Subgroups — C₂×C₇×#1420 #13190840

Order 1792.

286 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×M₆₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#368	C₂	1	1	yes	maximal, characteristic
4	C₇×#1420	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₇×Q₁₆	—	2	1	no	—
6	C₂×C₇×Q₃₂	—	2	1	no	—
7	C₂×C₇×Q₃₂	—	2	1	no	—
8	C₂×C₇×Q₃₂	—	2	1	no	—
9	C₂×C₇×Q₃₂	C₄	1	2	yes	—
10	C₂×C₇×Q₃₂	C₄	1	2	yes	—
11	C₂×C₇×Q₃₂	C₂²	1	2	yes	—
12	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₇×#108	—	2	1	no	—
14	C₇×M₆₄	—	2	1	no	—
15	C₇×M₆₄	C₂²	1	2	yes	—
16	C₂×C₇×C₃₂	—	2	1	no	—
17	C₇×#368	—	2	1	no	—
18	C₇×#368	C₂²	1	2	yes	—
19	C₂×#1420	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₇×Q₈	—	4	1	no	—
21	C₂×C₇×Q₁₆	—	2	2	no	—
22	C₂×C₇×Q₁₆	—	2	2	no	—
23	C₂×C₇×Q₁₆	—	2	2	no	—
24	C₂×C₇×Q₁₆	—	4	1	no	—
25	C₂×C₇×Q₁₆	—	4	1	no	—
26	C₂×C₇×Q₁₆	—	4	1	no	—
27	C₂×C₇×M₁₆	—	4	1	no	—
28	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
29	C₇×Q₃₂	—	2	2	no	—
30	C₇×Q₃₂	—	2	2	no	—
31	C₇×Q₃₂	—	2	2	no	—
32	C₇×Q₃₂	C₂×C₄	1	4	yes	—
33	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
36	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
37	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
38	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
39	C₇×#108	—	2	2	no	—
40	C₇×#108	—	4	1	no	—
41	C₇×C₃₂	—	2	2	no	—
42	C₂²×Q₃₂	C₂×C₇	1	1	yes	characteristic
43	C₂×M₆₄	C₂×C₇	1	1	yes	characteristic
44	C₂×#368	C₂×C₇	1	1	yes	characteristic
45	#1420	C₂×C₇	1	4	yes	direct product factor
46	C₂×C₇×Q₈	—	4	2	no	—
47	C₂×C₇×Q₈	—	4	2	no	—
48	C₂×C₇×Q₈	—	4	2	no	—
49	C₂×C₇×Q₈	—	8	1	no	—
50	C₂×C₇×Q₈	—	8	1	no	—
51	C₂×C₇×Q₈	—	8	1	no	—
52	C₂²×C₄×C₇	—	8	1	no	—
53	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
54	C₇×Q₁₆	—	2	4	no	—
55	C₇×Q₁₆	—	4	2	no	—
56	C₇×Q₁₆	—	4	2	no	—
57	C₇×Q₁₆	—	4	2	no	—
58	C₇×M₁₆	—	4	2	no	—
59	C₇×M₁₆	—	8	1	no	—
60	C₂×C₇×C₈	—	8	1	no	—
61	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₇×C₈	D₈	1	1	yes	characteristic
64	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
65	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
66	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
67	C₇×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
68	C₇×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
69	C₇×C₁₆	C₂²×C₄	1	1	yes	characteristic
70	C₇×C₁₆	C₂×D₄	1	1	yes	characteristic
71	C₂²×Q₁₆	—	2	1	no	—
72	C₂×Q₃₂	—	2	1	no	—
73	C₂×Q₃₂	—	2	1	no	—
74	C₂×Q₃₂	—	2	1	no	—
75	C₂×Q₃₂	C₄×C₇	1	2	yes	—
76	C₂×Q₃₂	C₄×C₇	1	2	yes	—
77	C₂×Q₃₂	C₂²×C₇	1	2	yes	—
78	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
79	C₂×#108	—	2	1	no	—
80	M₆₄	—	2	1	no	—
81	M₆₄	C₂²×C₇	1	2	yes	—
82	C₂×C₃₂	—	2	1	no	—
83	#368	—	2	1	no	—
84	#368	C₂²×C₇	1	2	yes	—
85	C₂³×C₇	SD₃₂	1	1	yes	characteristic
86	C₇×Q₈	—	4	4	no	—
87	C₇×Q₈	—	8	2	no	—
88	C₇×Q₈	—	8	2	no	—
89	C₇×Q₈	—	8	2	no	—
90	C₂×C₄×C₇	—	8	2	no	—
91	C₂×C₄×C₇	—	8	2	no	—
92	C₂×C₄×C₇	—	8	2	no	—
93	C₂×C₄×C₇	#102	1	1	yes	characteristic
94	C₂×C₄×C₇	#102	1	1	yes	characteristic
95	C₂×C₄×C₇	D₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇	D₁₆	1	1	yes	characteristic
97	C₂×C₄×C₇	SD₃₂	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
99	C₇×C₈	—	8	2	no	—
100	C₇×C₈	#102	1	1	yes	characteristic
101	C₇×C₈	#102	1	1	yes	characteristic
102	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
104	C₂²×Q₈	—	4	1	no	—
105	C₂×Q₁₆	—	2	2	no	—
106	C₂×Q₁₆	—	2	2	no	—
107	C₂×Q₁₆	—	2	2	no	—
108	C₂×Q₁₆	—	4	1	no	—
109	C₂×Q₁₆	—	4	1	no	—
110	C₂×Q₁₆	—	4	1	no	—
111	C₂×M₁₆	—	4	1	no	—
112	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
113	Q₃₂	—	2	2	no	—
114	Q₃₂	—	2	2	no	—
115	Q₃₂	—	2	2	no	—
116	Q₃₂	C₂×C₄×C₇	1	4	yes	—
117	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
118	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
119	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
120	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
121	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
122	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
123	#108	—	2	2	no	—
124	#108	—	4	1	no	—
125	C₃₂	—	2	2	no	—
126	C₂²×C₇	—	2	2	no	—
127	C₂²×C₇	#357	1	1	yes	characteristic
128	C₂²×C₇	#357	1	1	yes	characteristic
129	C₂²×C₇	C₂×SD₃₂	1	1	yes	characteristic
130	C₄×C₇	—	8	4	no	—
131	C₄×C₇	#357	1	1	yes	characteristic
132	C₄×C₇	#357	1	1	yes	characteristic
133	C₄×C₇	C₂×#102	1	1	yes	characteristic
134	C₄×C₇	C₂×D₁₆	1	1	yes	characteristic
135	C₂×Q₈	—	4	2	no	—
136	C₂×Q₈	—	4	2	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	—	8	1	no	—
139	C₂×Q₈	—	8	1	no	—
140	C₂×Q₈	—	8	1	no	—
141	C₂²×C₄	—	8	1	no	—
142	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
143	Q₁₆	—	2	4	no	—
144	Q₁₆	—	4	2	no	—
145	Q₁₆	—	4	2	no	—
146	Q₁₆	—	4	2	no	—
147	M₁₆	—	4	2	no	—
148	M₁₆	—	8	1	no	—
149	C₂×C₈	—	8	1	no	—
150	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₈	C₇×D₈	1	1	yes	characteristic
153	C₂×C₈	C₇×SD₁₆	1	1	yes	characteristic
154	C₂×C₈	C₇×SD₁₆	1	1	yes	characteristic
155	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
156	C₁₆	C₇×C₂²⋊C₄	1	1	yes	characteristic
157	C₁₆	C₇×C₂²⋊C₄	1	1	yes	characteristic
158	C₁₆	C₂²×C₄×C₇	1	1	yes	characteristic
159	C₁₆	C₂×C₇×D₄	1	1	yes	characteristic
160	C₂×C₇	—	2	1	no	—
161	C₂×C₇	—	2	1	no	—
162	C₂×C₇	#1420	1	2	yes	direct product factor
163	C₂×C₇	C₂×#357	1	1	yes	characteristic
164	C₂³	C₇×SD₃₂	1	1	yes	characteristic
165	Q₈	—	4	4	no	—
166	Q₈	—	8	2	no	—
167	Q₈	—	8	2	no	—
168	Q₈	—	8	2	no	—
169	C₂×C₄	—	8	2	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	C₇×#102	1	1	yes	characteristic
173	C₂×C₄	C₇×#102	1	1	yes	characteristic
174	C₂×C₄	C₇×D₁₆	1	1	yes	characteristic
175	C₂×C₄	C₇×D₁₆	1	1	yes	characteristic
176	C₂×C₄	C₇×SD₃₂	1	1	yes	characteristic
177	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
178	C₈	—	8	2	no	—
179	C₈	C₇×#102	1	1	yes	characteristic
180	C₈	C₇×#102	1	1	yes	characteristic
181	C₈	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
182	C₈	C₂×C₇×SD₁₆	1	1	yes	characteristic
183	C₇	C₂×#1420	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²	—	2	2	no	—
185	C₂²	C₇×#357	1	1	yes	characteristic
186	C₂²	C₇×#357	1	1	yes	characteristic
187	C₂²	C₂×C₇×SD₃₂	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	C₇×#357	1	1	yes	characteristic
190	C₄	C₇×#357	1	1	yes	characteristic
191	C₄	C₂×C₇×#102	1	1	yes	characteristic
192	C₄	C₂×C₇×D₁₆	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	C₇×#1420	1	2	yes	direct product factor
196	C₂	C₂×C₇×#357	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	#357
commutator	C₁₆	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
prime radical	C₂³×C₇	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1420	C₂	direct (N × H)
19. C₂×#1420	C₇	direct (N × H)
45. #1420	C₂×C₇	direct (N × H)
162. C₂×C₇	#1420	direct (N × H)
183. C₇	C₂×#1420	direct (N × H)
195. C₂	C₇×#1420	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×Q₃₂	2
C₂×C₇×M₆₄	2
C₂×C₇×#368	2
C₇×#1420	2
C₂×#1420	7

Subgroups — C2×C7×#1420 #13190840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1420 #13190840