Subgroups — C₇×#12807 #13190023

Order 1792.

428 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2931	C₂	1	1	yes	maximal, characteristic
2	C₇×#2126	C₂	1	2	yes	maximal
3	C₇×#1812	C₂	1	2	yes	maximal, complemented
4	C₇×#1391	C₂	1	2	yes	maximal, complemented
5	C₂×C₇×#131	C₄	1	1	yes	characteristic, complemented
6	C₂×C₇×#131	C₂²	1	1	yes	characteristic
7	C₇×#532	C₂²	1	1	yes	characteristic
8	C₇×#436	—	2	2	no	—
9	C₇×C₈×D₄	—	2	2	no	—
10	C₇×#427	—	2	2	no	—
11	C₇×#423	C₄	1	1	yes	characteristic, complemented
12	C₇×#410	C₂²	1	4	yes	—
13	C₇×#408	C₄	1	1	yes	characteristic, complemented
14	C₇×#408	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic, complemented
16	C₇×#354	—	2	2	no	—
17	C₇×#353	—	2	2	no	—
18	C₇×H(3,ℤ/4)	—	2	2	no	—
19	#12807	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂×C₇×#41	C₂×C₄	1	1	yes	characteristic
21	C₂×C₇×#41	C₂³	1	1	yes	characteristic
22	C₇×#131	—	4	1	no	—
23	C₇×#131	—	4	1	no	—
24	C₇×#131	—	4	1	no	—
25	C₇×#131	—	4	1	no	—
26	C₂×C₇×M₁₆	—	2	1	no	—
27	C₂×C₇×M₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₇×M₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₇×M₁₆	D₄	1	2	yes	—
30	C₂²×C₇×C₈	—	2	1	no	—
31	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₇×C₈	D₄	1	2	yes	—
34	C₇×C₄⋊Q₈	Q₈	1	1	yes	characteristic
35	C₇×#127	D₄	1	1	yes	characteristic
36	C₇×#124	D₄	1	1	yes	characteristic
37	C₇×#124	Q₈	1	1	yes	characteristic
38	C₇×#121	—	2	2	no	—
39	C₄×C₇×D₄	—	2	2	no	—
40	C₇×#117	—	2	4	no	—
41	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
42	C₇×#107	—	4	2	no	—
43	C₇×#106	—	4	2	no	—
44	C₇×C₄⋊C₈	—	2	2	no	—
45	C₇×#99	—	2	4	no	—
46	C₇×#99	—	2	4	no	—
47	C₇×C₂²⋊C₈	—	2	2	no	—
48	C₇×C₂²⋊C₈	—	2	2	no	—
49	C₄×C₇×C₈	—	2	1	no	—
50	C₄×C₇×C₈	—	2	1	no	—
51	#2931	C₂×C₇	1	1	yes	characteristic
52	#2126	C₂×C₇	1	2	yes	—
53	#1812	C₂×C₇	1	2	yes	complemented
54	#1391	C₂×C₇	1	2	yes	complemented
55	C₇×#41	—	4	1	no	—
56	C₇×#41	—	4	1	no	—
57	C₇×#41	—	4	1	no	—
58	C₇×#41	—	4	1	no	—
59	C₂×C₇×Q₈	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₇×Q₈	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₇×D₄	—	2	1	no	—
62	C₂×C₇×D₄	—	2	2	no	—
63	C₂×C₇×D₄	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
65	C₂×C₇×D₄	#41	1	2	yes	—
66	C₂²×C₄×C₇	—	2	1	no	—
67	C₂²×C₄×C₇	—	2	1	no	—
68	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
70	C₇×M₁₆	—	4	1	no	—
71	C₇×M₁₆	—	4	2	no	—
72	C₇×M₁₆	—	4	2	no	—
73	C₇×M₁₆	—	4	1	no	—
74	C₂×C₇×C₈	—	2	2	no	—
75	C₂×C₇×C₈	—	2	2	no	—
76	C₂×C₇×C₈	—	2	1	no	—
77	C₂×C₇×C₈	—	2	1	no	—
78	C₂×C₇×C₈	—	2	1	no	—
79	C₂×C₇×C₈	—	2	1	no	—
80	C₂×C₇×C₈	—	4	1	no	—
81	C₂×C₇×C₈	—	4	1	no	—
82	C₂×C₇×C₈	—	4	1	no	—
83	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
84	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₇×C₄⋊C₄	—	2	2	no	—
87	C₇×C₄⋊C₄	—	4	4	no	—
88	C₇×C₂²⋊C₄	—	2	2	no	—
89	C₇×C₂²⋊C₄	—	2	2	no	—
90	C₇×C₂²⋊C₄	—	4	4	no	—
91	C₄²×C₇	—	2	1	no	—
92	C₄²×C₇	—	4	4	no	—
93	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
94	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
95	C₂×#131	C₄×C₇	1	1	yes	characteristic, complemented
96	C₂×#131	C₂²×C₇	1	1	yes	characteristic
97	#532	C₂²×C₇	1	1	yes	characteristic
98	#436	—	2	2	no	—
99	C₈×D₄	—	2	2	no	—
100	#427	—	2	2	no	—
101	#423	C₄×C₇	1	1	yes	characteristic, complemented
102	#410	C₂²×C₇	1	4	yes	—
103	#408	C₄×C₇	1	1	yes	characteristic, complemented
104	#408	C₂²×C₇	1	1	yes	characteristic
105	C₂×C₄×C₈	C₄×C₇	1	1	yes	characteristic, complemented
106	#354	—	2	2	no	—
107	#353	—	2	2	no	—
108	H(3,ℤ/4)	—	2	2	no	—
109	C₂³×C₇	—	2	1	no	—
110	C₂³×C₇	#117	1	1	yes	characteristic
111	C₂³×C₇	#121	1	2	yes	—
112	C₇×Q₈	—	4	1	no	—
113	C₇×Q₈	—	4	1	no	—
114	C₇×D₄	—	4	1	no	—
115	C₇×D₄	—	4	2	no	—
116	C₇×D₄	—	4	1	no	—
117	C₇×D₄	—	4	4	no	—
118	C₇×D₄	—	4	2	no	—
119	C₂×C₄×C₇	—	2	1	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	2	1	no	—
122	C₂×C₄×C₇	—	2	2	no	—
123	C₂×C₄×C₇	—	2	1	no	—
124	C₂×C₄×C₇	—	2	1	no	—
125	C₂×C₄×C₇	—	2	1	no	—
126	C₂×C₄×C₇	—	4	2	no	—
127	C₂×C₄×C₇	—	4	1	no	—
128	C₂×C₄×C₇	—	4	1	no	—
129	C₂×C₄×C₇	—	8	4	no	—
130	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
133	C₇×C₈	—	2	2	no	—
134	C₇×C₈	—	2	1	no	—
135	C₇×C₈	—	4	1	no	—
136	C₇×C₈	—	4	1	no	—
137	C₇×C₈	#99	1	1	yes	characteristic
138	C₇×C₈	#99	1	1	yes	characteristic
139	C₂×#41	C₂×C₄×C₇	1	1	yes	characteristic
140	C₂×#41	C₂³×C₇	1	1	yes	characteristic
141	#131	—	4	1	no	—
142	#131	—	4	1	no	—
143	#131	—	4	1	no	—
144	#131	—	4	1	no	—
145	C₂×M₁₆	—	2	1	no	—
146	C₂×M₁₆	C₂×C₄×C₇	1	1	yes	characteristic
147	C₂×M₁₆	C₂×C₄×C₇	1	1	yes	characteristic
148	C₂×M₁₆	C₇×D₄	1	2	yes	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
151	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
152	C₂²×C₈	C₇×D₄	1	2	yes	—
153	C₄⋊Q₈	C₇×Q₈	1	1	yes	characteristic
154	#127	C₇×D₄	1	1	yes	characteristic
155	#124	C₇×D₄	1	1	yes	characteristic
156	#124	C₇×Q₈	1	1	yes	characteristic
157	#121	—	2	2	no	—
158	C₄×D₄	—	2	2	no	—
159	#117	—	2	4	no	—
160	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
161	#107	—	4	2	no	—
162	#106	—	4	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	#99	—	2	4	no	—
165	#99	—	2	4	no	—
166	C₂²⋊C₈	—	2	2	no	—
167	C₂²⋊C₈	—	2	2	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	—	2	1	no	—
170	C₂²×C₇	—	2	2	no	—
171	C₂²×C₇	—	2	1	no	—
172	C₂²×C₇	—	4	1	no	—
173	C₂²×C₇	—	4	2	no	—
174	C₂²×C₇	—	4	2	no	—
175	C₂²×C₇	—	4	1	no	—
176	C₂²×C₇	#380	1	1	yes	characteristic
177	C₄×C₇	—	2	2	no	—
178	C₄×C₇	—	2	1	no	—
179	C₄×C₇	—	2	2	no	—
180	C₄×C₇	—	4	1	no	—
181	C₄×C₇	—	4	1	no	—
182	C₄×C₇	—	16	4	no	—
183	C₄×C₇	C₂×#99	1	1	yes	characteristic
184	C₄×C₇	#410	1	1	yes	characteristic
185	#41	—	4	1	no	—
186	#41	—	4	1	no	—
187	#41	—	4	1	no	—
188	#41	—	4	1	no	—
189	C₂×Q₈	C₇×C₄⋊C₄	1	1	yes	characteristic
190	C₂×Q₈	C₂×C₇×Q₈	1	1	yes	characteristic
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	C₇×C₄⋊C₄	1	1	yes	characteristic
194	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
195	C₂×D₄	C₇×#41	1	2	yes	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
200	M₁₆	—	4	1	no	—
201	M₁₆	—	4	2	no	—
202	M₁₆	—	4	2	no	—
203	M₁₆	—	4	1	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	4	1	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₈	C₇×C₂²⋊C₄	1	2	yes	—
215	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	4	4	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	4	4	no	—
223	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
224	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
225	C₂×C₇	—	2	1	no	—
226	C₂×C₇	—	4	2	no	—
227	C₂×C₇	—	4	1	no	—
228	C₂×C₇	—	8	1	no	—
229	C₂×C₇	#1799	1	1	yes	characteristic
230	C₂³	—	2	1	no	—
231	C₂³	C₇×#117	1	1	yes	characteristic
232	C₂³	C₇×#121	1	2	yes	—
233	Q₈	—	4	1	no	—
234	Q₈	—	4	1	no	—
235	D₄	—	4	1	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	1	no	—
238	D₄	—	4	4	no	—
239	D₄	—	4	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
252	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
253	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
254	C₈	—	2	2	no	—
255	C₈	—	2	1	no	—
256	C₈	—	4	1	no	—
257	C₈	—	4	1	no	—
258	C₈	C₇×#99	1	1	yes	characteristic
259	C₈	C₇×#99	1	1	yes	characteristic
260	C₇	#12807	1	1	yes	characteristic, direct product factor, Sylow
261	C₂²	—	2	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	C₇×#380	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	16	4	no	—
274	C₄	C₂×C₇×#99	1	1	yes	characteristic
275	C₄	C₇×#410	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	4	2	no	—
278	C₂	—	4	1	no	—
279	C₂	—	8	1	no	—
280	C₂	C₇×#1799	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	#410
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×#41	C₂³×C₇
socle	C₂×C₇	#1799
prime radical	C₇×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₇×C₈ ◁ C₂×C₇×#131 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1812	C₂	semidirect
4. C₇×#1391	C₂	semidirect
5. C₂×C₇×#131	C₄	semidirect
11. C₇×#423	C₄	semidirect
13. C₇×#408	C₄	semidirect
15. C₂×C₄×C₇×C₈	C₄	semidirect
19. #12807	C₇	direct (N × H)
53. #1812	C₂×C₇	semidirect
54. #1391	C₂×C₇	semidirect
95. C₂×#131	C₄×C₇	semidirect
101. #423	C₄×C₇	semidirect
103. #408	C₄×C₇	semidirect
105. C₂×C₄×C₈	C₄×C₇	semidirect
260. C₇	#12807	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×#2931	2
C₇×#2126	2
C₇×#1812	2
C₇×#1391	2
#12807	7

Subgroups — C7×#12807 #13190023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12807 #13190023