Subgroups — C₇×#21995 #13199211

Order 1792.

658 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2921	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#433	C₂	1	1	yes	maximal, characteristic
3	C₇×#2905	C₂	1	1	yes	maximal, characteristic
4	C₇×#1950	C₂	1	4	yes	maximal, complemented
5	C₇×#1855	C₂	1	4	yes	maximal
6	C₇×#1849	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1838	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1837	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
11	C₂×C₇×#117	C₂²	1	1	yes	characteristic
12	C₇×#433	—	2	4	no	—
13	C₇×#433	C₄	1	8	yes	—
14	C₂×C₇×#108	C₂²	1	4	yes	—
15	C₂×C₇×#108	C₂²	1	4	yes	—
16	C₇×#424	—	2	4	no	—
17	C₇×#423	C₂²	1	1	yes	characteristic
18	C₇×#423	C₂²	1	1	yes	characteristic
19	C₇×#423	C₂²	1	2	yes	—
20	C₇×#423	C₂²	1	1	yes	characteristic
21	C₇×#423	C₂²	1	1	yes	characteristic
22	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
23	C₇×#407	C₂²	1	2	yes	—
24	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
25	C₄×C₇×M₁₆	C₂²	1	2	yes	—
26	C₂×C₇×#97	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
28	C₇×#341	C₂²	1	4	yes	—
29	C₇×#341	C₂²	1	4	yes	—
30	#21995	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
32	C₂×C₇×M₁₆	—	2	2	no	—
33	C₂×C₇×M₁₆	—	4	4	no	—
34	C₂×C₇×M₁₆	C₂³	1	2	yes	—
35	C₂×C₇×M₁₆	C₂³	1	2	yes	—
36	C₂×C₇×M₁₆	C₂³	1	2	yes	—
37	C₂×C₇×M₁₆	C₂³	1	2	yes	—
38	C₂²×C₇×C₈	—	2	2	no	—
39	C₂²×C₇×C₈	C₂³	1	2	yes	—
40	C₂²×C₇×C₈	C₂³	1	2	yes	—
41	C₇×#117	C₂×C₄	1	4	yes	—
42	C₇×#117	D₄	1	4	yes	—
43	C₂×C₇×C₄⋊C₄	Q₈	1	2	yes	—
44	C₂×C₇×C₂²⋊C₄	Q₈	1	2	yes	—
45	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
47	C₇×#108	—	4	4	no	—
48	C₇×#108	—	4	4	no	—
49	C₇×C₄⋊C₈	—	2	4	no	—
50	C₇×C₄⋊C₈	—	2	2	no	—
51	C₇×C₄⋊C₈	—	2	2	no	—
52	C₇×C₄⋊C₈	—	2	2	no	—
53	C₇×C₄⋊C₈	—	2	2	no	—
54	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
55	C₇×C₂²⋊C₈	—	2	4	no	—
56	C₇×C₂²⋊C₈	C₂×C₄	1	8	yes	—
57	C₇×#97	—	2	2	no	—
58	C₇×#97	C₂×C₄	1	4	yes	—
59	C₄×C₇×C₈	—	2	2	no	—
60	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
61	#2921	C₂×C₇	1	1	yes	characteristic
62	C₂×#433	C₂×C₇	1	1	yes	characteristic
63	#2905	C₂×C₇	1	1	yes	characteristic
64	#1950	C₂×C₇	1	4	yes	complemented
65	#1855	C₂×C₇	1	4	yes	—
66	#1849	C₂×C₇	1	1	yes	characteristic, complemented
67	#1848	C₂×C₇	1	1	yes	characteristic, complemented
68	#1838	C₂×C₇	1	1	yes	characteristic, complemented
69	#1837	C₂×C₇	1	1	yes	characteristic, complemented
70	C₂⁴×C₇	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₇	—	2	2	no	—
72	C₂²×C₄×C₇	—	2	2	no	—
73	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
79	C₇×M₁₆	—	2	4	no	—
80	C₇×M₁₆	—	2	4	no	—
81	C₇×M₁₆	—	2	4	no	—
82	C₇×M₁₆	—	2	4	no	—
83	C₇×M₁₆	—	4	4	no	—
84	C₂×C₇×C₈	—	2	2	no	—
85	C₂×C₇×C₈	—	2	2	no	—
86	C₂×C₇×C₈	—	2	2	no	—
87	C₂×C₇×C₈	—	2	2	no	—
88	C₂×C₇×C₈	—	2	2	no	—
89	C₂×C₇×C₈	—	2	2	no	—
90	C₂×C₇×C₈	—	2	2	no	—
91	C₂×C₇×C₈	—	2	2	no	—
92	C₂×C₇×C₈	—	4	4	no	—
93	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
94	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
95	C₇×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
96	C₇×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
97	C₄²×C₇	C₂²×C₄	1	2	yes	—
98	C₄²×C₇	C₂²×C₄	1	2	yes	—
99	C₄²×C₇	C₂×D₄	1	2	yes	—
100	C₄²×C₇	C₂×D₄	1	2	yes	—
101	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
102	C₂×#117	C₂²×C₇	1	1	yes	characteristic
103	#433	—	2	4	no	—
104	#433	C₄×C₇	1	8	yes	—
105	C₂×#108	C₂²×C₇	1	4	yes	—
106	C₂×#108	C₂²×C₇	1	4	yes	—
107	#424	—	2	4	no	—
108	#423	C₂²×C₇	1	1	yes	characteristic
109	#423	C₂²×C₇	1	1	yes	characteristic
110	#423	C₂²×C₇	1	1	yes	characteristic
111	#423	C₂²×C₇	1	1	yes	characteristic
112	#423	C₂²×C₇	1	2	yes	—
113	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
114	#407	C₂²×C₇	1	2	yes	—
115	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
116	C₄×M₁₆	C₂²×C₇	1	2	yes	—
117	C₂×#97	C₂²×C₇	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
119	#341	C₂²×C₇	1	4	yes	—
120	#341	C₂²×C₇	1	4	yes	—
121	C₂³×C₇	—	2	2	no	—
122	C₂³×C₇	—	4	2	no	—
123	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
124	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₇	—	2	2	no	—
126	C₂×C₄×C₇	—	2	4	no	—
127	C₂×C₄×C₇	—	2	2	no	—
128	C₂×C₄×C₇	—	2	2	no	—
129	C₂×C₄×C₇	—	4	2	no	—
130	C₂×C₄×C₇	—	4	2	no	—
131	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
132	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
133	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
137	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
138	C₇×C₈	—	2	4	no	—
139	C₇×C₈	—	2	4	no	—
140	C₇×C₈	—	4	2	no	—
141	C₇×C₈	—	4	2	no	—
142	C₇×C₈	—	4	2	no	—
143	C₇×C₈	—	4	2	no	—
144	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	—	4	4	no	—
147	C₂×M₁₆	C₂³×C₇	1	2	yes	—
148	C₂×M₁₆	C₂³×C₇	1	2	yes	—
149	C₂×M₁₆	C₂³×C₇	1	2	yes	—
150	C₂×M₁₆	C₂³×C₇	1	2	yes	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	C₂³×C₇	1	2	yes	—
153	C₂²×C₈	C₂³×C₇	1	2	yes	—
154	#117	C₂×C₄×C₇	1	4	yes	—
155	#117	C₇×D₄	1	4	yes	—
156	C₂×C₄⋊C₄	C₇×Q₈	1	2	yes	—
157	C₂×C₂²⋊C₄	C₇×Q₈	1	2	yes	—
158	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
159	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
160	#108	—	4	4	no	—
161	#108	—	4	4	no	—
162	C₄⋊C₈	—	2	4	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	—
168	C₂²⋊C₈	—	2	4	no	—
169	C₂²⋊C₈	C₂×C₄×C₇	1	8	yes	—
170	#97	—	2	2	no	—
171	#97	C₂×C₄×C₇	1	4	yes	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
174	C₂²×C₇	—	2	2	no	—
175	C₂²×C₇	—	2	2	no	—
176	C₂²×C₇	—	2	4	no	—
177	C₂²×C₇	—	8	2	no	—
178	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
179	C₂²×C₇	#544	1	2	yes	—
180	C₄×C₇	—	2	2	no	—
181	C₄×C₇	—	2	4	no	—
182	C₄×C₇	—	2	4	no	—
183	C₄×C₇	#520	1	2	yes	—
184	C₄×C₇	#535	1	1	yes	characteristic
185	C₄×C₇	#536	1	1	yes	characteristic
186	C₂⁴	C₂×C₇×Q₈	1	1	yes	characteristic
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
190	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
191	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
193	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
194	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
195	M₁₆	—	2	4	no	—
196	M₁₆	—	2	4	no	—
197	M₁₆	—	2	4	no	—
198	M₁₆	—	2	4	no	—
199	M₁₆	—	4	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	4	4	no	—
209	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
210	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
211	C₄⋊C₄	C₇×C₄⋊C₄	1	8	yes	—
212	C₂²⋊C₄	C₇×C₄⋊C₄	1	8	yes	—
213	C₄²	C₂²×C₄×C₇	1	2	yes	—
214	C₄²	C₂²×C₄×C₇	1	2	yes	—
215	C₄²	C₂×C₇×D₄	1	2	yes	—
216	C₄²	C₂×C₇×D₄	1	2	yes	—
217	C₂×C₇	—	2	2	no	—
218	C₂×C₇	—	4	2	no	—
219	C₂×C₇	#2317	1	1	yes	characteristic
220	C₂×C₇	#2916	1	2	yes	—
221	C₂³	—	2	2	no	—
222	C₂³	—	4	2	no	—
223	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
224	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
232	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
233	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
234	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
236	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
237	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
238	C₈	—	2	4	no	—
239	C₈	—	2	4	no	—
240	C₈	—	4	2	no	—
241	C₈	—	4	2	no	—
242	C₈	—	4	2	no	—
243	C₈	—	4	2	no	—
244	C₇	#21995	1	1	yes	characteristic, direct product factor, Sylow
245	C₂²	—	2	2	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
250	C₂²	C₇×#544	1	2	yes	—
251	C₄	—	2	4	no	—
252	C₄	—	2	4	no	—
253	C₄	—	2	2	no	—
254	C₄	C₇×#520	1	2	yes	—
255	C₄	C₇×#535	1	1	yes	characteristic
256	C₄	C₇×#536	1	1	yes	characteristic
257	C₂	—	2	2	no	—
258	C₂	—	4	2	no	—
259	C₂	C₇×#2317	1	1	yes	characteristic
260	C₂	C₇×#2916	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#117 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1950	C₂	semidirect
6. C₇×#1849	C₂	semidirect
7. C₇×#1848	C₂	semidirect
8. C₇×#1838	C₂	semidirect
9. C₇×#1837	C₂	semidirect
30. #21995	C₇	direct (N × H)
64. #1950	C₂×C₇	semidirect
66. #1849	C₂×C₇	semidirect
67. #1848	C₂×C₇	semidirect
68. #1838	C₂×C₇	semidirect
69. #1837	C₂×C₇	semidirect
244. C₇	#21995	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×#2921	2
C₂×C₇×#433	2
C₇×#2905	2
C₇×#1950	2
C₇×#1855	2
C₇×#1849	2
C₇×#1848	2
C₇×#1838	2
C₇×#1837	2
#21995	7

Subgroups — C7×#21995 #13199211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21995 #13199211