Subgroups — C₇×#18090 #13195306

Order 1792.

564 conjugacy classes of subgroups.

The subgroup lattice has 280 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×#2986	C₂	1	2	yes	maximal
3	C₇×#1593	C₂	1	2	yes	maximal, complemented
4	C₇×#1584	C₂	1	4	yes	maximal, complemented
5	C₇×#1579	C₂	1	2	yes	maximal, complemented
6	C₇×#1564	C₂	1	1	yes	maximal, characteristic
7	C₇×#1485	C₂	1	2	yes	maximal
8	C₇×#1451	C₂	1	1	yes	maximal, characteristic
9	C₇×#533	C₂²	1	2	yes	—
10	C₇×#500	C₄	1	1	yes	characteristic
11	C₇×#494	C₄	1	1	yes	characteristic
12	C₇×#487	C₄	1	2	yes	—
13	C₇×#467	C₄	1	2	yes	—
14	C₇×#467	C₄	1	2	yes	—
15	C₇×#446	—	2	2	no	—
16	C₇×#446	C₂²	1	4	yes	—
17	C₇×C₈×Q₈	—	2	2	no	—
18	C₇×C₈×Q₈	C₂²	1	4	yes	—
19	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
20	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
21	C₇×#433	—	2	4	no	—
22	C₇×#431	C₂²	1	2	yes	—
23	C₇×#431	C₂²	1	1	yes	characteristic
24	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
26	C₇×#335	C₂²	1	2	yes	—
27	C₇×#334	C₂²	1	1	yes	characteristic
28	C₇×#334	C₂²	1	1	yes	characteristic
29	C₇×Q₈⋊C₈	C₂²	1	4	yes	—
30	C₇×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₇×#322	C₂²	1	2	yes	—
32	C₇×#322	C₂²	1	2	yes	—
33	#18090	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₇×M₁₆	—	2	1	no	—
37	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
39	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
41	C₇×#125	—	2	2	no	—
42	C₇×C₂²⋊Q₈	—	2	2	no	—
43	C₇×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
44	C₄×C₇×Q₈	—	2	2	no	—
45	C₄×C₇×Q₈	C₂³	1	4	yes	—
46	C₇×#117	—	2	2	no	—
47	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
48	C₇×#107	C₂×C₄	1	2	yes	—
49	C₇×#107	C₂×C₄	1	2	yes	—
50	C₇×C₄⋊C₈	—	2	4	no	—
51	C₇×C₄⋊C₈	—	2	4	no	—
52	C₇×C₄⋊C₈	D₄	1	2	yes	—
53	C₇×C₄⋊C₈	C₂³	1	2	yes	—
54	C₇×C₄⋊C₈	C₂³	1	2	yes	—
55	C₇×#103	C₂×C₄	1	4	yes	—
56	C₇×#103	C₂×C₄	1	4	yes	—
57	C₇×C₂²⋊C₈	—	2	2	no	—
58	C₇×C₂²⋊C₈	D₄	1	2	yes	—
59	C₇×#97	—	2	4	no	—
60	C₇×#97	—	2	1	no	—
61	C₄×C₇×C₈	—	2	4	no	—
62	C₄×C₇×C₈	—	2	2	no	—
63	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
64	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
65	C₄×C₇×C₈	C₂³	1	2	yes	—
66	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
68	C₄×C₇×C₈	C₂³	1	2	yes	—
69	#3170	C₂×C₇	1	1	yes	characteristic
70	#2986	C₂×C₇	1	2	yes	—
71	#1593	C₂×C₇	1	2	yes	complemented
72	#1584	C₂×C₇	1	4	yes	complemented
73	#1579	C₂×C₇	1	2	yes	complemented
74	#1564	C₂×C₇	1	1	yes	characteristic
75	#1485	C₂×C₇	1	2	yes	—
76	#1451	C₂×C₇	1	1	yes	characteristic
77	C₂×C₇×Q₈	—	2	2	no	—
78	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	#41	1	1	yes	characteristic
82	C₇×Q₁₆	—	2	4	no	—
83	C₇×Q₁₆	—	4	2	no	—
84	C₇×M₁₆	—	4	2	no	—
85	C₂×C₇×C₈	—	2	2	no	—
86	C₂×C₇×C₈	—	2	2	no	—
87	C₂×C₇×C₈	—	2	2	no	—
88	C₂×C₇×C₈	—	2	2	no	—
89	C₂×C₇×C₈	—	4	1	no	—
90	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
95	C₇×C₄⋊C₄	—	2	2	no	—
96	C₇×C₄⋊C₄	—	2	2	no	—
97	C₇×C₄⋊C₄	—	2	2	no	—
98	C₇×C₄⋊C₄	—	2	4	no	—
99	C₇×C₄⋊C₄	—	2	4	no	—
100	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
101	C₇×C₂²⋊C₄	—	2	4	no	—
102	C₄²×C₇	—	2	4	no	—
103	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇	#41	1	1	yes	characteristic
106	C₄²×C₇	C₂⁴	1	1	yes	characteristic
107	#533	C₂²×C₇	1	2	yes	—
108	#500	C₄×C₇	1	1	yes	characteristic
109	#494	C₄×C₇	1	1	yes	characteristic
110	#487	C₄×C₇	1	2	yes	—
111	#467	C₄×C₇	1	2	yes	—
112	#467	C₄×C₇	1	2	yes	—
113	#446	—	2	2	no	—
114	#446	C₂²×C₇	1	4	yes	—
115	C₈×Q₈	—	2	2	no	—
116	C₈×Q₈	C₂²×C₇	1	4	yes	—
117	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
118	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
119	#433	—	2	4	no	—
120	#431	C₂²×C₇	1	2	yes	—
121	#431	C₂²×C₇	1	1	yes	characteristic
122	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
124	#335	C₂²×C₇	1	2	yes	—
125	#334	C₂²×C₇	1	1	yes	characteristic
126	#334	C₂²×C₇	1	1	yes	characteristic
127	Q₈⋊C₈	C₂²×C₇	1	4	yes	—
128	Q₈⋊C₈	C₂²×C₇	1	4	yes	—
129	#322	C₂²×C₇	1	2	yes	—
130	#322	C₂²×C₇	1	2	yes	—
131	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
132	C₇×Q₈	—	2	4	no	—
133	C₇×Q₈	—	2	4	no	—
134	C₇×Q₈	—	4	4	no	—
135	C₂×C₄×C₇	—	2	1	no	—
136	C₂×C₄×C₇	—	2	1	no	—
137	C₂×C₄×C₇	—	2	2	no	—
138	C₂×C₄×C₇	—	2	2	no	—
139	C₂×C₄×C₇	—	2	4	no	—
140	C₂×C₄×C₇	—	2	4	no	—
141	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
147	C₇×C₈	—	2	1	no	—
148	C₇×C₈	—	2	1	no	—
149	C₇×C₈	—	2	1	no	—
150	C₇×C₈	—	2	4	no	—
151	C₇×C₈	—	4	2	no	—
152	C₇×C₈	—	4	2	no	—
153	C₇×C₈	—	4	2	no	—
154	C₇×C₈	#131	1	2	yes	—
155	C₂×Q₁₆	C₂×C₄×C₇	1	2	yes	—
156	C₂×Q₁₆	C₂×C₄×C₇	1	2	yes	—
157	C₂×M₁₆	—	2	1	no	—
158	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
159	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
160	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
161	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
162	#125	—	2	2	no	—
163	C₂²⋊Q₈	—	2	2	no	—
164	C₂²⋊Q₈	C₂×C₄×C₇	1	4	yes	—
165	C₄×Q₈	—	2	2	no	—
166	C₄×Q₈	C₂³×C₇	1	4	yes	—
167	#117	—	2	2	no	—
168	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
169	#107	C₂×C₄×C₇	1	2	yes	—
170	#107	C₂×C₄×C₇	1	2	yes	—
171	C₄⋊C₈	—	2	4	no	—
172	C₄⋊C₈	—	2	4	no	—
173	C₄⋊C₈	C₇×D₄	1	2	yes	—
174	C₄⋊C₈	C₂³×C₇	1	2	yes	—
175	C₄⋊C₈	C₂³×C₇	1	2	yes	—
176	#103	C₂×C₄×C₇	1	4	yes	—
177	#103	C₂×C₄×C₇	1	4	yes	—
178	C₂²⋊C₈	—	2	2	no	—
179	C₂²⋊C₈	C₇×D₄	1	2	yes	—
180	#97	—	2	4	no	—
181	#97	—	2	1	no	—
182	C₄×C₈	—	2	4	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
185	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
186	C₄×C₈	C₂³×C₇	1	2	yes	—
187	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
189	C₄×C₈	C₂³×C₇	1	2	yes	—
190	C₂²×C₇	—	2	1	no	—
191	C₂²×C₇	—	2	1	no	—
192	C₂²×C₇	—	2	1	no	—
193	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
194	C₄×C₇	—	2	1	no	—
195	C₄×C₇	—	2	1	no	—
196	C₄×C₇	—	2	1	no	—
197	C₄×C₇	—	4	4	no	—
198	C₄×C₇	—	4	4	no	—
199	C₄×C₇	#444	1	2	yes	—
200	C₄×C₇	C₂×#131	1	1	yes	characteristic
201	C₄×C₇	#568	1	1	yes	characteristic
202	C₄×C₇	#578	1	2	yes	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
205	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
207	C₂²×C₄	C₇×#41	1	1	yes	characteristic
208	Q₁₆	—	2	4	no	—
209	Q₁₆	—	4	2	no	—
210	M₁₆	—	4	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
217	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
218	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
219	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
220	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₄²	—	2	4	no	—
229	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
230	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
231	C₄²	C₇×#41	1	1	yes	characteristic
232	C₄²	C₂⁴×C₇	1	1	yes	characteristic
233	C₂×C₇	—	4	1	no	—
234	C₂×C₇	#2935	1	1	yes	characteristic
235	C₂×C₇	#2950	1	1	yes	characteristic
236	C₂×C₇	C₂×#444	1	1	yes	characteristic
237	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
238	Q₈	—	2	4	no	—
239	Q₈	—	2	4	no	—
240	Q₈	—	4	4	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
248	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
249	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
250	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
252	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	4	no	—
257	C₈	—	4	2	no	—
258	C₈	—	4	2	no	—
259	C₈	—	4	2	no	—
260	C₈	C₇×#131	1	2	yes	—
261	C₇	#18090	1	1	yes	characteristic, direct product factor, Sylow
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	4	4	no	—
270	C₄	—	4	4	no	—
271	C₄	C₇×#444	1	2	yes	—
272	C₄	C₂×C₇×#131	1	1	yes	characteristic
273	C₄	C₇×#568	1	1	yes	characteristic
274	C₄	C₇×#578	1	2	yes	—
275	C₂	—	4	1	no	—
276	C₂	C₇×#2935	1	1	yes	characteristic
277	C₂	C₇×#2950	1	1	yes	characteristic
278	C₂	C₂×C₇×#444	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 ◁ C₇×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1593	C₂	semidirect
4. C₇×#1584	C₂	semidirect
5. C₇×#1579	C₂	semidirect
33. #18090	C₇	direct (N × H)
71. #1593	C₂×C₇	semidirect
72. #1584	C₂×C₇	semidirect
73. #1579	C₂×C₇	semidirect
261. C₇	#18090	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×#2986	2
C₇×#1593	2
C₇×#1584	2
C₇×#1579	2
C₇×#1564	2
C₇×#1485	2
C₇×#1451	2
#18090	7

Subgroups — C7×#18090 #13195306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#18090 #13195306