Subgroups — C₇×#23157 #13200373

Order 1792.

806 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic
2	C₇×#2365	C₂	1	2	yes	maximal
3	C₇×#1988	C₂	1	2	yes	maximal
4	C₇×#1942	C₂	1	2	yes	maximal
5	C₇×#1918	C₂	1	4	yes	maximal
6	C₄×C₇×#107	C₂	1	2	yes	maximal
7	C₄×C₇×#107	C₂	1	1	yes	maximal, characteristic
8	C₇×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×C₇×Q₈	—	2	2	no	—
11	C₇×#500	C₄	1	8	yes	complemented
12	C₂×C₇×#107	C₂²	1	1	yes	characteristic
13	C₂×C₇×#107	C₂²	1	2	yes	—
14	C₂×C₇×#107	C₂²	1	2	yes	—
15	C₂×C₇×#107	C₂²	1	1	yes	characteristic
16	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₇×#391	—	2	2	no	—
20	C₇×#389	C₂²	1	4	yes	—
21	C₇×#387	—	2	4	no	—
22	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
23	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
24	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₇×#340	C₂²	1	4	yes	—
26	C₇×#340	C₂²	1	4	yes	—
27	C₇×#340	C₂²	1	2	yes	—
28	C₇×#336	C₂²	1	2	yes	—
29	#23157	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₇×Q₈	—	2	2	no	—
31	C₂²×C₇×C₈	C₂³	1	2	yes	—
32	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
34	C₇×C₄⋊Q₈	—	2	4	no	—
35	C₇×C₄⋊Q₈	C₂×C₄	1	8	yes	—
36	C₄×C₇×Q₈	—	4	8	no	—
37	C₂×C₇×C₄⋊C₄	—	2	2	no	—
38	C₂×C₇×C₄⋊C₄	—	2	4	no	—
39	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
40	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
41	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
43	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
44	C₂×C₄²×C₇	—	2	4	no	—
45	C₂×C₄²×C₇	—	2	2	no	—
46	C₂×C₄²×C₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
48	C₇×#107	—	2	2	no	—
49	C₇×#107	C₂×C₄	1	8	yes	—
50	C₇×#107	C₂×C₄	1	8	yes	—
51	C₇×#107	Q₈	1	4	yes	complemented
52	C₇×C₄⋊C₈	—	2	2	no	—
53	C₄×C₇×C₈	—	2	4	no	—
54	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
55	C₇×#95	—	2	2	no	—
56	C₇×#95	—	2	4	no	—
57	C₇×#95	—	2	4	no	—
58	C₇×#95	D₄	1	2	yes	—
59	C₂×#500	C₂×C₇	1	1	yes	characteristic
60	#2365	C₂×C₇	1	2	yes	—
61	#1988	C₂×C₇	1	2	yes	—
62	#1942	C₂×C₇	1	2	yes	—
63	#1918	C₂×C₇	1	4	yes	—
64	C₄×#107	C₂×C₇	1	1	yes	characteristic
65	C₄×#107	C₂×C₇	1	2	yes	—
66	C₈×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
67	C₂×C₇×Q₈	—	2	8	no	—
68	C₂×C₇×Q₈	—	4	4	no	—
69	C₂×C₇×Q₈	—	4	4	no	—
70	C₂²×C₄×C₇	—	2	2	no	—
71	C₂²×C₄×C₇	—	2	4	no	—
72	C₂²×C₄×C₇	—	2	2	no	—
73	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
78	C₂×C₇×C₈	—	2	1	no	—
79	C₂×C₇×C₈	—	2	1	no	—
80	C₂×C₇×C₈	—	2	2	no	—
81	C₂×C₇×C₈	—	2	2	no	—
82	C₂×C₇×C₈	—	2	2	no	—
83	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
86	C₂×C₇×C₈	#41	1	2	yes	—
87	C₇×C₄⋊C₄	—	2	8	no	—
88	C₇×C₄⋊C₄	—	2	2	no	—
89	C₇×C₄⋊C₄	—	2	4	no	—
90	C₇×C₄⋊C₄	—	2	2	no	—
91	C₇×C₄⋊C₄	—	2	2	no	—
92	C₇×C₄⋊C₄	—	2	2	no	—
93	C₇×C₄⋊C₄	—	4	8	no	—
94	C₇×C₄⋊C₄	D₈	1	4	yes	—
95	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
96	C₇×C₄⋊C₄	C₂×Q₈	1	4	yes	—
97	C₄²×C₇	—	2	2	no	—
98	C₄²×C₇	—	2	2	no	—
99	C₄²×C₇	—	4	8	no	—
100	C₄²×C₇	—	4	4	no	—
101	C₄²×C₇	C₂²×C₄	1	2	yes	—
102	C₄²×C₇	#41	1	2	yes	—
103	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
104	C₂×C₄×Q₈	—	2	2	no	—
105	#500	C₄×C₇	1	8	yes	complemented
106	C₂×#107	C₂²×C₇	1	1	yes	characteristic
107	C₂×#107	C₂²×C₇	1	2	yes	—
108	C₂×#107	C₂²×C₇	1	2	yes	—
109	C₂×#107	C₂²×C₇	1	1	yes	characteristic
110	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
112	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
113	#391	—	2	2	no	—
114	#389	C₂²×C₇	1	4	yes	—
115	#387	—	2	4	no	—
116	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
117	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
118	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
119	#340	C₂²×C₇	1	2	yes	—
120	#340	C₂²×C₇	1	4	yes	—
121	#340	C₂²×C₇	1	4	yes	—
122	#336	C₂²×C₇	1	2	yes	—
123	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
124	C₇×Q₈	—	4	16	no	—
125	C₂×C₄×C₇	—	2	2	no	—
126	C₂×C₄×C₇	—	2	2	no	—
127	C₂×C₄×C₇	—	2	1	no	—
128	C₂×C₄×C₇	—	2	1	no	—
129	C₂×C₄×C₇	—	2	8	no	—
130	C₂×C₄×C₇	—	2	2	no	—
131	C₂×C₄×C₇	—	2	4	no	—
132	C₂×C₄×C₇	—	2	2	no	—
133	C₂×C₄×C₇	—	2	2	no	—
134	C₂×C₄×C₇	—	4	2	no	—
135	C₂×C₄×C₇	—	4	4	no	—
136	C₂×C₄×C₇	—	4	4	no	—
137	C₂×C₄×C₇	—	4	2	no	—
138	C₂×C₄×C₇	—	4	2	no	—
139	C₂×C₄×C₇	—	4	2	no	—
140	C₂×C₄×C₇	—	4	2	no	—
141	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
143	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
144	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₇	#135	1	2	yes	—
146	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
151	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
152	C₇×C₈	—	2	2	no	—
153	C₇×C₈	—	4	2	no	—
154	C₇×C₈	C₄×Q₈	1	4	yes	complemented
155	C₂²×Q₈	—	2	2	no	—
156	C₂²×C₈	C₂³×C₇	1	2	yes	—
157	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
158	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
159	C₄⋊Q₈	—	2	4	no	—
160	C₄⋊Q₈	C₂×C₄×C₇	1	8	yes	—
161	C₄×Q₈	—	4	8	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	4	no	—
164	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
166	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
167	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
168	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
169	C₂×C₄²	—	2	4	no	—
170	C₂×C₄²	—	2	2	no	—
171	C₂×C₄²	C₂³×C₇	1	2	yes	—
172	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
173	#107	—	2	2	no	—
174	#107	C₂×C₄×C₇	1	8	yes	—
175	#107	C₂×C₄×C₇	1	8	yes	—
176	#107	C₇×Q₈	1	4	yes	complemented
177	C₄⋊C₈	—	2	2	no	—
178	C₄×C₈	—	2	4	no	—
179	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
180	#95	—	2	2	no	—
181	#95	—	2	4	no	—
182	#95	—	2	4	no	—
183	#95	C₇×D₄	1	2	yes	—
184	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
185	C₂²×C₇	#547	1	1	yes	characteristic
186	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
187	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
188	C₂²×C₇	C₂×#135	1	1	yes	characteristic
189	C₂²×C₇	#578	1	1	yes	characteristic
190	C₂²×C₇	#578	1	1	yes	characteristic
191	C₄×C₇	—	2	4	no	—
192	C₄×C₇	—	2	2	no	—
193	C₄×C₇	—	4	2	no	—
194	C₄×C₇	—	4	4	no	—
195	C₄×C₇	—	4	8	no	—
196	C₄×C₇	—	8	2	no	—
197	C₄×C₇	C₄×D₈	1	4	yes	—
198	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
199	C₄×C₇	#519	1	1	yes	characteristic
200	C₄×C₇	#556	1	1	yes	characteristic
201	C₄×C₇	#557	1	1	yes	characteristic
202	C₂×Q₈	—	2	8	no	—
203	C₂×Q₈	—	4	4	no	—
204	C₂×Q₈	—	4	4	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
209	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
211	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
212	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
219	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
220	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
221	C₂×C₈	C₇×#41	1	2	yes	—
222	C₄⋊C₄	—	2	8	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	4	8	no	—
229	C₄⋊C₄	C₇×D₈	1	4	yes	—
230	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
231	C₄⋊C₄	C₂×C₇×Q₈	1	4	yes	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	4	8	no	—
235	C₄²	—	4	4	no	—
236	C₄²	C₂²×C₄×C₇	1	2	yes	—
237	C₄²	C₇×#41	1	2	yes	—
238	C₂×C₇	#2365	1	1	yes	characteristic
239	C₂×C₇	C₂×C₄×D₈	1	1	yes	characteristic
240	C₂×C₇	#2950	1	1	yes	characteristic
241	C₂×C₇	#3334	1	1	yes	characteristic
242	C₂×C₇	#3342	1	1	yes	characteristic
243	C₂×C₇	Q₈×D₈	1	1	yes	characteristic
244	C₂×C₇	#3383	1	1	yes	characteristic
245	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
246	Q₈	—	4	16	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
264	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
265	C₂×C₄	C₂×C₇×D₈	1	4	yes	—
266	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
267	C₂×C₄	C₇×#135	1	2	yes	—
268	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
270	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
271	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
272	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
273	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	4	2	no	—
276	C₈	C₄×C₇×Q₈	1	4	yes	complemented
277	C₇	#23157	1	1	yes	characteristic, direct product factor, Sylow
278	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
279	C₂²	C₇×#547	1	1	yes	characteristic
280	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
281	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
282	C₂²	C₂×C₇×#135	1	1	yes	characteristic
283	C₂²	C₇×#578	1	1	yes	characteristic
284	C₂²	C₇×#578	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	4	no	—
287	C₄	—	4	8	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	4	no	—
290	C₄	—	8	2	no	—
291	C₄	C₄×C₇×D₈	1	4	yes	—
292	C₄	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
293	C₄	C₇×#519	1	1	yes	characteristic
294	C₄	C₇×#556	1	1	yes	characteristic
295	C₄	C₇×#557	1	1	yes	characteristic
296	C₂	C₇×#2365	1	1	yes	characteristic
297	C₂	C₂×C₄×C₇×D₈	1	1	yes	characteristic
298	C₂	C₇×#2950	1	1	yes	characteristic
299	C₂	C₇×#3334	1	1	yes	characteristic
300	C₂	C₇×#3342	1	1	yes	characteristic
301	C₂	C₇×Q₈×D₈	1	1	yes	characteristic
302	C₂	C₇×#3383	1	1	yes	characteristic

	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×#500	C₄	semidirect
29. #23157	C₇	direct (N × H)
51. C₇×#107	Q₈	semidirect
105. #500	C₄×C₇	semidirect
154. C₇×C₈	C₄×Q₈	semidirect
176. #107	C₇×Q₈	semidirect
276. C₈	C₄×C₇×Q₈	semidirect
277. C₇	#23157	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×#500	2
C₇×#2365	2
C₇×#1988	2
C₇×#1942	2
C₇×#1918	2
C₄×C₇×#107	2
C₇×C₈×C₄⋊C₄	2
#23157	7

Subgroups — C7×#23157 #13200373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23157 #13200373