Subgroups — C₇×#32870 #13210086

Order 1792.

502 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2103	C₂	1	2	yes	maximal, complemented
2	C₇×#2098	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2081	C₂	1	2	yes	maximal, complemented
4	C₇×#2069	C₂	1	2	yes	maximal, complemented
5	C₇×#2048	C₂	1	2	yes	maximal, complemented
6	C₇×#2045	C₂	1	2	yes	maximal, complemented
7	C₇×#2018	C₂	1	1	yes	maximal, characteristic
8	C₇×#1903	C₂	1	2	yes	maximal, complemented
9	C₇×#1858	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	2	yes	—
11	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
12	C₇×#568	C₂²	1	1	yes	characteristic
13	C₇×#430	C₂²	1	2	yes	—
14	C₇×#430	C₂²	1	1	yes	characteristic
15	C₇×#430	C₂²	1	2	yes	—
16	C₇×#424	C₂²	1	2	yes	—
17	C₇×#421	C₂²	1	2	yes	—
18	C₇×#418	C₂²	1	2	yes	—
19	C₇×#418	C₂²	1	2	yes	—
20	C₇×#413	C₂²	1	1	yes	characteristic
21	C₂×C₇×#100	C₂²	1	2	yes	—
22	C₂×C₇×#100	C₂²	1	2	yes	—
23	C₂×C₇×#100	C₂²	1	2	yes	—
24	C₇×#408	C₂²	1	1	yes	characteristic
25	C₇×#344	C₂²	1	2	yes	—
26	C₇×#344	C₂²	1	2	yes	—
27	C₇×#341	C₂²	1	1	yes	characteristic
28	C₇×#341	C₂²	1	2	yes	—
29	C₇×#338	C₂²	1	2	yes	—
30	C₇×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
31	#32870	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂×C₇×#41	C₂³	1	1	yes	characteristic
33	C₂²×C₇×D₄	C₂³	1	2	yes	—
34	C₇×AGL(1,ℤ/8)	—	4	2	no	—
35	C₇×AGL(1,ℤ/8)	—	4	2	no	—
36	C₂×C₇×SD₁₆	—	2	2	no	—
37	C₂×C₇×D₈	—	2	2	no	—
38	C₂×C₇×D₈	—	4	1	no	—
39	C₂×C₇×D₈	—	4	2	no	—
40	C₇×#131	—	4	2	no	—
41	C₂×C₇×M₁₆	—	4	1	no	—
42	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
43	C₂×C₇×M₁₆	C₂³	1	2	yes	—
44	C₂×C₇×M₁₆	C₂³	1	2	yes	—
45	C₂×C₇×M₁₆	C₂³	1	2	yes	—
46	C₂×C₇×M₁₆	C₂³	1	2	yes	—
47	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
48	C₇×#117	C₂³	1	2	yes	—
49	C₇×#108	—	4	1	no	—
50	C₇×#108	—	4	2	no	—
51	C₇×#108	—	4	2	no	—
52	C₇×C₄⋊C₈	—	4	2	no	—
53	C₇×#104	—	4	2	no	—
54	C₇×#102	—	4	2	no	—
55	C₇×#102	—	4	2	no	—
56	C₇×#101	—	2	1	no	—
57	C₇×#100	—	2	1	no	—
58	C₇×#100	—	4	2	no	—
59	C₇×#100	—	4	2	no	—
60	C₇×#100	—	4	2	no	—
61	C₇×C₂²⋊C₈	—	4	1	no	—
62	#2103	C₂×C₇	1	2	yes	complemented
63	#2098	C₂×C₇	1	1	yes	characteristic, complemented
64	#2081	C₂×C₇	1	2	yes	complemented
65	#2069	C₂×C₇	1	2	yes	complemented
66	#2048	C₂×C₇	1	2	yes	complemented
67	#2045	C₂×C₇	1	2	yes	complemented
68	#2018	C₂×C₇	1	1	yes	characteristic
69	#1903	C₂×C₇	1	2	yes	complemented
70	#1858	C₂×C₇	1	1	yes	characteristic, complemented
71	C₂⁴×C₇	—	2	2	no	—
72	C₇×#41	—	2	2	no	—
73	C₇×#41	—	2	2	no	—
74	C₂×C₇×Q₈	#41	1	1	yes	characteristic
75	C₂×C₇×D₄	—	2	1	no	—
76	C₂×C₇×D₄	—	2	2	no	—
77	C₂×C₇×D₄	—	2	2	no	—
78	C₂×C₇×D₄	—	2	2	no	—
79	C₂×C₇×D₄	—	2	2	no	—
80	C₂×C₇×D₄	—	2	2	no	—
81	C₂×C₇×D₄	—	2	2	no	—
82	C₂×C₇×D₄	#41	1	1	yes	characteristic
83	C₂²×C₄×C₇	—	2	1	no	—
84	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
85	C₇×SD₁₆	—	4	4	no	—
86	C₇×D₈	—	4	4	no	—
87	C₇×D₈	—	4	4	no	—
88	C₇×M₁₆	—	2	2	no	—
89	C₇×M₁₆	—	2	2	no	—
90	C₇×M₁₆	—	2	2	no	—
91	C₇×M₁₆	—	2	2	no	—
92	C₇×M₁₆	—	2	2	no	—
93	C₇×M₁₆	—	2	2	no	—
94	C₇×M₁₆	—	2	2	no	—
95	C₇×M₁₆	—	2	2	no	—
96	C₇×M₁₆	—	2	2	no	—
97	C₇×M₁₆	—	4	2	no	—
98	C₂×C₇×C₈	—	2	2	no	—
99	C₂×C₇×C₈	—	2	1	no	—
100	C₂×C₇×C₈	—	2	1	no	—
101	C₂×C₇×C₈	—	2	2	no	—
102	C₂×C₇×C₈	—	2	2	no	—
103	C₂×C₇×C₈	—	2	2	no	—
104	C₂×C₇×C₈	—	2	2	no	—
105	C₂×C₇×C₈	—	4	2	no	—
106	C₇×C₄⋊C₄	—	2	2	no	—
107	C₇×C₂²⋊C₄	—	2	2	no	—
108	C₄²×C₇	—	2	2	no	—
109	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	2	yes	—
110	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
111	#568	C₂²×C₇	1	1	yes	characteristic
112	#430	C₂²×C₇	1	2	yes	—
113	#430	C₂²×C₇	1	1	yes	characteristic
114	#430	C₂²×C₇	1	2	yes	—
115	#424	C₂²×C₇	1	2	yes	—
116	#421	C₂²×C₇	1	2	yes	—
117	#418	C₂²×C₇	1	2	yes	—
118	#418	C₂²×C₇	1	2	yes	—
119	#413	C₂²×C₇	1	1	yes	characteristic
120	C₂×#100	C₂²×C₇	1	2	yes	—
121	C₂×#100	C₂²×C₇	1	2	yes	—
122	C₂×#100	C₂²×C₇	1	2	yes	—
123	#408	C₂²×C₇	1	1	yes	characteristic
124	#344	C₂²×C₇	1	2	yes	—
125	#344	C₂²×C₇	1	2	yes	—
126	#341	C₂²×C₇	1	1	yes	characteristic
127	#341	C₂²×C₇	1	2	yes	—
128	#338	C₂²×C₇	1	2	yes	—
129	H(3,ℤ/4)	C₂²×C₇	1	1	yes	characteristic
130	C₂³×C₇	—	2	1	no	—
131	C₂³×C₇	—	4	2	no	—
132	C₂³×C₇	—	4	2	no	—
133	C₂³×C₇	—	4	2	no	—
134	C₂³×C₇	—	16	2	no	—
135	C₂³×C₇	2₋¹⁺⁴	1	1	yes	characteristic
136	C₇×Q₈	—	2	2	no	—
137	C₇×D₄	—	2	2	no	—
138	C₇×D₄	—	4	2	no	—
139	C₇×D₄	—	4	2	no	—
140	C₇×D₄	—	4	2	no	—
141	C₇×D₄	—	4	2	no	—
142	C₇×D₄	—	4	2	no	—
143	C₂×C₄×C₇	—	2	1	no	—
144	C₂×C₄×C₇	—	4	1	no	—
145	C₂×C₄×C₇	—	4	1	no	—
146	C₂×C₄×C₇	—	4	2	no	—
147	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
149	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
150	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
151	C₇×C₈	—	4	2	no	—
152	C₇×C₈	—	4	2	no	—
153	C₇×C₈	—	4	1	no	—
154	C₇×C₈	—	4	2	no	—
155	C₇×C₈	—	4	1	no	—
156	C₇×C₈	—	4	2	no	—
157	C₂×#41	C₂³×C₇	1	1	yes	characteristic
158	C₂²×D₄	C₂³×C₇	1	2	yes	—
159	AGL(1,ℤ/8)	—	4	2	no	—
160	AGL(1,ℤ/8)	—	4	2	no	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×D₈	—	2	2	no	—
163	C₂×D₈	—	4	1	no	—
164	C₂×D₈	—	4	2	no	—
165	#131	—	4	2	no	—
166	C₂×M₁₆	—	4	1	no	—
167	C₂×M₁₆	C₂³×C₇	1	2	yes	—
168	C₂×M₁₆	C₂³×C₇	1	2	yes	—
169	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
170	C₂×M₁₆	C₂³×C₇	1	2	yes	—
171	C₂×M₁₆	C₂³×C₇	1	2	yes	—
172	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
173	#117	C₂³×C₇	1	2	yes	—
174	#108	—	4	1	no	—
175	#108	—	4	2	no	—
176	#108	—	4	2	no	—
177	C₄⋊C₈	—	4	2	no	—
178	#104	—	4	2	no	—
179	#102	—	4	2	no	—
180	#102	—	4	2	no	—
181	#101	—	2	1	no	—
182	#100	—	2	1	no	—
183	#100	—	4	2	no	—
184	#100	—	4	2	no	—
185	#100	—	4	2	no	—
186	C₂²⋊C₈	—	4	1	no	—
187	C₂²×C₇	—	4	1	no	—
188	C₂²×C₇	—	4	1	no	—
189	C₂²×C₇	—	8	1	no	—
190	C₂²×C₇	—	8	2	no	—
191	C₂²×C₇	—	16	2	no	—
192	C₂²×C₇	—	16	2	no	—
193	C₂²×C₇	—	16	2	no	—
194	C₂²×C₇	#539	1	1	yes	characteristic
195	C₂²×C₇	#562	1	2	yes	—
196	C₄×C₇	—	4	1	no	—
197	C₄×C₇	—	8	2	no	—
198	C₄×C₇	#538	1	2	yes	—
199	C₄×C₇	#560	1	1	yes	characteristic
200	C₄×C₇	#560	1	1	yes	characteristic
201	C₂⁴	—	2	2	no	—
202	#41	—	2	2	no	—
203	#41	—	2	2	no	—
204	C₂×Q₈	C₇×#41	1	1	yes	characteristic
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	C₇×#41	1	1	yes	characteristic
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
215	SD₁₆	—	4	4	no	—
216	D₈	—	4	4	no	—
217	D₈	—	4	4	no	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	M₁₆	—	2	2	no	—
227	M₁₆	—	4	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₂×C₇	—	2	2	no	—
240	C₂×C₇	—	2	1	no	—
241	C₂×C₇	—	8	1	no	—
242	C₂×C₇	—	16	2	no	—
243	C₂×C₇	#2826	1	1	yes	characteristic
244	C₂³	—	2	1	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	16	2	no	—
249	C₂³	C₇×2₋¹⁺⁴	1	1	yes	characteristic
250	Q₈	—	2	2	no	—
251	D₄	—	2	2	no	—
252	D₄	—	4	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
262	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
263	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
264	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
265	C₈	—	4	2	no	—
266	C₈	—	4	2	no	—
267	C₈	—	4	1	no	—
268	C₈	—	4	2	no	—
269	C₈	—	4	1	no	—
270	C₈	—	4	2	no	—
271	C₇	#32870	1	1	yes	characteristic, direct product factor, Sylow
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	8	1	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	16	2	no	—
277	C₂²	—	16	2	no	—
278	C₂²	—	16	2	no	—
279	C₂²	C₇×#539	1	1	yes	characteristic
280	C₂²	C₇×#562	1	2	yes	—
281	C₄	—	4	1	no	—
282	C₄	—	8	2	no	—
283	C₄	C₇×#538	1	2	yes	—
284	C₄	C₇×#560	1	1	yes	characteristic
285	C₄	C₇×#560	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	2	1	no	—
288	C₂	—	8	1	no	—
289	C₂	—	16	2	no	—
290	C₂	C₇×#2826	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#2826
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₇×#2018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#2103	C₂	semidirect
2. C₇×#2098	C₂	semidirect
3. C₇×#2081	C₂	semidirect
4. C₇×#2069	C₂	semidirect
5. C₇×#2048	C₂	semidirect
6. C₇×#2045	C₂	semidirect
8. C₇×#1903	C₂	semidirect
9. C₇×#1858	C₂	semidirect
31. #32870	C₇	direct (N × H)
62. #2103	C₂×C₇	semidirect
63. #2098	C₂×C₇	semidirect
64. #2081	C₂×C₇	semidirect
65. #2069	C₂×C₇	semidirect
66. #2048	C₂×C₇	semidirect
67. #2045	C₂×C₇	semidirect
69. #1903	C₂×C₇	semidirect
70. #1858	C₂×C₇	semidirect
271. C₇	#32870	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×#2103	2
C₇×#2098	2
C₇×#2081	2
C₇×#2069	2
C₇×#2048	2
C₇×#2045	2
C₇×#2018	2
C₇×#1903	2
C₇×#1858	2
#32870	7

Subgroups — C7×#32870 #13210086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#32870 #13210086