Subgroups — C₇×#7901 #13185117

Order 1792.

818 conjugacy classes of subgroups.

The subgroup lattice has 338 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	C₂×C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₇×#1279	C₂	1	4	yes	maximal, complemented
5	C₂²×C₇×M₁₆	—	2	1	no	—
6	C₂²×C₇×C₄⋊C₄	C₄	1	2	yes	—
7	C₂²×C₄²×C₇	—	2	1	no	—
8	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₇×#431	—	2	2	no	—
10	C₇×#431	C₄	1	4	yes	complemented
11	C₇×#423	—	2	2	no	—
12	C₇×#423	C₄	1	4	yes	complemented
13	C₂×C₇×C₄⋊C₈	—	2	1	no	—
14	C₂×C₇×C₄⋊C₈	—	2	1	no	—
15	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₇×C₂²⋊C₈	—	2	1	no	—
17	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
18	C₄×C₇×C₄⋊C₄	—	2	1	no	—
19	C₄×C₇×C₄⋊C₄	—	2	1	no	—
20	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
21	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₇×#95	—	2	1	no	—
23	#7901	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₇	—	2	2	no	—
25	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
28	C₂×C₇×M₁₆	—	2	4	no	—
29	C₂×C₇×M₁₆	—	4	2	no	—
30	C₂×C₇×M₁₆	—	4	2	no	—
31	C₂²×C₇×C₈	—	2	2	no	—
32	C₂²×C₇×C₈	—	2	2	no	—
33	C₂×C₇×C₄⋊C₄	—	2	2	no	—
34	C₂×C₇×C₄⋊C₄	—	2	8	no	—
35	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₇	—	2	2	no	—
37	C₂×C₄²×C₇	—	2	2	no	—
38	C₂×C₄²×C₇	—	4	2	no	—
39	C₂×C₄²×C₇	—	4	2	no	—
40	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₇	D₄	1	2	yes	—
46	C₂×C₄²×C₇	Q₈	1	2	yes	—
47	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
48	C₇×C₄⋊C₈	—	2	4	no	—
49	C₇×C₄⋊C₈	—	2	4	no	—
50	C₇×C₄⋊C₈	—	2	2	no	—
51	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
52	C₇×C₂²⋊C₈	—	2	4	no	—
53	C₄×C₇×C₈	—	2	2	no	—
54	C₄×C₇×C₈	C₂×C₄	1	4	yes	complemented
55	C₇×#95	—	2	2	no	—
56	C₇×#95	—	2	2	no	—
57	C₂×#431	C₂×C₇	1	1	yes	characteristic
58	C₂×#423	C₂×C₇	1	1	yes	characteristic
59	C₂×C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
60	#1279	C₂×C₇	1	4	yes	complemented
61	C₂⁴×C₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	—	2	4	no	—
63	C₂²×C₄×C₇	—	2	8	no	—
64	C₂²×C₄×C₇	—	2	4	no	—
65	C₂²×C₄×C₇	—	4	2	no	—
66	C₂²×C₄×C₇	—	4	4	no	—
67	C₂²×C₄×C₇	—	4	4	no	—
68	C₂²×C₄×C₇	C₄²	1	2	yes	—
69	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₇	D₈	1	2	yes	—
78	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
79	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
80	C₂²×C₄×C₇	Q₁₆	1	2	yes	—
81	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
84	C₇×M₁₆	—	4	8	no	—
85	C₂×C₇×C₈	—	2	4	no	—
86	C₂×C₇×C₈	—	2	4	no	—
87	C₂×C₇×C₈	—	4	4	no	—
88	C₂×C₇×C₈	—	4	2	no	—
89	C₂×C₇×C₈	—	4	2	no	—
90	C₇×C₄⋊C₄	—	2	8	no	—
91	C₇×C₄⋊C₄	—	2	4	no	—
92	C₇×C₄⋊C₄	M₁₆	1	8	yes	complemented
93	C₄²×C₇	—	4	4	no	—
94	C₄²×C₇	—	4	2	no	—
95	C₄²×C₇	—	4	2	no	—
96	C₄²×C₇	C₄²	1	2	yes	—
97	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
98	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
99	C₄²×C₇	C₄⋊C₄	1	2	yes	—
100	C₄²×C₇	C₄⋊C₄	1	2	yes	—
101	C₄²×C₇	C₄⋊C₄	1	2	yes	—
102	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
106	C₂²×M₁₆	—	2	1	no	—
107	C₂²×C₄⋊C₄	C₄×C₇	1	2	yes	—
108	C₂²×C₄²	—	2	1	no	—
109	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
110	#431	—	2	2	no	—
111	#431	C₄×C₇	1	4	yes	complemented
112	#423	—	2	2	no	—
113	#423	C₄×C₇	1	4	yes	complemented
114	C₂×C₄⋊C₈	—	2	1	no	—
115	C₂×C₄⋊C₈	—	2	1	no	—
116	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
117	C₂×C₂²⋊C₈	—	2	1	no	—
118	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
119	C₄×C₄⋊C₄	—	2	1	no	—
120	C₄×C₄⋊C₄	—	2	1	no	—
121	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
122	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
123	C₂×#95	—	2	1	no	—
124	C₂³×C₇	—	2	2	no	—
125	C₂³×C₇	—	2	2	no	—
126	C₂³×C₇	#95	1	2	yes	—
127	C₂³×C₇	#102	1	2	yes	—
128	C₂³×C₇	#103	1	2	yes	—
129	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₇	—	2	2	no	—
131	C₂×C₄×C₇	—	2	2	no	—
132	C₂×C₄×C₇	—	2	2	no	—
133	C₂×C₄×C₇	—	2	4	no	—
134	C₂×C₄×C₇	—	2	4	no	—
135	C₂×C₄×C₇	—	2	8	no	—
136	C₂×C₄×C₇	—	4	2	no	—
137	C₂×C₄×C₇	—	4	2	no	—
138	C₂×C₄×C₇	—	4	16	no	—
139	C₂×C₄×C₇	—	4	4	no	—
140	C₂×C₄×C₇	—	4	4	no	—
141	C₂×C₄×C₇	#95	1	2	yes	—
142	C₂×C₄×C₇	#95	1	2	yes	—
143	C₂×C₄×C₇	#95	1	2	yes	—
144	C₂×C₄×C₇	#102	1	2	yes	—
145	C₂×C₄×C₇	#102	1	4	yes	—
146	C₂×C₄×C₇	#103	1	2	yes	—
147	C₂×C₄×C₇	#103	1	4	yes	—
148	C₂×C₄×C₇	#106	1	4	yes	—
149	C₂×C₄×C₇	#107	1	4	yes	—
150	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
151	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
157	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
159	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
161	C₇×C₈	—	4	4	no	—
162	C₇×C₈	—	4	4	no	—
163	C₂³×C₄	—	2	2	no	—
164	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
165	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
166	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
167	C₂×M₁₆	—	2	4	no	—
168	C₂×M₁₆	—	4	2	no	—
169	C₂×M₁₆	—	4	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	8	no	—
174	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	—	2	2	no	—
177	C₂×C₄²	—	4	2	no	—
178	C₂×C₄²	—	4	2	no	—
179	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
180	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
181	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
182	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
183	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
184	C₂×C₄²	C₇×D₄	1	2	yes	—
185	C₂×C₄²	C₇×Q₈	1	2	yes	—
186	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
187	C₄⋊C₈	—	2	4	no	—
188	C₄⋊C₈	—	2	4	no	—
189	C₄⋊C₈	—	2	2	no	—
190	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	complemented
191	C₂²⋊C₈	—	2	4	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	C₂×C₄×C₇	1	4	yes	complemented
194	#95	—	2	2	no	—
195	#95	—	2	2	no	—
196	C₂²×C₇	—	2	2	no	—
197	C₂²×C₇	—	2	2	no	—
198	C₂²×C₇	—	2	4	no	—
199	C₂²×C₇	—	2	4	no	—
200	C₂²×C₇	#340	1	4	yes	—
201	C₂²×C₇	C₂×#95	1	1	yes	characteristic
202	C₂²×C₇	#407	1	2	yes	—
203	C₂²×C₇	C₂×#102	1	1	yes	characteristic
204	C₂²×C₇	C₂×#103	1	1	yes	characteristic
205	C₂²×C₇	#421	1	2	yes	—
206	C₄×C₇	—	2	2	no	—
207	C₄×C₇	—	2	1	no	—
208	C₄×C₇	—	2	2	no	—
209	C₄×C₇	—	2	1	no	—
210	C₄×C₇	—	4	8	no	—
211	C₄×C₇	#340	1	4	yes	—
212	C₄×C₇	C₄×M₁₆	1	2	yes	—
213	C₄×C₇	C₂×#102	1	1	yes	characteristic
214	C₄×C₇	C₂×#103	1	1	yes	characteristic
215	C₄×C₇	#423	1	2	yes	—
216	C₄×C₇	C₂×#106	1	1	yes	characteristic
217	C₄×C₇	C₂×#107	1	1	yes	characteristic
218	C₂⁴	C₇×C₂²⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	C₄²×C₇	1	2	yes	—
226	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
227	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
228	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
229	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
230	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
231	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
232	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
233	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
234	C₂²×C₄	C₇×D₈	1	2	yes	—
235	C₂²×C₄	C₇×SD₁₆	1	2	yes	—
236	C₂²×C₄	C₇×SD₁₆	1	2	yes	—
237	C₂²×C₄	C₇×Q₁₆	1	2	yes	—
238	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
241	M₁₆	—	4	8	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	4	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₄⋊C₄	—	2	8	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	C₇×M₁₆	1	8	yes	complemented
250	C₄²	—	4	4	no	—
251	C₄²	—	4	2	no	—
252	C₄²	—	4	2	no	—
253	C₄²	C₄²×C₇	1	2	yes	—
254	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
255	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
256	C₄²	C₇×C₄⋊C₄	1	2	yes	—
257	C₄²	C₇×C₄⋊C₄	1	2	yes	—
258	C₄²	C₇×C₄⋊C₄	1	2	yes	—
259	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
260	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
261	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
262	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
263	C₂×C₇	—	2	4	no	—
264	C₂×C₇	#1486	1	2	yes	—
265	C₂×C₇	#1487	1	2	yes	—
266	C₂×C₇	#1728	1	1	yes	characteristic
267	C₂×C₇	#1733	1	1	yes	characteristic
268	C₂×C₇	C₂×#340	1	1	yes	characteristic
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	C₇×#95	1	2	yes	—
272	C₂³	C₇×#102	1	2	yes	—
273	C₂³	C₇×#103	1	2	yes	—
274	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	16	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	C₇×#95	1	2	yes	—
287	C₂×C₄	C₇×#95	1	2	yes	—
288	C₂×C₄	C₇×#95	1	2	yes	—
289	C₂×C₄	C₇×#102	1	2	yes	—
290	C₂×C₄	C₇×#102	1	4	yes	—
291	C₂×C₄	C₇×#103	1	2	yes	—
292	C₂×C₄	C₇×#103	1	4	yes	—
293	C₂×C₄	C₇×#106	1	4	yes	—
294	C₂×C₄	C₇×#107	1	4	yes	—
295	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
296	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
297	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
298	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
299	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
300	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
301	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
302	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
303	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
304	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
305	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
306	C₈	—	4	4	no	—
307	C₈	—	4	4	no	—
308	C₇	#7901	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	2	4	no	—
313	C₂²	C₇×#340	1	4	yes	—
314	C₂²	C₂×C₇×#95	1	1	yes	characteristic
315	C₂²	C₇×#407	1	2	yes	—
316	C₂²	C₂×C₇×#102	1	1	yes	characteristic
317	C₂²	C₂×C₇×#103	1	1	yes	characteristic
318	C₂²	C₇×#421	1	2	yes	—
319	C₄	—	2	2	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	1	no	—
323	C₄	—	4	8	no	—
324	C₄	C₇×#340	1	4	yes	—
325	C₄	C₄×C₇×M₁₆	1	2	yes	—
326	C₄	C₂×C₇×#102	1	1	yes	characteristic
327	C₄	C₂×C₇×#103	1	1	yes	characteristic
328	C₄	C₇×#423	1	2	yes	—
329	C₄	C₂×C₇×#106	1	1	yes	characteristic
330	C₄	C₂×C₇×#107	1	1	yes	characteristic
331	C₂	—	2	4	no	—
332	C₂	C₇×#1486	1	2	yes	—
333	C₂	C₇×#1487	1	2	yes	—
334	C₂	C₇×#1728	1	1	yes	characteristic
335	C₂	C₇×#1733	1	1	yes	characteristic
336	C₂	C₂×C₇×#340	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₇×#1279	C₂	semidirect
10. C₇×#431	C₄	semidirect
12. C₇×#423	C₄	semidirect
23. #7901	C₇	direct (N × H)
51. C₇×C₄⋊C₈	C₂×C₄	semidirect
54. C₄×C₇×C₈	C₂×C₄	semidirect
60. #1279	C₂×C₇	semidirect
92. C₇×C₄⋊C₄	M₁₆	semidirect
111. #431	C₄×C₇	semidirect
113. #423	C₄×C₇	semidirect
190. C₄⋊C₈	C₂×C₄×C₇	semidirect
193. C₄×C₈	C₂×C₄×C₇	semidirect
249. C₄⋊C₄	C₇×M₁₆	semidirect
308. C₇	#7901	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
C₂×C₄×C₇×C₄⋊C₄	2
C₇×#1279	2
#7901	7

Subgroups — C7×#7901 #13185117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#7901 #13185117