Subgroups — C₇×#20817 #13198033

Order 1792.

982 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#2285	C₂	1	2	yes	maximal
3	C₇×#1845	C₂	1	2	yes	maximal, complemented
4	C₇×#1834	C₂	1	2	yes	maximal, complemented
5	C₇×#1808	C₂	1	4	yes	maximal
6	C₄×C₇×#107	C₂	1	4	yes	maximal, complemented
7	C₂³×C₇×C₈	C₂²	1	2	yes	—
8	C₂×C₇×#117	—	2	2	no	—
9	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₇×#107	C₂²	1	8	yes	—
11	C₂×C₄×C₇×C₈	C₄	1	8	yes	complemented
12	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₇×#384	C₂²	1	4	yes	—
16	C₇×#382	—	2	2	no	—
17	C₇×#380	—	2	2	no	—
18	C₇×#380	C₂²	1	4	yes	—
19	C₄×C₇×C₄⋊C₄	—	2	2	no	—
20	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
21	C₇×#340	C₂²	1	8	yes	—
22	#20817	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₂²×C₇×C₈	—	2	8	no	—
27	C₂²×C₇×C₈	C₂×C₄	1	8	yes	—
28	C₂²×C₇×C₈	C₂³	1	2	yes	—
29	C₂²×C₇×C₈	C₂³	1	2	yes	—
30	C₇×#117	—	4	8	no	—
31	C₂×C₇×C₄⋊C₄	—	2	4	no	—
32	C₂×C₇×C₄⋊C₄	—	2	4	no	—
33	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
34	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
35	C₂×C₄²×C₇	—	2	4	no	—
36	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇	Q₈	1	4	yes	—
40	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
42	C₇×#107	—	2	16	no	—
43	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
44	C₄×C₇×C₈	C₂×C₄	1	8	yes	—
45	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
46	C₇×#95	—	2	4	no	—
47	C₇×#95	—	2	4	no	—
48	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
49	#2285	C₂×C₇	1	2	yes	—
50	#1845	C₂×C₇	1	2	yes	complemented
51	#1834	C₂×C₇	1	2	yes	complemented
52	#1808	C₂×C₇	1	4	yes	—
53	C₄×#107	C₂×C₇	1	4	yes	complemented
54	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	—	2	4	no	—
56	C₂²×C₄×C₇	—	2	4	no	—
57	C₂²×C₄×C₇	—	2	8	no	—
58	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
59	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
60	C₂²×C₄×C₇	D₈	1	4	yes	—
61	C₂²×C₄×C₇	Q₁₆	1	4	yes	—
62	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
70	C₂×C₇×C₈	—	2	4	no	—
71	C₂×C₇×C₈	—	2	16	no	—
72	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
73	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
74	C₂×C₇×C₈	#41	1	8	yes	—
75	C₇×C₄⋊C₄	—	2	16	no	—
76	C₇×C₄⋊C₄	—	4	8	no	—
77	C₇×C₄⋊C₄	—	4	8	no	—
78	C₇×C₂²⋊C₄	—	4	8	no	—
79	C₄²×C₇	—	4	8	no	—
80	C₄²×C₇	C₄⋊C₄	1	4	yes	—
81	C₄²×C₇	C₄⋊C₄	1	4	yes	—
82	C₄²×C₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₇	C₂²×C₄	1	2	yes	—
84	C₄²×C₇	C₂×Q₈	1	2	yes	—
85	C₄²×C₇	C₂×Q₈	1	2	yes	—
86	C₂³×C₈	C₂²×C₇	1	2	yes	—
87	C₂×#117	—	2	2	no	—
88	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
89	C₂×#107	C₂²×C₇	1	8	yes	—
90	C₂×C₄×C₈	C₄×C₇	1	8	yes	complemented
91	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
92	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
94	#384	C₂²×C₇	1	4	yes	—
95	#382	—	2	2	no	—
96	#380	—	2	2	no	—
97	#380	C₂²×C₇	1	4	yes	—
98	C₄×C₄⋊C₄	—	2	2	no	—
99	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
100	#340	C₂²×C₇	1	8	yes	—
101	C₂³×C₇	—	2	2	no	—
102	C₂³×C₇	—	2	2	no	—
103	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂³×C₇	C₂×D₈	1	2	yes	—
105	C₂³×C₇	C₂×Q₁₆	1	2	yes	—
106	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	—	2	2	no	—
109	C₂×C₄×C₇	—	2	4	no	—
110	C₂×C₄×C₇	—	2	4	no	—
111	C₂×C₄×C₇	—	2	8	no	—
112	C₂×C₄×C₇	—	4	8	no	—
113	C₂×C₄×C₇	—	4	8	no	—
114	C₂×C₄×C₇	—	4	8	no	—
115	C₂×C₄×C₇	#107	1	16	yes	complemented
116	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
125	C₂×C₄×C₇	#135	1	4	yes	—
126	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂×#41	1	4	yes	—
129	C₇×C₈	—	2	4	no	—
130	C₇×C₈	#117	1	8	yes	complemented
131	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
132	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
133	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
134	C₂²×C₈	—	2	8	no	—
135	C₂²×C₈	C₂×C₄×C₇	1	8	yes	—
136	C₂²×C₈	C₂³×C₇	1	2	yes	—
137	C₂²×C₈	C₂³×C₇	1	2	yes	—
138	#117	—	4	8	no	—
139	C₂×C₄⋊C₄	—	2	4	no	—
140	C₂×C₄⋊C₄	—	2	4	no	—
141	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
142	C₂×C₂²⋊C₄	—	2	4	no	—
143	C₂×C₄²	—	2	4	no	—
144	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
145	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
146	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
147	C₂×C₄²	C₇×Q₈	1	4	yes	—
148	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
149	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
150	#107	—	2	16	no	—
151	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
152	C₄×C₈	C₂×C₄×C₇	1	8	yes	—
153	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
154	#95	—	2	4	no	—
155	#95	—	2	4	no	—
156	C₂²×C₇	—	2	2	no	—
157	C₂²×C₇	—	2	4	no	—
158	C₂²×C₇	—	2	4	no	—
159	C₂²×C₇	—	2	2	no	—
160	C₂²×C₇	C₂×#107	1	4	yes	—
161	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
162	C₂²×C₇	#532	1	2	yes	—
163	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
164	C₂²×C₇	C₂²×Q₁₆	1	1	yes	characteristic
165	C₂²×C₇	C₂×#135	1	2	yes	—
166	C₄×C₇	—	2	2	no	—
167	C₄×C₇	—	2	2	no	—
168	C₄×C₇	—	8	8	no	—
169	C₄×C₇	C₂×#107	1	4	yes	—
170	C₄×C₇	C₂×#107	1	4	yes	—
171	C₄×C₇	#427	1	4	yes	—
172	C₄×C₇	C₂×#117	1	2	yes	—
173	C₄×C₇	#533	1	2	yes	—
174	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
179	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
180	C₂²×C₄	C₇×D₈	1	4	yes	—
181	C₂²×C₄	C₇×Q₁₆	1	4	yes	—
182	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
183	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
189	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	16	no	—
192	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
193	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
194	C₂×C₈	C₇×#41	1	8	yes	—
195	C₄⋊C₄	—	2	16	no	—
196	C₄⋊C₄	—	4	8	no	—
197	C₄⋊C₄	—	4	8	no	—
198	C₂²⋊C₄	—	4	8	no	—
199	C₄²	—	4	8	no	—
200	C₄²	C₇×C₄⋊C₄	1	4	yes	—
201	C₄²	C₇×C₄⋊C₄	1	4	yes	—
202	C₄²	C₂²×C₄×C₇	1	2	yes	—
203	C₄²	C₂²×C₄×C₇	1	2	yes	—
204	C₄²	C₂×C₇×Q₈	1	2	yes	—
205	C₄²	C₂×C₇×Q₈	1	2	yes	—
206	C₂×C₇	—	2	4	no	—
207	C₂×C₇	#2285	1	1	yes	characteristic
208	C₂×C₇	C₂²×#107	1	1	yes	characteristic
209	C₂×C₇	C₂×#427	1	1	yes	characteristic
210	C₂×C₇	#3169	1	2	yes	—
211	C₂×C₇	#3170	1	2	yes	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
215	C₂³	C₂×C₇×D₈	1	2	yes	—
216	C₂³	C₂×C₇×Q₁₆	1	2	yes	—
217	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	—	4	8	no	—
226	C₂×C₄	C₇×#107	1	16	yes	complemented
227	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
228	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
229	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
230	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
231	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
232	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
233	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
234	C₂×C₄	C₂×C₇×Q₁₆	1	2	yes	—
235	C₂×C₄	C₂×C₇×Q₁₆	1	2	yes	—
236	C₂×C₄	C₇×#135	1	4	yes	—
237	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
239	C₂×C₄	C₂×C₇×#41	1	4	yes	—
240	C₈	—	2	4	no	—
241	C₈	C₇×#117	1	8	yes	complemented
242	C₇	#20817	1	1	yes	characteristic, direct product factor, Sylow
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	2	no	—
247	C₂²	C₂×C₇×#107	1	4	yes	—
248	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
249	C₂²	C₇×#532	1	2	yes	—
250	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
251	C₂²	C₂²×C₇×Q₁₆	1	1	yes	characteristic
252	C₂²	C₂×C₇×#135	1	2	yes	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	8	8	no	—
256	C₄	C₂×C₇×#107	1	4	yes	—
257	C₄	C₂×C₇×#107	1	4	yes	—
258	C₄	C₇×#427	1	4	yes	—
259	C₄	C₂×C₇×#117	1	2	yes	—
260	C₄	C₇×#533	1	2	yes	—
261	C₂	—	2	4	no	—
262	C₂	C₇×#2285	1	1	yes	characteristic
263	C₂	C₂²×C₇×#107	1	1	yes	characteristic
264	C₂	C₂×C₇×#427	1	1	yes	characteristic
265	C₂	C₇×#3169	1	2	yes	—
266	C₂	C₇×#3170	1	2	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	C₂⁴×C₇	C₂×D₄
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
3. C₇×#1845	C₂	semidirect
4. C₇×#1834	C₂	semidirect
6. C₄×C₇×#107	C₂	semidirect
11. C₂×C₄×C₇×C₈	C₄	semidirect
22. #20817	C₇	direct (N × H)
50. #1845	C₂×C₇	semidirect
51. #1834	C₂×C₇	semidirect
53. C₄×#107	C₂×C₇	semidirect
90. C₂×C₄×C₈	C₄×C₇	semidirect
115. C₂×C₄×C₇	#107	semidirect
130. C₇×C₈	#117	semidirect
226. C₂×C₄	C₇×#107	semidirect
241. C₈	C₇×#117	semidirect
242. C₇	#20817	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄×C₇×C₈	2
C₇×#2285	2
C₇×#1845	2
C₇×#1834	2
C₇×#1808	2
C₄×C₇×#107	2
#20817	7

Subgroups — C7×#20817 #13198033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20817 #13198033