Subgroups — C₇×#21976 #13199192

Order 1792.

910 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#432	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₇×#117	C₂	1	1	yes	maximal, characteristic
4	C₇×#1940	C₂	1	4	yes	maximal, complemented
5	C₇×#1850	C₂	1	4	yes	maximal
6	C₇×#1842	C₂	1	2	yes	maximal, complemented
7	C₇×#1831	C₂	1	2	yes	maximal, complemented
8	C₂³×C₇×C₈	C₂²	1	2	yes	—
9	C₂×C₇×#117	C₄	1	2	yes	—
10	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₇×#432	—	2	4	no	—
12	C₂×C₇×C₄⋊C₈	—	2	4	no	—
13	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₈	C₂²	1	8	yes	—
18	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
19	C₂×C₇×#97	C₂²	1	2	yes	—
20	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
21	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
22	C₄×C₇×C₂²⋊C₄	C₄	1	1	yes	characteristic
23	C₄×C₇×C₂²⋊C₄	C₄	1	1	yes	characteristic
24	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₇×#376	C₄	1	2	yes	—
26	C₄³×C₇	C₂²	1	2	yes	—
27	C₇×#336	C₂²	1	8	yes	—
28	#21976	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
32	C₂²×C₇×C₈	—	2	4	no	—
33	C₂²×C₇×C₈	—	4	4	no	—
34	C₂²×C₇×C₈	C₂³	1	2	yes	—
35	C₂²×C₇×C₈	C₂³	1	4	yes	—
36	C₂²×C₇×C₈	C₂³	1	2	yes	—
37	C₇×#117	—	2	8	no	—
38	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
39	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₇	D₄	1	4	yes	—
47	C₂×C₄²×C₇	Q₈	1	4	yes	—
48	C₂×C₄²×C₇	C₂³	1	4	yes	—
49	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
51	C₇×C₄⋊C₈	—	2	4	no	—
52	C₇×C₄⋊C₈	—	2	4	no	—
53	C₇×C₄⋊C₈	—	2	2	no	—
54	C₇×C₄⋊C₈	—	2	2	no	—
55	C₇×C₄⋊C₈	—	4	8	no	—
56	C₇×C₂²⋊C₈	—	2	4	no	—
57	C₇×#97	—	2	4	no	—
58	C₇×#95	C₂×C₄	1	2	yes	—
59	C₇×#95	C₂×C₄	1	2	yes	—
60	C₇×#95	C₂×C₄	1	4	yes	—
61	C₂×#432	C₂×C₇	1	1	yes	characteristic
62	C₂²×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
63	C₄×#117	C₂×C₇	1	1	yes	characteristic
64	#1940	C₂×C₇	1	4	yes	complemented
65	#1850	C₂×C₇	1	4	yes	—
66	#1842	C₂×C₇	1	2	yes	complemented
67	#1831	C₂×C₇	1	2	yes	complemented
68	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇	—	2	4	no	—
70	C₂²×C₄×C₇	—	2	4	no	—
71	C₂²×C₄×C₇	—	2	2	no	—
72	C₂²×C₄×C₇	—	2	2	no	—
73	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
74	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
75	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₇×C₈	—	2	2	no	—
86	C₂×C₇×C₈	—	2	2	no	—
87	C₂×C₇×C₈	—	2	4	no	—
88	C₂×C₇×C₈	—	2	4	no	—
89	C₂×C₇×C₈	—	2	8	no	—
90	C₂×C₇×C₈	—	2	4	no	—
91	C₂×C₇×C₈	—	4	8	no	—
92	C₇×C₄⋊C₄	—	2	8	no	—
93	C₇×C₂²⋊C₄	—	2	4	no	—
94	C₇×C₂²⋊C₄	M₁₆	1	8	yes	—
95	C₄²×C₇	—	2	8	no	—
96	C₄²×C₇	—	2	4	no	—
97	C₄²×C₇	—	2	4	no	—
98	C₄²×C₇	—	2	4	no	—
99	C₄²×C₇	—	2	4	no	—
100	C₄²×C₇	C₄⋊C₄	1	4	yes	—
101	C₄²×C₇	C₄⋊C₄	1	4	yes	—
102	C₄²×C₇	C₂×D₄	1	2	yes	—
103	C₄²×C₇	C₂×D₄	1	2	yes	—
104	C₄²×C₇	C₂×Q₈	1	2	yes	—
105	C₄²×C₇	C₂×Q₈	1	2	yes	—
106	C₂³×C₈	C₂²×C₇	1	2	yes	—
107	C₂×#117	C₄×C₇	1	2	yes	—
108	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
109	#432	—	2	4	no	—
110	C₂×C₄⋊C₈	—	2	4	no	—
111	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
112	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
113	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
114	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
115	C₂×C₄⋊C₈	C₂²×C₇	1	8	yes	—
116	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
117	C₂×#97	C₂²×C₇	1	2	yes	—
118	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
119	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
120	C₄×C₂²⋊C₄	C₄×C₇	1	1	yes	characteristic
121	C₄×C₂²⋊C₄	C₄×C₇	1	1	yes	characteristic
122	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
123	#376	C₄×C₇	1	2	yes	—
124	C₄³	C₂²×C₇	1	2	yes	—
125	#336	C₂²×C₇	1	8	yes	—
126	C₂³×C₇	—	2	2	no	—
127	C₂³×C₇	—	2	2	no	—
128	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
129	C₂³×C₇	C₂×M₁₆	1	4	yes	—
130	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₇	—	2	4	no	—
132	C₂×C₄×C₇	—	2	2	no	—
133	C₂×C₄×C₇	—	2	2	no	—
134	C₂×C₄×C₇	—	2	2	no	—
135	C₂×C₄×C₇	—	2	4	no	—
136	C₂×C₄×C₇	—	2	2	no	—
137	C₂×C₄×C₇	—	2	8	no	—
138	C₂×C₄×C₇	—	2	2	no	—
139	C₂×C₄×C₇	—	2	2	no	—
140	C₂×C₄×C₇	—	2	4	no	—
141	C₂×C₄×C₇	—	2	8	no	—
142	C₂×C₄×C₇	—	2	4	no	—
143	C₂×C₄×C₇	—	2	4	no	—
144	C₂×C₄×C₇	—	2	4	no	—
145	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
146	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
147	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
149	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
150	C₂×C₄×C₇	C₂×M₁₆	1	8	yes	—
151	C₂×C₄×C₇	#131	1	4	yes	—
152	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
155	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
156	C₇×C₈	—	4	2	no	—
157	C₇×C₈	—	4	4	no	—
158	C₇×C₈	—	4	2	no	—
159	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
160	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
161	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
162	C₂²×C₈	—	2	4	no	—
163	C₂²×C₈	—	4	4	no	—
164	C₂²×C₈	C₂³×C₇	1	4	yes	—
165	C₂²×C₈	C₂³×C₇	1	2	yes	—
166	C₂²×C₈	C₂³×C₇	1	2	yes	—
167	#117	—	2	8	no	—
168	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
169	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
170	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
171	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
172	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
173	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
174	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
176	C₂×C₄²	C₇×D₄	1	4	yes	—
177	C₂×C₄²	C₇×Q₈	1	4	yes	—
178	C₂×C₄²	C₂³×C₇	1	4	yes	—
179	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
181	C₄⋊C₈	—	2	4	no	—
182	C₄⋊C₈	—	2	4	no	—
183	C₄⋊C₈	—	2	2	no	—
184	C₄⋊C₈	—	2	2	no	—
185	C₄⋊C₈	—	4	8	no	—
186	C₂²⋊C₈	—	2	4	no	—
187	#97	—	2	4	no	—
188	#95	C₂×C₄×C₇	1	2	yes	—
189	#95	C₂×C₄×C₇	1	2	yes	—
190	#95	C₂×C₄×C₇	1	4	yes	—
191	C₂²×C₇	—	2	2	no	—
192	C₂²×C₇	—	2	2	no	—
193	C₂²×C₇	—	2	8	no	—
194	C₂²×C₇	#423	1	4	yes	—
195	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
196	C₂²×C₇	#520	1	2	yes	—
197	C₂²×C₇	C₂²×M₁₆	1	2	yes	—
198	C₂²×C₇	C₂×#131	1	2	yes	—
199	C₄×C₇	—	2	2	no	—
200	C₄×C₇	—	2	2	no	—
201	C₄×C₇	—	2	2	no	—
202	C₄×C₇	—	2	2	no	—
203	C₄×C₇	—	2	8	no	—
204	C₄×C₇	—	4	4	no	—
205	C₄×C₇	#424	1	4	yes	—
206	C₄×C₇	#535	1	1	yes	characteristic
207	C₄×C₇	#536	1	1	yes	characteristic
208	C₄×C₇	#544	1	2	yes	—
209	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
215	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
216	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
217	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
224	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
225	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	8	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	4	8	no	—
233	C₄⋊C₄	—	2	8	no	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₂²⋊C₄	C₇×M₁₆	1	8	yes	—
236	C₄²	—	2	8	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	4	no	—
241	C₄²	C₇×C₄⋊C₄	1	4	yes	—
242	C₄²	C₇×C₄⋊C₄	1	4	yes	—
243	C₄²	C₂×C₇×D₄	1	2	yes	—
244	C₄²	C₂×C₇×D₄	1	2	yes	—
245	C₄²	C₂×C₇×Q₈	1	2	yes	—
246	C₄²	C₂×C₇×Q₈	1	2	yes	—
247	C₂×C₇	—	2	4	no	—
248	C₂×C₇	#2317	1	1	yes	characteristic
249	C₂×C₇	C₂×#423	1	1	yes	characteristic
250	C₂×C₇	C₂×#424	1	1	yes	characteristic
251	C₂×C₇	#2990	1	4	yes	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
255	C₂³	C₂×C₇×M₁₆	1	4	yes	—
256	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
272	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
273	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
274	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
275	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
276	C₂×C₄	C₂×C₇×M₁₆	1	8	yes	—
277	C₂×C₄	C₇×#131	1	4	yes	—
278	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
280	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
281	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
282	C₈	—	4	2	no	—
283	C₈	—	4	4	no	—
284	C₈	—	4	2	no	—
285	C₇	#21976	1	1	yes	characteristic, direct product factor, Sylow
286	C₂²	—	2	2	no	—
287	C₂²	—	2	8	no	—
288	C₂²	—	2	2	no	—
289	C₂²	C₇×#423	1	4	yes	—
290	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
291	C₂²	C₇×#520	1	2	yes	—
292	C₂²	C₂²×C₇×M₁₆	1	2	yes	—
293	C₂²	C₂×C₇×#131	1	2	yes	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	8	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	4	no	—
300	C₄	C₇×#424	1	4	yes	—
301	C₄	C₇×#535	1	1	yes	characteristic
302	C₄	C₇×#536	1	1	yes	characteristic
303	C₄	C₇×#544	1	2	yes	—
304	C₂	—	2	4	no	—
305	C₂	C₇×#2317	1	1	yes	characteristic
306	C₂	C₂×C₇×#423	1	1	yes	characteristic
307	C₂	C₂×C₇×#424	1	1	yes	characteristic
308	C₂	C₇×#2990	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₇×#1940	C₂	semidirect
6. C₇×#1842	C₂	semidirect
7. C₇×#1831	C₂	semidirect
28. #21976	C₇	direct (N × H)
64. #1940	C₂×C₇	semidirect
66. #1842	C₂×C₇	semidirect
67. #1831	C₂×C₇	semidirect
285. C₇	#21976	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×#432	2
C₂²×C₇×C₄⋊C₈	2
C₄×C₇×#117	2
C₇×#1940	2
C₇×#1850	2
C₇×#1842	2
C₇×#1831	2
#21976	7

Subgroups — C7×#21976 #13199192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21976 #13199192