Subgroups — C₇×#13363 #13190579

Order 1792.

450 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	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#357	C₂	1	2	yes	maximal
3	C₇×#1970	C₂	1	1	yes	maximal, characteristic
4	C₇×#1380	C₂	1	2	yes	maximal, complemented
5	C₇×#1374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
8	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
9	C₇×#500	—	2	4	no	—
10	C₂×C₇×#107	—	2	1	no	—
11	C₂×C₇×#107	C₂²	1	1	yes	characteristic
12	C₂×C₇×#107	C₂²	1	2	yes	—
13	C₂×C₇×#102	—	2	1	no	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₇×#390	—	2	1	no	—
16	C₇×#357	—	2	4	no	—
17	#13363	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₇×Q₈	—	4	1	no	—
19	C₂²×C₇×D₄	—	2	1	no	—
20	C₂×C₇×D₈	—	2	2	no	—
21	C₂×C₇×D₈	—	2	2	no	—
22	C₂×C₇×D₈	—	2	1	no	—
23	C₂×C₇×D₈	Q₈	1	2	yes	complemented
24	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
26	C₇×C₄⋊Q₈	—	2	4	no	—
27	C₇×C₄⋊Q₈	—	4	2	no	—
28	C₂×C₇×C₄⋊C₄	—	2	2	no	—
29	C₂×C₇×C₄⋊C₄	—	2	1	no	—
30	C₂×C₇×C₄⋊C₄	—	4	1	no	—
31	C₂×C₇×C₂²⋊C₄	—	4	1	no	—
32	C₂×C₇×C₂²⋊C₄	—	4	1	no	—
33	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
34	C₂×C₇×C₁₆	—	2	2	no	—
35	C₂×C₇×C₁₆	—	2	2	no	—
36	C₂×C₇×C₁₆	—	2	2	no	—
37	C₇×#107	—	2	4	no	—
38	C₇×#107	—	2	2	no	—
39	C₇×#107	—	2	2	no	—
40	C₇×#107	D₄	1	4	yes	—
41	C₇×#102	—	4	2	no	—
42	C₄×C₇×C₈	—	2	2	no	—
43	C₂×#500	C₂×C₇	1	1	yes	characteristic, complemented
44	C₂×#357	C₂×C₇	1	2	yes	—
45	#1970	C₂×C₇	1	1	yes	characteristic
46	#1380	C₂×C₇	1	2	yes	complemented
47	#1374	C₂×C₇	1	1	yes	characteristic, complemented
48	C₂⁴×C₇	—	4	1	no	—
49	C₂×C₇×Q₈	—	4	4	no	—
50	C₂×C₇×Q₈	—	8	2	no	—
51	C₂×C₇×Q₈	—	8	2	no	—
52	C₂×C₇×D₄	—	2	2	no	—
53	C₂×C₇×D₄	—	4	1	no	—
54	C₂×C₇×D₄	—	4	1	no	—
55	C₂×C₇×D₄	—	4	1	no	—
56	C₂×C₇×D₄	—	4	1	no	—
57	C₂×C₇×D₄	—	4	1	no	—
58	C₂²×C₄×C₇	—	4	2	no	—
59	C₂²×C₄×C₇	—	4	1	no	—
60	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
63	C₇×D₈	—	2	4	no	—
64	C₇×D₈	—	4	2	no	—
65	C₂×C₇×C₈	—	2	1	no	—
66	C₂×C₇×C₈	—	2	1	no	—
67	C₂×C₇×C₈	SD₁₆	1	2	yes	—
68	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
69	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₇×C₈	#41	1	2	yes	—
71	C₂×C₇×C₈	#41	1	1	yes	characteristic
72	C₇×C₄⋊C₄	—	2	4	no	—
73	C₇×C₄⋊C₄	—	4	4	no	—
74	C₇×C₄⋊C₄	—	4	2	no	—
75	C₇×C₄⋊C₄	—	4	1	no	—
76	C₇×C₄⋊C₄	—	4	1	no	—
77	C₇×C₂²⋊C₄	—	8	2	no	—
78	C₇×C₂²⋊C₄	—	8	2	no	—
79	C₄²×C₇	—	2	1	no	—
80	C₄²×C₇	—	2	1	no	—
81	C₇×C₁₆	—	4	2	no	—
82	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
83	C₂×C₄⋊Q₈	—	2	1	no	—
84	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
85	#500	—	2	4	no	—
86	C₂×#107	—	2	1	no	—
87	C₂×#107	C₂²×C₇	1	1	yes	characteristic
88	C₂×#107	C₂²×C₇	1	2	yes	—
89	C₂×#102	—	2	1	no	—
90	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
91	#390	—	2	1	no	—
92	#357	—	2	4	no	—
93	C₂³×C₇	—	4	2	no	—
94	C₂³×C₇	—	8	1	no	—
95	C₂³×C₇	—	8	1	no	—
96	C₂³×C₇	—	8	1	no	—
97	C₂³×C₇	—	8	1	no	—
98	C₂³×C₇	—	8	1	no	—
99	C₂³×C₇	—	8	1	no	—
100	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
101	C₇×Q₈	—	8	8	no	—
102	C₇×D₄	—	4	2	no	—
103	C₇×D₄	—	4	2	no	—
104	C₇×D₄	—	8	1	no	—
105	C₇×D₄	—	8	1	no	—
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	2	1	no	—
110	C₂×C₄×C₇	—	4	4	no	—
111	C₂×C₄×C₇	—	8	2	no	—
112	C₂×C₄×C₇	—	8	2	no	—
113	C₂×C₄×C₇	—	8	1	no	—
114	C₂×C₄×C₇	—	8	1	no	—
115	C₂×C₄×C₇	—	8	1	no	—
116	C₂×C₄×C₇	D₁₆	1	2	yes	—
117	C₂×C₄×C₇	SD₃₂	1	2	yes	—
118	C₂×C₄×C₇	#120	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₇	#124	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₇	#137	1	1	yes	characteristic
124	C₇×C₈	—	2	2	no	—
125	C₇×C₈	C₂²⋊Q₈	1	2	yes	—
126	C₇×C₈	#124	1	2	yes	—
127	C₂²×Q₈	—	4	1	no	—
128	C₂²×D₄	—	2	1	no	—
129	C₂×D₈	—	2	2	no	—
130	C₂×D₈	—	2	2	no	—
131	C₂×D₈	—	2	1	no	—
132	C₂×D₈	C₇×Q₈	1	2	yes	complemented
133	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
134	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
135	C₄⋊Q₈	—	2	4	no	—
136	C₄⋊Q₈	—	4	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	4	1	no	—
140	C₂×C₂²⋊C₄	—	4	1	no	—
141	C₂×C₂²⋊C₄	—	4	1	no	—
142	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	#107	—	2	4	no	—
147	#107	—	2	2	no	—
148	#107	—	2	2	no	—
149	#107	C₇×D₄	1	4	yes	—
150	#102	—	4	2	no	—
151	C₄×C₈	—	2	2	no	—
152	C₂²×C₇	—	8	2	no	—
153	C₂²×C₇	—	8	2	no	—
154	C₂²×C₇	—	8	2	no	—
155	C₂²×C₇	—	16	1	no	—
156	C₂²×C₇	—	16	1	no	—
157	C₂²×C₇	—	16	1	no	—
158	C₂²×C₇	—	16	1	no	—
159	C₂²×C₇	#447	1	1	yes	characteristic
160	C₂²×C₇	#474	1	1	yes	characteristic
161	C₂²×C₇	#486	1	1	yes	characteristic
162	C₂²×C₇	C₂×D₁₆	1	1	yes	characteristic
163	C₂²×C₇	C₂×SD₃₂	1	1	yes	characteristic
164	C₂²×C₇	#509	1	1	yes	characteristic
165	C₂²×C₇	#510	1	1	yes	characteristic
166	C₄×C₇	—	2	2	no	—
167	C₄×C₇	—	2	2	no	—
168	C₄×C₇	—	8	4	no	—
169	C₄×C₇	—	16	1	no	—
170	C₄×C₇	#393	1	1	yes	characteristic
171	C₄×C₇	#448	1	1	yes	characteristic
172	C₄×C₇	#476	1	1	yes	characteristic
173	C₄×C₇	#489	1	1	yes	characteristic
174	C₂⁴	—	4	1	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	8	2	no	—
177	C₂×Q₈	—	8	2	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	4	1	no	—
180	C₂×D₄	—	4	1	no	—
181	C₂×D₄	—	4	1	no	—
182	C₂×D₄	—	4	1	no	—
183	C₂×D₄	—	4	1	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
187	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
189	D₈	—	2	4	no	—
190	D₈	—	4	2	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	C₇×SD₁₆	1	2	yes	—
194	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
195	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
196	C₂×C₈	C₇×#41	1	2	yes	—
197	C₂×C₈	C₇×#41	1	1	yes	characteristic
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	1	no	—
202	C₄⋊C₄	—	4	1	no	—
203	C₂²⋊C₄	—	8	2	no	—
204	C₂²⋊C₄	—	8	2	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₁₆	—	4	2	no	—
208	C₂×C₇	—	16	2	no	—
209	C₂×C₇	#2023	1	1	yes	characteristic
210	C₂×C₇	#2184	1	1	yes	characteristic
211	C₂×C₇	#2185	1	1	yes	characteristic
212	C₂×C₇	#2224	1	1	yes	characteristic
213	C₂×C₇	#2226	1	1	yes	characteristic
214	C₂×C₇	#2240	1	1	yes	characteristic
215	C₂×C₇	#2243	1	1	yes	characteristic
216	C₂³	—	4	2	no	—
217	C₂³	—	8	1	no	—
218	C₂³	—	8	1	no	—
219	C₂³	—	8	1	no	—
220	C₂³	—	8	1	no	—
221	C₂³	—	8	1	no	—
222	C₂³	—	8	1	no	—
223	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
224	Q₈	—	8	8	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	8	1	no	—
228	D₄	—	8	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	1	no	—
237	C₂×C₄	—	8	1	no	—
238	C₂×C₄	—	8	1	no	—
239	C₂×C₄	C₇×D₁₆	1	2	yes	—
240	C₂×C₄	C₇×SD₃₂	1	2	yes	—
241	C₂×C₄	C₇×#120	1	1	yes	characteristic
242	C₂×C₄	C₇×C₂²⋊Q₈	1	1	yes	characteristic
243	C₂×C₄	C₇×#124	1	1	yes	characteristic
244	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
245	C₂×C₄	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
246	C₂×C₄	C₇×#137	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	C₇×C₂²⋊Q₈	1	2	yes	—
249	C₈	C₇×#124	1	2	yes	—
250	C₇	#13363	1	1	yes	characteristic, direct product factor, Sylow
251	C₂²	—	8	2	no	—
252	C₂²	—	8	2	no	—
253	C₂²	—	8	2	no	—
254	C₂²	—	16	1	no	—
255	C₂²	—	16	1	no	—
256	C₂²	—	16	1	no	—
257	C₂²	—	16	1	no	—
258	C₂²	C₇×#447	1	1	yes	characteristic
259	C₂²	C₇×#474	1	1	yes	characteristic
260	C₂²	C₇×#486	1	1	yes	characteristic
261	C₂²	C₂×C₇×D₁₆	1	1	yes	characteristic
262	C₂²	C₂×C₇×SD₃₂	1	1	yes	characteristic
263	C₂²	C₇×#509	1	1	yes	characteristic
264	C₂²	C₇×#510	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	8	4	no	—
268	C₄	—	16	1	no	—
269	C₄	C₇×#393	1	1	yes	characteristic
270	C₄	C₇×#448	1	1	yes	characteristic
271	C₄	C₇×#476	1	1	yes	characteristic
272	C₄	C₇×#489	1	1	yes	characteristic
273	C₂	—	16	2	no	—
274	C₂	C₇×#2023	1	1	yes	characteristic
275	C₂	C₇×#2184	1	1	yes	characteristic
276	C₂	C₇×#2185	1	1	yes	characteristic
277	C₂	C₇×#2224	1	1	yes	characteristic
278	C₂	C₇×#2226	1	1	yes	characteristic
279	C₂	C₇×#2240	1	1	yes	characteristic
280	C₂	C₇×#2243	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂²×C₇×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#500	C₂	semidirect
4. C₇×#1380	C₂	semidirect
5. C₇×#1374	C₂	semidirect
17. #13363	C₇	direct (N × H)
23. C₂×C₇×D₈	Q₈	semidirect
43. C₂×#500	C₂×C₇	semidirect
46. #1380	C₂×C₇	semidirect
47. #1374	C₂×C₇	semidirect
132. C₂×D₈	C₇×Q₈	semidirect
250. C₇	#13363	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#500	2
C₂×C₇×#357	2
C₇×#1970	2
C₇×#1380	2
C₇×#1374	2
#13363	7

Subgroups — C7×#13363 #13190579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13363 #13190579