Subgroups — C₇×#21974 #13199190

Order 1792.

898 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#423	C₂	1	1	yes	maximal, characteristic
3	C₄×C₇×#117	C₂	1	1	yes	maximal, characteristic
4	C₇×#1939	C₂	1	4	yes	maximal, complemented
5	C₇×#1851	C₂	1	4	yes	maximal
6	C₇×#1842	C₂	1	2	yes	maximal, complemented
7	C₇×#1831	C₂	1	2	yes	maximal, complemented
8	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
9	C₂×C₇×#117	C₄	1	2	yes	—
10	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₇×#431	—	2	4	no	—
12	C₇×#423	—	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×C₄×C₇×C₈	C₂²	1	2	yes	—
20	C₄×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
21	C₄×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
22	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
23	C₄×C₇×C₂²⋊C₄	C₄	1	2	yes	—
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	#21974	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	C₂×C₇×M₁₆	—	2	4	no	—
33	C₂×C₇×M₁₆	—	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	2	yes	—
39	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	4	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	C₄×C₇×C₈	—	2	4	no	—
58	C₇×#95	C₂×C₄	1	4	yes	—
59	C₇×#95	C₂×C₄	1	2	yes	—
60	C₇×#95	C₂×C₄	1	2	yes	—
61	C₂×#431	C₂×C₇	1	1	yes	characteristic
62	C₂×#423	C₂×C₇	1	1	yes	characteristic
63	C₄×#117	C₂×C₇	1	1	yes	characteristic
64	#1939	C₂×C₇	1	4	yes	complemented
65	#1851	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	2	no	—
71	C₂²×C₄×C₇	—	2	2	no	—
72	C₂²×C₄×C₇	—	2	2	no	—
73	C₂²×C₄×C₇	—	2	2	no	—
74	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
75	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
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	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
86	C₇×M₁₆	—	4	8	no	—
87	C₂×C₇×C₈	—	2	2	no	—
88	C₂×C₇×C₈	—	2	2	no	—
89	C₂×C₇×C₈	—	2	4	no	—
90	C₂×C₇×C₈	—	2	4	no	—
91	C₂×C₇×C₈	—	2	8	no	—
92	C₂×C₇×C₈	—	2	4	no	—
93	C₇×C₄⋊C₄	—	2	4	no	—
94	C₇×C₄⋊C₄	M₁₆	1	8	yes	complemented
95	C₇×C₂²⋊C₄	—	2	8	no	—
96	C₄²×C₇	—	2	8	no	—
97	C₄²×C₇	—	2	4	no	—
98	C₄²×C₇	—	2	4	no	—
99	C₄²×C₇	—	2	4	no	—
100	C₄²×C₇	—	2	4	no	—
101	C₄²×C₇	C₄⋊C₄	1	4	yes	—
102	C₄²×C₇	C₄⋊C₄	1	4	yes	—
103	C₄²×C₇	C₂×D₄	1	2	yes	—
104	C₄²×C₇	C₂×D₄	1	2	yes	—
105	C₄²×C₇	C₂×Q₈	1	2	yes	—
106	C₄²×C₇	C₂×Q₈	1	2	yes	—
107	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
108	C₂×#117	C₄×C₇	1	2	yes	—
109	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
110	#431	—	2	4	no	—
111	#423	—	2	4	no	—
112	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
113	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
114	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
115	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
116	C₂×C₄⋊C₈	C₂²×C₇	1	8	yes	—
117	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
119	C₄×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
120	C₄×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
121	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
122	C₄×C₂²⋊C₄	C₄×C₇	1	2	yes	—
123	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
124	#376	C₄×C₇	1	2	yes	—
125	C₄³	C₂²×C₇	1	2	yes	—
126	#336	C₂²×C₇	1	8	yes	—
127	C₂³×C₇	—	2	2	no	—
128	C₂³×C₇	—	2	2	no	—
129	C₂³×C₇	—	2	1	no	—
130	C₂³×C₇	—	2	1	no	—
131	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
132	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₇	—	2	4	no	—
134	C₂×C₄×C₇	—	2	2	no	—
135	C₂×C₄×C₇	—	2	2	no	—
136	C₂×C₄×C₇	—	2	4	no	—
137	C₂×C₄×C₇	—	2	1	no	—
138	C₂×C₄×C₇	—	2	1	no	—
139	C₂×C₄×C₇	—	2	2	no	—
140	C₂×C₄×C₇	—	2	2	no	—
141	C₂×C₄×C₇	—	2	4	no	—
142	C₂×C₄×C₇	—	2	8	no	—
143	C₂×C₄×C₇	—	2	8	no	—
144	C₂×C₄×C₇	—	2	4	no	—
145	C₂×C₄×C₇	—	2	4	no	—
146	C₂×C₄×C₇	—	2	4	no	—
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×C₄⋊C₄	1	2	yes	—
151	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
152	C₂×C₄×C₇	C₂×M₁₆	1	8	yes	—
153	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
154	C₂×C₄×C₇	#131	1	2	yes	—
155	C₂×C₄×C₇	#131	1	2	yes	—
156	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
158	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
159	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
160	C₇×C₈	—	4	2	no	—
161	C₇×C₈	—	4	4	no	—
162	C₇×C₈	—	4	2	no	—
163	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
164	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
165	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
166	C₂×M₁₆	—	2	4	no	—
167	C₂×M₁₆	—	4	4	no	—
168	C₂²×C₈	C₂³×C₇	1	4	yes	—
169	C₂²×C₈	C₂³×C₇	1	2	yes	—
170	C₂²×C₈	C₂³×C₇	1	2	yes	—
171	#117	—	2	8	no	—
172	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
174	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	4	yes	—
175	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
176	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
177	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
178	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
179	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
180	C₂×C₄²	C₇×D₄	1	4	yes	—
181	C₂×C₄²	C₇×Q₈	1	4	yes	—
182	C₂×C₄²	C₂³×C₇	1	4	yes	—
183	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
185	C₄⋊C₈	—	2	4	no	—
186	C₄⋊C₈	—	2	4	no	—
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	2	2	no	—
189	C₄⋊C₈	—	4	8	no	—
190	C₂²⋊C₈	—	2	4	no	—
191	C₄×C₈	—	2	4	no	—
192	#95	C₂×C₄×C₇	1	4	yes	—
193	#95	C₂×C₄×C₇	1	2	yes	—
194	#95	C₂×C₄×C₇	1	2	yes	—
195	C₂²×C₇	—	2	2	no	—
196	C₂²×C₇	—	2	2	no	—
197	C₂²×C₇	—	2	2	no	—
198	C₂²×C₇	—	4	2	no	—
199	C₂²×C₇	—	4	2	no	—
200	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
201	C₂²×C₇	#520	1	2	yes	—
202	C₂²×C₇	C₂²×M₁₆	1	2	yes	—
203	C₂²×C₇	C₂×#131	1	1	yes	characteristic
204	C₂²×C₇	C₂×#131	1	1	yes	characteristic
205	C₄×C₇	—	2	2	no	—
206	C₄×C₇	—	2	2	no	—
207	C₄×C₇	—	2	2	no	—
208	C₄×C₇	—	2	8	no	—
209	C₄×C₇	—	4	4	no	—
210	C₄×C₇	#423	1	4	yes	—
211	C₄×C₇	#424	1	4	yes	—
212	C₄×C₇	#535	1	1	yes	characteristic
213	C₄×C₇	#536	1	1	yes	characteristic
214	C₄×C₇	#544	1	2	yes	—
215	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
222	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
223	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
224	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
228	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
229	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
231	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
232	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
233	M₁₆	—	4	8	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	8	no	—
239	C₂×C₈	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	C₇×M₁₆	1	8	yes	complemented
242	C₂²⋊C₄	—	2	8	no	—
243	C₄²	—	2	8	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	4	no	—
247	C₄²	—	2	4	no	—
248	C₄²	C₇×C₄⋊C₄	1	4	yes	—
249	C₄²	C₇×C₄⋊C₄	1	4	yes	—
250	C₄²	C₂×C₇×D₄	1	2	yes	—
251	C₄²	C₂×C₇×D₄	1	2	yes	—
252	C₄²	C₂×C₇×Q₈	1	2	yes	—
253	C₄²	C₂×C₇×Q₈	1	2	yes	—
254	C₂×C₇	—	4	2	no	—
255	C₂×C₇	#2317	1	1	yes	characteristic
256	C₂×C₇	C₂×#423	1	1	yes	characteristic
257	C₂×C₇	C₂×#424	1	1	yes	characteristic
258	C₂×C₇	#2990	1	2	yes	—
259	C₂×C₇	#2991	1	2	yes	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	1	no	—
264	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
265	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
281	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
282	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
283	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
284	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
285	C₂×C₄	C₂×C₇×M₁₆	1	8	yes	—
286	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
287	C₂×C₄	C₇×#131	1	2	yes	—
288	C₂×C₄	C₇×#131	1	2	yes	—
289	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
291	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
292	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
293	C₈	—	4	2	no	—
294	C₈	—	4	4	no	—
295	C₈	—	4	2	no	—
296	C₇	#21974	1	1	yes	characteristic, direct product factor, Sylow
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
303	C₂²	C₇×#520	1	2	yes	—
304	C₂²	C₂²×C₇×M₁₆	1	2	yes	—
305	C₂²	C₂×C₇×#131	1	1	yes	characteristic
306	C₂²	C₂×C₇×#131	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	8	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	4	no	—
312	C₄	C₇×#423	1	4	yes	—
313	C₄	C₇×#424	1	4	yes	—
314	C₄	C₇×#535	1	1	yes	characteristic
315	C₄	C₇×#536	1	1	yes	characteristic
316	C₄	C₇×#544	1	2	yes	—
317	C₂	—	4	2	no	—
318	C₂	C₇×#2317	1	1	yes	characteristic
319	C₂	C₂×C₇×#423	1	1	yes	characteristic
320	C₂	C₂×C₇×#424	1	1	yes	characteristic
321	C₂	C₇×#2990	1	2	yes	—
322	C₂	C₇×#2991	1	2	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×#1939	C₂	semidirect
6. C₇×#1842	C₂	semidirect
7. C₇×#1831	C₂	semidirect
28. #21974	C₇	direct (N × H)
64. #1939	C₂×C₇	semidirect
66. #1842	C₂×C₇	semidirect
67. #1831	C₂×C₇	semidirect
94. C₇×C₄⋊C₄	M₁₆	semidirect
241. C₄⋊C₄	C₇×M₁₆	semidirect
296. C₇	#21974	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×#431	2
C₂×C₇×#423	2
C₄×C₇×#117	2
C₇×#1939	2
C₇×#1851	2
C₇×#1842	2
C₇×#1831	2
#21974	7

Subgroups — C7×#21974 #13199190

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21974 #13199190