Subgroups — C₇×#21972 #13199188

Order 1792.

946 conjugacy classes of subgroups.

The subgroup lattice has 328 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	C₂²×C₇×C₄⋊C₈	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×#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	1	yes	characteristic
10	C₂×C₇×#117	C₄	1	1	yes	characteristic
11	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
12	C₇×#431	—	2	4	no	—
13	C₂×C₇×C₄⋊C₈	—	2	4	no	—
14	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₄⋊C₈	C₂²	1	8	yes	—
19	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
21	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
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	#21972	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	4	no	—
38	C₇×#117	C₈	1	8	yes	complemented
39	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₇	D₄	1	4	yes	—
49	C₂×C₄²×C₇	Q₈	1	4	yes	—
50	C₂×C₄²×C₇	C₂³	1	4	yes	—
51	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
53	C₇×C₄⋊C₈	—	2	4	no	—
54	C₇×C₄⋊C₈	—	2	4	no	—
55	C₇×C₄⋊C₈	—	2	2	no	—
56	C₇×C₄⋊C₈	—	2	2	no	—
57	C₇×C₄⋊C₈	—	4	8	no	—
58	C₇×C₂²⋊C₈	—	2	4	no	—
59	C₄×C₇×C₈	—	2	4	no	—
60	C₇×#95	C₂×C₄	1	4	yes	—
61	C₇×#95	C₂×C₄	1	4	yes	—
62	C₂×#431	C₂×C₇	1	1	yes	characteristic
63	C₂²×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
64	C₄×#117	C₂×C₇	1	1	yes	characteristic
65	#1939	C₂×C₇	1	4	yes	complemented
66	#1850	C₂×C₇	1	4	yes	—
67	#1842	C₂×C₇	1	2	yes	complemented
68	#1831	C₂×C₇	1	2	yes	complemented
69	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	—	2	2	no	—
71	C₂²×C₄×C₇	—	2	2	no	—
72	C₂²×C₄×C₇	—	4	2	no	—
73	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
74	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
75	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
76	C₂²×C₄×C₇	M₁₆	1	4	yes	—
77	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
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	1	yes	characteristic
81	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
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	C₂×C₇×C₈	—	4	8	no	—
94	C₇×C₄⋊C₄	—	2	4	no	—
95	C₇×C₄⋊C₄	C₂×C₈	1	8	yes	complemented
96	C₇×C₂²⋊C₄	—	2	4	no	—
97	C₇×C₂²⋊C₄	C₂×C₈	1	8	yes	—
98	C₄²×C₇	—	2	4	no	—
99	C₄²×C₇	—	2	4	no	—
100	C₄²×C₇	—	2	4	no	—
101	C₄²×C₇	—	4	4	no	—
102	C₄²×C₇	C₄⋊C₄	1	4	yes	—
103	C₄²×C₇	C₄⋊C₄	1	4	yes	—
104	C₄²×C₇	C₂×C₈	1	8	yes	complemented
105	C₄²×C₇	C₂×D₄	1	2	yes	—
106	C₄²×C₇	C₂×D₄	1	2	yes	—
107	C₄²×C₇	C₂×Q₈	1	2	yes	—
108	C₄²×C₇	C₂×Q₈	1	2	yes	—
109	C₂³×C₈	C₂²×C₇	1	2	yes	—
110	C₂×#117	C₄×C₇	1	1	yes	characteristic
111	C₂×#117	C₄×C₇	1	1	yes	characteristic
112	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
113	#431	—	2	4	no	—
114	C₂×C₄⋊C₈	—	2	4	no	—
115	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
116	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
117	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
118	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
119	C₂×C₄⋊C₈	C₂²×C₇	1	8	yes	—
120	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
121	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
122	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
123	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
124	C₄×C₂²⋊C₄	C₄×C₇	1	2	yes	—
125	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
126	#376	C₄×C₇	1	2	yes	—
127	C₄³	C₂²×C₇	1	2	yes	—
128	#336	C₂²×C₇	1	8	yes	—
129	C₂³×C₇	—	2	2	no	—
130	C₂³×C₇	—	2	2	no	—
131	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
132	C₂³×C₇	C₂²×C₈	1	2	yes	—
133	C₂³×C₇	C₂×M₁₆	1	2	yes	—
134	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₇	—	2	4	no	—
136	C₂×C₄×C₇	—	2	2	no	—
137	C₂×C₄×C₇	—	2	2	no	—
138	C₂×C₄×C₇	—	2	4	no	—
139	C₂×C₄×C₇	—	2	2	no	—
140	C₂×C₄×C₇	—	2	2	no	—
141	C₂×C₄×C₇	—	2	2	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	—	2	4	no	—
146	C₂×C₄×C₇	—	4	4	no	—
147	C₂×C₄×C₇	C₄⋊C₈	1	16	yes	complemented
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×C₄⋊C₄	1	2	yes	—
153	C₂×C₄×C₇	C₂²×C₈	1	8	yes	—
154	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
155	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
156	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
157	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
158	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
161	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
162	C₇×C₈	—	4	2	no	—
163	C₇×C₈	—	4	4	no	—
164	C₇×C₈	—	4	2	no	—
165	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
166	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
167	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
168	C₂²×C₈	—	2	4	no	—
169	C₂²×C₈	—	4	4	no	—
170	C₂²×C₈	C₂³×C₇	1	4	yes	—
171	C₂²×C₈	C₂³×C₇	1	2	yes	—
172	C₂²×C₈	C₂³×C₇	1	2	yes	—
173	#117	—	2	4	no	—
174	#117	C₇×C₈	1	8	yes	complemented
175	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
176	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
177	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
178	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
179	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
180	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
181	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
182	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
183	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
184	C₂×C₄²	C₇×D₄	1	4	yes	—
185	C₂×C₄²	C₇×Q₈	1	4	yes	—
186	C₂×C₄²	C₂³×C₇	1	4	yes	—
187	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
188	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
189	C₄⋊C₈	—	2	4	no	—
190	C₄⋊C₈	—	2	4	no	—
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	—	4	8	no	—
194	C₂²⋊C₈	—	2	4	no	—
195	C₄×C₈	—	2	4	no	—
196	#95	C₂×C₄×C₇	1	4	yes	—
197	#95	C₂×C₄×C₇	1	4	yes	—
198	C₂²×C₇	—	2	2	no	—
199	C₂²×C₇	—	2	2	no	—
200	C₂²×C₇	—	2	4	no	—
201	C₂²×C₇	—	2	4	no	—
202	C₂²×C₇	C₂×C₄⋊C₈	1	4	yes	—
203	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
204	C₂²×C₇	#520	1	2	yes	—
205	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
206	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
207	C₂²×C₇	#568	1	2	yes	—
208	C₄×C₇	—	2	2	no	—
209	C₄×C₇	—	2	2	no	—
210	C₄×C₇	—	2	2	no	—
211	C₄×C₇	—	2	8	no	—
212	C₄×C₇	—	4	4	no	—
213	C₄×C₇	C₂×C₄⋊C₈	1	4	yes	—
214	C₄×C₇	C₂×C₄⋊C₈	1	4	yes	—
215	C₄×C₇	#535	1	1	yes	characteristic
216	C₄×C₇	#536	1	1	yes	characteristic
217	C₄×C₇	#544	1	2	yes	—
218	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
223	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
224	C₂²×C₄	C₂×C₇×C₈	1	4	yes	—
225	C₂²×C₄	C₇×M₁₆	1	4	yes	—
226	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
227	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
229	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
230	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
231	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
234	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
235	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	8	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	4	8	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	C₂×C₇×C₈	1	8	yes	complemented
245	C₂²⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	C₂×C₇×C₈	1	8	yes	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	4	no	—
249	C₄²	—	2	4	no	—
250	C₄²	—	4	4	no	—
251	C₄²	C₇×C₄⋊C₄	1	4	yes	—
252	C₄²	C₇×C₄⋊C₄	1	4	yes	—
253	C₄²	C₂×C₇×C₈	1	8	yes	complemented
254	C₄²	C₂×C₇×D₄	1	2	yes	—
255	C₄²	C₂×C₇×D₄	1	2	yes	—
256	C₄²	C₂×C₇×Q₈	1	2	yes	—
257	C₄²	C₂×C₇×Q₈	1	2	yes	—
258	C₂×C₇	—	2	4	no	—
259	C₂×C₇	#2317	1	1	yes	characteristic
260	C₂×C₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
261	C₂×C₇	#2905	1	1	yes	characteristic
262	C₂×C₇	#2988	1	2	yes	—
263	C₂×C₇	#2989	1	2	yes	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
267	C₂³	C₂²×C₇×C₈	1	2	yes	—
268	C₂³	C₂×C₇×M₁₆	1	2	yes	—
269	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	C₇×C₄⋊C₈	1	16	yes	complemented
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×C₄⋊C₄	1	2	yes	—
286	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
287	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
288	C₂×C₄	C₂²×C₇×C₈	1	8	yes	—
289	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
290	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
291	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
292	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
293	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
295	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
296	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
297	C₈	—	4	2	no	—
298	C₈	—	4	4	no	—
299	C₈	—	4	2	no	—
300	C₇	#21972	1	1	yes	characteristic, direct product factor, Sylow
301	C₂²	—	2	2	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	2	2	no	—
305	C₂²	C₂×C₇×C₄⋊C₈	1	4	yes	—
306	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
307	C₂²	C₇×#520	1	2	yes	—
308	C₂²	C₂³×C₇×C₈	1	1	yes	characteristic
309	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
310	C₂²	C₇×#568	1	2	yes	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	8	no	—
314	C₄	—	2	2	no	—
315	C₄	—	4	4	no	—
316	C₄	C₂×C₇×C₄⋊C₈	1	4	yes	—
317	C₄	C₂×C₇×C₄⋊C₈	1	4	yes	—
318	C₄	C₇×#535	1	1	yes	characteristic
319	C₄	C₇×#536	1	1	yes	characteristic
320	C₄	C₇×#544	1	2	yes	—
321	C₂	—	2	4	no	—
322	C₂	C₇×#2317	1	1	yes	characteristic
323	C₂	C₂²×C₇×C₄⋊C₈	1	1	yes	characteristic
324	C₂	C₇×#2905	1	1	yes	characteristic
325	C₂	C₇×#2988	1	2	yes	—
326	C₂	C₇×#2989	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂²	C₂³×C₇×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂³×C₄
prime radical	C₂⁴×C₇	C₂²×C₄
LCS	G ▷ C₂² ▷ 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. #21972	C₇	direct (N × H)
38. C₇×#117	C₈	semidirect
65. #1939	C₂×C₇	semidirect
67. #1842	C₂×C₇	semidirect
68. #1831	C₂×C₇	semidirect
95. C₇×C₄⋊C₄	C₂×C₈	semidirect
104. C₄²×C₇	C₂×C₈	semidirect
147. C₂×C₄×C₇	C₄⋊C₈	semidirect
174. #117	C₇×C₈	semidirect
244. C₄⋊C₄	C₂×C₇×C₈	semidirect
253. C₄²	C₂×C₇×C₈	semidirect
282. C₂×C₄	C₇×C₄⋊C₈	semidirect
300. C₇	#21972	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
C₂²×C₇×C₄⋊C₈	2
C₄×C₇×#117	2
C₇×#1939	2
C₇×#1850	2
C₇×#1842	2
C₇×#1831	2
#21972	7

Subgroups — C7×#21972 #13199188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21972 #13199188