Subgroups — C₇×#20660 #13197876

Order 1792.

1518 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#427	C₂	1	2	yes	maximal, complemented
3	C₇×#2280	C₂	1	2	yes	maximal
4	C₇×#1809	C₂	1	8	yes	maximal, complemented
5	C₇×#1735	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂²×C₇×M₁₆	—	2	8	no	—
8	C₂²×C₇×M₁₆	C₄	1	8	yes	complemented
9	C₂²×C₇×M₁₆	C₂²	1	4	yes	—
10	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₇×#117	C₂²	1	4	yes	—
13	C₇×#427	—	2	8	no	—
14	C₂×C₇×#107	C₂²	1	4	yes	complemented
15	C₂×C₇×#106	C₂²	1	4	yes	complemented
16	C₇×#381	—	2	2	no	—
17	C₇×#380	C₂²	1	8	yes	—
18	C₇×#379	—	2	2	no	—
19	C₄×C₇×C₂²⋊C₄	—	2	4	no	—
20	C₇×#340	C₂²	1	8	yes	complemented
21	#20660	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂⁵×C₇	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₇	—	2	4	no	—
24	C₂³×C₄×C₇	—	2	4	no	—
25	C₂³×C₄×C₇	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₇	D₄	1	2	yes	—
28	C₂³×C₄×C₇	Q₈	1	4	yes	—
29	C₂³×C₄×C₇	C₂³	1	2	yes	—
30	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₂×C₇×M₁₆	—	2	32	no	—
32	C₂×C₇×M₁₆	—	2	8	no	—
33	C₂×C₇×M₁₆	—	2	8	no	—
34	C₂×C₇×M₁₆	C₂×C₄	1	16	yes	—
35	C₂²×C₇×C₈	—	2	4	no	—
36	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
37	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
38	C₂²×C₇×C₈	D₄	1	8	yes	—
39	C₂²×C₇×C₈	C₂³	1	2	yes	—
40	C₂²×C₇×C₈	C₂³	1	2	yes	—
41	C₇×#117	—	2	8	no	—
42	C₇×#117	—	2	8	no	—
43	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
44	C₂×C₇×C₂²⋊C₄	—	2	8	no	—
45	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
46	C₂×C₄²×C₇	—	2	4	no	—
47	C₇×#107	—	2	8	no	—
48	C₇×#106	—	2	8	no	—
49	C₇×#95	—	2	8	no	—
50	C₂³×M₁₆	C₂×C₇	1	1	yes	characteristic
51	C₂×#427	C₂×C₇	1	2	yes	complemented
52	#2280	C₂×C₇	1	2	yes	—
53	#1809	C₂×C₇	1	8	yes	complemented
54	#1735	C₂×C₇	1	2	yes	complemented
55	C₂⁴×C₇	—	2	4	no	—
56	C₂⁴×C₇	—	4	4	no	—
57	C₂⁴×C₇	C₄⋊C₄	1	4	yes	—
58	C₂⁴×C₇	C₂×D₄	1	2	yes	—
59	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	—	2	4	no	—
61	C₂²×C₄×C₇	—	2	8	no	—
62	C₂²×C₄×C₇	—	2	8	no	—
63	C₂²×C₄×C₇	—	2	8	no	—
64	C₂²×C₄×C₇	—	2	16	no	—
65	C₂²×C₄×C₇	—	2	4	no	—
66	C₂²×C₄×C₇	—	2	4	no	—
67	C₂²×C₄×C₇	—	2	8	no	—
68	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₇	C₄⋊C₄	1	8	yes	—
70	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₇	#41	1	4	yes	—
78	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
79	C₇×M₁₆	—	2	32	no	—
80	C₂×C₇×C₈	—	2	8	no	—
81	C₂×C₇×C₈	—	2	4	no	—
82	C₂×C₇×C₈	C₂²⋊C₄	1	16	yes	complemented
83	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
84	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
85	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
86	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
87	C₇×C₄⋊C₄	—	2	8	no	—
88	C₇×C₄⋊C₄	—	2	8	no	—
89	C₇×C₂²⋊C₄	—	4	16	no	—
90	C₇×C₂²⋊C₄	—	4	8	no	—
91	C₄²×C₇	—	4	8	no	—
92	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
93	C₂²×M₁₆	—	2	8	no	—
94	C₂²×M₁₆	C₄×C₇	1	8	yes	complemented
95	C₂²×M₁₆	C₂²×C₇	1	4	yes	—
96	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
97	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
98	C₂×#117	C₂²×C₇	1	4	yes	—
99	#427	—	2	8	no	—
100	C₂×#107	C₂²×C₇	1	4	yes	complemented
101	C₂×#106	C₂²×C₇	1	4	yes	complemented
102	#381	—	2	2	no	—
103	#380	C₂²×C₇	1	8	yes	—
104	#379	—	2	2	no	—
105	C₄×C₂²⋊C₄	—	2	4	no	—
106	#340	C₂²×C₇	1	8	yes	complemented
107	C₂³×C₇	—	2	4	no	—
108	C₂³×C₇	—	2	4	no	—
109	C₂³×C₇	—	2	8	no	—
110	C₂³×C₇	—	2	8	no	—
111	C₂³×C₇	—	4	8	no	—
112	C₂³×C₇	—	4	16	no	—
113	C₂³×C₇	C₂×C₄⋊C₄	1	4	yes	—
114	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂³×C₇	#121	1	4	yes	—
116	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇	—	2	4	no	—
118	C₂×C₄×C₇	—	2	8	no	—
119	C₂×C₄×C₇	—	2	4	no	—
120	C₂×C₄×C₇	—	2	4	no	—
121	C₂×C₄×C₇	—	2	8	no	—
122	C₂×C₄×C₇	—	2	16	no	—
123	C₂×C₄×C₇	—	4	8	no	—
124	C₂×C₄×C₇	—	4	16	no	—
125	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
126	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
127	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₇	#117	1	4	yes	—
129	C₂×C₄×C₇	#121	1	4	yes	—
130	C₂×C₄×C₇	C₂²⋊Q₈	1	8	yes	—
131	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
133	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂×#41	1	2	yes	—
135	C₇×C₈	—	2	4	no	—
136	C₇×C₈	C₂×C₂²⋊C₄	1	8	yes	—
137	C₂⁵	C₇×D₄	1	1	yes	characteristic
138	C₂³×C₄	—	2	4	no	—
139	C₂³×C₄	—	2	4	no	—
140	C₂³×C₄	C₂×C₄×C₇	1	4	yes	—
141	C₂³×C₄	C₇×D₄	1	2	yes	—
142	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
143	C₂³×C₄	C₇×Q₈	1	4	yes	—
144	C₂³×C₄	C₂³×C₇	1	2	yes	—
145	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
146	C₂×M₁₆	—	2	32	no	—
147	C₂×M₁₆	—	2	8	no	—
148	C₂×M₁₆	—	2	8	no	—
149	C₂×M₁₆	C₂×C₄×C₇	1	16	yes	—
150	C₂²×C₈	—	2	4	no	—
151	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
152	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
153	C₂²×C₈	C₇×D₄	1	8	yes	—
154	C₂²×C₈	C₂³×C₇	1	2	yes	—
155	C₂²×C₈	C₂³×C₇	1	2	yes	—
156	#117	—	2	8	no	—
157	#117	—	2	8	no	—
158	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
159	C₂×C₂²⋊C₄	—	2	4	no	—
160	C₂×C₂²⋊C₄	—	2	8	no	—
161	C₂×C₄²	—	2	4	no	—
162	#107	—	2	8	no	—
163	#106	—	2	8	no	—
164	#95	—	2	8	no	—
165	C₂²×C₇	—	2	4	no	—
166	C₂²×C₇	—	2	8	no	—
167	C₂²×C₇	—	2	8	no	—
168	C₂²×C₇	—	2	4	no	—
169	C₂²×C₇	—	4	16	no	—
170	C₂²×C₇	—	4	8	no	—
171	C₂²×C₇	#380	1	4	yes	—
172	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
173	C₂²×C₇	C₂×#121	1	2	yes	—
174	C₂²×C₇	#575	1	4	yes	—
175	C₄×C₇	—	2	4	no	—
176	C₄×C₇	—	8	8	no	—
177	C₄×C₇	#380	1	4	yes	—
178	C₄×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
179	C₄×C₇	C₂×#117	1	1	yes	characteristic
180	C₄×C₇	C₂×C₂²⋊Q₈	1	2	yes	—
181	C₂⁴	—	2	4	no	—
182	C₂⁴	—	4	4	no	—
183	C₂⁴	C₇×C₄⋊C₄	1	4	yes	—
184	C₂⁴	C₂×C₇×D₄	1	2	yes	—
185	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	8	no	—
188	C₂²×C₄	—	2	8	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	2	16	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
195	C₂²×C₄	C₇×C₄⋊C₄	1	8	yes	—
196	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
197	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
198	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
199	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
200	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
202	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
203	C₂²×C₄	C₇×#41	1	4	yes	—
204	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
205	M₁₆	—	2	32	no	—
206	C₂×C₈	—	2	8	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	C₇×C₂²⋊C₄	1	16	yes	complemented
209	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
210	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
211	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
212	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
213	C₄⋊C₄	—	2	8	no	—
214	C₄⋊C₄	—	2	8	no	—
215	C₂²⋊C₄	—	4	16	no	—
216	C₂²⋊C₄	—	4	8	no	—
217	C₄²	—	4	8	no	—
218	C₂×C₇	—	2	4	no	—
219	C₂×C₇	—	4	4	no	—
220	C₂×C₇	C₂×#380	1	1	yes	characteristic
221	C₂×C₇	#2911	1	2	yes	—
222	C₂×C₇	#3054	1	4	yes	—
223	C₂³	—	2	4	no	—
224	C₂³	—	2	8	no	—
225	C₂³	—	2	4	no	—
226	C₂³	—	2	8	no	—
227	C₂³	—	4	8	no	—
228	C₂³	—	4	16	no	—
229	C₂³	C₂×C₇×C₄⋊C₄	1	4	yes	—
230	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
231	C₂³	C₇×#121	1	4	yes	—
232	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	16	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	4	8	no	—
240	C₂×C₄	—	4	16	no	—
241	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
242	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
243	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
244	C₂×C₄	C₇×#117	1	4	yes	—
245	C₂×C₄	C₇×#121	1	4	yes	—
246	C₂×C₄	C₇×C₂²⋊Q₈	1	8	yes	—
247	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
249	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
250	C₂×C₄	C₂×C₇×#41	1	2	yes	—
251	C₈	—	2	4	no	—
252	C₈	C₂×C₇×C₂²⋊C₄	1	8	yes	—
253	C₇	#20660	1	1	yes	characteristic, direct product factor, Sylow
254	C₂²	—	2	4	no	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	4	16	no	—
259	C₂²	—	4	8	no	—
260	C₂²	C₇×#380	1	4	yes	—
261	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
262	C₂²	C₂×C₇×#121	1	2	yes	—
263	C₂²	C₇×#575	1	4	yes	—
264	C₄	—	2	4	no	—
265	C₄	—	8	8	no	—
266	C₄	C₇×#380	1	4	yes	—
267	C₄	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
268	C₄	C₂×C₇×#117	1	1	yes	characteristic
269	C₄	C₂×C₇×C₂²⋊Q₈	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	—	4	4	no	—
272	C₂	C₂×C₇×#380	1	1	yes	characteristic
273	C₂	C₇×#2911	1	2	yes	—
274	C₂	C₇×#3054	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂²×D₄
prime radical	C₂⁵×C₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂⁴×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#427	C₂	semidirect
4. C₇×#1809	C₂	semidirect
5. C₇×#1735	C₂	semidirect
8. C₂²×C₇×M₁₆	C₄	semidirect
14. C₂×C₇×#107	C₂²	semidirect
15. C₂×C₇×#106	C₂²	semidirect
20. C₇×#340	C₂²	semidirect
21. #20660	C₇	direct (N × H)
51. C₂×#427	C₂×C₇	semidirect
53. #1809	C₂×C₇	semidirect
54. #1735	C₂×C₇	semidirect
82. C₂×C₇×C₈	C₂²⋊C₄	semidirect
94. C₂²×M₁₆	C₄×C₇	semidirect
100. C₂×#107	C₂²×C₇	semidirect
101. C₂×#106	C₂²×C₇	semidirect
106. #340	C₂²×C₇	semidirect
208. C₂×C₈	C₇×C₂²⋊C₄	semidirect
253. C₇	#20660	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇×M₁₆	2
C₂×C₇×#427	2
C₇×#2280	2
C₇×#1809	2
C₇×#1735	2
#20660	7

Subgroups — C7×#20660 #13197876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20660 #13197876