Subgroups — C₇×#21163 #13198379

Order 1792.

882 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#427	C₂	1	1	yes	maximal, characteristic
3	C₇×#2905	C₂	1	2	yes	maximal
4	C₂×C₇×#405	C₂	1	1	yes	maximal, characteristic
5	C₇×#1946	C₂	1	4	yes	maximal, complemented
6	C₇×#1945	C₂	1	4	yes	maximal, complemented
7	C₇×#1733	C₂	1	2	yes	maximal
8	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₇×#117	C₂²	1	2	yes	—
12	C₂×C₇×#117	C₂²	1	1	yes	characteristic
13	C₂×C₇×#108	—	2	4	no	—
14	C₂×C₇×#108	C₂²	1	2	yes	—
15	C₂×C₇×#108	C₂²	1	2	yes	—
16	C₇×#427	C₄	1	2	yes	complemented
17	C₇×#427	C₄	1	2	yes	complemented
18	C₇×#427	C₂²	1	2	yes	—
19	C₇×#427	C₂²	1	2	yes	—
20	C₂×C₇×#107	C₄	1	2	yes	complemented
21	C₂×C₇×#106	C₄	1	2	yes	complemented
22	C₇×#424	—	2	4	no	—
23	C₇×#424	C₂²	1	8	yes	—
24	C₇×#423	—	2	4	no	—
25	C₇×#405	—	2	4	no	—
26	C₂×C₇×#97	C₂²	1	2	yes	—
27	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
28	C₇×#339	C₂²	1	8	yes	—
29	#21163	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₂×C₇×M₁₆	—	2	2	no	—
32	C₂×C₇×M₁₆	—	2	2	no	—
33	C₂×C₇×M₁₆	—	2	2	no	—
34	C₂×C₇×M₁₆	—	2	4	no	—
35	C₂×C₇×M₁₆	—	2	4	no	—
36	C₂×C₇×M₁₆	D₄	1	2	yes	—
37	C₂×C₇×M₁₆	D₄	1	2	yes	—
38	C₂×C₇×M₁₆	C₂³	1	4	yes	—
39	C₂²×C₇×C₈	—	2	2	no	—
40	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₇×C₈	D₄	1	4	yes	—
45	C₂²×C₇×C₈	Q₈	1	4	yes	—
46	C₂²×C₇×C₈	C₂³	1	2	yes	—
47	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
48	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
49	C₇×#117	—	2	2	no	—
50	C₇×#117	—	2	2	no	—
51	C₇×#117	—	2	2	no	—
52	C₇×#117	C₂×C₄	1	4	yes	—
53	C₇×#117	Q₈	1	4	yes	—
54	C₇×#117	C₂³	1	4	yes	—
55	C₂×C₇×C₄⋊C₄	—	2	1	no	—
56	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
57	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
58	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
59	C₂×C₄²×C₇	—	2	2	no	—
60	C₂×C₄²×C₇	C₂³	1	2	yes	—
61	C₇×#108	—	2	4	no	—
62	C₇×#108	—	2	4	no	—
63	C₇×#107	C₂×C₄	1	4	yes	—
64	C₇×#107	C₂×C₄	1	4	yes	—
65	C₇×#106	C₂×C₄	1	4	yes	—
66	C₇×#106	C₂×C₄	1	4	yes	—
67	C₇×C₄⋊C₈	—	2	16	no	—
68	C₇×#97	—	2	4	no	—
69	C₄×C₇×C₈	—	2	4	no	—
70	C₂²×#108	C₂×C₇	1	1	yes	characteristic
71	C₂×#427	C₂×C₇	1	1	yes	characteristic
72	#2905	C₂×C₇	1	2	yes	—
73	C₂×#405	C₂×C₇	1	1	yes	characteristic
74	#1946	C₂×C₇	1	4	yes	complemented
75	#1945	C₂×C₇	1	4	yes	complemented
76	#1733	C₂×C₇	1	2	yes	—
77	C₂⁴×C₇	#41	1	1	yes	characteristic
78	C₂²×C₄×C₇	—	2	2	no	—
79	C₂²×C₄×C₇	—	2	4	no	—
80	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
87	C₂²×C₄×C₇	#41	1	1	yes	characteristic
88	C₂²×C₄×C₇	#41	1	2	yes	—
89	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
90	C₇×M₁₆	—	2	4	no	—
91	C₇×M₁₆	—	2	4	no	—
92	C₇×M₁₆	—	2	4	no	—
93	C₇×M₁₆	—	2	4	no	—
94	C₇×M₁₆	—	4	4	no	—
95	C₂×C₇×C₈	—	2	2	no	—
96	C₂×C₇×C₈	—	2	2	no	—
97	C₂×C₇×C₈	—	2	4	no	—
98	C₂×C₇×C₈	—	4	4	no	—
99	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
100	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
101	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
102	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
103	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
108	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
109	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
110	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
111	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
112	C₇×C₄⋊C₄	—	2	4	no	—
113	C₇×C₄⋊C₄	—	2	4	no	—
114	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
115	C₇×C₂²⋊C₄	—	2	4	no	—
116	C₇×C₂²⋊C₄	—	2	4	no	—
117	C₇×C₂²⋊C₄	—	2	4	no	—
118	C₄²×C₇	—	2	4	no	—
119	C₄²×C₇	—	2	2	no	—
120	C₄²×C₇	—	2	4	no	—
121	C₄²×C₇	C₂×Q₈	1	4	yes	—
122	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
123	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
124	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
125	C₂×#117	C₂²×C₇	1	2	yes	—
126	C₂×#117	C₂²×C₇	1	1	yes	characteristic
127	C₂×#108	—	2	4	no	—
128	C₂×#108	C₂²×C₇	1	2	yes	—
129	C₂×#108	C₂²×C₇	1	2	yes	—
130	#427	C₄×C₇	1	2	yes	complemented
131	#427	C₄×C₇	1	2	yes	complemented
132	#427	C₂²×C₇	1	2	yes	—
133	#427	C₂²×C₇	1	2	yes	—
134	C₂×#107	C₄×C₇	1	2	yes	complemented
135	C₂×#106	C₄×C₇	1	2	yes	complemented
136	#424	—	2	4	no	—
137	#424	C₂²×C₇	1	8	yes	—
138	#423	—	2	4	no	—
139	#405	—	2	4	no	—
140	C₂×#97	C₂²×C₇	1	2	yes	—
141	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
142	#339	C₂²×C₇	1	8	yes	—
143	C₂³×C₇	—	4	2	no	—
144	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
145	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
146	C₂³×C₇	C₂²⋊Q₈	1	2	yes	—
147	C₂³×C₇	#125	1	2	yes	—
148	C₂³×C₇	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₇	—	2	8	no	—
150	C₂×C₄×C₇	—	2	4	no	—
151	C₂×C₄×C₇	—	4	4	no	—
152	C₂×C₄×C₇	—	8	4	no	—
153	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
154	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
155	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
158	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
159	C₂×C₄×C₇	#121	1	2	yes	—
160	C₂×C₄×C₇	#121	1	2	yes	—
161	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
162	C₂×C₄×C₇	#125	1	2	yes	—
163	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
164	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
165	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
171	C₇×C₈	—	2	4	no	—
172	C₇×C₈	—	4	4	no	—
173	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
174	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
175	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
176	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
177	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	—	2	4	no	—
183	C₂×M₁₆	C₇×D₄	1	2	yes	—
184	C₂×M₁₆	C₇×D₄	1	2	yes	—
185	C₂×M₁₆	C₂³×C₇	1	4	yes	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
188	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
189	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
190	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
191	C₂²×C₈	C₇×D₄	1	4	yes	—
192	C₂²×C₈	C₇×Q₈	1	4	yes	—
193	C₂²×C₈	C₂³×C₇	1	2	yes	—
194	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	#117	C₂×C₄×C₇	1	4	yes	—
200	#117	C₇×Q₈	1	4	yes	—
201	#117	C₂³×C₇	1	4	yes	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₂²⋊C₄	—	2	1	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	C₂³×C₇	1	2	yes	—
208	#108	—	2	4	no	—
209	#108	—	2	4	no	—
210	#107	C₂×C₄×C₇	1	4	yes	—
211	#107	C₂×C₄×C₇	1	4	yes	—
212	#106	C₂×C₄×C₇	1	4	yes	—
213	#106	C₂×C₄×C₇	1	4	yes	—
214	C₄⋊C₈	—	2	16	no	—
215	#97	—	2	4	no	—
216	C₄×C₈	—	2	4	no	—
217	C₂²×C₇	—	2	2	no	—
218	C₂²×C₇	—	2	2	no	—
219	C₂²×C₇	—	2	2	no	—
220	C₂²×C₇	—	4	2	no	—
221	C₂²×C₇	—	4	2	no	—
222	C₂²×C₇	#389	1	2	yes	—
223	C₂²×C₇	#389	1	2	yes	—
224	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
225	C₂²×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
226	C₂²×C₇	C₂×#125	1	1	yes	characteristic
227	C₄×C₇	—	4	4	no	—
228	C₄×C₇	—	4	8	no	—
229	C₄×C₇	#389	1	2	yes	—
230	C₄×C₇	#389	1	2	yes	—
231	C₄×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
232	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
233	C₄×C₇	C₂×#121	1	1	yes	characteristic
234	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
235	C₂⁴	C₇×#41	1	1	yes	characteristic
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
239	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
241	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
244	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
245	C₂²×C₄	C₇×#41	1	1	yes	characteristic
246	C₂²×C₄	C₇×#41	1	2	yes	—
247	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
248	M₁₆	—	2	4	no	—
249	M₁₆	—	2	4	no	—
250	M₁₆	—	2	4	no	—
251	M₁₆	—	2	4	no	—
252	M₁₆	—	4	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	4	4	no	—
257	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
258	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
259	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
260	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
261	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
262	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
263	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
264	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
265	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
266	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
267	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
268	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
269	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
273	C₂²⋊C₄	—	2	4	no	—
274	C₂²⋊C₄	—	2	4	no	—
275	C₂²⋊C₄	—	2	4	no	—
276	C₄²	—	2	4	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	4	no	—
279	C₄²	C₂×C₇×Q₈	1	4	yes	—
280	C₂×C₇	—	2	1	no	—
281	C₂×C₇	—	2	1	no	—
282	C₂×C₇	—	2	2	no	—
283	C₂×C₇	—	2	2	no	—
284	C₂×C₇	C₂×#389	1	1	yes	characteristic
285	C₂×C₇	#2957	1	2	yes	—
286	C₂³	—	4	2	no	—
287	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
288	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
289	C₂³	C₇×C₂²⋊Q₈	1	2	yes	—
290	C₂³	C₇×#125	1	2	yes	—
291	C₂³	C₂×C₇×#41	1	1	yes	characteristic
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	8	4	no	—
296	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
297	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
298	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
299	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
300	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
301	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
302	C₂×C₄	C₇×#121	1	2	yes	—
303	C₂×C₄	C₇×#121	1	2	yes	—
304	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
305	C₂×C₄	C₇×#125	1	2	yes	—
306	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
307	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
308	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
312	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
313	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
314	C₈	—	2	4	no	—
315	C₈	—	4	4	no	—
316	C₈	C₂×C₇×C₄⋊C₄	1	2	yes	—
317	C₈	C₂×C₇×C₄⋊C₄	1	2	yes	—
318	C₈	C₂×C₇×C₄⋊C₄	1	2	yes	—
319	C₈	C₂×C₇×C₄⋊C₄	1	2	yes	—
320	C₇	#21163	1	1	yes	characteristic, direct product factor, Sylow
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	C₇×#389	1	2	yes	—
327	C₂²	C₇×#389	1	2	yes	—
328	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
329	C₂²	C₂×C₇×C₂²⋊Q₈	1	1	yes	characteristic
330	C₂²	C₂×C₇×#125	1	1	yes	characteristic
331	C₄	—	4	4	no	—
332	C₄	—	4	8	no	—
333	C₄	C₇×#389	1	2	yes	—
334	C₄	C₇×#389	1	2	yes	—
335	C₄	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
336	C₄	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
337	C₄	C₂×C₇×#121	1	1	yes	characteristic
338	C₄	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	2	no	—
341	C₂	—	2	2	no	—
342	C₂	—	2	1	no	—
343	C₂	C₂×C₇×#389	1	1	yes	characteristic
344	C₂	C₇×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂⁴×C₇	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₇×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₇×#1946	C₂	semidirect
6. C₇×#1945	C₂	semidirect
16. C₇×#427	C₄	semidirect
17. C₇×#427	C₄	semidirect
20. C₂×C₇×#107	C₄	semidirect
21. C₂×C₇×#106	C₄	semidirect
29. #21163	C₇	direct (N × H)
74. #1946	C₂×C₇	semidirect
75. #1945	C₂×C₇	semidirect
130. #427	C₄×C₇	semidirect
131. #427	C₄×C₇	semidirect
134. C₂×#107	C₄×C₇	semidirect
135. C₂×#106	C₄×C₇	semidirect
320. C₇	#21163	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₇×#108	2
C₂×C₇×#427	2
C₇×#2905	2
C₂×C₇×#405	2
C₇×#1946	2
C₇×#1945	2
C₇×#1733	2
#21163	7

Subgroups — C7×#21163 #13198379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21163 #13198379