Subgroups — C₇×#21923 #13199139

Order 1792.

668 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2298	C₂	1	1	yes	maximal, characteristic
2	C₇×#1923	C₂	1	4	yes	maximal
3	C₇×#1916	C₂	1	4	yes	maximal, complemented
4	C₇×#1843	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1832	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1798	C₂	1	2	yes	maximal
7	C₄×C₇×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₂×C₇×#117	C₂²	1	1	yes	characteristic
9	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₇×C₂²⋊C₈	C₂²	1	4	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	4	yes	—
14	C₇×#388	C₄	1	4	yes	—
15	C₇×#387	C₄	1	4	yes	—
16	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
17	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₇×#376	C₂²	1	1	yes	characteristic
19	C₄³×C₇	C₂²	1	1	yes	characteristic
20	C₇×#336	C₂²	1	4	yes	—
21	C₇×#336	C₂²	1	4	yes	—
22	C₇×#336	C₂²	1	4	yes	—
23	#21923	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂²×C₇×C₈	C₂³	1	4	yes	—
26	C₂²×C₇×C₈	C₂³	1	4	yes	—
27	C₇×#117	—	2	2	no	—
28	C₇×#117	—	2	2	no	—
29	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	4	yes	—
32	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₇	C₂³	1	2	yes	—
36	C₂×C₄²×C₇	C₂³	1	2	yes	—
37	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
38	C₇×C₄⋊C₈	—	2	4	no	—
39	C₇×C₄⋊C₈	—	2	4	no	—
40	C₇×C₄⋊C₈	—	4	4	no	—
41	C₇×C₂²⋊C₈	—	4	4	no	—
42	C₄×C₇×C₈	—	4	4	no	—
43	C₇×#95	C₂×C₄	1	4	yes	—
44	C₇×#95	C₂×C₄	1	4	yes	—
45	C₇×#95	C₂×C₄	1	4	yes	—
46	#2298	C₂×C₇	1	1	yes	characteristic
47	#1923	C₂×C₇	1	4	yes	—
48	#1916	C₂×C₇	1	4	yes	complemented
49	#1843	C₂×C₇	1	1	yes	characteristic, complemented
50	#1832	C₂×C₇	1	1	yes	characteristic, complemented
51	#1798	C₂×C₇	1	2	yes	—
52	C₄×C₄⋊C₈	C₂×C₇	1	2	yes	complemented
53	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	—	2	1	no	—
55	C₂²×C₄×C₇	—	2	1	no	—
56	C₂²×C₄×C₇	—	2	2	no	—
57	C₂²×C₄×C₇	—	2	2	no	—
58	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
64	C₂×C₇×C₈	—	2	4	no	—
65	C₂×C₇×C₈	—	2	4	no	—
66	C₂×C₇×C₈	—	2	4	no	—
67	C₂×C₇×C₈	—	2	4	no	—
68	C₂×C₇×C₈	—	2	4	no	—
69	C₂×C₇×C₈	—	2	4	no	—
70	C₇×C₄⋊C₄	—	2	4	no	—
71	C₇×C₄⋊C₄	—	2	8	no	—
72	C₇×C₂²⋊C₄	—	2	4	no	—
73	C₇×C₂²⋊C₄	—	2	8	no	—
74	C₄²×C₇	—	2	8	no	—
75	C₄²×C₇	—	2	1	no	—
76	C₄²×C₇	—	2	1	no	—
77	C₄²×C₇	—	2	2	no	—
78	C₄²×C₇	—	2	2	no	—
79	C₄²×C₇	—	2	2	no	—
80	C₄²×C₇	—	2	2	no	—
81	C₄²×C₇	#41	1	2	yes	—
82	C₄²×C₇	#41	1	2	yes	—
83	C₂×#117	C₂²×C₇	1	1	yes	characteristic
84	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
85	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
86	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
87	C₂×C₂²⋊C₈	C₂²×C₇	1	4	yes	—
88	C₂×C₄×C₈	C₂²×C₇	1	4	yes	—
89	#388	C₄×C₇	1	4	yes	—
90	#387	C₄×C₇	1	4	yes	—
91	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
92	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
93	#376	C₂²×C₇	1	1	yes	characteristic
94	C₄³	C₂²×C₇	1	1	yes	characteristic
95	#336	C₂²×C₇	1	4	yes	—
96	#336	C₂²×C₇	1	4	yes	—
97	#336	C₂²×C₇	1	4	yes	—
98	C₂³×C₇	—	2	1	no	—
99	C₂³×C₇	—	2	2	no	—
100	C₂³×C₇	—	2	4	no	—
101	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₇	—	2	4	no	—
103	C₂×C₄×C₇	—	2	4	no	—
104	C₂×C₄×C₇	—	2	4	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	—	2	4	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	4	no	—
112	C₂×C₄×C₇	—	4	1	no	—
113	C₂×C₄×C₇	—	4	1	no	—
114	C₂×C₄×C₇	—	4	2	no	—
115	C₂×C₄×C₇	#117	1	4	yes	—
116	C₂×C₄×C₇	#131	1	8	yes	—
117	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
120	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
121	C₇×C₈	—	4	4	no	—
122	C₇×C₈	—	4	4	no	—
123	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
124	C₂²×C₈	C₂³×C₇	1	4	yes	—
125	C₂²×C₈	C₂³×C₇	1	4	yes	—
126	#117	—	2	2	no	—
127	#117	—	2	2	no	—
128	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
129	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
130	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
131	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	4	yes	—
132	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
133	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
134	C₂×C₄²	C₂³×C₇	1	2	yes	—
135	C₂×C₄²	C₂³×C₇	1	2	yes	—
136	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
137	C₄⋊C₈	—	2	4	no	—
138	C₄⋊C₈	—	2	4	no	—
139	C₄⋊C₈	—	4	4	no	—
140	C₂²⋊C₈	—	4	4	no	—
141	C₄×C₈	—	4	4	no	—
142	#95	C₂×C₄×C₇	1	4	yes	—
143	#95	C₂×C₄×C₇	1	4	yes	—
144	#95	C₂×C₄×C₇	1	4	yes	—
145	C₂²×C₇	—	4	1	no	—
146	C₂²×C₇	—	4	2	no	—
147	C₂²×C₇	—	4	4	no	—
148	C₂²×C₇	C₂×#117	1	1	yes	characteristic
149	C₂²×C₇	#520	1	2	yes	—
150	C₂²×C₇	C₂×#131	1	4	yes	—
151	C₄×C₇	—	2	4	no	—
152	C₄×C₇	—	2	8	no	—
153	C₄×C₇	—	4	1	no	—
154	C₄×C₇	—	4	2	no	—
155	C₄×C₇	—	4	4	no	—
156	C₄×C₇	#540	1	1	yes	characteristic
157	C₄×C₇	#541	1	1	yes	characteristic
158	C₄×C₇	#542	1	2	yes	—
159	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
165	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
166	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
167	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
168	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
169	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	2	4	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	8	no	—
178	C₂²⋊C₄	—	2	4	no	—
179	C₂²⋊C₄	—	2	8	no	—
180	C₄²	—	2	8	no	—
181	C₄²	—	2	1	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	C₇×#41	1	2	yes	—
188	C₄²	C₇×#41	1	2	yes	—
189	C₂×C₇	—	8	1	no	—
190	C₂×C₇	#2314	1	1	yes	characteristic
191	C₂×C₇	#2922	1	2	yes	—
192	C₂×C₇	#2994	1	4	yes	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	4	no	—
196	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	C₇×#117	1	4	yes	—
211	C₂×C₄	C₇×#131	1	8	yes	—
212	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
213	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
214	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
215	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
216	C₈	—	4	4	no	—
217	C₈	—	4	4	no	—
218	C₇	#21923	1	1	yes	characteristic, direct product factor, Sylow
219	C₂²	—	4	2	no	—
220	C₂²	—	4	4	no	—
221	C₂²	—	4	1	no	—
222	C₂²	C₂×C₇×#117	1	1	yes	characteristic
223	C₂²	C₇×#520	1	2	yes	—
224	C₂²	C₂×C₇×#131	1	4	yes	—
225	C₄	—	2	4	no	—
226	C₄	—	2	8	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	4	no	—
229	C₄	—	4	1	no	—
230	C₄	C₇×#540	1	1	yes	characteristic
231	C₄	C₇×#541	1	1	yes	characteristic
232	C₄	C₇×#542	1	2	yes	—
233	C₂	—	8	1	no	—
234	C₂	C₇×#2314	1	1	yes	characteristic
235	C₂	C₇×#2922	1	2	yes	—
236	C₂	C₇×#2994	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₇×#1916	C₂	semidirect
4. C₇×#1843	C₂	semidirect
5. C₇×#1832	C₂	semidirect
7. C₄×C₇×C₄⋊C₈	C₂	semidirect
23. #21923	C₇	direct (N × H)
48. #1916	C₂×C₇	semidirect
49. #1843	C₂×C₇	semidirect
50. #1832	C₂×C₇	semidirect
52. C₄×C₄⋊C₈	C₂×C₇	semidirect
218. C₇	#21923	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#2298	2
C₇×#1923	2
C₇×#1916	2
C₇×#1843	2
C₇×#1832	2
C₇×#1798	2
C₄×C₇×C₄⋊C₈	2
#21923	7

Subgroups — C7×#21923 #13199139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21923 #13199139