Subgroups — C₇×#11963 #13189179

Order 1792.

396 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×#3147	C₂	1	1	yes	maximal, characteristic
2	C₇×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#1568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1362	C₂	1	2	yes	maximal, complemented
5	C₇×#1361	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#135	—	2	1	no	—
7	C₂×C₇×#135	C₄	1	2	yes	—
8	C₇×#532	—	2	1	no	—
9	C₇×#495	—	2	1	no	—
10	C₇×#495	C₄	1	2	yes	—
11	C₇×#494	C₂²	1	1	yes	characteristic
12	C₇×#493	C₂²	1	1	yes	characteristic
13	C₇×#492	—	2	1	no	—
14	C₇×#432	—	2	1	no	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₇×C₂²⋊C₁₆	—	2	1	no	—
18	C₄×C₇×C₁₆	—	2	1	no	—
19	C₇×#335	C₂²	1	2	yes	—
20	C₇×#334	—	2	1	no	—
21	#11963	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₇×#41	—	2	2	no	—
23	C₇×#135	—	4	4	no	—
24	C₇×#135	—	4	4	no	—
25	C₂×C₇×Q₁₆	—	2	1	no	—
26	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₇×SD₁₆	—	2	2	no	—
28	C₂×C₇×SD₁₆	—	2	2	no	—
29	C₂×C₇×D₈	—	2	1	no	—
30	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
31	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
33	C₇×C₄⋊Q₈	—	2	1	no	—
34	C₇×#127	—	2	1	no	—
35	C₇×#124	—	2	2	no	—
36	C₇×#121	—	4	1	no	—
37	C₇×#121	—	4	1	no	—
38	C₄×C₇×D₄	—	4	1	no	—
39	C₄×C₇×D₄	—	4	1	no	—
40	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
41	C₂×C₇×C₁₆	—	2	2	no	—
42	C₇×C₄⋊C₈	—	2	2	no	—
43	C₇×C₂²⋊C₈	—	4	1	no	—
44	C₇×#97	—	4	1	no	—
45	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₇×C₈	D₄	1	1	yes	characteristic
47	C₄×C₇×C₈	D₄	1	1	yes	characteristic
48	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
49	#3147	C₂×C₇	1	1	yes	characteristic
50	#2162	C₂×C₇	1	1	yes	characteristic, complemented
51	#1568	C₂×C₇	1	1	yes	characteristic, complemented
52	#1362	C₂×C₇	1	2	yes	complemented
53	#1361	C₂×C₇	1	2	yes	complemented
54	C₇×#41	—	4	4	no	—
55	C₇×#41	—	4	4	no	—
56	C₂×C₇×Q₈	—	2	2	no	—
57	C₂×C₇×D₄	—	2	2	no	—
58	C₂×C₇×D₄	—	4	2	no	—
59	C₂×C₇×D₄	—	4	1	no	—
60	C₂×C₇×D₄	—	4	1	no	—
61	C₂²×C₄×C₇	—	4	2	no	—
62	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
65	C₇×Q₁₆	—	4	2	no	—
66	C₇×Q₁₆	—	4	2	no	—
67	C₇×SD₁₆	—	4	4	no	—
68	C₇×SD₁₆	—	4	4	no	—
69	C₇×D₈	—	4	2	no	—
70	C₇×D₈	—	4	2	no	—
71	C₂×C₇×C₈	—	2	2	no	—
72	C₂×C₇×C₈	—	2	2	no	—
73	C₂×C₇×C₈	—	4	2	no	—
74	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
78	C₇×C₄⋊C₄	—	4	1	no	—
79	C₇×C₄⋊C₄	—	4	1	no	—
80	C₇×C₂²⋊C₄	—	4	2	no	—
81	C₇×C₂²⋊C₄	—	4	2	no	—
82	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₇	D₈	1	1	yes	characteristic
84	C₄²×C₇	SD₁₆	1	1	yes	characteristic
85	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
86	C₇×C₁₆	—	4	2	no	—
87	C₂×#135	—	2	1	no	—
88	C₂×#135	C₄×C₇	1	2	yes	—
89	#532	—	2	1	no	—
90	#495	—	2	1	no	—
91	#495	C₄×C₇	1	2	yes	—
92	#494	C₂²×C₇	1	1	yes	characteristic
93	#493	C₂²×C₇	1	1	yes	characteristic
94	#492	—	2	1	no	—
95	#432	—	2	1	no	—
96	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
97	C₄⋊C₁₆	C₂²×C₇	1	2	yes	—
98	C₂²⋊C₁₆	—	2	1	no	—
99	C₄×C₁₆	—	2	1	no	—
100	#335	C₂²×C₇	1	2	yes	—
101	#334	—	2	1	no	—
102	C₂³×C₇	—	4	2	no	—
103	C₂³×C₇	#102	1	1	yes	characteristic
104	C₇×Q₈	—	4	2	no	—
105	C₇×Q₈	—	4	2	no	—
106	C₇×D₄	—	4	2	no	—
107	C₇×D₄	—	4	2	no	—
108	C₇×D₄	—	8	2	no	—
109	C₇×D₄	—	8	4	no	—
110	C₇×D₄	—	8	2	no	—
111	C₂×C₄×C₇	—	2	1	no	—
112	C₂×C₄×C₇	—	2	1	no	—
113	C₂×C₄×C₇	—	2	2	no	—
114	C₂×C₄×C₇	—	4	1	no	—
115	C₂×C₄×C₇	—	4	2	no	—
116	C₂×C₄×C₇	—	8	4	no	—
117	C₂×C₄×C₇	#102	1	1	yes	characteristic
118	C₂×C₄×C₇	#102	1	1	yes	characteristic
119	C₂×C₄×C₇	#102	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
123	C₇×C₈	—	2	1	no	—
124	C₇×C₈	—	2	1	no	—
125	C₇×C₈	—	8	2	no	—
126	C₇×C₈	#100	1	2	yes	—
127	C₇×C₈	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×#41	—	2	2	no	—
129	#135	—	4	4	no	—
130	#135	—	4	4	no	—
131	C₂×Q₁₆	—	2	1	no	—
132	C₂×Q₁₆	C₂×C₄×C₇	1	2	yes	—
133	C₂×SD₁₆	—	2	2	no	—
134	C₂×SD₁₆	—	2	2	no	—
135	C₂×D₈	—	2	1	no	—
136	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
137	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
138	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
139	C₄⋊Q₈	—	2	1	no	—
140	#127	—	2	1	no	—
141	#124	—	2	2	no	—
142	#121	—	4	1	no	—
143	#121	—	4	1	no	—
144	C₄×D₄	—	4	1	no	—
145	C₄×D₄	—	4	1	no	—
146	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
147	C₂×C₁₆	—	2	2	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₂²⋊C₈	—	4	1	no	—
150	#97	—	4	1	no	—
151	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
152	C₄×C₈	C₇×D₄	1	1	yes	characteristic
153	C₄×C₈	C₇×D₄	1	1	yes	characteristic
154	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
155	C₂²×C₇	—	2	1	no	—
156	C₂²×C₇	—	2	1	no	—
157	C₂²×C₇	—	2	1	no	—
158	C₂²×C₇	—	8	2	no	—
159	C₂²×C₇	—	8	2	no	—
160	C₂²×C₇	—	8	2	no	—
161	C₂²×C₇	C₂×#102	1	1	yes	characteristic
162	C₄×C₇	—	2	1	no	—
163	C₄×C₇	—	2	1	no	—
164	C₄×C₇	—	2	1	no	—
165	C₄×C₇	—	2	1	no	—
166	C₄×C₇	—	8	2	no	—
167	C₄×C₇	C₂×#100	1	1	yes	characteristic
168	C₄×C₇	#418	1	1	yes	characteristic
169	C₄×C₇	#508	1	2	yes	—
170	#41	—	4	4	no	—
171	#41	—	4	4	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	4	2	no	—
175	C₂×D₄	—	4	1	no	—
176	C₂×D₄	—	4	1	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
180	C₂²×C₄	C₇×SD₁₆	1	1	yes	characteristic
181	Q₁₆	—	4	2	no	—
182	Q₁₆	—	4	2	no	—
183	SD₁₆	—	4	4	no	—
184	SD₁₆	—	4	4	no	—
185	D₈	—	4	2	no	—
186	D₈	—	4	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
191	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
192	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
193	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	4	1	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
199	C₄²	C₇×D₈	1	1	yes	characteristic
200	C₄²	C₇×SD₁₆	1	1	yes	characteristic
201	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
202	C₁₆	—	4	2	no	—
203	C₂×C₇	—	4	1	no	—
204	C₂×C₇	—	16	2	no	—
205	C₂×C₇	#1545	1	1	yes	characteristic
206	C₂×C₇	#2138	1	1	yes	characteristic
207	C₂×C₇	#2148	1	1	yes	characteristic
208	C₂³	—	4	2	no	—
209	C₂³	C₇×#102	1	1	yes	characteristic
210	Q₈	—	4	2	no	—
211	Q₈	—	4	2	no	—
212	D₄	—	4	2	no	—
213	D₄	—	4	2	no	—
214	D₄	—	8	2	no	—
215	D₄	—	8	4	no	—
216	D₄	—	8	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	4	no	—
223	C₂×C₄	C₇×#102	1	1	yes	characteristic
224	C₂×C₄	C₇×#102	1	1	yes	characteristic
225	C₂×C₄	C₇×#102	1	1	yes	characteristic
226	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
227	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
228	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	2	1	no	—
231	C₈	—	8	2	no	—
232	C₈	C₇×#100	1	2	yes	—
233	C₈	C₇×AGL(1,ℤ/8)	1	2	yes	—
234	C₇	#11963	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	8	2	no	—
239	C₂²	—	8	2	no	—
240	C₂²	—	8	2	no	—
241	C₂²	C₂×C₇×#102	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	8	2	no	—
247	C₄	C₂×C₇×#100	1	1	yes	characteristic
248	C₄	C₇×#418	1	1	yes	characteristic
249	C₄	C₇×#508	1	2	yes	—
250	C₂	—	4	1	no	—
251	C₂	—	16	2	no	—
252	C₂	C₇×#1545	1	1	yes	characteristic
253	C₂	C₇×#2138	1	1	yes	characteristic
254	C₂	C₇×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2162	C₂	semidirect
3. C₇×#1568	C₂	semidirect
4. C₇×#1362	C₂	semidirect
5. C₇×#1361	C₂	semidirect
21. #11963	C₇	direct (N × H)
50. #2162	C₂×C₇	semidirect
51. #1568	C₂×C₇	semidirect
52. #1362	C₂×C₇	semidirect
53. #1361	C₂×C₇	semidirect
234. C₇	#11963	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×#3147	2
C₇×#2162	2
C₇×#1568	2
C₇×#1362	2
C₇×#1361	2
#11963	7

Subgroups — C7×#11963 #13189179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11963 #13189179