Subgroups — C₇×#23361 #13200577

Order 1792.

756 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3151	C₂	1	1	yes	maximal, characteristic
2	C₇×#1904	C₂	1	4	yes	maximal, complemented
3	C₇×#1903	C₂	1	4	yes	maximal, complemented
4	C₇×#1887	C₂	1	2	yes	maximal, complemented
5	C₇×#1884	C₂	1	2	yes	maximal, complemented
6	C₇×#1841	C₂	1	2	yes	maximal, complemented
7	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	4	yes	—
8	C₇×#568	C₂²	1	2	yes	—
9	C₇×#532	C₂²	1	1	yes	characteristic
10	C₂×C₇×#127	C₂²	1	1	yes	characteristic
11	C₇×#518	C₂²	1	2	yes	—
12	C₇×#496	C₄	1	4	yes	complemented
13	C₇×#493	C₄	1	2	yes	complemented
14	C₇×#492	C₄	1	2	yes	complemented
15	C₇×#421	C₂²	1	4	yes	—
16	C₇×#418	C₂²	1	4	yes	—
17	C₂×C₇×#100	C₂²	1	2	yes	—
18	C₇×#410	C₂²	1	2	yes	—
19	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
20	C₇×#344	C₂²	1	4	yes	—
21	C₇×#339	C₂²	1	4	yes	—
22	C₇×#338	C₂²	1	2	yes	—
23	C₇×H(3,ℤ/4)	C₂²	1	2	yes	—
24	#23361	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₇×#41	C₂³	1	2	yes	—
26	C₂²×C₇×D₄	—	4	1	no	—
27	C₂²×C₇×D₄	C₂³	1	2	yes	—
28	C₇×AGL(1,ℤ/8)	—	4	4	no	—
29	C₇×AGL(1,ℤ/8)	—	4	4	no	—
30	C₂×C₇×SD₁₆	C₂×C₄	1	8	yes	complemented
31	C₂×C₇×D₈	C₂×C₄	1	8	yes	complemented
32	C₇×#131	—	4	4	no	—
33	C₂×C₇×M₁₆	—	4	2	no	—
34	C₂×C₇×M₁₆	C₂³	1	4	yes	—
35	C₂×C₇×M₁₆	C₂³	1	2	yes	—
36	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
37	C₇×#127	—	4	1	no	—
38	C₇×#127	C₂×C₄	1	2	yes	—
39	C₇×#127	C₂×C₄	1	2	yes	—
40	C₇×#127	C₂×C₄	1	1	yes	characteristic, complemented
41	C₇×#124	C₂×C₄	1	2	yes	complemented
42	C₇×#121	—	4	1	no	—
43	C₄×C₇×D₄	—	4	4	no	—
44	C₄×C₇×D₄	—	4	1	no	—
45	C₇×#117	C₂³	1	4	yes	—
46	C₂×C₇×C₂²⋊C₄	—	4	2	no	—
47	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
48	C₇×#104	—	4	4	no	—
49	C₇×#102	—	4	4	no	—
50	C₇×#100	—	4	2	no	—
51	C₇×#99	—	4	2	no	—
52	C₇×#97	C₂×C₄	1	2	yes	complemented
53	C₄×C₇×C₈	C₂×C₄	1	2	yes	complemented
54	#3151	C₂×C₇	1	1	yes	characteristic
55	#1904	C₂×C₇	1	4	yes	complemented
56	#1903	C₂×C₇	1	4	yes	complemented
57	#1887	C₂×C₇	1	2	yes	complemented
58	#1884	C₂×C₇	1	2	yes	complemented
59	#1841	C₂×C₇	1	2	yes	complemented
60	C₂⁴×C₇	—	2	2	no	—
61	C₇×#41	—	2	4	no	—
62	C₇×#41	—	2	4	no	—
63	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
64	C₂×C₇×D₄	—	2	2	no	—
65	C₂×C₇×D₄	—	2	4	no	—
66	C₂×C₇×D₄	—	2	4	no	—
67	C₂×C₇×D₄	—	2	2	no	—
68	C₂×C₇×D₄	—	4	1	no	—
69	C₂×C₇×D₄	—	4	2	no	—
70	C₂×C₇×D₄	—	4	2	no	—
71	C₂×C₇×D₄	—	8	4	no	—
72	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
73	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₇	—	2	1	no	—
75	C₂²×C₄×C₇	—	2	2	no	—
76	C₂²×C₄×C₇	—	4	4	no	—
77	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
78	C₇×SD₁₆	—	4	8	no	—
79	C₇×D₈	—	4	8	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	4	no	—
85	C₂×C₇×C₈	—	4	4	no	—
86	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
87	C₇×C₄⋊C₄	—	2	4	no	—
88	C₇×C₄⋊C₄	—	4	1	no	—
89	C₇×C₂²⋊C₄	—	2	4	no	—
90	C₇×C₂²⋊C₄	—	4	2	no	—
91	C₇×C₂²⋊C₄	—	4	4	no	—
92	C₄²×C₇	—	2	1	no	—
93	C₄²×C₇	—	2	4	no	—
94	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	4	yes	—
97	#568	C₂²×C₇	1	2	yes	—
98	#532	C₂²×C₇	1	1	yes	characteristic
99	C₂×#127	C₂²×C₇	1	1	yes	characteristic
100	#518	C₂²×C₇	1	2	yes	—
101	#496	C₄×C₇	1	4	yes	complemented
102	#493	C₄×C₇	1	2	yes	complemented
103	#492	C₄×C₇	1	2	yes	complemented
104	#421	C₂²×C₇	1	4	yes	—
105	#418	C₂²×C₇	1	4	yes	—
106	C₂×#100	C₂²×C₇	1	2	yes	—
107	#410	C₂²×C₇	1	2	yes	—
108	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
109	#344	C₂²×C₇	1	4	yes	—
110	#339	C₂²×C₇	1	4	yes	—
111	#338	C₂²×C₇	1	2	yes	—
112	H(3,ℤ/4)	C₂²×C₇	1	2	yes	—
113	C₂³×C₇	—	2	2	no	—
114	C₂³×C₇	—	2	4	no	—
115	C₂³×C₇	—	4	2	no	—
116	C₂³×C₇	—	4	2	no	—
117	C₂³×C₇	—	8	4	no	—
118	C₂³×C₇	H(5,𝔽₂)	1	1	yes	characteristic
119	C₇×Q₈	—	2	4	no	—
120	C₇×D₄	—	2	4	no	—
121	C₇×D₄	—	2	8	no	—
122	C₇×D₄	—	4	4	no	—
123	C₇×D₄	—	4	4	no	—
124	C₇×D₄	—	8	2	no	—
125	C₇×D₄	—	8	4	no	—
126	C₇×D₄	—	8	4	no	—
127	C₂×C₄×C₇	—	2	1	no	—
128	C₂×C₄×C₇	—	2	1	no	—
129	C₂×C₄×C₇	—	2	2	no	—
130	C₂×C₄×C₇	—	4	2	no	—
131	C₂×C₄×C₇	—	4	2	no	—
132	C₂×C₄×C₇	—	4	2	no	—
133	C₂×C₄×C₇	—	4	4	no	—
134	C₂×C₄×C₇	—	16	4	no	—
135	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
137	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
138	C₇×C₈	—	2	4	no	—
139	C₇×C₈	—	4	4	no	—
140	C₂×#41	C₂³×C₇	1	2	yes	—
141	C₂²×D₄	—	4	1	no	—
142	C₂²×D₄	C₂³×C₇	1	2	yes	—
143	AGL(1,ℤ/8)	—	4	4	no	—
144	AGL(1,ℤ/8)	—	4	4	no	—
145	C₂×SD₁₆	C₂×C₄×C₇	1	8	yes	complemented
146	C₂×D₈	C₂×C₄×C₇	1	8	yes	complemented
147	#131	—	4	4	no	—
148	C₂×M₁₆	—	4	2	no	—
149	C₂×M₁₆	C₂³×C₇	1	4	yes	—
150	C₂×M₁₆	C₂³×C₇	1	2	yes	—
151	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic, complemented
152	#127	—	4	1	no	—
153	#127	C₂×C₄×C₇	1	2	yes	—
154	#127	C₂×C₄×C₇	1	2	yes	—
155	#127	C₂×C₄×C₇	1	1	yes	characteristic, complemented
156	#124	C₂×C₄×C₇	1	2	yes	complemented
157	#121	—	4	1	no	—
158	C₄×D₄	—	4	4	no	—
159	C₄×D₄	—	4	1	no	—
160	#117	C₂³×C₇	1	4	yes	—
161	C₂×C₂²⋊C₄	—	4	2	no	—
162	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
163	#104	—	4	4	no	—
164	#102	—	4	4	no	—
165	#100	—	4	2	no	—
166	#99	—	4	2	no	—
167	#97	C₂×C₄×C₇	1	2	yes	complemented
168	C₄×C₈	C₂×C₄×C₇	1	2	yes	complemented
169	C₂²×C₇	—	4	1	no	—
170	C₂²×C₇	—	4	2	no	—
171	C₂²×C₇	—	4	4	no	—
172	C₂²×C₇	—	8	4	no	—
173	C₂²×C₇	—	8	2	no	—
174	C₂²×C₇	—	8	4	no	—
175	C₂²×C₇	—	16	2	no	—
176	C₂²×C₇	#518	1	1	yes	characteristic
177	C₂²×C₇	#560	1	2	yes	—
178	C₄×C₇	—	2	2	no	—
179	C₄×C₇	—	4	1	no	—
180	C₄×C₇	—	4	2	no	—
181	C₄×C₇	—	8	4	no	—
182	C₄×C₇	#518	1	2	yes	—
183	C₄×C₇	#560	1	2	yes	—
184	C₂⁴	—	2	2	no	—
185	#41	—	2	4	no	—
186	#41	—	2	4	no	—
187	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂×D₄	—	8	4	no	—
196	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
197	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
202	SD₁₆	—	4	8	no	—
203	D₈	—	4	8	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	4	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	4	4	no	—
210	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	4	1	no	—
213	C₂²⋊C₄	—	2	4	no	—
214	C₂²⋊C₄	—	4	2	no	—
215	C₂²⋊C₄	—	4	4	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	2	4	no	—
218	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
219	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
220	C₂×C₇	—	2	1	no	—
221	C₂×C₇	—	2	2	no	—
222	C₂×C₇	—	8	2	no	—
223	C₂×C₇	—	8	4	no	—
224	C₂×C₇	#2380	1	1	yes	characteristic
225	C₂³	—	2	2	no	—
226	C₂³	—	2	4	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	8	4	no	—
230	C₂³	C₇×H(5,𝔽₂)	1	1	yes	characteristic
231	Q₈	—	2	4	no	—
232	D₄	—	2	4	no	—
233	D₄	—	2	8	no	—
234	D₄	—	4	4	no	—
235	D₄	—	4	4	no	—
236	D₄	—	8	2	no	—
237	D₄	—	8	4	no	—
238	D₄	—	8	4	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	16	4	no	—
247	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
248	C₂×C₄	C₇×H(5,𝔽₂)	1	4	yes	—
249	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
250	C₈	—	2	4	no	—
251	C₈	—	4	4	no	—
252	C₇	#23361	1	1	yes	characteristic, direct product factor, Sylow
253	C₂²	—	4	1	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	4	no	—
259	C₂²	—	16	2	no	—
260	C₂²	C₇×#518	1	1	yes	characteristic
261	C₂²	C₇×#560	1	2	yes	—
262	C₄	—	2	2	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	2	no	—
265	C₄	—	8	4	no	—
266	C₄	C₇×#518	1	2	yes	—
267	C₄	C₇×#560	1	2	yes	—
268	C₂	—	2	1	no	—
269	C₂	—	2	2	no	—
270	C₂	—	8	4	no	—
271	C₂	—	8	2	no	—
272	C₂	C₇×#2380	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#2380
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₇×#3151	C₂
LCS	G ▷ C₂×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
2. C₇×#1904	C₂	semidirect
3. C₇×#1903	C₂	semidirect
4. C₇×#1887	C₂	semidirect
5. C₇×#1884	C₂	semidirect
6. C₇×#1841	C₂	semidirect
12. C₇×#496	C₄	semidirect
13. C₇×#493	C₄	semidirect
14. C₇×#492	C₄	semidirect
24. #23361	C₇	direct (N × H)
30. C₂×C₇×SD₁₆	C₂×C₄	semidirect
31. C₂×C₇×D₈	C₂×C₄	semidirect
36. C₇×C₄⋊Q₈	C₂×C₄	semidirect
40. C₇×#127	C₂×C₄	semidirect
41. C₇×#124	C₂×C₄	semidirect
52. C₇×#97	C₂×C₄	semidirect
53. C₄×C₇×C₈	C₂×C₄	semidirect
55. #1904	C₂×C₇	semidirect
56. #1903	C₂×C₇	semidirect
57. #1887	C₂×C₇	semidirect
58. #1884	C₂×C₇	semidirect
59. #1841	C₂×C₇	semidirect
101. #496	C₄×C₇	semidirect
102. #493	C₄×C₇	semidirect
103. #492	C₄×C₇	semidirect
145. C₂×SD₁₆	C₂×C₄×C₇	semidirect
146. C₂×D₈	C₂×C₄×C₇	semidirect
151. C₄⋊Q₈	C₂×C₄×C₇	semidirect
155. #127	C₂×C₄×C₇	semidirect
156. #124	C₂×C₄×C₇	semidirect
167. #97	C₂×C₄×C₇	semidirect
168. C₄×C₈	C₂×C₄×C₇	semidirect
252. C₇	#23361	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×#3151	2
C₇×#1904	2
C₇×#1903	2
C₇×#1887	2
C₇×#1884	2
C₇×#1841	2
#23361	7

Subgroups — C7×#23361 #13200577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23361 #13200577