Subgroups — C₇×#23360 #13200576

Order 1792.

688 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3152	C₂	1	1	yes	maximal, characteristic
2	C₇×#1905	C₂	1	4	yes	maximal, complemented
3	C₇×#1902	C₂	1	4	yes	maximal, complemented
4	C₇×#1886	C₂	1	2	yes	maximal, complemented
5	C₇×#1884	C₂	1	2	yes	maximal
6	C₇×#1841	C₂	1	2	yes	maximal, complemented
7	C₂×C₇×#137	C₂²	1	4	yes	—
8	C₇×#568	C₂²	1	2	yes	—
9	C₇×#532	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₇×#519	C₂²	1	2	yes	—
12	C₇×#497	C₄	1	4	yes	complemented
13	C₇×#494	C₄	1	2	yes	complemented
14	C₇×#492	C₄	1	2	yes	complemented
15	C₇×#421	C₂²	1	4	yes	—
16	C₇×#419	C₂²	1	4	yes	—
17	C₂×C₇×#101	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	#23360	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₇×#41	C₂³	1	2	yes	—
26	C₂²×C₇×Q₈	C₂³	1	2	yes	—
27	C₇×#137	—	4	4	no	—
28	C₇×#137	—	4	4	no	—
29	C₂×C₇×Q₁₆	C₂×C₄	1	8	yes	—
30	C₂×C₇×SD₁₆	C₂×C₄	1	8	yes	—
31	C₇×#131	—	4	4	no	—
32	C₂×C₇×M₁₆	—	4	2	no	—
33	C₂×C₇×M₁₆	C₂³	1	4	yes	—
34	C₂×C₇×M₁₆	C₂³	1	2	yes	—
35	C₇×C₄⋊Q₈	—	4	1	no	—
36	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
37	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
38	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₇×#127	C₂×C₄	1	1	yes	characteristic
40	C₇×#124	C₂×C₄	1	2	yes	—
41	C₇×#121	—	4	1	no	—
42	C₄×C₇×Q₈	—	4	4	no	—
43	C₄×C₇×D₄	—	4	1	no	—
44	C₇×#117	—	4	2	no	—
45	C₇×#117	C₂³	1	4	yes	—
46	C₂×C₇×C₄⋊C₄	—	4	1	no	—
47	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
48	C₇×#104	—	4	4	no	—
49	C₇×#103	—	4	4	no	—
50	C₇×#101	—	4	2	no	—
51	C₇×#99	—	4	2	no	—
52	C₇×#97	C₂×C₄	1	2	yes	—
53	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
54	#3152	C₂×C₇	1	1	yes	characteristic
55	#1905	C₂×C₇	1	4	yes	complemented
56	#1902	C₂×C₇	1	4	yes	complemented
57	#1886	C₂×C₇	1	2	yes	complemented
58	#1884	C₂×C₇	1	2	yes	—
59	#1841	C₂×C₇	1	2	yes	complemented
60	C₇×#41	—	2	4	no	—
61	C₇×#41	—	2	4	no	—
62	C₂×C₇×Q₈	—	2	2	no	—
63	C₂×C₇×Q₈	—	2	4	no	—
64	C₂×C₇×Q₈	—	2	4	no	—
65	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
66	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
67	C₂×C₇×D₄	—	2	2	no	—
68	C₂×C₇×D₄	—	4	1	no	—
69	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₇	—	2	1	no	—
71	C₂²×C₄×C₇	—	2	2	no	—
72	C₂²×C₄×C₇	—	2	2	no	—
73	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
74	C₇×Q₁₆	—	4	8	no	—
75	C₇×SD₁₆	—	4	8	no	—
76	C₇×M₁₆	—	2	4	no	—
77	C₇×M₁₆	—	2	4	no	—
78	C₇×M₁₆	—	2	4	no	—
79	C₇×M₁₆	—	4	4	no	—
80	C₂×C₇×C₈	—	2	4	no	—
81	C₂×C₇×C₈	—	4	4	no	—
82	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
83	C₇×C₄⋊C₄	—	2	4	no	—
84	C₇×C₄⋊C₄	—	4	2	no	—
85	C₇×C₄⋊C₄	—	4	2	no	—
86	C₇×C₄⋊C₄	—	4	1	no	—
87	C₇×C₄⋊C₄	—	4	4	no	—
88	C₇×C₂²⋊C₄	—	2	4	no	—
89	C₇×C₂²⋊C₄	—	4	2	no	—
90	C₄²×C₇	—	2	1	no	—
91	C₄²×C₇	—	2	4	no	—
92	C₄²×C₇	—	4	4	no	—
93	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×#137	C₂²×C₇	1	4	yes	—
96	#568	C₂²×C₇	1	2	yes	—
97	#532	C₂²×C₇	1	1	yes	characteristic
98	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
99	#519	C₂²×C₇	1	2	yes	—
100	#497	C₄×C₇	1	4	yes	complemented
101	#494	C₄×C₇	1	2	yes	complemented
102	#492	C₄×C₇	1	2	yes	complemented
103	#421	C₂²×C₇	1	4	yes	—
104	#419	C₂²×C₇	1	4	yes	—
105	C₂×#101	C₂²×C₇	1	2	yes	—
106	#410	C₂²×C₇	1	2	yes	—
107	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
108	#344	C₂²×C₇	1	4	yes	—
109	#339	C₂²×C₇	1	4	yes	—
110	#338	C₂²×C₇	1	2	yes	—
111	H(3,ℤ/4)	C₂²×C₇	1	2	yes	—
112	C₂³×C₇	—	2	2	no	—
113	C₂³×C₇	H(5,𝔽₂)	1	1	yes	characteristic
114	C₇×Q₈	—	2	8	no	—
115	C₇×Q₈	—	2	4	no	—
116	C₇×Q₈	—	4	4	no	—
117	C₇×D₄	—	2	4	no	—
118	C₇×D₄	—	4	4	no	—
119	C₇×D₄	—	8	2	no	—
120	C₂×C₄×C₇	—	2	1	no	—
121	C₂×C₄×C₇	—	2	1	no	—
122	C₂×C₄×C₇	—	2	2	no	—
123	C₂×C₄×C₇	—	2	4	no	—
124	C₂×C₄×C₇	—	4	2	no	—
125	C₂×C₄×C₇	—	4	2	no	—
126	C₂×C₄×C₇	—	4	2	no	—
127	C₂×C₄×C₇	—	4	2	no	—
128	C₂×C₄×C₇	—	4	2	no	—
129	C₂×C₄×C₇	—	4	4	no	—
130	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
132	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
133	C₇×C₈	—	2	4	no	—
134	C₇×C₈	—	4	4	no	—
135	C₂×#41	C₂³×C₇	1	2	yes	—
136	C₂²×Q₈	C₂³×C₇	1	2	yes	—
137	#137	—	4	4	no	—
138	#137	—	4	4	no	—
139	C₂×Q₁₆	C₂×C₄×C₇	1	8	yes	—
140	C₂×SD₁₆	C₂×C₄×C₇	1	8	yes	—
141	#131	—	4	4	no	—
142	C₂×M₁₆	—	4	2	no	—
143	C₂×M₁₆	C₂³×C₇	1	4	yes	—
144	C₂×M₁₆	C₂³×C₇	1	2	yes	—
145	C₄⋊Q₈	—	4	1	no	—
146	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
147	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
148	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic
149	#127	C₂×C₄×C₇	1	1	yes	characteristic
150	#124	C₂×C₄×C₇	1	2	yes	—
151	#121	—	4	1	no	—
152	C₄×Q₈	—	4	4	no	—
153	C₄×D₄	—	4	1	no	—
154	#117	—	4	2	no	—
155	#117	C₂³×C₇	1	4	yes	—
156	C₂×C₄⋊C₄	—	4	1	no	—
157	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
158	#104	—	4	4	no	—
159	#103	—	4	4	no	—
160	#101	—	4	2	no	—
161	#99	—	4	2	no	—
162	#97	C₂×C₄×C₇	1	2	yes	—
163	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
164	C₂²×C₇	—	4	1	no	—
165	C₂²×C₇	—	4	2	no	—
166	C₂²×C₇	—	8	2	no	—
167	C₂²×C₇	#518	1	1	yes	characteristic
168	C₂²×C₇	#560	1	2	yes	—
169	C₄×C₇	—	2	2	no	—
170	C₄×C₇	—	4	1	no	—
171	C₄×C₇	—	4	2	no	—
172	C₄×C₇	—	4	4	no	—
173	C₄×C₇	—	8	4	no	—
174	C₄×C₇	#518	1	2	yes	—
175	C₄×C₇	#560	1	2	yes	—
176	#41	—	2	4	no	—
177	#41	—	2	4	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
182	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	4	1	no	—
185	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
190	Q₁₆	—	4	8	no	—
191	SD₁₆	—	4	8	no	—
192	M₁₆	—	2	4	no	—
193	M₁₆	—	2	4	no	—
194	M₁₆	—	2	4	no	—
195	M₁₆	—	4	4	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	4	4	no	—
198	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	1	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	4	2	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	4	no	—
208	C₄²	—	4	4	no	—
209	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
210	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
211	C₂×C₇	—	2	1	no	—
212	C₂×C₇	—	2	2	no	—
213	C₂×C₇	—	8	2	no	—
214	C₂×C₇	#2380	1	1	yes	characteristic
215	C₂³	—	2	2	no	—
216	C₂³	C₇×H(5,𝔽₂)	1	1	yes	characteristic
217	Q₈	—	2	8	no	—
218	Q₈	—	2	4	no	—
219	Q₈	—	4	4	no	—
220	D₄	—	2	4	no	—
221	D₄	—	4	4	no	—
222	D₄	—	8	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
234	C₂×C₄	C₇×H(5,𝔽₂)	1	4	yes	—
235	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
236	C₈	—	2	4	no	—
237	C₈	—	4	4	no	—
238	C₇	#23360	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	4	1	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	C₇×#518	1	1	yes	characteristic
243	C₂²	C₇×#560	1	2	yes	—
244	C₄	—	2	2	no	—
245	C₄	—	4	1	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	4	no	—
248	C₄	—	8	4	no	—
249	C₄	C₇×#518	1	2	yes	—
250	C₄	C₇×#560	1	2	yes	—
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	8	2	no	—
254	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×#532	C₂²
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×#1905	C₂	semidirect
3. C₇×#1902	C₂	semidirect
4. C₇×#1886	C₂	semidirect
6. C₇×#1841	C₂	semidirect
12. C₇×#497	C₄	semidirect
13. C₇×#494	C₄	semidirect
14. C₇×#492	C₄	semidirect
24. #23360	C₇	direct (N × H)
55. #1905	C₂×C₇	semidirect
56. #1902	C₂×C₇	semidirect
57. #1886	C₂×C₇	semidirect
59. #1841	C₂×C₇	semidirect
100. #497	C₄×C₇	semidirect
101. #494	C₄×C₇	semidirect
102. #492	C₄×C₇	semidirect
238. C₇	#23360	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×#3152	2
C₇×#1905	2
C₇×#1902	2
C₇×#1886	2
C₇×#1884	2
C₇×#1841	2
#23360	7

Subgroups — C7×#23360 #13200576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23360 #13200576