Subgroups — C₇×#21277 #13198493

Order 1792.

950 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#2293	C₂	1	2	yes	maximal, complemented
3	C₇×#1972	C₂	1	2	yes	maximal
4	C₇×#1943	C₂	1	2	yes	maximal, complemented
5	C₇×#1936	C₂	1	4	yes	maximal, complemented
6	C₄×C₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₇×#102	C₂	1	2	yes	maximal, complemented
8	C₄×C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
11	C₂×C₇×#117	—	2	2	no	—
12	C₄×C₇×D₈	C₄	1	8	yes	complemented
13	C₂×C₇×#107	C₂²	1	1	yes	characteristic
14	C₂×C₇×#106	C₂²	1	2	yes	—
15	C₂×C₇×#102	C₂²	1	4	yes	—
16	C₂×C₇×#102	C₂²	1	2	yes	—
17	C₂×C₇×#97	C₂²	1	2	yes	—
18	C₂×C₇×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
21	C₇×#390	C₂²	1	4	yes	—
22	C₇×#389	C₂²	1	4	yes	—
23	C₇×#388	—	2	4	no	—
24	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄×C₇×C₂²⋊C₄	—	2	2	no	—
26	C₄³×C₇	C₂²	1	1	yes	characteristic
27	C₇×#340	C₂²	1	4	yes	—
28	C₇×#340	C₂²	1	2	yes	—
29	#21277	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₇×D₄	C₂³	1	2	yes	—
31	C₂³×C₄×C₇	—	2	2	no	—
32	C₂×C₇×D₈	—	2	4	no	—
33	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
34	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₇×C₈	C₂³	1	2	yes	—
36	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
37	C₄×C₇×D₄	—	2	4	no	—
38	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
39	C₇×#117	—	4	8	no	—
40	C₂×C₇×C₄⋊C₄	—	2	4	no	—
41	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
42	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
43	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
44	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
45	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
46	C₂×C₄²×C₇	—	2	2	no	—
47	C₂×C₄²×C₇	D₄	1	2	yes	—
48	C₂×C₄²×C₇	D₄	1	2	yes	—
49	C₂×C₄²×C₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
51	C₇×#107	C₂×C₄	1	4	yes	—
52	C₇×#106	—	2	4	no	—
53	C₇×#102	—	2	8	no	—
54	C₇×#102	C₂×C₄	1	8	yes	—
55	C₇×#97	—	2	4	no	—
56	C₇×#97	—	2	4	no	—
57	C₄×C₇×C₈	—	2	2	no	—
58	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
59	C₇×#95	—	2	4	no	—
60	C₇×#95	—	2	2	no	—
61	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic
62	#2293	C₂×C₇	1	2	yes	complemented
63	#1972	C₂×C₇	1	2	yes	—
64	#1943	C₂×C₇	1	2	yes	complemented
65	#1936	C₂×C₇	1	4	yes	complemented
66	C₄×#106	C₂×C₇	1	1	yes	characteristic, complemented
67	C₄×#102	C₂×C₇	1	2	yes	complemented
68	C₄×#97	C₂×C₇	1	1	yes	characteristic, complemented
69	C₂⁴×C₇	—	2	2	no	—
70	C₂×C₇×D₄	—	2	4	no	—
71	C₂×C₇×D₄	—	2	4	no	—
72	C₂×C₇×D₄	—	2	2	no	—
73	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₇	—	2	2	no	—
75	C₂²×C₄×C₇	—	2	8	no	—
76	C₂²×C₄×C₇	—	2	4	no	—
77	C₂²×C₄×C₇	—	4	4	no	—
78	C₂²×C₄×C₇	—	4	4	no	—
79	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
83	C₇×D₈	—	2	8	no	—
84	C₂×C₇×C₈	—	2	2	no	—
85	C₂×C₇×C₈	—	2	2	no	—
86	C₂×C₇×C₈	—	2	2	no	—
87	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₇×C₈	#41	1	4	yes	—
90	C₂×C₇×C₈	#41	1	4	yes	—
91	C₇×C₄⋊C₄	—	2	2	no	—
92	C₇×C₄⋊C₄	—	2	4	no	—
93	C₇×C₄⋊C₄	—	2	4	no	—
94	C₇×C₄⋊C₄	—	4	8	no	—
95	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
96	C₇×C₂²⋊C₄	—	2	8	no	—
97	C₇×C₂²⋊C₄	—	4	8	no	—
98	C₇×C₂²⋊C₄	—	4	8	no	—
99	C₄²×C₇	—	2	4	no	—
100	C₄²×C₇	—	2	2	no	—
101	C₄²×C₇	—	2	2	no	—
102	C₄²×C₇	—	4	2	no	—
103	C₄²×C₇	—	4	4	no	—
104	C₄²×C₇	C₂²×C₄	1	2	yes	—
105	C₄²×C₇	#41	1	2	yes	—
106	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
107	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
108	C₂×#117	—	2	2	no	—
109	C₄×D₈	C₄×C₇	1	8	yes	complemented
110	C₂×#107	C₂²×C₇	1	1	yes	characteristic
111	C₂×#106	C₂²×C₇	1	2	yes	—
112	C₂×#102	C₂²×C₇	1	4	yes	—
113	C₂×#102	C₂²×C₇	1	2	yes	—
114	C₂×#97	C₂²×C₇	1	2	yes	—
115	C₂×#97	C₂²×C₇	1	2	yes	—
116	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
118	#390	C₂²×C₇	1	4	yes	—
119	#389	C₂²×C₇	1	4	yes	—
120	#388	—	2	4	no	—
121	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
122	C₄×C₂²⋊C₄	—	2	2	no	—
123	C₄³	C₂²×C₇	1	1	yes	characteristic
124	#340	C₂²×C₇	1	2	yes	—
125	#340	C₂²×C₇	1	4	yes	—
126	C₂³×C₇	—	2	4	no	—
127	C₂³×C₇	—	4	4	no	—
128	C₂³×C₇	—	4	2	no	—
129	C₂³×C₇	—	4	2	no	—
130	C₂³×C₇	—	4	4	no	—
131	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
132	C₇×D₄	—	2	4	no	—
133	C₇×D₄	—	2	4	no	—
134	C₇×D₄	—	4	4	no	—
135	C₂×C₄×C₇	—	2	2	no	—
136	C₂×C₄×C₇	—	2	4	no	—
137	C₂×C₄×C₇	—	2	4	no	—
138	C₂×C₄×C₇	—	2	4	no	—
139	C₂×C₄×C₇	—	2	4	no	—
140	C₂×C₄×C₇	—	4	2	no	—
141	C₂×C₄×C₇	—	4	2	no	—
142	C₂×C₄×C₇	—	4	16	no	—
143	C₂×C₄×C₇	—	4	4	no	—
144	C₂×C₄×C₇	—	4	4	no	—
145	C₂×C₄×C₇	—	4	4	no	—
146	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
147	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
148	C₂×C₄×C₇	AGL(1,ℤ/8)	1	4	yes	—
149	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₇	C₂×#41	1	2	yes	—
151	C₂×C₄×C₇	C₂×#41	1	2	yes	—
152	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
153	C₇×C₈	—	2	2	no	—
154	C₇×C₈	—	2	4	no	—
155	C₇×C₈	#117	1	4	yes	complemented
156	C₂²×D₄	C₂³×C₇	1	2	yes	—
157	C₂³×C₄	—	2	2	no	—
158	C₂×D₈	—	2	4	no	—
159	C₂×D₈	C₂×C₄×C₇	1	4	yes	—
160	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₇	1	2	yes	—
162	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
163	C₄×D₄	—	2	4	no	—
164	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
165	#117	—	4	8	no	—
166	C₂×C₄⋊C₄	—	2	4	no	—
167	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
168	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
169	C₂×C₂²⋊C₄	—	2	2	no	—
170	C₂×C₂²⋊C₄	—	2	4	no	—
171	C₂×C₂²⋊C₄	—	2	4	no	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	C₇×D₄	1	2	yes	—
174	C₂×C₄²	C₇×D₄	1	2	yes	—
175	C₂×C₄²	C₂³×C₇	1	2	yes	—
176	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
177	#107	C₂×C₄×C₇	1	4	yes	—
178	#106	—	2	4	no	—
179	#102	—	2	8	no	—
180	#102	C₂×C₄×C₇	1	8	yes	—
181	#97	—	2	4	no	—
182	#97	—	2	4	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
185	#95	—	2	4	no	—
186	#95	—	2	2	no	—
187	C₂²×C₇	—	4	4	no	—
188	C₂²×C₇	—	4	4	no	—
189	C₂²×C₇	—	4	4	no	—
190	C₂²×C₇	—	8	4	no	—
191	C₂²×C₇	—	8	4	no	—
192	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
193	C₂²×C₇	#532	1	2	yes	—
194	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
195	C₂²×C₇	#575	1	2	yes	—
196	C₄×C₇	—	2	2	no	—
197	C₄×C₇	—	4	4	no	—
198	C₄×C₇	—	4	4	no	—
199	C₄×C₇	—	8	4	no	—
200	C₄×C₇	#442	1	4	yes	—
201	C₄×C₇	C₂×#117	1	1	yes	characteristic
202	C₄×C₇	C₄×#41	1	1	yes	characteristic
203	C₄×C₇	#529	1	2	yes	—
204	C₂⁴	—	2	2	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
215	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
217	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
218	D₈	—	2	8	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
223	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
224	C₂×C₈	C₇×#41	1	4	yes	—
225	C₂×C₈	C₇×#41	1	4	yes	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	4	8	no	—
230	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
231	C₂²⋊C₄	—	2	8	no	—
232	C₂²⋊C₄	—	4	8	no	—
233	C₂²⋊C₄	—	4	8	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	—	4	4	no	—
239	C₄²	C₂²×C₄×C₇	1	2	yes	—
240	C₄²	C₇×#41	1	2	yes	—
241	C₂×C₇	—	8	4	no	—
242	C₂×C₇	#2293	1	1	yes	characteristic
243	C₂×C₇	C₂×#442	1	1	yes	characteristic
244	C₂×C₇	#2943	1	1	yes	characteristic
245	C₂×C₇	#3171	1	4	yes	—
246	C₂³	—	2	4	no	—
247	C₂³	—	4	4	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	4	no	—
251	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
252	D₄	—	2	4	no	—
253	D₄	—	2	4	no	—
254	D₄	—	4	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	16	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
267	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
268	C₂×C₄	C₇×AGL(1,ℤ/8)	1	4	yes	—
269	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
270	C₂×C₄	C₂×C₇×#41	1	2	yes	—
271	C₂×C₄	C₂×C₇×#41	1	2	yes	—
272	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	—	2	4	no	—
275	C₈	C₇×#117	1	4	yes	complemented
276	C₇	#21277	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²	—	4	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	4	4	no	—
280	C₂²	—	8	4	no	—
281	C₂²	—	8	4	no	—
282	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
283	C₂²	C₇×#532	1	2	yes	—
284	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
285	C₂²	C₇×#575	1	2	yes	—
286	C₄	—	2	2	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	4	no	—
289	C₄	—	8	4	no	—
290	C₄	C₇×#442	1	4	yes	—
291	C₄	C₂×C₇×#117	1	1	yes	characteristic
292	C₄	C₄×C₇×#41	1	1	yes	characteristic
293	C₄	C₇×#529	1	2	yes	—
294	C₂	—	8	4	no	—
295	C₂	C₇×#2293	1	1	yes	characteristic
296	C₂	C₂×C₇×#442	1	1	yes	characteristic
297	C₂	C₇×#2943	1	1	yes	characteristic
298	C₂	C₇×#3171	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂²×D₄
prime radical	C₂²×C₇×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2293	C₂	semidirect
4. C₇×#1943	C₂	semidirect
5. C₇×#1936	C₂	semidirect
6. C₄×C₇×#106	C₂	semidirect
7. C₄×C₇×#102	C₂	semidirect
8. C₄×C₇×#97	C₂	semidirect
12. C₄×C₇×D₈	C₄	semidirect
29. #21277	C₇	direct (N × H)
62. #2293	C₂×C₇	semidirect
64. #1943	C₂×C₇	semidirect
65. #1936	C₂×C₇	semidirect
66. C₄×#106	C₂×C₇	semidirect
67. C₄×#102	C₂×C₇	semidirect
68. C₄×#97	C₂×C₇	semidirect
109. C₄×D₈	C₄×C₇	semidirect
155. C₇×C₈	#117	semidirect
275. C₈	C₇×#117	semidirect
276. C₇	#21277	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×C₄×C₇×D₈	2
C₇×#2293	2
C₇×#1972	2
C₇×#1943	2
C₇×#1936	2
C₄×C₇×#106	2
C₄×C₇×#102	2
C₄×C₇×#97	2
#21277	7

Subgroups — C7×#21277 #13198493

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21277 #13198493