Subgroups — C₇×#21172 #13198388

Order 1792.

1022 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#428	C₂	1	2	yes	maximal
2	C₂×C₇×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#382	C₂	1	1	yes	maximal, characteristic
4	C₄×C₇×#117	C₂	1	1	yes	maximal, characteristic
5	C₇×#1949	C₂	1	8	yes	maximal, complemented
6	C₇×#1736	C₂	1	2	yes	maximal
7	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
8	C₂×C₇×#117	C₂²	1	1	yes	characteristic
9	C₂×C₇×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
12	C₇×#428	—	2	8	no	—
13	C₂×C₇×#107	C₂²	1	4	yes	—
14	C₂×C₇×#106	C₂²	1	4	yes	—
15	C₇×#423	C₄	1	8	yes	complemented
16	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
17	C₇×#382	—	2	1	no	—
18	C₇×#382	—	2	1	no	—
19	C₇×#382	C₂²	1	4	yes	—
20	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
21	C₄×C₇×C₂²⋊C₄	—	2	2	no	—
22	C₇×#376	—	2	1	no	—
23	C₂×C₇×#95	—	2	2	no	—
24	C₄³×C₇	—	2	1	no	—
25	C₇×#340	C₂²	1	8	yes	—
26	#21172	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂³×C₄×C₇	—	2	2	no	—
28	C₂³×C₄×C₇	D₄	1	2	yes	—
29	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
30	C₂×C₇×M₁₆	—	2	4	no	—
31	C₂×C₇×M₁₆	—	2	4	no	—
32	C₂×C₇×M₁₆	C₂×C₄	1	8	yes	—
33	C₂²×C₇×C₈	C₂³	1	4	yes	—
34	C₇×#117	—	2	2	no	—
35	C₇×#117	—	2	2	no	—
36	C₇×#117	D₄	1	4	yes	—
37	C₇×#117	Q₈	1	4	yes	—
38	C₂×C₇×C₄⋊C₄	—	2	4	no	—
39	C₂×C₇×C₄⋊C₄	—	2	4	no	—
40	C₂×C₇×C₄⋊C₄	—	2	2	no	—
41	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
42	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
43	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
44	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
45	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
46	C₂×C₄²×C₇	—	2	1	no	—
47	C₂×C₄²×C₇	—	2	1	no	—
48	C₂×C₄²×C₇	—	2	4	no	—
49	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₇	Q₈	1	4	yes	—
53	C₂×C₄²×C₇	C₂³	1	2	yes	—
54	C₇×#107	—	2	8	no	—
55	C₇×#106	—	2	8	no	—
56	C₇×C₄⋊C₈	C₂×C₄	1	16	yes	complemented
57	C₇×#95	—	2	4	no	—
58	C₇×#95	—	2	4	no	—
59	C₇×#95	—	2	4	no	—
60	C₂×#428	C₂×C₇	1	2	yes	—
61	C₂×#423	C₂×C₇	1	1	yes	characteristic
62	C₂×#382	C₂×C₇	1	1	yes	characteristic
63	C₄×#117	C₂×C₇	1	1	yes	characteristic
64	#1949	C₂×C₇	1	8	yes	complemented
65	#1736	C₂×C₇	1	2	yes	—
66	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	—	2	2	no	—
68	C₂²×C₄×C₇	—	2	2	no	—
69	C₂²×C₄×C₇	—	2	4	no	—
70	C₂²×C₄×C₇	—	4	4	no	—
71	C₂²×C₄×C₇	—	4	8	no	—
72	C₂²×C₄×C₇	—	4	4	no	—
73	C₂²×C₄×C₇	C₄⋊C₄	1	8	yes	—
74	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₇	#41	1	2	yes	—
82	C₂²×C₄×C₇	#41	1	2	yes	—
83	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
84	C₇×M₁₆	—	2	16	no	—
85	C₂×C₇×C₈	—	2	4	no	—
86	C₂×C₇×C₈	—	2	4	no	—
87	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
88	C₇×C₄⋊C₄	—	2	2	no	—
89	C₇×C₄⋊C₄	—	2	2	no	—
90	C₇×C₄⋊C₄	—	2	4	no	—
91	C₇×C₄⋊C₄	—	2	4	no	—
92	C₇×C₄⋊C₄	—	4	4	no	—
93	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
94	C₇×C₄⋊C₄	C₂×Q₈	1	4	yes	—
95	C₇×C₂²⋊C₄	—	2	4	no	—
96	C₇×C₂²⋊C₄	—	2	4	no	—
97	C₄²×C₇	—	2	4	no	—
98	C₄²×C₇	—	2	4	no	—
99	C₄²×C₇	—	4	8	no	—
100	C₄²×C₇	C₄⋊C₄	1	4	yes	—
101	C₄²×C₇	C₄⋊C₄	1	4	yes	—
102	C₄²×C₇	C₂²×C₄	1	4	yes	—
103	C₄²×C₇	C₂×Q₈	1	4	yes	—
104	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
105	C₂×#117	C₂²×C₇	1	1	yes	characteristic
106	C₂×#117	C₂²×C₇	1	1	yes	characteristic
107	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
108	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
109	#428	—	2	8	no	—
110	C₂×#107	C₂²×C₇	1	4	yes	—
111	C₂×#106	C₂²×C₇	1	4	yes	—
112	#423	C₄×C₇	1	8	yes	complemented
113	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
114	#382	—	2	1	no	—
115	#382	—	2	1	no	—
116	#382	C₂²×C₇	1	4	yes	—
117	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
118	C₄×C₂²⋊C₄	—	2	2	no	—
119	#376	—	2	1	no	—
120	C₂×#95	—	2	2	no	—
121	C₄³	—	2	1	no	—
122	#340	C₂²×C₇	1	8	yes	—
123	C₂³×C₇	—	2	4	no	—
124	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
125	C₂³×C₇	#121	1	2	yes	—
126	C₂³×C₇	C₂²⋊Q₈	1	2	yes	—
127	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇	—	2	4	no	—
129	C₂×C₄×C₇	—	2	4	no	—
130	C₂×C₄×C₇	—	2	4	no	—
131	C₂×C₄×C₇	—	2	4	no	—
132	C₂×C₄×C₇	—	4	8	no	—
133	C₂×C₄×C₇	—	4	4	no	—
134	C₂×C₄×C₇	—	4	4	no	—
135	C₂×C₄×C₇	—	4	4	no	—
136	C₂×C₄×C₇	—	4	8	no	—
137	C₂×C₄×C₇	—	8	8	no	—
138	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
139	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
140	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₇	C₄×D₄	1	4	yes	—
142	C₂×C₄×C₇	C₄×Q₈	1	4	yes	—
143	C₂×C₄×C₇	#121	1	2	yes	—
144	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
145	C₂×C₄×C₇	#125	1	4	yes	—
146	C₂×C₄×C₇	C₄⋊Q₈	1	4	yes	—
147	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
150	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
151	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
153	C₇×C₈	—	2	8	no	—
154	C₂³×C₄	—	2	2	no	—
155	C₂³×C₄	C₇×D₄	1	2	yes	—
156	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂×M₁₆	C₂×C₄×C₇	1	8	yes	—
160	C₂²×C₈	C₂³×C₇	1	4	yes	—
161	#117	—	2	2	no	—
162	#117	—	2	2	no	—
163	#117	C₇×D₄	1	4	yes	—
164	#117	C₇×Q₈	1	4	yes	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄⋊C₄	—	2	4	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
169	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
170	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
171	C₂×C₂²⋊C₄	—	2	1	no	—
172	C₂×C₂²⋊C₄	—	2	1	no	—
173	C₂×C₄²	—	2	1	no	—
174	C₂×C₄²	—	2	1	no	—
175	C₂×C₄²	—	2	4	no	—
176	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
177	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
178	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
179	C₂×C₄²	C₇×Q₈	1	4	yes	—
180	C₂×C₄²	C₂³×C₇	1	2	yes	—
181	#107	—	2	8	no	—
182	#106	—	2	8	no	—
183	C₄⋊C₈	C₂×C₄×C₇	1	16	yes	complemented
184	#95	—	2	4	no	—
185	#95	—	2	4	no	—
186	#95	—	2	4	no	—
187	C₂²×C₇	—	2	4	no	—
188	C₂²×C₇	—	2	4	no	—
189	C₂²×C₇	—	2	4	no	—
190	C₂²×C₇	#389	1	4	yes	—
191	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
192	C₂²×C₇	C₂×#121	1	1	yes	characteristic
193	C₂²×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
194	C₂²×C₇	#575	1	4	yes	—
195	C₄×C₇	—	2	4	no	—
196	C₄×C₇	—	2	4	no	—
197	C₄×C₇	—	4	4	no	—
198	C₄×C₇	—	4	4	no	—
199	C₄×C₇	—	8	4	no	—
200	C₄×C₇	#389	1	4	yes	—
201	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
202	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
203	C₄×C₇	C₂×#125	1	1	yes	characteristic
204	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
205	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	8	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	C₇×C₄⋊C₄	1	8	yes	—
213	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
214	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
215	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
216	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
218	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
219	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
220	C₂²×C₄	C₇×#41	1	2	yes	—
221	C₂²×C₄	C₇×#41	1	2	yes	—
222	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
223	M₁₆	—	2	16	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	C₂²×C₄×C₇	1	8	yes	—
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	—	4	4	no	—
232	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
233	C₄⋊C₄	C₂×C₇×Q₈	1	4	yes	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	4	8	no	—
239	C₄²	C₇×C₄⋊C₄	1	4	yes	—
240	C₄²	C₇×C₄⋊C₄	1	4	yes	—
241	C₄²	C₂²×C₄×C₇	1	4	yes	—
242	C₄²	C₂×C₇×Q₈	1	4	yes	—
243	C₂×C₇	—	2	4	no	—
244	C₂×C₇	C₂×#389	1	1	yes	characteristic
245	C₂×C₇	#2911	1	2	yes	—
246	C₂×C₇	#3040	1	2	yes	—
247	C₂×C₇	#3080	1	2	yes	—
248	C₂³	—	2	4	no	—
249	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
250	C₂³	C₇×#121	1	2	yes	—
251	C₂³	C₇×C₂²⋊Q₈	1	2	yes	—
252	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	8	no	—
262	C₂×C₄	—	8	8	no	—
263	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
264	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
265	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
266	C₂×C₄	C₄×C₇×D₄	1	4	yes	—
267	C₂×C₄	C₄×C₇×Q₈	1	4	yes	—
268	C₂×C₄	C₇×#121	1	2	yes	—
269	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
270	C₂×C₄	C₇×#125	1	4	yes	—
271	C₂×C₄	C₇×C₄⋊Q₈	1	4	yes	—
272	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
273	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
276	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
277	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
278	C₈	—	2	8	no	—
279	C₇	#21172	1	1	yes	characteristic, direct product factor, Sylow
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	C₇×#389	1	4	yes	—
284	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
285	C₂²	C₂×C₇×#121	1	1	yes	characteristic
286	C₂²	C₂×C₇×C₂²⋊Q₈	1	1	yes	characteristic
287	C₂²	C₇×#575	1	4	yes	—
288	C₄	—	2	4	no	—
289	C₄	—	2	4	no	—
290	C₄	—	4	4	no	—
291	C₄	—	4	4	no	—
292	C₄	—	8	4	no	—
293	C₄	C₇×#389	1	4	yes	—
294	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
295	C₄	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
296	C₄	C₂×C₇×#125	1	1	yes	characteristic
297	C₄	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
298	C₂	—	2	4	no	—
299	C₂	C₂×C₇×#389	1	1	yes	characteristic
300	C₂	C₇×#2911	1	2	yes	—
301	C₂	C₇×#3040	1	2	yes	—
302	C₂	C₇×#3080	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. C₇×#1949	C₂	semidirect
15. C₇×#423	C₄	semidirect
26. #21172	C₇	direct (N × H)
56. C₇×C₄⋊C₈	C₂×C₄	semidirect
64. #1949	C₂×C₇	semidirect
112. #423	C₄×C₇	semidirect
183. C₄⋊C₈	C₂×C₄×C₇	semidirect
279. C₇	#21172	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×#428	2
C₂×C₇×#423	2
C₂×C₇×#382	2
C₄×C₇×#117	2
C₇×#1949	2
C₇×#1736	2
#21172	7

Subgroups — C7×#21172 #13198388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21172 #13198388