Subgroups — C₇×#7986 #13185202

Order 1792.

546 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#432	C₂	1	2	yes	maximal
2	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic
3	C₇×#1290	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
5	C₇×#432	—	2	2	no	—
6	C₇×#432	—	2	2	no	—
7	C₇×#432	C₄	1	4	yes	—
8	C₇×#432	C₂²	1	4	yes	—
9	C₇×#431	—	2	2	no	—
10	C₇×#431	—	2	2	no	—
11	C₂×C₇×C₄⋊C₈	—	2	2	no	—
12	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₇×C₂²⋊C₈	—	2	2	no	—
14	C₂×C₇×C₂²⋊C₈	C₄	1	2	yes	—
15	C₂×C₇×#97	C₄	1	4	yes	—
16	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
17	#7986	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₇	Q₈	1	1	yes	characteristic
21	C₂²×C₇×C₈	—	2	4	no	—
22	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₇	D₄	1	2	yes	—
28	C₂×C₄²×C₇	D₄	1	2	yes	—
29	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
31	C₇×C₄⋊C₈	—	2	4	no	—
32	C₇×C₄⋊C₈	—	2	4	no	—
33	C₇×C₄⋊C₈	—	2	4	no	—
34	C₇×C₂²⋊C₈	—	2	4	no	—
35	C₇×C₂²⋊C₈	—	2	4	no	—
36	C₇×C₂²⋊C₈	—	2	4	no	—
37	C₇×#97	—	2	4	no	—
38	C₇×#97	C₂×C₄	1	8	yes	—
39	C₄×C₇×C₈	—	2	2	no	—
40	C₄×C₇×C₈	—	2	2	no	—
41	C₂×#432	C₂×C₇	1	2	yes	—
42	C₂×#431	C₂×C₇	1	1	yes	characteristic
43	#1290	C₂×C₇	1	4	yes	complemented
44	C₂⁴×C₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇	—	2	4	no	—
46	C₂²×C₄×C₇	—	4	2	no	—
47	C₂²×C₄×C₇	—	4	2	no	—
48	C₂²×C₄×C₇	C₄²	1	2	yes	—
49	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₇×C₈	—	2	4	no	—
61	C₂×C₇×C₈	—	2	8	no	—
62	C₂×C₇×C₈	—	4	8	no	—
63	C₂×C₇×C₈	—	4	4	no	—
64	C₄²×C₇	C₄²	1	1	yes	characteristic
65	C₄²×C₇	C₄²	1	1	yes	characteristic
66	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₇	C₄⋊C₄	1	2	yes	—
71	C₄²×C₇	C₄⋊C₄	1	2	yes	—
72	C₄²×C₇	C₂²×C₄	1	2	yes	—
73	C₄²×C₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
75	#432	—	2	2	no	—
76	#432	—	2	2	no	—
77	#432	C₄×C₇	1	4	yes	—
78	#432	C₂²×C₇	1	4	yes	—
79	#431	—	2	2	no	—
80	#431	—	2	2	no	—
81	C₂×C₄⋊C₈	—	2	2	no	—
82	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
83	C₂×C₂²⋊C₈	—	2	2	no	—
84	C₂×C₂²⋊C₈	C₄×C₇	1	2	yes	—
85	C₂×#97	C₄×C₇	1	4	yes	—
86	C₂×C₄×C₈	C₄×C₇	1	2	yes	—
87	C₂³×C₇	—	4	1	no	—
88	C₂³×C₇	—	4	1	no	—
89	C₂³×C₇	#95	1	2	yes	—
90	C₂³×C₇	#108	1	4	yes	—
91	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₇	—	2	4	no	—
93	C₂×C₄×C₇	—	2	4	no	—
94	C₂×C₄×C₇	—	2	2	no	—
95	C₂×C₄×C₇	—	2	2	no	—
96	C₂×C₄×C₇	—	2	4	no	—
97	C₂×C₄×C₇	—	2	2	no	—
98	C₂×C₄×C₇	—	2	2	no	—
99	C₂×C₄×C₇	—	2	4	no	—
100	C₂×C₄×C₇	—	4	2	no	—
101	C₂×C₄×C₇	—	4	2	no	—
102	C₂×C₄×C₇	—	4	1	no	—
103	C₂×C₄×C₇	—	4	1	no	—
104	C₂×C₄×C₇	#95	1	1	yes	characteristic
105	C₂×C₄×C₇	#95	1	1	yes	characteristic
106	C₂×C₄×C₇	#95	1	1	yes	characteristic
107	C₂×C₄×C₇	#95	1	2	yes	—
108	C₂×C₄×C₇	#95	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
111	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₇×C₈	—	4	8	no	—
114	C₇×C₈	—	4	4	no	—
115	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
116	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
117	C₂³×C₄	C₇×Q₈	1	1	yes	characteristic
118	C₂²×C₈	—	2	4	no	—
119	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
120	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
121	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
122	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
123	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
124	C₂×C₄²	C₇×D₄	1	2	yes	—
125	C₂×C₄²	C₇×D₄	1	2	yes	—
126	C₂×C₄²	C₇×Q₈	1	1	yes	characteristic
127	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
128	C₄⋊C₈	—	2	4	no	—
129	C₄⋊C₈	—	2	4	no	—
130	C₄⋊C₈	—	2	4	no	—
131	C₂²⋊C₈	—	2	4	no	—
132	C₂²⋊C₈	—	2	4	no	—
133	C₂²⋊C₈	—	2	4	no	—
134	#97	—	2	4	no	—
135	#97	C₂×C₄×C₇	1	8	yes	—
136	C₄×C₈	—	2	2	no	—
137	C₄×C₈	—	2	2	no	—
138	C₂²×C₇	—	2	2	no	—
139	C₂²×C₇	—	2	1	no	—
140	C₂²×C₇	—	2	2	no	—
141	C₂²×C₇	—	2	1	no	—
142	C₂²×C₇	—	4	4	no	—
143	C₂²×C₇	#341	1	4	yes	—
144	C₂²×C₇	C₂×#95	1	1	yes	characteristic
145	C₂²×C₇	C₂×#108	1	2	yes	—
146	C₄×C₇	—	2	2	no	—
147	C₄×C₇	—	2	1	no	—
148	C₄×C₇	—	2	1	no	—
149	C₄×C₇	—	2	1	no	—
150	C₄×C₇	—	2	2	no	—
151	C₄×C₇	—	2	2	no	—
152	C₄×C₇	—	2	1	no	—
153	C₄×C₇	H(3,ℤ/4)	1	2	yes	—
154	C₄×C₇	#413	1	2	yes	—
155	C₂⁴	C₇×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	C₄²×C₇	1	2	yes	—
160	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
161	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
162	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
163	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
164	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
165	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
166	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
167	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
168	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	8	no	—
173	C₂×C₈	—	4	8	no	—
174	C₂×C₈	—	4	4	no	—
175	C₄²	C₄²×C₇	1	1	yes	characteristic
176	C₄²	C₄²×C₇	1	1	yes	characteristic
177	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
178	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
179	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
180	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
181	C₄²	C₇×C₄⋊C₄	1	2	yes	—
182	C₄²	C₇×C₄⋊C₄	1	2	yes	—
183	C₄²	C₂²×C₄×C₇	1	2	yes	—
184	C₄²	C₂×C₇×D₄	1	2	yes	—
185	C₂×C₇	—	2	4	no	—
186	C₂×C₇	—	2	2	no	—
187	C₂×C₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic
188	C₂×C₇	C₂×#341	1	1	yes	characteristic
189	C₂×C₇	#1742	1	1	yes	characteristic
190	C₂³	—	4	1	no	—
191	C₂³	—	4	1	no	—
192	C₂³	C₇×#95	1	2	yes	—
193	C₂³	C₇×#108	1	4	yes	—
194	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	C₇×#95	1	1	yes	characteristic
208	C₂×C₄	C₇×#95	1	1	yes	characteristic
209	C₂×C₄	C₇×#95	1	1	yes	characteristic
210	C₂×C₄	C₇×#95	1	2	yes	—
211	C₂×C₄	C₇×#95	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
213	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
214	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
216	C₈	—	4	8	no	—
217	C₈	—	4	4	no	—
218	C₇	#7986	1	1	yes	characteristic, direct product factor, Sylow
219	C₂²	—	2	2	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	4	4	no	—
224	C₂²	C₇×#341	1	4	yes	—
225	C₂²	C₂×C₇×#95	1	1	yes	characteristic
226	C₂²	C₂×C₇×#108	1	2	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	1	no	—
234	C₄	C₇×H(3,ℤ/4)	1	2	yes	—
235	C₄	C₇×#413	1	2	yes	—
236	C₂	—	2	4	no	—
237	C₂	—	2	2	no	—
238	C₂	C₂×C₇×H(3,ℤ/4)	1	1	yes	characteristic
239	C₂	C₂×C₇×#341	1	1	yes	characteristic
240	C₂	C₇×#1742	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	#95
commutator	C₂×C₄	C₂×C₄²×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	C₂×#95
prime radical	C₂⁴×C₇	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1290	C₂	semidirect
17. #7986	C₇	direct (N × H)
43. #1290	C₂×C₇	semidirect
218. C₇	#7986	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#432	2
C₂×C₇×#431	2
C₇×#1290	2
#7986	7

Subgroups — C7×#7986 #13185202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#7986 #13185202