Subgroups — C₇×#23159 #13200375

Order 1792.

718 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#499	C₂	1	1	yes	maximal, characteristic
2	C₇×#2370	C₂	1	2	yes	maximal
3	C₇×#1989	C₂	1	2	yes	maximal
4	C₇×#1941	C₂	1	2	yes	maximal
5	C₇×#1921	C₂	1	2	yes	maximal
6	C₇×#1918	C₂	1	2	yes	maximal
7	C₄×C₇×#106	C₂	1	2	yes	maximal
8	C₄×C₇×#106	C₂	1	1	yes	maximal, characteristic
9	C₇×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
10	C₂×C₇×#125	C₂²	1	2	yes	—
11	C₇×#499	C₄	1	8	yes	complemented
12	C₂×C₇×#107	C₂²	1	1	yes	characteristic
13	C₂×C₇×#107	C₂²	1	1	yes	characteristic
14	C₂×C₇×#106	C₂²	1	1	yes	characteristic
15	C₂×C₇×#106	C₂²	1	2	yes	—
16	C₂×C₇×#106	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₇×#389	—	2	2	no	—
21	C₇×#389	C₂²	1	2	yes	—
22	C₇×#389	C₂²	1	2	yes	—
23	C₇×#387	—	2	2	no	—
24	C₇×#387	—	2	2	no	—
25	C₄×C₇×C₄⋊C₄	—	2	2	no	—
26	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
27	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
28	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
29	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₇×#340	C₂²	1	2	yes	—
31	C₇×#340	C₂²	1	2	yes	—
32	C₇×#340	C₂²	1	2	yes	—
33	C₇×#340	C₂²	1	2	yes	—
34	C₇×#340	C₂²	1	2	yes	—
35	C₇×#336	C₂²	1	2	yes	—
36	#23159	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₇×C₈	C₂³	1	2	yes	—
38	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
40	C₇×#125	—	2	4	no	—
41	C₇×#125	C₂×C₄	1	8	yes	—
42	C₂×C₇×C₄⋊C₄	—	2	2	no	—
43	C₂×C₇×C₄⋊C₄	—	2	2	no	—
44	C₂×C₇×C₄⋊C₄	—	2	2	no	—
45	C₂×C₇×C₄⋊C₄	—	2	2	no	—
46	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
47	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₄²×C₇	—	2	2	no	—
52	C₂×C₄²×C₇	—	2	2	no	—
53	C₂×C₄²×C₇	—	2	2	no	—
54	C₂×C₄²×C₇	C₂³	1	2	yes	—
55	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
56	C₇×#107	C₂×C₄	1	4	yes	—
57	C₇×#107	C₂×C₄	1	4	yes	—
58	C₇×#106	—	2	2	no	—
59	C₇×#106	C₂×C₄	1	8	yes	—
60	C₇×#106	Q₈	1	4	yes	—
61	C₇×C₄⋊C₈	—	2	2	no	—
62	C₄×C₇×C₈	—	2	4	no	—
63	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
64	C₇×#95	—	2	2	no	—
65	C₇×#95	—	2	2	no	—
66	C₇×#95	—	2	2	no	—
67	C₇×#95	—	2	2	no	—
68	C₇×#95	—	2	2	no	—
69	C₇×#95	D₄	1	1	yes	characteristic
70	C₇×#95	D₄	1	1	yes	characteristic
71	C₂×#499	C₂×C₇	1	1	yes	characteristic
72	#2370	C₂×C₇	1	2	yes	—
73	#1989	C₂×C₇	1	2	yes	—
74	#1941	C₂×C₇	1	2	yes	—
75	#1921	C₂×C₇	1	2	yes	—
76	#1918	C₂×C₇	1	2	yes	—
77	C₄×#106	C₂×C₇	1	1	yes	characteristic
78	C₄×#106	C₂×C₇	1	2	yes	—
79	C₈×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
80	C₂²×C₄×C₇	—	2	2	no	—
81	C₂²×C₄×C₇	—	2	2	no	—
82	C₂²×C₄×C₇	—	2	2	no	—
83	C₂²×C₄×C₇	—	2	2	no	—
84	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
89	C₂×C₇×C₈	—	2	1	no	—
90	C₂×C₇×C₈	—	2	1	no	—
91	C₂×C₇×C₈	—	2	2	no	—
92	C₂×C₇×C₈	—	2	2	no	—
93	C₂×C₇×C₈	—	2	2	no	—
94	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
97	C₂×C₇×C₈	#41	1	2	yes	—
98	C₇×C₄⋊C₄	—	2	8	no	—
99	C₇×C₄⋊C₄	—	2	8	no	—
100	C₇×C₄⋊C₄	—	2	4	no	—
101	C₇×C₄⋊C₄	—	2	2	no	—
102	C₇×C₄⋊C₄	—	2	2	no	—
103	C₇×C₄⋊C₄	—	2	2	no	—
104	C₇×C₄⋊C₄	—	2	2	no	—
105	C₇×C₄⋊C₄	—	2	2	no	—
106	C₇×C₄⋊C₄	—	4	4	no	—
107	C₇×C₄⋊C₄	—	8	2	no	—
108	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
109	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
110	C₇×C₄⋊C₄	C₂×Q₈	1	4	yes	—
111	C₄²×C₇	—	2	2	no	—
112	C₄²×C₇	—	2	2	no	—
113	C₄²×C₇	—	4	4	no	—
114	C₄²×C₇	—	8	2	no	—
115	C₄²×C₇	—	8	2	no	—
116	C₄²×C₇	C₂²×C₄	1	2	yes	—
117	C₄²×C₇	#41	1	2	yes	—
118	C₂×#125	C₂²×C₇	1	2	yes	—
119	#499	C₄×C₇	1	8	yes	complemented
120	C₂×#107	C₂²×C₇	1	1	yes	characteristic
121	C₂×#107	C₂²×C₇	1	1	yes	characteristic
122	C₂×#106	C₂²×C₇	1	1	yes	characteristic
123	C₂×#106	C₂²×C₇	1	2	yes	—
124	C₂×#106	C₂²×C₇	1	1	yes	characteristic
125	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
127	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
128	#389	—	2	2	no	—
129	#389	C₂²×C₇	1	2	yes	—
130	#389	C₂²×C₇	1	2	yes	—
131	#387	—	2	2	no	—
132	#387	—	2	2	no	—
133	C₄×C₄⋊C₄	—	2	2	no	—
134	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
135	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
136	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
137	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
138	#340	C₂²×C₇	1	2	yes	—
139	#340	C₂²×C₇	1	2	yes	—
140	#340	C₂²×C₇	1	2	yes	—
141	#340	C₂²×C₇	1	2	yes	—
142	#340	C₂²×C₇	1	2	yes	—
143	#336	C₂²×C₇	1	2	yes	—
144	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇	—	2	2	no	—
146	C₂×C₄×C₇	—	2	2	no	—
147	C₂×C₄×C₇	—	2	2	no	—
148	C₂×C₄×C₇	—	2	2	no	—
149	C₂×C₄×C₇	—	2	4	no	—
150	C₂×C₄×C₇	—	2	4	no	—
151	C₂×C₄×C₇	—	2	2	no	—
152	C₂×C₄×C₇	—	2	4	no	—
153	C₂×C₄×C₇	—	2	2	no	—
154	C₂×C₄×C₇	—	2	2	no	—
155	C₂×C₄×C₇	—	4	2	no	—
156	C₂×C₄×C₇	—	4	2	no	—
157	C₂×C₄×C₇	—	4	2	no	—
158	C₂×C₄×C₇	—	4	2	no	—
159	C₂×C₄×C₇	—	4	2	no	—
160	C₂×C₄×C₇	—	4	2	no	—
161	C₂×C₄×C₇	—	4	2	no	—
162	C₂×C₄×C₇	—	4	2	no	—
163	C₂×C₄×C₇	—	4	2	no	—
164	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
165	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
166	C₂×C₄×C₇	#135	1	4	yes	—
167	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
171	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
172	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
173	C₇×C₈	—	2	2	no	—
174	C₇×C₈	—	4	2	no	—
175	C₇×C₈	C₄×Q₈	1	4	yes	—
176	C₂²×C₈	C₂³×C₇	1	2	yes	—
177	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
179	#125	—	2	4	no	—
180	#125	C₂×C₄×C₇	1	8	yes	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
186	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
187	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
188	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
189	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	C₂³×C₇	1	2	yes	—
194	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
195	#107	C₂×C₄×C₇	1	4	yes	—
196	#107	C₂×C₄×C₇	1	4	yes	—
197	#106	—	2	2	no	—
198	#106	C₂×C₄×C₇	1	8	yes	—
199	#106	C₇×Q₈	1	4	yes	—
200	C₄⋊C₈	—	2	2	no	—
201	C₄×C₈	—	2	4	no	—
202	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	#95	—	2	2	no	—
206	#95	—	2	2	no	—
207	#95	—	2	2	no	—
208	#95	C₇×D₄	1	1	yes	characteristic
209	#95	C₇×D₄	1	1	yes	characteristic
210	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
211	C₂²×C₇	#547	1	1	yes	characteristic
212	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
213	C₂²×C₇	C₂×#135	1	2	yes	—
214	C₂²×C₇	#577	1	2	yes	—
215	C₄×C₇	—	2	4	no	—
216	C₄×C₇	—	2	2	no	—
217	C₄×C₇	—	2	2	no	—
218	C₄×C₇	—	4	2	no	—
219	C₄×C₇	—	4	4	no	—
220	C₄×C₇	—	4	4	no	—
221	C₄×C₇	—	4	4	no	—
222	C₄×C₇	—	8	2	no	—
223	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
224	C₄×C₇	#519	1	1	yes	characteristic
225	C₄×C₇	#556	1	1	yes	characteristic
226	C₄×C₇	#557	1	1	yes	characteristic
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
232	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
235	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
242	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
243	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
244	C₂×C₈	C₇×#41	1	2	yes	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	2	8	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	8	2	no	—
255	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
256	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
257	C₄⋊C₄	C₂×C₇×Q₈	1	4	yes	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	4	4	no	—
261	C₄²	—	8	2	no	—
262	C₄²	—	8	2	no	—
263	C₄²	C₂²×C₄×C₇	1	2	yes	—
264	C₄²	C₇×#41	1	2	yes	—
265	C₂×C₇	#2365	1	1	yes	characteristic
266	C₂×C₇	C₄×#135	1	1	yes	characteristic
267	C₂×C₇	#2952	1	1	yes	characteristic
268	C₂×C₇	#3337	1	2	yes	—
269	C₂×C₇	#3381	1	2	yes	—
270	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
291	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
292	C₂×C₄	C₇×#135	1	4	yes	—
293	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
295	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
296	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
297	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
298	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
299	C₈	—	2	2	no	—
300	C₈	—	4	2	no	—
301	C₈	C₄×C₇×Q₈	1	4	yes	—
302	C₇	#23159	1	1	yes	characteristic, direct product factor, Sylow
303	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
304	C₂²	C₇×#547	1	1	yes	characteristic
305	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
306	C₂²	C₂×C₇×#135	1	2	yes	—
307	C₂²	C₇×#577	1	2	yes	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	4	no	—
311	C₄	—	4	4	no	—
312	C₄	—	4	4	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	4	no	—
315	C₄	—	8	2	no	—
316	C₄	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
317	C₄	C₇×#519	1	1	yes	characteristic
318	C₄	C₇×#556	1	1	yes	characteristic
319	C₄	C₇×#557	1	1	yes	characteristic
320	C₂	C₇×#2365	1	1	yes	characteristic
321	C₂	C₄×C₇×#135	1	1	yes	characteristic
322	C₂	C₇×#2952	1	1	yes	characteristic
323	C₂	C₇×#3337	1	2	yes	—
324	C₂	C₇×#3381	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₇×#499	C₄	semidirect
36. #23159	C₇	direct (N × H)
119. #499	C₄×C₇	semidirect
302. C₇	#23159	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×#499	2
C₇×#2370	2
C₇×#1989	2
C₇×#1941	2
C₇×#1921	2
C₇×#1918	2
C₄×C₇×#106	2
C₇×C₈×C₄⋊C₄	2
#23159	7

Subgroups — C7×#23159 #13200375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23159 #13200375