Subgroups — C₇×#17980 #13195196

Order 1792.

590 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	C₇×#3170	C₂	1	1	yes	maximal, characteristic
2	C₇×#2960	C₂	1	2	yes	maximal
3	C₇×#1587	C₂	1	8	yes	maximal, complemented
4	C₇×#1566	C₂	1	1	yes	maximal, characteristic
5	C₇×#1493	C₂	1	2	yes	maximal
6	C₂×C₇×#322	C₂	1	1	yes	maximal, characteristic
7	C₇×#533	C₂²	1	2	yes	—
8	C₇×#500	C₄	1	1	yes	characteristic
9	C₇×#494	C₄	1	1	yes	characteristic
10	C₇×#487	C₄	1	2	yes	—
11	C₇×#467	C₄	1	4	yes	—
12	C₇×#446	—	2	2	no	—
13	C₇×#446	C₂²	1	4	yes	—
14	C₇×C₈×Q₈	—	2	2	no	—
15	C₇×C₈×Q₈	C₂²	1	4	yes	—
16	C₄×C₇×Q₁₆	C₂²	1	4	yes	—
17	C₇×#424	—	2	2	no	—
18	C₇×#423	C₂²	1	2	yes	—
19	C₇×#405	—	2	2	no	—
20	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
23	C₇×#334	C₂²	1	4	yes	—
24	C₇×Q₈⋊C₈	C₂²	1	8	yes	—
25	C₇×#322	C₂²	1	4	yes	—
26	#17980	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	—
28	C₂×C₇×M₁₆	—	2	2	no	—
29	C₂²×C₇×C₈	—	2	1	no	—
30	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₇×C₈	D₄	1	2	yes	—
32	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
33	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
34	C₇×#125	—	2	2	no	—
35	C₇×C₂²⋊Q₈	—	2	2	no	—
36	C₇×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
37	C₄×C₇×Q₈	—	2	2	no	—
38	C₄×C₇×Q₈	C₂³	1	4	yes	—
39	C₇×#117	—	2	2	no	—
40	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
41	C₇×#107	C₂×C₄	1	4	yes	—
42	C₇×C₄⋊C₈	—	2	4	no	—
43	C₇×C₄⋊C₈	—	2	4	no	—
44	C₇×C₄⋊C₈	—	2	2	no	—
45	C₇×C₄⋊C₈	C₂³	1	4	yes	—
46	C₇×#103	C₂×C₄	1	8	yes	—
47	C₇×#97	—	2	4	no	—
48	C₄×C₇×C₈	—	2	4	no	—
49	C₄×C₇×C₈	—	2	1	no	—
50	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₇×C₈	D₄	1	1	yes	characteristic
53	C₄×C₇×C₈	D₄	1	1	yes	characteristic
54	C₄×C₇×C₈	C₂³	1	2	yes	—
55	C₄×C₇×C₈	C₂³	1	2	yes	—
56	C₄×C₇×C₈	C₂³	1	2	yes	—
57	#3170	C₂×C₇	1	1	yes	characteristic
58	#2960	C₂×C₇	1	2	yes	—
59	#1587	C₂×C₇	1	8	yes	complemented
60	#1566	C₂×C₇	1	1	yes	characteristic
61	#1493	C₂×C₇	1	2	yes	—
62	C₂×#322	C₂×C₇	1	1	yes	characteristic
63	C₂×C₇×Q₈	—	2	2	no	—
64	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	#41	1	1	yes	characteristic
68	C₇×Q₁₆	—	4	4	no	—
69	C₇×M₁₆	—	4	4	no	—
70	C₂×C₇×C₈	—	2	4	no	—
71	C₂×C₇×C₈	—	2	4	no	—
72	C₂×C₇×C₈	—	2	2	no	—
73	C₂×C₇×C₈	—	2	4	no	—
74	C₂×C₇×C₈	—	4	2	no	—
75	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
76	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
77	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
78	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
79	C₇×C₄⋊C₄	—	2	2	no	—
80	C₇×C₄⋊C₄	—	2	2	no	—
81	C₇×C₄⋊C₄	—	2	2	no	—
82	C₇×C₄⋊C₄	—	2	4	no	—
83	C₇×C₄⋊C₄	—	2	4	no	—
84	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
85	C₇×C₂²⋊C₄	—	2	4	no	—
86	C₄²×C₇	—	2	4	no	—
87	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₇	#41	1	1	yes	characteristic
90	C₄²×C₇	C₂⁴	1	1	yes	characteristic
91	#533	C₂²×C₇	1	2	yes	—
92	#500	C₄×C₇	1	1	yes	characteristic
93	#494	C₄×C₇	1	1	yes	characteristic
94	#487	C₄×C₇	1	2	yes	—
95	#467	C₄×C₇	1	4	yes	—
96	#446	—	2	2	no	—
97	#446	C₂²×C₇	1	4	yes	—
98	C₈×Q₈	—	2	2	no	—
99	C₈×Q₈	C₂²×C₇	1	4	yes	—
100	C₄×Q₁₆	C₂²×C₇	1	4	yes	—
101	#424	—	2	2	no	—
102	#423	C₂²×C₇	1	2	yes	—
103	#405	—	2	2	no	—
104	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
107	#334	C₂²×C₇	1	4	yes	—
108	Q₈⋊C₈	C₂²×C₇	1	8	yes	—
109	#322	C₂²×C₇	1	4	yes	—
110	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
111	C₇×Q₈	—	2	8	no	—
112	C₇×Q₈	—	4	4	no	—
113	C₂×C₄×C₇	—	2	2	no	—
114	C₂×C₄×C₇	—	2	2	no	—
115	C₂×C₄×C₇	—	2	4	no	—
116	C₂×C₄×C₇	—	2	4	no	—
117	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇	#137	1	4	yes	—
121	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
124	C₇×C₈	—	2	4	no	—
125	C₇×C₈	—	2	2	no	—
126	C₇×C₈	—	2	2	no	—
127	C₇×C₈	—	2	2	no	—
128	C₇×C₈	—	2	2	no	—
129	C₇×C₈	—	4	4	no	—
130	C₂×Q₁₆	C₂×C₄×C₇	1	4	yes	—
131	C₂×M₁₆	—	2	2	no	—
132	C₂²×C₈	—	2	1	no	—
133	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
134	C₂²×C₈	C₇×D₄	1	2	yes	—
135	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
136	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
137	#125	—	2	2	no	—
138	C₂²⋊Q₈	—	2	2	no	—
139	C₂²⋊Q₈	C₂×C₄×C₇	1	4	yes	—
140	C₄×Q₈	—	2	2	no	—
141	C₄×Q₈	C₂³×C₇	1	4	yes	—
142	#117	—	2	2	no	—
143	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
144	#107	C₂×C₄×C₇	1	4	yes	—
145	C₄⋊C₈	—	2	4	no	—
146	C₄⋊C₈	—	2	4	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	C₂³×C₇	1	4	yes	—
149	#103	C₂×C₄×C₇	1	8	yes	—
150	#97	—	2	4	no	—
151	C₄×C₈	—	2	4	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
154	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
155	C₄×C₈	C₇×D₄	1	1	yes	characteristic
156	C₄×C₈	C₇×D₄	1	1	yes	characteristic
157	C₄×C₈	C₂³×C₇	1	2	yes	—
158	C₄×C₈	C₂³×C₇	1	2	yes	—
159	C₄×C₈	C₂³×C₇	1	2	yes	—
160	C₂²×C₇	—	2	1	no	—
161	C₂²×C₇	—	2	1	no	—
162	C₂²×C₇	#441	1	2	yes	—
163	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
164	C₄×C₇	—	2	1	no	—
165	C₄×C₇	—	2	1	no	—
166	C₄×C₇	—	4	4	no	—
167	C₄×C₇	—	4	4	no	—
168	C₄×C₇	#441	1	2	yes	—
169	C₄×C₇	#443	1	2	yes	—
170	C₄×C₇	#568	1	2	yes	—
171	C₄×C₇	C₂×#137	1	2	yes	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
174	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
176	C₂²×C₄	C₇×#41	1	1	yes	characteristic
177	Q₁₆	—	4	4	no	—
178	M₁₆	—	4	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
185	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
186	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
187	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
194	C₂²⋊C₄	—	2	4	no	—
195	C₄²	—	2	4	no	—
196	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
197	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
198	C₄²	C₇×#41	1	1	yes	characteristic
199	C₄²	C₂⁴×C₇	1	1	yes	characteristic
200	C₂×C₇	—	2	2	no	—
201	C₂×C₇	#2932	1	1	yes	characteristic
202	C₂×C₇	C₂×#441	1	1	yes	characteristic
203	C₂×C₇	C₂×#443	1	1	yes	characteristic
204	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
205	Q₈	—	2	8	no	—
206	Q₈	—	4	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
212	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
213	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
214	C₂×C₄	C₇×#137	1	4	yes	—
215	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
216	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
217	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
218	C₈	—	2	4	no	—
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	4	4	no	—
224	C₇	#17980	1	1	yes	characteristic, direct product factor, Sylow
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	C₇×#441	1	2	yes	—
228	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	4	no	—
232	C₄	—	4	4	no	—
233	C₄	C₇×#441	1	2	yes	—
234	C₄	C₇×#443	1	2	yes	—
235	C₄	C₇×#568	1	2	yes	—
236	C₄	C₂×C₇×#137	1	2	yes	—
237	C₂	—	2	2	no	—
238	C₂	C₇×#2932	1	1	yes	characteristic
239	C₂	C₂×C₇×#441	1	1	yes	characteristic
240	C₂	C₂×C₇×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂³×C₇	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1587	C₂	semidirect
26. #17980	C₇	direct (N × H)
59. #1587	C₂×C₇	semidirect
224. C₇	#17980	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×#3170	2
C₇×#2960	2
C₇×#1587	2
C₇×#1566	2
C₇×#1493	2
C₂×C₇×#322	2
#17980	7

Subgroups — C7×#17980 #13195196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#17980 #13195196