Subgroups — C₇×#7903 #13185119

Order 1792.

794 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#423	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₇×#117	C₂	1	1	yes	maximal, characteristic
4	C₇×#1279	C₂	1	4	yes	maximal, complemented
5	C₂²×C₇×M₁₆	—	2	1	no	—
6	C₂²×C₇×M₁₆	—	2	1	no	—
7	C₂×C₇×#117	C₄	1	2	yes	—
8	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₇×#423	—	2	2	no	—
10	C₇×#423	C₄	1	4	yes	complemented
11	C₂×C₇×C₄⋊C₈	—	2	1	no	—
12	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
13	C₄×C₇×M₁₆	—	2	2	no	—
14	C₄×C₇×M₁₆	C₄	1	4	yes	complemented
15	C₂×C₇×#97	—	2	1	no	—
16	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
17	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
18	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₄×C₇×C₂²⋊C₄	—	2	1	no	—
20	C₄×C₇×C₂²⋊C₄	—	2	1	no	—
21	C₇×#376	—	2	1	no	—
22	C₄³×C₇	—	2	1	no	—
23	#7903	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
27	C₂×C₇×M₁₆	—	2	8	no	—
28	C₂×C₇×M₁₆	—	2	4	no	—
29	C₂×C₇×M₁₆	—	2	4	no	—
30	C₂×C₇×M₁₆	—	4	4	no	—
31	C₂²×C₇×C₈	—	2	2	no	—
32	C₂²×C₇×C₈	—	2	2	no	—
33	C₇×#117	—	2	8	no	—
34	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
35	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
36	C₂×C₄²×C₇	—	2	2	no	—
37	C₂×C₄²×C₇	—	2	2	no	—
38	C₂×C₄²×C₇	—	2	2	no	—
39	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇	D₄	1	2	yes	—
45	C₂×C₄²×C₇	Q₈	1	2	yes	—
46	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
47	C₇×C₄⋊C₈	—	2	4	no	—
48	C₇×C₄⋊C₈	—	2	2	no	—
49	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
50	C₇×#97	—	2	4	no	—
51	C₄×C₇×C₈	—	2	2	no	—
52	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
53	C₇×#95	—	2	2	no	—
54	C₇×#95	—	2	2	no	—
55	C₂×#423	C₂×C₇	1	1	yes	characteristic
56	C₂×C₄×M₁₆	C₂×C₇	1	1	yes	characteristic
57	C₄×#117	C₂×C₇	1	1	yes	characteristic
58	#1279	C₂×C₇	1	4	yes	complemented
59	C₂⁴×C₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	—	2	4	no	—
61	C₂²×C₄×C₇	—	2	4	no	—
62	C₂²×C₄×C₇	—	2	2	no	—
63	C₂²×C₄×C₇	—	2	2	no	—
64	C₂²×C₄×C₇	—	2	4	no	—
65	C₂²×C₄×C₇	C₄²	1	2	yes	—
66	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
77	C₇×M₁₆	—	2	16	no	—
78	C₇×M₁₆	—	4	8	no	—
79	C₂×C₇×C₈	—	2	8	no	—
80	C₂×C₇×C₈	—	2	4	no	—
81	C₂×C₇×C₈	—	2	4	no	—
82	C₂×C₇×C₈	—	4	4	no	—
83	C₇×C₄⋊C₄	—	2	4	no	—
84	C₇×C₄⋊C₄	M₁₆	1	8	yes	complemented
85	C₇×C₂²⋊C₄	—	2	8	no	—
86	C₄²×C₇	—	2	8	no	—
87	C₄²×C₇	—	4	4	no	—
88	C₄²×C₇	—	4	2	no	—
89	C₄²×C₇	—	4	2	no	—
90	C₄²×C₇	C₄²	1	2	yes	—
91	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₇	C₄⋊C₄	1	2	yes	—
94	C₄²×C₇	C₄⋊C₄	1	2	yes	—
95	C₄²×C₇	C₄⋊C₄	1	2	yes	—
96	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
100	C₂²×M₁₆	—	2	1	no	—
101	C₂²×M₁₆	—	2	1	no	—
102	C₂×#117	C₄×C₇	1	2	yes	—
103	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
104	#423	—	2	2	no	—
105	#423	C₄×C₇	1	4	yes	complemented
106	C₂×C₄⋊C₈	—	2	1	no	—
107	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
108	C₄×M₁₆	—	2	2	no	—
109	C₄×M₁₆	C₄×C₇	1	4	yes	complemented
110	C₂×#97	—	2	1	no	—
111	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
112	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
113	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
114	C₄×C₂²⋊C₄	—	2	1	no	—
115	C₄×C₂²⋊C₄	—	2	1	no	—
116	#376	—	2	1	no	—
117	C₄³	—	2	1	no	—
118	C₂³×C₇	—	2	2	no	—
119	C₂³×C₇	—	2	2	no	—
120	C₂³×C₇	—	2	2	no	—
121	C₂³×C₇	#95	1	2	yes	—
122	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇	—	2	2	no	—
124	C₂×C₄×C₇	—	2	2	no	—
125	C₂×C₄×C₇	—	2	2	no	—
126	C₂×C₄×C₇	—	2	4	no	—
127	C₂×C₄×C₇	—	2	4	no	—
128	C₂×C₄×C₇	—	2	4	no	—
129	C₂×C₄×C₇	—	2	2	no	—
130	C₂×C₄×C₇	—	2	2	no	—
131	C₂×C₄×C₇	—	2	4	no	—
132	C₂×C₄×C₇	—	2	4	no	—
133	C₂×C₄×C₇	—	2	8	no	—
134	C₂×C₄×C₇	—	4	4	no	—
135	C₂×C₄×C₇	—	4	4	no	—
136	C₂×C₄×C₇	#95	1	2	yes	—
137	C₂×C₄×C₇	#95	1	2	yes	—
138	C₂×C₄×C₇	#95	1	2	yes	—
139	C₂×C₄×C₇	#104	1	4	yes	—
140	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
147	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₇	#137	1	2	yes	—
149	C₇×C₈	—	2	8	no	—
150	C₇×C₈	—	4	4	no	—
151	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
152	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
153	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
154	C₂×M₁₆	—	2	8	no	—
155	C₂×M₁₆	—	2	4	no	—
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	—	4	4	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	#117	—	2	8	no	—
161	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
162	C₂×C₂²⋊C₄	—	2	2	no	—
163	C₂×C₄²	—	2	2	no	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
167	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
168	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
169	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
170	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
171	C₂×C₄²	C₇×D₄	1	2	yes	—
172	C₂×C₄²	C₇×Q₈	1	2	yes	—
173	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
174	C₄⋊C₈	—	2	4	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
177	#97	—	2	4	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
180	#95	—	2	2	no	—
181	#95	—	2	2	no	—
182	C₂²×C₇	—	2	2	no	—
183	C₂²×C₇	—	2	2	no	—
184	C₂²×C₇	—	2	2	no	—
185	C₂²×C₇	—	4	4	no	—
186	C₂²×C₇	C₂×#95	1	1	yes	characteristic
187	C₂²×C₇	#407	1	2	yes	—
188	C₂²×C₇	#417	1	2	yes	—
189	C₂²×C₇	C₂×#104	1	2	yes	—
190	C₄×C₇	—	2	2	no	—
191	C₄×C₇	—	2	1	no	—
192	C₄×C₇	—	2	1	no	—
193	C₄×C₇	—	2	2	no	—
194	C₄×C₇	—	4	8	no	—
195	C₄×C₇	#339	1	4	yes	—
196	C₄×C₇	C₄×M₁₆	1	2	yes	—
197	C₄×C₇	#418	1	1	yes	characteristic
198	C₄×C₇	#419	1	1	yes	characteristic
199	C₄×C₇	#423	1	2	yes	—
200	C₄×C₇	#428	1	2	yes	—
201	C₂⁴	C₇×C₂²⋊C₄	1	1	yes	characteristic
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	C₄²×C₇	1	2	yes	—
208	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
209	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
211	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
212	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
214	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
215	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
216	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
219	M₁₆	—	2	16	no	—
220	M₁₆	—	4	8	no	—
221	C₂×C₈	—	2	8	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	4	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	C₇×M₁₆	1	8	yes	complemented
227	C₂²⋊C₄	—	2	8	no	—
228	C₄²	—	2	8	no	—
229	C₄²	—	4	4	no	—
230	C₄²	—	4	2	no	—
231	C₄²	—	4	2	no	—
232	C₄²	C₄²×C₇	1	2	yes	—
233	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
234	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
235	C₄²	C₇×C₄⋊C₄	1	2	yes	—
236	C₄²	C₇×C₄⋊C₄	1	2	yes	—
237	C₄²	C₇×C₄⋊C₄	1	2	yes	—
238	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
239	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
240	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
241	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
242	C₂×C₇	—	4	2	no	—
243	C₂×C₇	#1489	1	4	yes	—
244	C₂×C₇	#1728	1	1	yes	characteristic
245	C₂×C₇	C₂×#339	1	1	yes	characteristic
246	C₂×C₇	#1736	1	1	yes	characteristic
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	C₇×#95	1	2	yes	—
251	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	C₇×#95	1	2	yes	—
266	C₂×C₄	C₇×#95	1	2	yes	—
267	C₂×C₄	C₇×#95	1	2	yes	—
268	C₂×C₄	C₇×#104	1	4	yes	—
269	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
270	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
272	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
274	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
275	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
276	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
277	C₂×C₄	C₇×#137	1	2	yes	—
278	C₈	—	2	8	no	—
279	C₈	—	4	4	no	—
280	C₇	#7903	1	1	yes	characteristic, direct product factor, Sylow
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	4	4	no	—
285	C₂²	C₂×C₇×#95	1	1	yes	characteristic
286	C₂²	C₇×#407	1	2	yes	—
287	C₂²	C₇×#417	1	2	yes	—
288	C₂²	C₂×C₇×#104	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	8	no	—
294	C₄	C₇×#339	1	4	yes	—
295	C₄	C₄×C₇×M₁₆	1	2	yes	—
296	C₄	C₇×#418	1	1	yes	characteristic
297	C₄	C₇×#419	1	1	yes	characteristic
298	C₄	C₇×#423	1	2	yes	—
299	C₄	C₇×#428	1	2	yes	—
300	C₂	—	4	2	no	—
301	C₂	C₇×#1489	1	4	yes	—
302	C₂	C₇×#1728	1	1	yes	characteristic
303	C₂	C₂×C₇×#339	1	1	yes	characteristic
304	C₂	C₇×#1736	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₇×#1279	C₂	semidirect
10. C₇×#423	C₄	semidirect
14. C₄×C₇×M₁₆	C₄	semidirect
23. #7903	C₇	direct (N × H)
58. #1279	C₂×C₇	semidirect
84. C₇×C₄⋊C₄	M₁₆	semidirect
105. #423	C₄×C₇	semidirect
109. C₄×M₁₆	C₄×C₇	semidirect
226. C₄⋊C₄	C₇×M₁₆	semidirect
280. C₇	#7903	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×#423	2
C₂×C₄×C₇×M₁₆	2
C₄×C₇×#117	2
C₇×#1279	2
#7903	7

Subgroups — C7×#7903 #13185119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#7903 #13185119