Subgroups — C₇×#21960 #13199176

Order 1792.

818 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#2283	C₂	1	2	yes	maximal
3	C₇×#1949	C₂	1	8	yes	maximal, complemented
4	C₇×#1808	C₂	1	2	yes	maximal
5	C₇×#1807	C₂	1	2	yes	maximal
6	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₇×#107	C₂²	1	4	yes	—
10	C₂×C₇×#106	C₂²	1	4	yes	—
11	C₂×C₇×C₄⋊C₈	C₄	1	8	yes	complemented
12	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
13	C₇×#383	—	2	2	no	—
14	C₇×#382	C₂²	1	4	yes	—
15	C₇×#381	—	2	2	no	—
16	C₇×#380	C₂²	1	4	yes	—
17	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
18	C₄×C₇×C₂²⋊C₄	—	2	2	no	—
19	C₇×#376	—	2	2	no	—
20	C₇×#340	C₂²	1	4	yes	—
21	C₇×#340	C₂²	1	4	yes	—
22	#21960	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	—	4	2	no	—
27	C₂²×C₇×C₈	—	4	2	no	—
28	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₇×C₈	C₂³	1	2	yes	—
31	C₂²×C₇×C₈	C₂³	1	2	yes	—
32	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
33	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
34	C₂×C₄²×C₇	—	2	4	no	—
35	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₇	D₄	1	2	yes	—
37	C₂×C₄²×C₇	Q₈	1	4	yes	—
38	C₂×C₄²×C₇	C₂³	1	2	yes	—
39	C₇×#107	—	4	4	no	—
40	C₇×#106	—	4	4	no	—
41	C₇×C₄⋊C₈	C₂×C₄	1	16	yes	—
42	C₇×#95	—	2	4	no	—
43	C₇×#95	—	2	4	no	—
44	C₇×#95	—	2	4	no	—
45	C₇×#95	—	2	4	no	—
46	C₂²×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
47	#2283	C₂×C₇	1	2	yes	—
48	#1949	C₂×C₇	1	8	yes	complemented
49	#1808	C₂×C₇	1	2	yes	—
50	#1807	C₂×C₇	1	2	yes	—
51	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	—	2	4	no	—
53	C₂²×C₄×C₇	—	2	2	no	—
54	C₂²×C₄×C₇	—	2	2	no	—
55	C₂²×C₄×C₇	—	2	8	no	—
56	C₂²×C₄×C₇	—	4	2	no	—
57	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
59	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
65	C₂×C₇×C₈	—	2	4	no	—
66	C₂×C₇×C₈	—	2	4	no	—
67	C₂×C₇×C₈	—	4	4	no	—
68	C₂×C₇×C₈	—	4	4	no	—
69	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
70	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
71	C₇×C₄⋊C₄	—	2	8	no	—
72	C₇×C₄⋊C₄	—	2	8	no	—
73	C₇×C₂²⋊C₄	—	4	8	no	—
74	C₄²×C₇	—	4	8	no	—
75	C₄²×C₇	C₄⋊C₄	1	8	yes	—
76	C₄²×C₇	C₂²×C₄	1	4	yes	—
77	C₄²×C₇	C₂×Q₈	1	4	yes	—
78	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
79	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
80	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
81	C₂×#107	C₂²×C₇	1	4	yes	—
82	C₂×#106	C₂²×C₇	1	4	yes	—
83	C₂×C₄⋊C₈	C₄×C₇	1	8	yes	complemented
84	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
85	#383	—	2	2	no	—
86	#382	C₂²×C₇	1	4	yes	—
87	#381	—	2	2	no	—
88	#380	C₂²×C₇	1	4	yes	—
89	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
90	C₄×C₂²⋊C₄	—	2	2	no	—
91	#376	—	2	2	no	—
92	#340	C₂²×C₇	1	4	yes	—
93	#340	C₂²×C₇	1	4	yes	—
94	C₂³×C₇	—	2	2	no	—
95	C₂³×C₇	—	2	2	no	—
96	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₇	#135	1	4	yes	—
98	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	2	4	no	—
101	C₂×C₄×C₇	—	2	4	no	—
102	C₂×C₄×C₇	—	2	8	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	2	4	no	—
106	C₂×C₄×C₇	—	4	4	no	—
107	C₂×C₄×C₇	—	4	8	no	—
108	C₂×C₄×C₇	—	4	8	no	—
109	C₂×C₄×C₇	—	4	8	no	—
110	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
113	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
116	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
117	C₇×C₈	—	2	4	no	—
118	C₇×C₈	—	2	4	no	—
119	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
120	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
121	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
122	C₂²×C₈	—	4	2	no	—
123	C₂²×C₈	—	4	2	no	—
124	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
125	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
126	C₂²×C₈	C₂³×C₇	1	2	yes	—
127	C₂²×C₈	C₂³×C₇	1	2	yes	—
128	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
129	C₂×C₂²⋊C₄	—	2	4	no	—
130	C₂×C₄²	—	2	4	no	—
131	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
132	C₂×C₄²	C₇×D₄	1	2	yes	—
133	C₂×C₄²	C₇×Q₈	1	4	yes	—
134	C₂×C₄²	C₂³×C₇	1	2	yes	—
135	#107	—	4	4	no	—
136	#106	—	4	4	no	—
137	C₄⋊C₈	C₂×C₄×C₇	1	16	yes	—
138	#95	—	2	4	no	—
139	#95	—	2	4	no	—
140	#95	—	2	4	no	—
141	#95	—	2	4	no	—
142	C₂²×C₇	—	2	2	no	—
143	C₂²×C₇	—	2	2	no	—
144	C₂²×C₇	—	2	8	no	—
145	C₂²×C₇	#427	1	4	yes	—
146	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₇	#537	1	2	yes	—
148	C₂²×C₇	C₂×#135	1	2	yes	—
149	C₂²×C₇	#575	1	2	yes	—
150	C₄×C₇	—	2	2	no	—
151	C₄×C₇	—	2	4	no	—
152	C₄×C₇	—	2	4	no	—
153	C₄×C₇	—	8	8	no	—
154	C₄×C₇	#518	1	1	yes	characteristic
155	C₄×C₇	#519	1	1	yes	characteristic
156	C₄×C₇	#544	1	2	yes	—
157	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
164	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
165	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
166	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
168	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
169	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
170	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	4	4	no	—
174	C₂×C₈	—	4	4	no	—
175	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
176	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
177	C₄⋊C₄	—	2	8	no	—
178	C₄⋊C₄	—	2	8	no	—
179	C₂²⋊C₄	—	4	8	no	—
180	C₄²	—	4	8	no	—
181	C₄²	C₇×C₄⋊C₄	1	8	yes	—
182	C₄²	C₂²×C₄×C₇	1	4	yes	—
183	C₄²	C₂×C₇×Q₈	1	4	yes	—
184	C₂×C₇	—	2	4	no	—
185	C₂×C₇	#2316	1	1	yes	characteristic
186	C₂×C₇	C₂×#427	1	1	yes	characteristic
187	C₂×C₇	#2911	1	1	yes	characteristic
188	C₂×C₇	#3250	1	4	yes	—
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
192	C₂³	C₇×#135	1	4	yes	—
193	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	8	no	—
203	C₂×C₄	—	4	8	no	—
204	C₂×C₄	—	4	8	no	—
205	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
206	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
207	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
208	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
210	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
211	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
212	C₈	—	2	4	no	—
213	C₈	—	2	4	no	—
214	C₇	#21960	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²	—	2	2	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	C₇×#427	1	4	yes	—
219	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
220	C₂²	C₇×#537	1	2	yes	—
221	C₂²	C₂×C₇×#135	1	2	yes	—
222	C₂²	C₇×#575	1	2	yes	—
223	C₄	—	2	4	no	—
224	C₄	—	2	4	no	—
225	C₄	—	2	2	no	—
226	C₄	—	8	8	no	—
227	C₄	C₇×#518	1	1	yes	characteristic
228	C₄	C₇×#519	1	1	yes	characteristic
229	C₄	C₇×#544	1	2	yes	—
230	C₂	—	2	4	no	—
231	C₂	C₇×#2316	1	1	yes	characteristic
232	C₂	C₂×C₇×#427	1	1	yes	characteristic
233	C₂	C₇×#2911	1	1	yes	characteristic
234	C₂	C₇×#3250	1	4	yes	—

	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×#1949	C₂	semidirect
11. C₂×C₇×C₄⋊C₈	C₄	semidirect
22. #21960	C₇	direct (N × H)
48. #1949	C₂×C₇	semidirect
83. C₂×C₄⋊C₈	C₄×C₇	semidirect
214. C₇	#21960	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×C₇×C₄⋊C₈	2
C₇×#2283	2
C₇×#1949	2
C₇×#1808	2
C₇×#1807	2
#21960	7

Subgroups — C7×#21960 #13199176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21960 #13199176