Subgroups — C₇×#21961 #13199177

Order 1792.

906 conjugacy classes of subgroups.

The subgroup lattice has 230 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×#2285	C₂	1	2	yes	maximal
3	C₇×#1947	C₂	1	8	yes	maximal, complemented
4	C₇×#1808	C₂	1	4	yes	maximal
5	C₂³×C₇×C₈	C₂²	1	2	yes	—
6	C₂×C₇×#117	—	2	2	no	—
7	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₇×#107	C₂²	1	8	yes	—
9	C₂×C₇×C₄⋊C₈	C₄	1	8	yes	complemented
10	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
11	C₇×#384	C₂²	1	4	yes	—
12	C₇×#382	—	2	2	no	—
13	C₇×#380	—	2	2	no	—
14	C₇×#380	C₂²	1	4	yes	—
15	C₄×C₇×C₄⋊C₄	—	2	2	no	—
16	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
17	C₇×#340	C₂²	1	8	yes	—
18	#21961	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂²×C₇×C₈	—	4	4	no	—
23	C₂²×C₇×C₈	C₂×C₄	1	8	yes	—
24	C₂²×C₇×C₈	C₂³	1	4	yes	—
25	C₇×#117	—	4	8	no	—
26	C₂×C₇×C₄⋊C₄	—	2	4	no	—
27	C₂×C₇×C₄⋊C₄	—	2	4	no	—
28	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
29	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
30	C₂×C₄²×C₇	—	2	4	no	—
31	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₇	D₄	1	2	yes	—
33	C₂×C₄²×C₇	Q₈	1	4	yes	—
34	C₂×C₄²×C₇	C₂³	1	2	yes	—
35	C₇×#107	—	4	8	no	—
36	C₇×C₄⋊C₈	C₂×C₄	1	16	yes	—
37	C₇×#95	—	2	4	no	—
38	C₇×#95	—	2	4	no	—
39	C₂²×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
40	#2285	C₂×C₇	1	2	yes	—
41	#1947	C₂×C₇	1	8	yes	complemented
42	#1808	C₂×C₇	1	4	yes	—
43	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₇	—	2	2	no	—
45	C₂²×C₄×C₇	—	2	2	no	—
46	C₂²×C₄×C₇	—	2	8	no	—
47	C₂²×C₄×C₇	—	4	2	no	—
48	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
49	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₇	D₈	1	4	yes	—
51	C₂²×C₄×C₇	Q₁₆	1	4	yes	—
52	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
58	C₂×C₇×C₈	—	2	8	no	—
59	C₂×C₇×C₈	—	4	8	no	—
60	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
61	C₇×C₄⋊C₄	—	2	8	no	—
62	C₇×C₄⋊C₄	—	2	8	no	—
63	C₇×C₄⋊C₄	—	4	8	no	—
64	C₇×C₄⋊C₄	—	4	8	no	—
65	C₇×C₂²⋊C₄	—	4	8	no	—
66	C₄²×C₇	—	4	8	no	—
67	C₄²×C₇	C₄⋊C₄	1	8	yes	—
68	C₄²×C₇	C₂²×C₄	1	4	yes	—
69	C₄²×C₇	C₂×Q₈	1	4	yes	—
70	C₂³×C₈	C₂²×C₇	1	2	yes	—
71	C₂×#117	—	2	2	no	—
72	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
73	C₂×#107	C₂²×C₇	1	8	yes	—
74	C₂×C₄⋊C₈	C₄×C₇	1	8	yes	complemented
75	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
76	#384	C₂²×C₇	1	4	yes	—
77	#382	—	2	2	no	—
78	#380	—	2	2	no	—
79	#380	C₂²×C₇	1	4	yes	—
80	C₄×C₄⋊C₄	—	2	2	no	—
81	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
82	#340	C₂²×C₇	1	8	yes	—
83	C₂³×C₇	—	2	2	no	—
84	C₂³×C₇	—	2	2	no	—
85	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
86	C₂³×C₇	C₂×D₈	1	2	yes	—
87	C₂³×C₇	C₂×Q₁₆	1	2	yes	—
88	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	2	4	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	—	2	4	no	—
94	C₂×C₄×C₇	—	4	4	no	—
95	C₂×C₄×C₇	—	4	8	no	—
96	C₂×C₄×C₇	—	4	8	no	—
97	C₂×C₄×C₇	—	4	8	no	—
98	C₂×C₄×C₇	#107	1	16	yes	complemented
99	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
101	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
103	C₂×C₄×C₇	C₂×Q₁₆	1	4	yes	—
104	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
107	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
108	C₇×C₈	—	2	8	no	—
109	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
110	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
111	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
112	C₂²×C₈	—	4	4	no	—
113	C₂²×C₈	C₂×C₄×C₇	1	8	yes	—
114	C₂²×C₈	C₂³×C₇	1	4	yes	—
115	#117	—	4	8	no	—
116	C₂×C₄⋊C₄	—	2	4	no	—
117	C₂×C₄⋊C₄	—	2	4	no	—
118	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
119	C₂×C₂²⋊C₄	—	2	4	no	—
120	C₂×C₄²	—	2	4	no	—
121	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
122	C₂×C₄²	C₇×D₄	1	2	yes	—
123	C₂×C₄²	C₇×Q₈	1	4	yes	—
124	C₂×C₄²	C₂³×C₇	1	2	yes	—
125	#107	—	4	8	no	—
126	C₄⋊C₈	C₂×C₄×C₇	1	16	yes	—
127	#95	—	2	4	no	—
128	#95	—	2	4	no	—
129	C₂²×C₇	—	2	2	no	—
130	C₂²×C₇	—	2	2	no	—
131	C₂²×C₇	—	2	4	no	—
132	C₂²×C₇	—	2	4	no	—
133	C₂²×C₇	C₂×#107	1	4	yes	—
134	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₇	#537	1	2	yes	—
136	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
137	C₂²×C₇	C₂²×Q₁₆	1	1	yes	characteristic
138	C₂²×C₇	#575	1	2	yes	—
139	C₄×C₇	—	2	2	no	—
140	C₄×C₇	—	2	4	no	—
141	C₄×C₇	—	8	8	no	—
142	C₄×C₇	C₂×#107	1	8	yes	—
143	C₄×C₇	#518	1	1	yes	characteristic
144	C₄×C₇	#519	1	1	yes	characteristic
145	C₄×C₇	#544	1	2	yes	—
146	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
152	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
153	C₂²×C₄	C₇×D₈	1	4	yes	—
154	C₂²×C₄	C₇×Q₁₆	1	4	yes	—
155	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
156	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
158	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
159	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
160	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
161	C₂×C₈	—	2	8	no	—
162	C₂×C₈	—	4	8	no	—
163	C₂×C₈	C₂²×C₄×C₇	1	8	yes	—
164	C₄⋊C₄	—	2	8	no	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	—	4	8	no	—
167	C₄⋊C₄	—	4	8	no	—
168	C₂²⋊C₄	—	4	8	no	—
169	C₄²	—	4	8	no	—
170	C₄²	C₇×C₄⋊C₄	1	8	yes	—
171	C₄²	C₂²×C₄×C₇	1	4	yes	—
172	C₄²	C₂×C₇×Q₈	1	4	yes	—
173	C₂×C₇	—	2	4	no	—
174	C₂×C₇	#2316	1	1	yes	characteristic
175	C₂×C₇	C₂²×#107	1	1	yes	characteristic
176	C₂×C₇	#2911	1	1	yes	characteristic
177	C₂×C₇	#3245	1	2	yes	—
178	C₂×C₇	#3249	1	2	yes	—
179	C₂³	—	2	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
182	C₂³	C₂×C₇×D₈	1	2	yes	—
183	C₂³	C₂×C₇×Q₁₆	1	2	yes	—
184	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	4	8	no	—
192	C₂×C₄	—	4	8	no	—
193	C₂×C₄	—	4	8	no	—
194	C₂×C₄	C₇×#107	1	16	yes	complemented
195	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
196	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
197	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
198	C₂×C₄	C₂×C₇×D₈	1	4	yes	—
199	C₂×C₄	C₂×C₇×Q₁₆	1	4	yes	—
200	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
201	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
202	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
203	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
204	C₈	—	2	8	no	—
205	C₇	#21961	1	1	yes	characteristic, direct product factor, Sylow
206	C₂²	—	2	2	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	2	no	—
210	C₂²	C₂×C₇×#107	1	4	yes	—
211	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
212	C₂²	C₇×#537	1	2	yes	—
213	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
214	C₂²	C₂²×C₇×Q₁₆	1	1	yes	characteristic
215	C₂²	C₇×#575	1	2	yes	—
216	C₄	—	2	4	no	—
217	C₄	—	2	2	no	—
218	C₄	—	8	8	no	—
219	C₄	C₂×C₇×#107	1	8	yes	—
220	C₄	C₇×#518	1	1	yes	characteristic
221	C₄	C₇×#519	1	1	yes	characteristic
222	C₄	C₇×#544	1	2	yes	—
223	C₂	—	2	4	no	—
224	C₂	C₇×#2316	1	1	yes	characteristic
225	C₂	C₂²×C₇×#107	1	1	yes	characteristic
226	C₂	C₇×#2911	1	1	yes	characteristic
227	C₂	C₇×#3245	1	2	yes	—
228	C₂	C₇×#3249	1	2	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×#1947	C₂	semidirect
9. C₂×C₇×C₄⋊C₈	C₄	semidirect
18. #21961	C₇	direct (N × H)
41. #1947	C₂×C₇	semidirect
74. C₂×C₄⋊C₈	C₄×C₇	semidirect
98. C₂×C₄×C₇	#107	semidirect
194. C₂×C₄	C₇×#107	semidirect
205. C₇	#21961	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×#2285	2
C₇×#1947	2
C₇×#1808	2
#21961	7

Subgroups — C7×#21961 #13199177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21961 #13199177