Subgroups — C₇×#20824 #13198040

Order 1792.

930 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₇×#2285	C₂	1	2	yes	maximal
3	C₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1834	C₂	1	2	yes	maximal, complemented
6	C₇×#1809	C₂	1	4	yes	maximal
7	C₇×#1776	C₂	1	4	yes	maximal, complemented
8	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
9	C₂×C₇×#117	—	2	2	no	—
10	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₇×#107	C₂²	1	4	yes	—
12	C₂×C₇×#106	C₂²	1	4	yes	—
13	C₄×C₇×M₁₆	C₄	1	8	yes	complemented
14	C₂×C₇×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
17	C₇×#384	C₂²	1	2	yes	—
18	C₇×#384	C₂²	1	2	yes	—
19	C₇×#382	—	2	2	no	—
20	C₇×#380	—	2	2	no	—
21	C₇×#380	C₂²	1	4	yes	—
22	C₄×C₇×C₄⋊C₄	—	2	2	no	—
23	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
24	C₇×#340	C₂²	1	8	yes	—
25	#20824	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂³×C₄×C₇	D₄	1	2	yes	—
27	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₂×C₇×M₁₆	—	2	4	no	—
29	C₂×C₇×M₁₆	—	2	4	no	—
30	C₂×C₇×M₁₆	C₂×C₄	1	8	yes	—
31	C₂²×C₇×C₈	C₂³	1	2	yes	—
32	C₂²×C₇×C₈	C₂³	1	2	yes	—
33	C₇×#117	—	4	8	no	—
34	C₂×C₇×C₄⋊C₄	—	2	2	no	—
35	C₂×C₇×C₄⋊C₄	—	2	2	no	—
36	C₂×C₇×C₄⋊C₄	—	2	2	no	—
37	C₂×C₇×C₄⋊C₄	—	2	2	no	—
38	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
39	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
40	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
41	C₂×C₄²×C₇	—	2	4	no	—
42	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₇	D₄	1	2	yes	—
44	C₂×C₄²×C₇	Q₈	1	4	yes	—
45	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
47	C₇×#107	—	2	8	no	—
48	C₇×#106	—	2	8	no	—
49	C₇×#97	C₂×C₄	1	8	yes	—
50	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
51	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
52	C₇×#95	—	2	4	no	—
53	C₇×#95	—	2	4	no	—
54	C₂×C₄×M₁₆	C₂×C₇	1	1	yes	characteristic
55	#2285	C₂×C₇	1	2	yes	—
56	#1845	C₂×C₇	1	1	yes	characteristic, complemented
57	#1844	C₂×C₇	1	1	yes	characteristic, complemented
58	#1834	C₂×C₇	1	2	yes	complemented
59	#1809	C₂×C₇	1	4	yes	—
60	#1776	C₂×C₇	1	4	yes	complemented
61	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	—	2	4	no	—
63	C₂²×C₄×C₇	—	2	4	no	—
64	C₂²×C₄×C₇	—	2	2	no	—
65	C₂²×C₄×C₇	—	2	2	no	—
66	C₂²×C₄×C₇	—	2	4	no	—
67	C₂²×C₄×C₇	—	2	4	no	—
68	C₂²×C₄×C₇	C₄⋊C₄	1	8	yes	—
69	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
75	C₇×M₁₆	—	2	16	no	—
76	C₂×C₇×C₈	—	2	2	no	—
77	C₂×C₇×C₈	—	2	2	no	—
78	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
79	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
80	C₂×C₇×C₈	#41	1	4	yes	—
81	C₂×C₇×C₈	#41	1	4	yes	—
82	C₇×C₄⋊C₄	—	2	8	no	—
83	C₇×C₄⋊C₄	—	2	8	no	—
84	C₇×C₄⋊C₄	—	4	4	no	—
85	C₇×C₄⋊C₄	—	4	4	no	—
86	C₇×C₄⋊C₄	—	4	4	no	—
87	C₇×C₄⋊C₄	—	4	4	no	—
88	C₇×C₂²⋊C₄	—	4	8	no	—
89	C₄²×C₇	—	4	8	no	—
90	C₄²×C₇	C₄⋊C₄	1	8	yes	—
91	C₄²×C₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₇	C₂²×C₄	1	2	yes	—
93	C₄²×C₇	C₂×Q₈	1	2	yes	—
94	C₄²×C₇	C₂×Q₈	1	2	yes	—
95	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
96	C₂×#117	—	2	2	no	—
97	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
98	C₂×#107	C₂²×C₇	1	4	yes	—
99	C₂×#106	C₂²×C₇	1	4	yes	—
100	C₄×M₁₆	C₄×C₇	1	8	yes	complemented
101	C₂×#97	C₂²×C₇	1	2	yes	—
102	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
104	#384	C₂²×C₇	1	2	yes	—
105	#384	C₂²×C₇	1	2	yes	—
106	#382	—	2	2	no	—
107	#380	—	2	2	no	—
108	#380	C₂²×C₇	1	4	yes	—
109	C₄×C₄⋊C₄	—	2	2	no	—
110	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
111	#340	C₂²×C₇	1	8	yes	—
112	C₂³×C₇	—	2	1	no	—
113	C₂³×C₇	—	2	1	no	—
114	C₂³×C₇	—	2	2	no	—
115	C₂³×C₇	—	2	2	no	—
116	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇	—	2	1	no	—
119	C₂×C₄×C₇	—	2	1	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	2	2	no	—
122	C₂×C₄×C₇	—	2	4	no	—
123	C₂×C₄×C₇	—	2	4	no	—
124	C₂×C₄×C₇	—	2	16	no	—
125	C₂×C₄×C₇	—	4	4	no	—
126	C₂×C₄×C₇	—	4	4	no	—
127	C₂×C₄×C₇	—	4	4	no	—
128	C₂×C₄×C₇	—	4	4	no	—
129	C₂×C₄×C₇	—	4	4	no	—
130	C₂×C₄×C₇	—	4	4	no	—
131	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
133	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
134	C₂×C₄×C₇	AGL(1,ℤ/8)	1	4	yes	—
135	C₂×C₄×C₇	#137	1	4	yes	—
136	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₇	C₂×#41	1	2	yes	—
139	C₂×C₄×C₇	C₂×#41	1	2	yes	—
140	C₇×C₈	—	2	4	no	—
141	C₇×C₈	#117	1	8	yes	complemented
142	C₂³×C₄	C₇×D₄	1	2	yes	—
143	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
144	C₂×M₁₆	—	2	4	no	—
145	C₂×M₁₆	—	2	4	no	—
146	C₂×M₁₆	C₂×C₄×C₇	1	8	yes	—
147	C₂²×C₈	C₂³×C₇	1	2	yes	—
148	C₂²×C₈	C₂³×C₇	1	2	yes	—
149	#117	—	4	8	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
155	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
156	C₂×C₂²⋊C₄	—	2	4	no	—
157	C₂×C₄²	—	2	4	no	—
158	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
159	C₂×C₄²	C₇×D₄	1	2	yes	—
160	C₂×C₄²	C₇×Q₈	1	4	yes	—
161	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
162	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
163	#107	—	2	8	no	—
164	#106	—	2	8	no	—
165	#97	C₂×C₄×C₇	1	8	yes	—
166	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
167	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
168	#95	—	2	4	no	—
169	#95	—	2	4	no	—
170	C₂²×C₇	—	2	4	no	—
171	C₂²×C₇	—	2	2	no	—
172	C₂²×C₇	—	4	2	no	—
173	C₂²×C₇	—	4	1	no	—
174	C₂²×C₇	—	4	1	no	—
175	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
176	C₂²×C₇	#532	1	1	yes	characteristic
177	C₂²×C₇	#532	1	1	yes	characteristic
178	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
179	C₂²×C₇	C₂×#137	1	2	yes	—
180	C₄×C₇	—	2	4	no	—
181	C₄×C₇	—	2	2	no	—
182	C₄×C₇	—	8	4	no	—
183	C₄×C₇	—	8	4	no	—
184	C₄×C₇	#428	1	8	yes	—
185	C₄×C₇	C₂×#117	1	2	yes	—
186	C₄×C₇	#533	1	2	yes	—
187	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	C₇×C₄⋊C₄	1	8	yes	—
195	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
196	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
198	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
199	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
200	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
201	M₁₆	—	2	16	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
205	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
206	C₂×C₈	C₇×#41	1	4	yes	—
207	C₂×C₈	C₇×#41	1	4	yes	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	—	2	8	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₂²⋊C₄	—	4	8	no	—
215	C₄²	—	4	8	no	—
216	C₄²	C₇×C₄⋊C₄	1	8	yes	—
217	C₄²	C₂²×C₄×C₇	1	2	yes	—
218	C₄²	C₂²×C₄×C₇	1	2	yes	—
219	C₄²	C₂×C₇×Q₈	1	2	yes	—
220	C₄²	C₂×C₇×Q₈	1	2	yes	—
221	C₂×C₇	—	4	2	no	—
222	C₂×C₇	#2285	1	1	yes	characteristic
223	C₂×C₇	C₂×#428	1	2	yes	—
224	C₂×C₇	#3182	1	2	yes	—
225	C₂×C₇	#3184	1	1	yes	characteristic
226	C₂×C₇	#3185	1	1	yes	characteristic
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
232	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	16	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
247	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
248	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
249	C₂×C₄	C₇×AGL(1,ℤ/8)	1	4	yes	—
250	C₂×C₄	C₇×#137	1	4	yes	—
251	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
253	C₂×C₄	C₂×C₇×#41	1	2	yes	—
254	C₂×C₄	C₂×C₇×#41	1	2	yes	—
255	C₈	—	2	4	no	—
256	C₈	C₇×#117	1	8	yes	complemented
257	C₇	#20824	1	1	yes	characteristic, direct product factor, Sylow
258	C₂²	—	2	4	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	1	no	—
263	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
264	C₂²	C₇×#532	1	1	yes	characteristic
265	C₂²	C₇×#532	1	1	yes	characteristic
266	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
267	C₂²	C₂×C₇×#137	1	2	yes	—
268	C₄	—	2	4	no	—
269	C₄	—	2	2	no	—
270	C₄	—	8	4	no	—
271	C₄	—	8	4	no	—
272	C₄	C₇×#428	1	8	yes	—
273	C₄	C₂×C₇×#117	1	2	yes	—
274	C₄	C₇×#533	1	2	yes	—
275	C₂	—	4	2	no	—
276	C₂	C₇×#2285	1	1	yes	characteristic
277	C₂	C₂×C₇×#428	1	2	yes	—
278	C₂	C₇×#3182	1	2	yes	—
279	C₂	C₇×#3184	1	1	yes	characteristic
280	C₂	C₇×#3185	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₇×#1845	C₂	semidirect
4. C₇×#1844	C₂	semidirect
5. C₇×#1834	C₂	semidirect
7. C₇×#1776	C₂	semidirect
13. C₄×C₇×M₁₆	C₄	semidirect
25. #20824	C₇	direct (N × H)
56. #1845	C₂×C₇	semidirect
57. #1844	C₂×C₇	semidirect
58. #1834	C₂×C₇	semidirect
60. #1776	C₂×C₇	semidirect
100. C₄×M₁₆	C₄×C₇	semidirect
141. C₇×C₈	#117	semidirect
256. C₈	C₇×#117	semidirect
257. C₇	#20824	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×M₁₆	2
C₇×#2285	2
C₇×#1845	2
C₇×#1844	2
C₇×#1834	2
C₇×#1809	2
C₇×#1776	2
#20824	7

Subgroups — C7×#20824 #13198040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20824 #13198040