yes	—
46	C₇×#117	—	2	2	no	—
47	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
48	C₇×#107	C₂×C₄	1	2	yes	—
49	C₇×#107	C₂×C₄	1	2	yes	—
50	C₇×C₄⋊C₈	—	2	4	no	—
51	C₇×C₄⋊C₈	—	2	4	no	—
52	C₇×C₄⋊C₈	—	2	1	no	—
53	C₇×C₄⋊C₈	D₄	1	2	yes	—
54	C₇×C₄⋊C₈	C₂³	1	4	yes	—
55	C₇×#103	C₂×C₄	1	4	yes	—
56	C₇×#103	C₂×C₄	1	4	yes	—
57	C₇×C₂²⋊C₈	—	2	2	no	—
58	C₇×C₂²⋊C₈	—	2	1	no	—
59	C₇×C₂²⋊C₈	D₄	1	2	yes	—
60	C₇×#97	—	2	4	no	—
61	C₇×#97	—	2	2	no	—
62	C₄×C₇×C₈	—	2	4	no	—
63	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
64	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
65	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
66	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
68	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
69	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
70	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
71	#3170	C₂×C₇	1	1	yes	characteristic
72	#2985	C₂×C₇	1	2	yes	—
73	#1593	C₂×C₇	1	2	yes	complemented
74	#1591	C₂×C₇	1	2	yes	complemented
75	#1579	C₂×C₇	1	2	yes	complemented
76	C₈×Q₁₆	C₂×C₇	1	2	yes	complemented
77	#1568	C₂×C₇	1	1	yes	characteristic
78	#1497	C₂×C₇	1	2	yes	—
79	#1450	C₂×C₇	1	1	yes	characteristic
80	C₂×C₇×Q₈	—	2	2	no	—
81	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	#41	1	1	yes	characteristic
85	C₇×Q₁₆	—	2	4	no	—
86	C₇×Q₁₆	—	2	4	no	—
87	C₂×C₇×C₈	—	2	2	no	—
88	C₂×C₇×C₈	—	2	2	no	—
89	C₂×C₇×C₈	—	2	4	no	—
90	C₂×C₇×C₈	—	2	1	no	—
91	C₂×C₇×C₈	—	2	1	no	—
92	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
97	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
98	C₇×C₄⋊C₄	—	2	2	no	—
99	C₇×C₄⋊C₄	—	2	2	no	—
100	C₇×C₄⋊C₄	—	2	2	no	—
101	C₇×C₄⋊C₄	—	2	4	no	—
102	C₇×C₄⋊C₄	—	2	4	no	—
103	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
104	C₇×C₂²⋊C₄	—	2	4	no	—
105	C₄²×C₇	—	2	4	no	—
106	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₇	#41	1	1	yes	characteristic
109	C₄²×C₇	C₂⁴	1	1	yes	characteristic
110	#533	C₂²×C₇	1	2	yes	—
111	#500	C₄×C₇	1	1	yes	characteristic
112	#494	C₄×C₇	1	1	yes	characteristic
113	#487	C₄×C₇	1	2	yes	—
114	#467	C₄×C₇	1	2	yes	—
115	#467	C₄×C₇	1	2	yes	—
116	#446	—	2	4	no	—
117	#446	C₂²×C₇	1	4	yes	—
118	C₈×Q₈	C₂²×C₇	1	4	yes	—
119	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
120	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
121	#433	—	2	4	no	—
122	#432	C₂²×C₇	1	2	yes	—
123	#431	C₂²×C₇	1	1	yes	characteristic
124	#431	C₂²×C₇	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
126	#335	C₂²×C₇	1	2	yes	—
127	#334	C₂²×C₇	1	2	yes	—
128	Q₈⋊C₈	C₂²×C₇	1	4	yes	—
129	Q₈⋊C₈	C₂²×C₇	1	4	yes	—
130	#322	C₂²×C₇	1	1	yes	characteristic
131	#322	C₂²×C₇	1	1	yes	characteristic
132	C₈²	C₂²×C₇	1	1	yes	characteristic
133	C₈²	C₂²×C₇	1	1	yes	characteristic
134	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
135	C₇×Q₈	—	2	4	no	—
136	C₇×Q₈	—	2	4	no	—
137	C₇×Q₈	—	8	2	no	—
138	C₂×C₄×C₇	—	2	1	no	—
139	C₂×C₄×C₇	—	2	1	no	—
140	C₂×C₄×C₇	—	2	2	no	—
141	C₂×C₄×C₇	—	2	4	no	—
142	C₂×C₄×C₇	—	2	4	no	—
143	C₂×C₄×C₇	—	4	1	no	—
144	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
150	C₇×C₈	—	2	1	no	—
151	C₇×C₈	—	2	1	no	—
152	C₇×C₈	—	2	2	no	—
153	C₇×C₈	—	2	2	no	—
154	C₇×C₈	—	4	1	no	—
155	C₇×C₈	—	4	1	no	—
156	C₇×C₈	—	4	4	no	—
157	C₇×C₈	#131	1	2	yes	—
158	C₇×C₈	#131	1	2	yes	—
159	C₂×Q₁₆	C₂×C₄×C₇	1	2	yes	—
160	C₂×Q₁₆	C₂×C₄×C₇	1	2	yes	—
161	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
162	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
163	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
164	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
165	#125	—	2	2	no	—
166	C₂²⋊Q₈	—	2	2	no	—
167	C₂²⋊Q₈	C₂×C₄×C₇	1	4	yes	—
168	C₄×Q₈	—	2	2	no	—
169	C₄×Q₈	C₂³×C₇	1	4	yes	—
170	#117	—	2	2	no	—
171	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
172	#107	C₂×C₄×C₇	1	2	yes	—
173	#107	C₂×C₄×C₇	1	2	yes	—
174	C₄⋊C₈	—	2	4	no	—
175	C₄⋊C₈	—	2	4	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	C₇×D₄	1	2	yes	—
178	C₄⋊C₈	C₂³×C₇	1	4	yes	—
179	#103	C₂×C₄×C₇	1	4	yes	—
180	#103	C₂×C₄×C₇	1	4	yes	—
181	C₂²⋊C₈	—	2	2	no	—
182	C₂²⋊C₈	—	2	1	no	—
183	C₂²⋊C₈	C₇×D₄	1	2	yes	—
184	#97	—	2	4	no	—
185	#97	—	2	2	no	—
186	C₄×C₈	—	2	4	no	—
187	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
188	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
189	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
190	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
191	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
192	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
193	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
194	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
195	C₂²×C₇	—	2	1	no	—
196	C₂²×C₇	—	2	1	no	—
197	C₂²×C₇	—	2	1	no	—
198	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
199	C₄×C₇	—	2	1	no	—
200	C₄×C₇	—	2	1	no	—
201	C₄×C₇	—	2	1	no	—
202	C₄×C₇	—	2	1	no	—
203	C₄×C₇	—	4	4	no	—
204	C₄×C₇	—	4	4	no	—
205	C₄×C₇	C₂×#131	1	1	yes	characteristic
206	C₄×C₇	C₂×#131	1	1	yes	characteristic
207	C₄×C₇	#578	1	2	yes	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
210	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
212	C₂²×C₄	C₇×#41	1	1	yes	characteristic
213	Q₁₆	—	2	4	no	—
214	Q₁₆	—	2	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
221	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
222	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
223	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
224	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
225	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
235	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
236	C₄²	C₇×#41	1	1	yes	characteristic
237	C₄²	C₂⁴×C₇	1	1	yes	characteristic
238	C₂×C₇	—	4	1	no	—
239	C₂×C₇	#2931	1	1	yes	characteristic
240	C₂×C₇	#2950	1	1	yes	characteristic
241	C₂×C₇	#2957	1	1	yes	characteristic
242	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
243	Q₈	—	2	4	no	—
244	Q₈	—	2	4	no	—
245	Q₈	—	8	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
253	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
254	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
255	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
257	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₈	—	4	1	no	—
263	C₈	—	4	1	no	—
264	C₈	—	4	4	no	—
265	C₈	C₇×#131	1	2	yes	—
266	C₈	C₇×#131	1	2	yes	—
267	C₇	#17972	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	4	no	—
277	C₄	—	4	4	no	—
278	C₄	C₂×C₇×#131	1	1	yes	characteristic
279	C₄	C₂×C₇×#131	1	1	yes	characteristic
280	C₄	C₇×#578	1	2	yes	—
281	C₂	—	4	1	no	—
282	C₂	C₇×#2931	1	1	yes	characteristic
283	C₂	C₇×#2950	1	1	yes	characteristic
284	C₂	C₇×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂³×C₇	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₇×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1593	C₂	semidirect
4. C₇×#1591	C₂	semidirect
5. C₇×#1579	C₂	semidirect
6. C₇×C₈×Q₁₆	C₂	semidirect
34. #17972	C₇	direct (N × H)
73. #1593	C₂×C₇	semidirect
74. #1591	C₂×C₇	semidirect
75. #1579	C₂×C₇	semidirect
76. C₈×Q₁₆	C₂×C₇	semidirect
267. C₇	#17972	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×#3170	2
C₇×#2985	2
C₇×#1593	2
C₇×#1591	2
C₇×#1579	2
C₇×C₈×Q₁₆	2
C₇×#1568	2
C₇×#1497	2
C₇×#1450	2
#17972	7

Subgroups — C7×#17972 #13195188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#17972 #13195188