Subgroups — C₇×#21965 #13199181

Order 1792.

786 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#423	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×#1809	C₂	1	4	yes	maximal
5	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
6	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₇×#107	C₂²	1	4	yes	—
8	C₂×C₇×#106	C₂²	1	4	yes	—
9	C₇×#423	C₄	1	8	yes	complemented
10	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
11	C₇×#383	—	2	2	no	—
12	C₇×#382	C₂²	1	4	yes	—
13	C₇×#381	—	2	2	no	—
14	C₇×#380	C₂²	1	4	yes	—
15	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
16	C₄×C₇×C₂²⋊C₄	—	2	2	no	—
17	C₇×#376	—	2	2	no	—
18	C₇×#340	C₂²	1	8	yes	—
19	#21965	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₇	D₄	1	2	yes	—
21	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂×C₇×M₁₆	—	4	2	no	—
23	C₂×C₇×M₁₆	—	4	2	no	—
24	C₂×C₇×M₁₆	C₂×C₄	1	8	yes	—
25	C₂²×C₇×C₈	C₂³	1	4	yes	—
26	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
27	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
28	C₂×C₄²×C₇	—	2	4	no	—
29	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇	Q₈	1	4	yes	—
33	C₂×C₄²×C₇	C₂³	1	2	yes	—
34	C₇×#107	—	4	4	no	—
35	C₇×#106	—	4	4	no	—
36	C₇×C₄⋊C₈	C₂×C₄	1	16	yes	—
37	C₇×#95	—	2	4	no	—
38	C₇×#95	—	2	4	no	—
39	C₇×#95	—	2	4	no	—
40	C₇×#95	—	2	4	no	—
41	C₂×#423	C₂×C₇	1	1	yes	characteristic
42	#2283	C₂×C₇	1	2	yes	—
43	#1949	C₂×C₇	1	8	yes	complemented
44	#1809	C₂×C₇	1	4	yes	—
45	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	—	2	2	no	—
47	C₂²×C₄×C₇	—	2	2	no	—
48	C₂²×C₄×C₇	—	2	8	no	—
49	C₂²×C₄×C₇	—	4	4	no	—
50	C₂²×C₄×C₇	C₄⋊C₄	1	8	yes	—
51	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
57	C₇×M₁₆	—	4	8	no	—
58	C₂×C₇×C₈	—	2	4	no	—
59	C₂×C₇×C₈	—	2	4	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×C₂²⋊C₄	—	4	8	no	—
64	C₄²×C₇	—	4	8	no	—
65	C₄²×C₇	C₄⋊C₄	1	4	yes	—
66	C₄²×C₇	C₄⋊C₄	1	4	yes	—
67	C₄²×C₇	C₂²×C₄	1	4	yes	—
68	C₄²×C₇	C₂×Q₈	1	4	yes	—
69	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
70	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
71	C₂×#107	C₂²×C₇	1	4	yes	—
72	C₂×#106	C₂²×C₇	1	4	yes	—
73	#423	C₄×C₇	1	8	yes	complemented
74	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
75	#383	—	2	2	no	—
76	#382	C₂²×C₇	1	4	yes	—
77	#381	—	2	2	no	—
78	#380	C₂²×C₇	1	4	yes	—
79	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
80	C₄×C₂²⋊C₄	—	2	2	no	—
81	#376	—	2	2	no	—
82	#340	C₂²×C₇	1	8	yes	—
83	C₂³×C₇	—	2	2	no	—
84	C₂³×C₇	—	2	4	no	—
85	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
86	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	2	4	no	—
89	C₂×C₄×C₇	—	2	4	no	—
90	C₂×C₄×C₇	—	2	2	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	—	2	4	no	—
93	C₂×C₄×C₇	—	4	8	no	—
94	C₂×C₄×C₇	—	4	8	no	—
95	C₂×C₄×C₇	—	4	8	no	—
96	C₂×C₄×C₇	—	4	8	no	—
97	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
103	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
104	C₇×C₈	—	2	8	no	—
105	C₂³×C₄	C₇×D₄	1	2	yes	—
106	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
107	C₂×M₁₆	—	4	2	no	—
108	C₂×M₁₆	—	4	2	no	—
109	C₂×M₁₆	C₂×C₄×C₇	1	8	yes	—
110	C₂²×C₈	C₂³×C₇	1	4	yes	—
111	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
112	C₂×C₂²⋊C₄	—	2	4	no	—
113	C₂×C₄²	—	2	4	no	—
114	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
115	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
116	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
117	C₂×C₄²	C₇×Q₈	1	4	yes	—
118	C₂×C₄²	C₂³×C₇	1	2	yes	—
119	#107	—	4	4	no	—
120	#106	—	4	4	no	—
121	C₄⋊C₈	C₂×C₄×C₇	1	16	yes	—
122	#95	—	2	4	no	—
123	#95	—	2	4	no	—
124	#95	—	2	4	no	—
125	#95	—	2	4	no	—
126	C₂²×C₇	—	2	2	no	—
127	C₂²×C₇	—	2	4	no	—
128	C₂²×C₇	—	4	4	no	—
129	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₇	#537	1	2	yes	—
131	C₂²×C₇	#575	1	4	yes	—
132	C₄×C₇	—	2	2	no	—
133	C₄×C₇	—	2	4	no	—
134	C₄×C₇	—	2	4	no	—
135	C₄×C₇	—	8	8	no	—
136	C₄×C₇	#518	1	1	yes	characteristic
137	C₄×C₇	#519	1	1	yes	characteristic
138	C₄×C₇	#544	1	2	yes	—
139	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	2	8	no	—
143	C₂²×C₄	—	4	4	no	—
144	C₂²×C₄	C₇×C₄⋊C₄	1	8	yes	—
145	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
146	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
148	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
149	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
150	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
151	M₁₆	—	4	8	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	C₂²×C₄×C₇	1	8	yes	—
155	C₄⋊C₄	—	2	8	no	—
156	C₄⋊C₄	—	2	8	no	—
157	C₂²⋊C₄	—	4	8	no	—
158	C₄²	—	4	8	no	—
159	C₄²	C₇×C₄⋊C₄	1	4	yes	—
160	C₄²	C₇×C₄⋊C₄	1	4	yes	—
161	C₄²	C₂²×C₄×C₇	1	4	yes	—
162	C₄²	C₂×C₇×Q₈	1	4	yes	—
163	C₂×C₇	—	4	2	no	—
164	C₂×C₇	#2316	1	1	yes	characteristic
165	C₂×C₇	#2911	1	2	yes	—
166	C₂×C₇	#3256	1	4	yes	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	4	no	—
169	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
170	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	4	8	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	8	no	—
180	C₂×C₄	—	4	8	no	—
181	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
182	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
183	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
184	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
185	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
186	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
187	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
188	C₈	—	2	8	no	—
189	C₇	#21965	1	1	yes	characteristic, direct product factor, Sylow
190	C₂²	—	2	4	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	4	4	no	—
193	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
194	C₂²	C₇×#537	1	2	yes	—
195	C₂²	C₇×#575	1	4	yes	—
196	C₄	—	2	4	no	—
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	—	8	8	no	—
200	C₄	C₇×#518	1	1	yes	characteristic
201	C₄	C₇×#519	1	1	yes	characteristic
202	C₄	C₇×#544	1	2	yes	—
203	C₂	—	4	2	no	—
204	C₂	C₇×#2316	1	1	yes	characteristic
205	C₂	C₇×#2911	1	2	yes	—
206	C₂	C₇×#3256	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
9. C₇×#423	C₄	semidirect
19. #21965	C₇	direct (N × H)
43. #1949	C₂×C₇	semidirect
73. #423	C₄×C₇	semidirect
189. C₇	#21965	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×C₇×#423	2
C₇×#2283	2
C₇×#1949	2
C₇×#1809	2
#21965	7

Subgroups — C7×#21965 #13199181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21965 #13199181