Subgroups — C₇×#11947 #13189163

Order 1792.

354 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3169	C₂	1	1	yes	maximal, characteristic
2	C₇×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1364	C₂	1	2	yes	maximal, complemented
5	C₇×#1361	C₂	1	2	yes	maximal, complemented
6	C₇×#532	—	2	1	no	—
7	C₇×#500	C₂²	1	1	yes	characteristic
8	C₇×#493	C₂²	1	1	yes	characteristic
9	C₇×#486	—	2	1	no	—
10	C₇×#466	—	2	1	no	—
11	C₇×#466	C₄	1	2	yes	—
12	C₄×C₇×D₈	—	2	1	no	—
13	C₄×C₇×D₈	C₄	1	2	yes	—
14	C₂×C₇×C₄⋊C₈	—	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	—	2	1	no	—
20	C₇×#335	C₂²	1	2	yes	—
21	#11947	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₇×#41	—	4	1	no	—
23	C₂×C₇×D₈	—	2	1	no	—
24	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
25	C₂²×C₇×C₈	—	4	1	no	—
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
28	C₇×C₄⋊Q₈	—	2	1	no	—
29	C₇×#127	—	2	1	no	—
30	C₇×#124	—	4	1	no	—
31	C₇×#121	—	2	2	no	—
32	C₇×#121	—	4	1	no	—
33	C₄×C₇×D₄	—	2	2	no	—
34	C₄×C₇×D₄	—	4	1	no	—
35	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
36	C₂×C₇×C₁₆	—	2	2	no	—
37	C₇×#107	—	2	1	no	—
38	C₇×#107	C₂×C₄	1	2	yes	—
39	C₇×C₄⋊C₈	—	2	2	no	—
40	C₇×C₄⋊C₈	—	4	1	no	—
41	C₇×#102	—	2	2	no	—
42	C₇×#102	—	2	2	no	—
43	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₇×C₈	D₄	1	1	yes	characteristic
45	C₄×C₇×C₈	D₄	1	1	yes	characteristic
46	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
47	#3169	C₂×C₇	1	1	yes	characteristic
48	#2162	C₂×C₇	1	1	yes	characteristic, complemented
49	C₂×#335	C₂×C₇	1	1	yes	characteristic, complemented
50	#1364	C₂×C₇	1	2	yes	complemented
51	#1361	C₂×C₇	1	2	yes	complemented
52	C₇×#41	—	8	2	no	—
53	C₇×#41	—	8	2	no	—
54	C₂×C₇×Q₈	—	4	1	no	—
55	C₂×C₇×D₄	—	2	2	no	—
56	C₂×C₇×D₄	—	4	2	no	—
57	C₂×C₇×D₄	—	4	1	no	—
58	C₂×C₇×D₄	—	4	1	no	—
59	C₂²×C₄×C₇	—	4	2	no	—
60	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
63	C₇×D₈	—	4	2	no	—
64	C₇×D₈	—	4	2	no	—
65	C₂×C₇×C₈	—	2	2	no	—
66	C₂×C₇×C₈	—	4	1	no	—
67	C₂×C₇×C₈	—	4	2	no	—
68	C₂×C₇×C₈	—	8	2	no	—
69	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
73	C₇×C₄⋊C₄	—	2	2	no	—
74	C₇×C₄⋊C₄	—	4	1	no	—
75	C₇×C₂²⋊C₄	—	4	2	no	—
76	C₇×C₂²⋊C₄	—	4	2	no	—
77	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₇	D₈	1	1	yes	characteristic
79	C₄²×C₇	SD₁₆	1	1	yes	characteristic
80	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
81	C₇×C₁₆	—	4	2	no	—
82	#532	—	2	1	no	—
83	#500	C₂²×C₇	1	1	yes	characteristic
84	#493	C₂²×C₇	1	1	yes	characteristic
85	#486	—	2	1	no	—
86	#466	—	2	1	no	—
87	#466	C₄×C₇	1	2	yes	—
88	C₄×D₈	—	2	1	no	—
89	C₄×D₈	C₄×C₇	1	2	yes	—
90	C₂×C₄⋊C₈	—	2	1	no	—
91	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
92	C₄⋊C₁₆	C₂²×C₇	1	2	yes	—
93	C₂²⋊C₁₆	—	2	1	no	—
94	C₄×C₁₆	—	2	1	no	—
95	#335	—	2	1	no	—
96	#335	C₂²×C₇	1	2	yes	—
97	C₂³×C₇	—	4	2	no	—
98	C₂³×C₇	#102	1	1	yes	characteristic
99	C₇×Q₈	—	8	1	no	—
100	C₇×Q₈	—	8	1	no	—
101	C₇×D₄	—	4	2	no	—
102	C₇×D₄	—	4	2	no	—
103	C₇×D₄	—	8	2	no	—
104	C₇×D₄	—	8	4	no	—
105	C₇×D₄	—	8	1	no	—
106	C₇×D₄	—	8	1	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	4	1	no	—
110	C₂×C₄×C₇	—	4	2	no	—
111	C₂×C₄×C₇	—	8	4	no	—
112	C₂×C₄×C₇	#102	1	1	yes	characteristic
113	C₂×C₄×C₇	#102	1	1	yes	characteristic
114	C₂×C₄×C₇	#102	1	1	yes	characteristic
115	C₂×C₄×C₇	D₁₆	1	2	yes	—
116	C₂×C₄×C₇	SD₃₂	1	2	yes	—
117	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
120	C₇×C₈	—	2	1	no	—
121	C₇×C₈	—	2	1	no	—
122	C₇×C₈	—	2	1	no	—
123	C₇×C₈	—	8	2	no	—
124	C₇×C₈	#135	1	2	yes	—
125	C₂×#41	—	4	1	no	—
126	C₂×D₈	—	2	1	no	—
127	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
128	C₂²×C₈	—	4	1	no	—
129	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
130	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
131	C₄⋊Q₈	—	2	1	no	—
132	#127	—	2	1	no	—
133	#124	—	4	1	no	—
134	#121	—	2	2	no	—
135	#121	—	4	1	no	—
136	C₄×D₄	—	2	2	no	—
137	C₄×D₄	—	4	1	no	—
138	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
139	C₂×C₁₆	—	2	2	no	—
140	#107	—	2	1	no	—
141	#107	C₂×C₄×C₇	1	2	yes	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	4	1	no	—
144	#102	—	2	2	no	—
145	#102	—	2	2	no	—
146	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
147	C₄×C₈	C₇×D₄	1	1	yes	characteristic
148	C₄×C₈	C₇×D₄	1	1	yes	characteristic
149	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
150	C₂²×C₇	—	2	1	no	—
151	C₂²×C₇	—	2	1	no	—
152	C₂²×C₇	—	8	2	no	—
153	C₂²×C₇	—	8	2	no	—
154	C₂²×C₇	—	8	2	no	—
155	C₂²×C₇	#357	1	2	yes	—
156	C₂²×C₇	C₂×#102	1	1	yes	characteristic
157	C₄×C₇	—	2	1	no	—
158	C₄×C₇	—	2	1	no	—
159	C₄×C₇	—	2	1	no	—
160	C₄×C₇	—	8	2	no	—
161	C₄×C₇	#357	1	2	yes	—
162	C₄×C₇	#413	1	1	yes	characteristic
163	C₄×C₇	#416	1	1	yes	characteristic
164	C₄×C₇	C₂×D₁₆	1	1	yes	characteristic
165	C₄×C₇	C₂×SD₃₂	1	1	yes	characteristic
166	#41	—	8	2	no	—
167	#41	—	8	2	no	—
168	C₂×Q₈	—	4	1	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂×D₄	—	4	1	no	—
172	C₂×D₄	—	4	1	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
175	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
176	C₂²×C₄	C₇×SD₁₆	1	1	yes	characteristic
177	D₈	—	4	2	no	—
178	D₈	—	4	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	8	2	no	—
183	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
184	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
186	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₂²⋊C₄	—	4	2	no	—
190	C₂²⋊C₄	—	4	2	no	—
191	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
192	C₄²	C₇×D₈	1	1	yes	characteristic
193	C₄²	C₇×SD₁₆	1	1	yes	characteristic
194	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
195	C₁₆	—	4	2	no	—
196	C₂×C₇	—	2	2	no	—
197	C₂×C₇	—	16	2	no	—
198	C₂×C₇	#1540	1	1	yes	characteristic
199	C₂×C₇	C₂×#357	1	1	yes	characteristic
200	C₂×C₇	#2148	1	1	yes	characteristic
201	C₂³	—	4	2	no	—
202	C₂³	C₇×#102	1	1	yes	characteristic
203	Q₈	—	8	1	no	—
204	Q₈	—	8	1	no	—
205	D₄	—	4	2	no	—
206	D₄	—	4	2	no	—
207	D₄	—	8	2	no	—
208	D₄	—	8	4	no	—
209	D₄	—	8	1	no	—
210	D₄	—	8	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	8	4	no	—
216	C₂×C₄	C₇×#102	1	1	yes	characteristic
217	C₂×C₄	C₇×#102	1	1	yes	characteristic
218	C₂×C₄	C₇×#102	1	1	yes	characteristic
219	C₂×C₄	C₇×D₁₆	1	2	yes	—
220	C₂×C₄	C₇×SD₃₂	1	2	yes	—
221	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
223	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	8	2	no	—
228	C₈	C₇×#135	1	2	yes	—
229	C₇	#11947	1	1	yes	characteristic, direct product factor, Sylow
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	8	2	no	—
233	C₂²	—	8	2	no	—
234	C₂²	—	8	2	no	—
235	C₂²	C₇×#357	1	2	yes	—
236	C₂²	C₂×C₇×#102	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	8	2	no	—
241	C₄	C₇×#357	1	2	yes	—
242	C₄	C₇×#413	1	1	yes	characteristic
243	C₄	C₇×#416	1	1	yes	characteristic
244	C₄	C₂×C₇×D₁₆	1	1	yes	characteristic
245	C₄	C₂×C₇×SD₃₂	1	1	yes	characteristic
246	C₂	—	2	2	no	—
247	C₂	—	16	2	no	—
248	C₂	C₇×#1540	1	1	yes	characteristic
249	C₂	C₂×C₇×#357	1	1	yes	characteristic
250	C₂	C₇×#2148	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×#102
prime radical	C₇×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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. C₂×C₇×#335	C₂	semidirect
4. C₇×#1364	C₂	semidirect
5. C₇×#1361	C₂	semidirect
21. #11947	C₇	direct (N × H)
48. #2162	C₂×C₇	semidirect
49. C₂×#335	C₂×C₇	semidirect
50. #1364	C₂×C₇	semidirect
51. #1361	C₂×C₇	semidirect
229. C₇	#11947	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×#3169	2
C₇×#2162	2
C₂×C₇×#335	2
C₇×#1364	2
C₇×#1361	2
#11947	7

Subgroups — C7×#11947 #13189163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11947 #13189163