Subgroups — C₂×C₇×#2070 #13200837

Order 1792.

950 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#430	C₂	1	3	yes	maximal
2	C₂²×C₇×#101	C₂	1	3	yes	maximal
3	C₇×#2070	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₇×#338	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₇×Q₈	C₂²	1	1	yes	characteristic
6	C₂²×C₇×M₁₆	C₂²	1	3	yes	—
7	C₂²×C₇×M₁₆	C₂²	1	3	yes	—
8	C₇×#430	—	2	6	no	—
9	C₇×#430	C₂²	1	12	yes	—
10	C₂×C₇×#108	—	2	6	no	—
11	C₂×C₇×#101	—	2	6	no	—
12	C₂×C₇×#101	—	2	6	no	—
13	C₂×C₇×#101	C₂²	1	12	yes	—
14	C₇×#338	C₂²	1	4	yes	—
15	C₂×#2070	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₇×Q₈	—	2	3	no	—
17	C₂²×C₇×Q₈	—	2	3	no	—
18	C₂²×C₇×Q₈	—	2	3	no	—
19	C₂²×C₇×Q₈	—	2	3	no	—
20	C₂²×C₇×Q₈	Q₈	1	2	yes	—
21	C₂²×C₇×Q₈	C₂³	1	2	yes	—
22	C₂³×C₄×C₇	Q₈	1	2	yes	—
23	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂×C₇×M₁₆	—	2	3	no	—
25	C₂×C₇×M₁₆	—	2	3	no	—
26	C₂×C₇×M₁₆	—	2	3	no	—
27	C₂×C₇×M₁₆	—	2	3	no	—
28	C₂×C₇×M₁₆	—	2	3	no	—
29	C₂×C₇×M₁₆	—	2	3	no	—
30	C₂×C₇×M₁₆	—	2	3	no	—
31	C₂×C₇×M₁₆	—	2	3	no	—
32	C₂×C₇×M₁₆	D₄	1	6	yes	—
33	C₂×C₇×M₁₆	D₄	1	6	yes	—
34	C₂×C₇×M₁₆	C₂³	1	6	yes	—
35	C₂×C₇×M₁₆	C₂³	1	6	yes	—
36	C₂²×C₇×C₈	—	2	3	no	—
37	C₂²×C₇×C₈	—	2	3	no	—
38	C₇×#108	—	2	24	no	—
39	C₇×#101	—	2	24	no	—
40	C₂×#430	C₂×C₇	1	3	yes	—
41	C₂²×#101	C₂×C₇	1	3	yes	—
42	#2070	C₂×C₇	1	8	yes	direct product factor
43	C₂×#338	C₂×C₇	1	1	yes	characteristic
44	C₂⁴×C₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₇×Q₈	—	2	6	no	—
46	C₂×C₇×Q₈	—	2	6	no	—
47	C₂×C₇×Q₈	—	2	6	no	—
48	C₂×C₇×Q₈	—	2	6	no	—
49	C₂×C₇×Q₈	—	4	3	no	—
50	C₂×C₇×Q₈	—	4	3	no	—
51	C₂×C₇×Q₈	—	4	3	no	—
52	C₂×C₇×Q₈	—	4	3	no	—
53	C₂×C₇×Q₈	—	4	3	no	—
54	C₂×C₇×Q₈	—	4	1	no	—
55	C₂²×C₄×C₇	—	2	6	no	—
56	C₂²×C₄×C₇	—	2	6	no	—
57	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
59	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
60	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₇	#41	1	3	yes	—
62	C₂²×C₄×C₇	#41	1	3	yes	—
63	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
64	C₇×M₁₆	—	2	6	no	—
65	C₇×M₁₆	—	2	6	no	—
66	C₇×M₁₆	—	2	6	no	—
67	C₇×M₁₆	—	2	6	no	—
68	C₇×M₁₆	—	2	6	no	—
69	C₇×M₁₆	—	4	3	no	—
70	C₇×M₁₆	—	4	3	no	—
71	C₇×M₁₆	C₂×D₄	1	12	yes	—
72	C₂×C₇×C₈	—	2	6	no	—
73	C₂×C₇×C₈	—	2	6	no	—
74	C₂×C₇×C₈	—	2	6	no	—
75	C₂×C₇×C₈	—	2	6	no	—
76	C₂×C₇×C₈	—	4	3	no	—
77	C₂×C₇×C₈	—	4	3	no	—
78	C₂³×Q₈	C₂²×C₇	1	1	yes	characteristic
79	C₂²×M₁₆	C₂²×C₇	1	3	yes	—
80	C₂²×M₁₆	C₂²×C₇	1	3	yes	—
81	#430	—	2	6	no	—
82	#430	C₂²×C₇	1	12	yes	—
83	C₂×#108	—	2	6	no	—
84	C₂×#101	—	2	6	no	—
85	C₂×#101	—	2	6	no	—
86	C₂×#101	C₂²×C₇	1	12	yes	—
87	#338	C₂²×C₇	1	4	yes	—
88	C₂³×C₇	—	4	2	no	—
89	C₂³×C₇	C₂²⋊Q₈	1	3	yes	—
90	C₂³×C₇	C₂²⋊Q₈	1	3	yes	—
91	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
92	C₇×Q₈	—	4	6	no	—
93	C₇×Q₈	—	4	2	no	—
94	C₇×Q₈	—	4	6	no	—
95	C₇×Q₈	—	4	2	no	—
96	C₂×C₄×C₇	—	2	12	no	—
97	C₂×C₄×C₇	—	4	6	no	—
98	C₂×C₄×C₇	—	4	2	no	—
99	C₂×C₄×C₇	#121	1	3	yes	—
100	C₂×C₄×C₇	#121	1	3	yes	—
101	C₂×C₄×C₇	#121	1	3	yes	—
102	C₂×C₄×C₇	#121	1	3	yes	—
103	C₂×C₄×C₇	C₂²⋊Q₈	1	3	yes	—
104	C₂×C₄×C₇	C₂²⋊Q₈	1	3	yes	—
105	C₂×C₄×C₇	#126	1	3	yes	—
106	C₂×C₄×C₇	#126	1	1	yes	characteristic
107	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
108	C₂×C₄×C₇	C₂×#41	1	3	yes	—
109	C₇×C₈	—	2	12	no	—
110	C₇×C₈	—	4	6	no	—
111	C₂²×Q₈	—	2	3	no	—
112	C₂²×Q₈	—	2	3	no	—
113	C₂²×Q₈	—	2	3	no	—
114	C₂²×Q₈	—	2	3	no	—
115	C₂²×Q₈	C₇×Q₈	1	2	yes	—
116	C₂²×Q₈	C₂³×C₇	1	2	yes	—
117	C₂³×C₄	C₇×Q₈	1	2	yes	—
118	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
119	C₂×M₁₆	—	2	3	no	—
120	C₂×M₁₆	—	2	3	no	—
121	C₂×M₁₆	—	2	3	no	—
122	C₂×M₁₆	—	2	3	no	—
123	C₂×M₁₆	—	2	3	no	—
124	C₂×M₁₆	—	2	3	no	—
125	C₂×M₁₆	—	2	3	no	—
126	C₂×M₁₆	—	2	3	no	—
127	C₂×M₁₆	C₇×D₄	1	6	yes	—
128	C₂×M₁₆	C₇×D₄	1	6	yes	—
129	C₂×M₁₆	C₂³×C₇	1	6	yes	—
130	C₂×M₁₆	C₂³×C₇	1	6	yes	—
131	C₂²×C₈	—	2	3	no	—
132	C₂²×C₈	—	2	3	no	—
133	#108	—	2	24	no	—
134	#101	—	2	24	no	—
135	C₂²×C₇	—	2	6	no	—
136	C₂²×C₇	—	4	3	no	—
137	C₂²×C₇	—	4	1	no	—
138	C₂²×C₇	#396	1	3	yes	—
139	C₂²×C₇	#396	1	1	yes	characteristic
140	C₂²×C₇	C₂×C₂²⋊Q₈	1	3	yes	—
141	C₄×C₇	—	4	4	no	—
142	C₄×C₇	#396	1	3	yes	—
143	C₄×C₇	#396	1	1	yes	characteristic
144	C₄×C₇	C₂×#121	1	3	yes	—
145	C₄×C₇	C₂×#126	1	1	yes	characteristic
146	C₂⁴	C₂×C₇×Q₈	1	1	yes	characteristic
147	C₂×Q₈	—	2	6	no	—
148	C₂×Q₈	—	2	6	no	—
149	C₂×Q₈	—	2	6	no	—
150	C₂×Q₈	—	2	6	no	—
151	C₂×Q₈	—	4	3	no	—
152	C₂×Q₈	—	4	3	no	—
153	C₂×Q₈	—	4	3	no	—
154	C₂×Q₈	—	4	3	no	—
155	C₂×Q₈	—	4	3	no	—
156	C₂×Q₈	—	4	1	no	—
157	C₂²×C₄	—	2	6	no	—
158	C₂²×C₄	—	2	6	no	—
159	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
160	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
161	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
162	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
163	C₂²×C₄	C₇×#41	1	3	yes	—
164	C₂²×C₄	C₇×#41	1	3	yes	—
165	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
166	M₁₆	—	2	6	no	—
167	M₁₆	—	2	6	no	—
168	M₁₆	—	2	6	no	—
169	M₁₆	—	2	6	no	—
170	M₁₆	—	2	6	no	—
171	M₁₆	—	4	3	no	—
172	M₁₆	—	4	3	no	—
173	M₁₆	C₂×C₇×D₄	1	12	yes	—
174	C₂×C₈	—	2	6	no	—
175	C₂×C₈	—	2	6	no	—
176	C₂×C₈	—	2	6	no	—
177	C₂×C₈	—	2	6	no	—
178	C₂×C₈	—	4	3	no	—
179	C₂×C₈	—	4	3	no	—
180	C₂×C₇	—	2	3	no	—
181	C₂×C₇	—	2	3	no	—
182	C₂×C₇	#2070	1	2	yes	direct product factor
183	C₂×C₇	C₂×#396	1	1	yes	characteristic
184	C₂³	—	4	2	no	—
185	C₂³	C₇×C₂²⋊Q₈	1	3	yes	—
186	C₂³	C₇×C₂²⋊Q₈	1	3	yes	—
187	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
188	Q₈	—	4	6	no	—
189	Q₈	—	4	2	no	—
190	Q₈	—	4	6	no	—
191	Q₈	—	4	2	no	—
192	C₂×C₄	—	2	12	no	—
193	C₂×C₄	—	4	6	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	C₇×#121	1	3	yes	—
196	C₂×C₄	C₇×#121	1	3	yes	—
197	C₂×C₄	C₇×#121	1	3	yes	—
198	C₂×C₄	C₇×#121	1	3	yes	—
199	C₂×C₄	C₇×C₂²⋊Q₈	1	3	yes	—
200	C₂×C₄	C₇×C₂²⋊Q₈	1	3	yes	—
201	C₂×C₄	C₇×#126	1	3	yes	—
202	C₂×C₄	C₇×#126	1	1	yes	characteristic
203	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
204	C₂×C₄	C₂×C₇×#41	1	3	yes	—
205	C₈	—	2	12	no	—
206	C₈	—	4	6	no	—
207	C₇	C₂×#2070	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²	—	2	6	no	—
209	C₂²	—	4	3	no	—
210	C₂²	—	4	1	no	—
211	C₂²	C₇×#396	1	3	yes	—
212	C₂²	C₇×#396	1	1	yes	characteristic
213	C₂²	C₂×C₇×C₂²⋊Q₈	1	3	yes	—
214	C₄	—	4	4	no	—
215	C₄	C₇×#396	1	3	yes	—
216	C₄	C₇×#396	1	1	yes	characteristic
217	C₄	C₂×C₇×#121	1	3	yes	—
218	C₄	C₂×C₇×#126	1	1	yes	characteristic
219	C₂	—	2	3	no	—
220	C₂	—	2	3	no	—
221	C₂	C₇×#2070	1	2	yes	direct product factor
222	C₂	C₂×C₇×#396	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	#396
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂⁴×C₇	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#2070	C₂	direct (N × H)
15. C₂×#2070	C₇	direct (N × H)
42. #2070	C₂×C₇	direct (N × H)
182. C₂×C₇	#2070	direct (N × H)
207. C₇	C₂×#2070	direct (N × H)
221. C₂	C₇×#2070	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×#430	2
C₂²×C₇×#101	2
C₇×#2070	2
C₂×C₇×#338	2
C₂×#2070	7

Subgroups — C2×C7×#2070 #13200837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2070 #13200837