Subgroups — C₂×C₇×#2090 #13200889

Order 1792.

810 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#428	C₂	1	2	yes	maximal
2	C₂²×C₇×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#424	C₂	1	1	yes	maximal, characteristic
4	C₇×#2090	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#343	C₂	1	2	yes	maximal
6	C₂×C₇×#341	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
9	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×#117	C₂²	1	1	yes	characteristic
11	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₇×#428	—	2	4	no	—
13	C₇×#428	C₂²	1	8	yes	—
14	C₂×C₇×#107	—	2	2	no	—
15	C₂×C₇×#106	—	2	2	no	—
16	C₂×C₇×#106	—	2	2	no	—
17	C₂×C₇×#106	—	2	2	no	—
18	C₂×C₇×#106	C₂²	1	4	yes	—
19	C₇×#424	—	2	2	no	—
20	C₇×#424	C₂²	1	4	yes	—
21	C₂×C₇×C₄⋊C₈	—	2	2	no	—
22	C₇×#343	C₂²	1	8	yes	—
23	C₇×#341	C₂²	1	4	yes	—
24	C₂×#2090	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂³×C₄×C₇	—	2	2	no	—
26	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂×C₇×M₁₆	—	2	2	no	—
28	C₂×C₇×M₁₆	—	2	4	no	—
29	C₂×C₇×M₁₆	—	2	1	no	—
30	C₂×C₇×M₁₆	—	2	4	no	—
31	C₂×C₇×M₁₆	Q₈	1	8	yes	—
32	C₂×C₇×M₁₆	C₂³	1	2	yes	—
33	C₂×C₇×M₁₆	C₂³	1	4	yes	—
34	C₂²×C₇×C₈	—	2	2	no	—
35	C₂²×C₇×C₈	—	2	4	no	—
36	C₂²×C₇×C₈	—	2	1	no	—
37	C₂²×C₇×C₈	D₄	1	2	yes	—
38	C₂²×C₇×C₈	D₄	1	2	yes	—
39	C₂²×C₇×C₈	C₂³	1	2	yes	—
40	C₇×#117	—	2	1	no	—
41	C₇×#117	—	2	1	no	—
42	C₇×#117	D₄	1	2	yes	—
43	C₇×#117	C₂³	1	2	yes	—
44	C₂×C₇×C₄⋊C₄	—	2	2	no	—
45	C₂×C₇×C₄⋊C₄	—	2	2	no	—
46	C₂×C₇×C₄⋊C₄	—	2	2	no	—
47	C₂×C₇×C₄⋊C₄	—	2	2	no	—
48	C₂×C₇×C₄⋊C₄	—	2	2	no	—
49	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
50	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
51	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
52	C₂×C₄²×C₇	—	2	1	no	—
53	C₇×#107	—	2	8	no	—
54	C₇×#106	—	2	8	no	—
55	C₇×#106	—	2	8	no	—
56	C₇×C₄⋊C₈	—	2	8	no	—
57	C₂×#428	C₂×C₇	1	2	yes	—
58	C₂²×#106	C₂×C₇	1	1	yes	characteristic
59	C₂×#424	C₂×C₇	1	1	yes	characteristic
60	#2090	C₂×C₇	1	8	yes	direct product factor
61	C₂×#343	C₂×C₇	1	2	yes	—
62	C₂×#341	C₂×C₇	1	1	yes	characteristic
63	C₂⁴×C₇	#41	1	1	yes	characteristic
64	C₂²×C₄×C₇	—	2	2	no	—
65	C₂²×C₄×C₇	—	2	4	no	—
66	C₂²×C₄×C₇	—	4	2	no	—
67	C₂²×C₄×C₇	—	4	2	no	—
68	C₂²×C₄×C₇	—	8	4	no	—
69	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
74	C₂²×C₄×C₇	#41	1	1	yes	characteristic
75	C₂²×C₄×C₇	#41	1	4	yes	—
76	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
77	C₇×M₁₆	—	2	8	no	—
78	C₇×M₁₆	—	2	4	no	—
79	C₇×M₁₆	—	2	4	no	—
80	C₇×M₁₆	—	4	2	no	—
81	C₇×M₁₆	C₂×Q₈	1	8	yes	—
82	C₂×C₇×C₈	—	2	8	no	—
83	C₂×C₇×C₈	—	2	4	no	—
84	C₂×C₇×C₈	—	2	4	no	—
85	C₂×C₇×C₈	—	2	4	no	—
86	C₂×C₇×C₈	—	2	4	no	—
87	C₂×C₇×C₈	—	2	2	no	—
88	C₂×C₇×C₈	—	4	2	no	—
89	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
90	C₇×C₄⋊C₄	—	2	2	no	—
91	C₇×C₄⋊C₄	—	2	4	no	—
92	C₇×C₄⋊C₄	—	2	4	no	—
93	C₇×C₄⋊C₄	—	4	2	no	—
94	C₇×C₄⋊C₄	—	4	2	no	—
95	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
96	C₇×C₂²⋊C₄	—	2	2	no	—
97	C₇×C₂²⋊C₄	—	2	2	no	—
98	C₄²×C₇	—	2	2	no	—
99	C₄²×C₇	—	2	2	no	—
100	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
101	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
102	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
103	C₂×#117	C₂²×C₇	1	1	yes	characteristic
104	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
105	#428	—	2	4	no	—
106	#428	C₂²×C₇	1	8	yes	—
107	C₂×#107	—	2	2	no	—
108	C₂×#106	—	2	2	no	—
109	C₂×#106	—	2	2	no	—
110	C₂×#106	—	2	2	no	—
111	C₂×#106	C₂²×C₇	1	4	yes	—
112	#424	—	2	2	no	—
113	#424	C₂²×C₇	1	4	yes	—
114	C₂×C₄⋊C₈	—	2	2	no	—
115	#343	C₂²×C₇	1	8	yes	—
116	#341	C₂²×C₇	1	4	yes	—
117	C₂³×C₇	—	2	4	no	—
118	C₂³×C₇	#121	1	1	yes	characteristic
119	C₂³×C₇	#121	1	1	yes	characteristic
120	C₂³×C₇	#125	1	4	yes	—
121	C₂³×C₇	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₇	—	2	4	no	—
123	C₂×C₄×C₇	—	4	4	no	—
124	C₂×C₄×C₇	—	4	4	no	—
125	C₂×C₄×C₇	—	8	4	no	—
126	C₂×C₄×C₇	—	8	4	no	—
127	C₂×C₄×C₇	—	8	4	no	—
128	C₂×C₄×C₇	#121	1	1	yes	characteristic
129	C₂×C₄×C₇	#121	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂²⋊Q₈	1	4	yes	—
131	C₂×C₄×C₇	C₂²⋊Q₈	1	4	yes	—
132	C₂×C₄×C₇	#123	1	2	yes	—
133	C₂×C₄×C₇	#123	1	2	yes	—
134	C₂×C₄×C₇	#125	1	4	yes	—
135	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
136	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
137	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
140	C₂×C₄×C₇	C₂×#41	1	2	yes	—
141	C₇×C₈	—	2	8	no	—
142	C₇×C₈	—	2	4	no	—
143	C₇×C₈	—	4	2	no	—
144	C₂³×C₄	—	2	2	no	—
145	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	—	2	4	no	—
148	C₂×M₁₆	—	2	1	no	—
149	C₂×M₁₆	—	2	4	no	—
150	C₂×M₁₆	C₇×Q₈	1	8	yes	—
151	C₂×M₁₆	C₂³×C₇	1	2	yes	—
152	C₂×M₁₆	C₂³×C₇	1	4	yes	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	4	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	C₇×D₄	1	2	yes	—
157	C₂²×C₈	C₇×D₄	1	2	yes	—
158	C₂²×C₈	C₂³×C₇	1	2	yes	—
159	#117	—	2	1	no	—
160	#117	—	2	1	no	—
161	#117	C₇×D₄	1	2	yes	—
162	#117	C₂³×C₇	1	2	yes	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
169	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
170	C₂×C₂²⋊C₄	—	2	1	no	—
171	C₂×C₄²	—	2	1	no	—
172	#107	—	2	8	no	—
173	#106	—	2	8	no	—
174	#106	—	2	8	no	—
175	C₄⋊C₈	—	2	8	no	—
176	C₂²×C₇	—	2	8	no	—
177	C₂²×C₇	—	2	2	no	—
178	C₂²×C₇	—	2	2	no	—
179	C₂²×C₇	#398	1	2	yes	—
180	C₂²×C₇	#398	1	2	yes	—
181	C₂²×C₇	#471	1	4	yes	—
182	C₂²×C₇	C₂×#121	1	1	yes	characteristic
183	C₂²×C₇	C₂×#125	1	2	yes	—
184	C₄×C₇	—	4	4	no	—
185	C₄×C₇	—	8	4	no	—
186	C₄×C₇	#398	1	2	yes	—
187	C₄×C₇	#398	1	2	yes	—
188	C₄×C₇	C₂×C₂²⋊Q₈	1	2	yes	—
189	C₄×C₇	C₂×#123	1	1	yes	characteristic
190	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
191	C₂⁴	C₇×#41	1	1	yes	characteristic
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	8	4	no	—
197	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
202	C₂²×C₄	C₇×#41	1	1	yes	characteristic
203	C₂²×C₄	C₇×#41	1	4	yes	—
204	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
205	M₁₆	—	2	8	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	C₂×C₇×Q₈	1	8	yes	—
210	C₂×C₈	—	2	8	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₂×C₇	—	2	2	no	—
229	C₂×C₇	—	2	2	no	—
230	C₂×C₇	#2090	1	4	yes	direct product factor
231	C₂×C₇	C₂×#398	1	1	yes	characteristic
232	C₂×C₇	C₂×#471	1	2	yes	—
233	C₂³	—	2	4	no	—
234	C₂³	C₇×#121	1	1	yes	characteristic
235	C₂³	C₇×#121	1	1	yes	characteristic
236	C₂³	C₇×#125	1	4	yes	—
237	C₂³	C₂×C₇×#41	1	1	yes	characteristic
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	8	4	no	—
242	C₂×C₄	—	8	4	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	C₇×#121	1	1	yes	characteristic
245	C₂×C₄	C₇×#121	1	1	yes	characteristic
246	C₂×C₄	C₇×C₂²⋊Q₈	1	4	yes	—
247	C₂×C₄	C₇×C₂²⋊Q₈	1	4	yes	—
248	C₂×C₄	C₇×#123	1	2	yes	—
249	C₂×C₄	C₇×#123	1	2	yes	—
250	C₂×C₄	C₇×#125	1	4	yes	—
251	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
252	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
253	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
256	C₂×C₄	C₂×C₇×#41	1	2	yes	—
257	C₈	—	2	8	no	—
258	C₈	—	2	4	no	—
259	C₈	—	4	2	no	—
260	C₇	C₂×#2090	1	1	yes	characteristic, direct product factor, Sylow
261	C₂²	—	2	8	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	C₇×#398	1	2	yes	—
265	C₂²	C₇×#398	1	2	yes	—
266	C₂²	C₇×#471	1	4	yes	—
267	C₂²	C₂×C₇×#121	1	1	yes	characteristic
268	C₂²	C₂×C₇×#125	1	2	yes	—
269	C₄	—	4	4	no	—
270	C₄	—	8	4	no	—
271	C₄	C₇×#398	1	2	yes	—
272	C₄	C₇×#398	1	2	yes	—
273	C₄	C₂×C₇×C₂²⋊Q₈	1	2	yes	—
274	C₄	C₂×C₇×#123	1	1	yes	characteristic
275	C₄	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
276	C₂	—	2	2	no	—
277	C₂	—	2	2	no	—
278	C₂	C₇×#2090	1	4	yes	direct product factor
279	C₂	C₂×C₇×#398	1	1	yes	characteristic
280	C₂	C₂×C₇×#471	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂⁴×C₇	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂³×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#2090	C₂	direct (N × H)
24. C₂×#2090	C₇	direct (N × H)
60. #2090	C₂×C₇	direct (N × H)
230. C₂×C₇	#2090	direct (N × H)
260. C₇	C₂×#2090	direct (N × H)
278. C₂	C₇×#2090	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×#428	2
C₂²×C₇×#106	2
C₂×C₇×#424	2
C₇×#2090	2
C₂×C₇×#343	2
C₂×C₇×#341	2
C₂×#2090	7

Subgroups — C2×C7×#2090 #13200889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2090 #13200889