Subgroups — C₂×C₇×#2245 #13210916

Order 1792.

582 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic
3	C₇×#2245	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₇×#358	C₂	1	4	yes	maximal
5	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₁₆	—	2	1	no	—
7	C₂²×C₇×Q₁₆	C₂²	1	2	yes	—
8	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₇×#125	—	2	1	no	—
10	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
11	C₇×#499	—	2	2	no	—
12	C₇×#499	C₂²	1	4	yes	—
13	C₇×#494	—	2	2	no	—
14	C₇×#494	C₂²	1	4	yes	—
15	C₂×C₇×#107	C₂²	1	2	yes	—
16	C₂×C₇×#106	—	2	1	no	—
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₇×#358	C₂²	1	16	yes	—
19	C₇×#346	C₂²	1	4	yes	—
20	C₂×#2245	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₇×Q₈	—	2	2	no	—
22	C₂×C₇×Q₁₆	—	2	2	no	—
23	C₂×C₇×Q₁₆	—	2	4	no	—
24	C₂×C₇×Q₁₆	—	2	8	no	—
25	C₂×C₇×Q₁₆	—	4	4	no	—
26	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
27	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
29	C₇×C₄⋊Q₈	—	2	4	no	—
30	C₇×C₄⋊Q₈	—	4	2	no	—
31	C₇×#125	—	2	4	no	—
32	C₇×#125	—	4	2	no	—
33	C₂×C₇×C₄⋊C₄	—	2	2	no	—
34	C₂×C₇×C₄⋊C₄	—	4	2	no	—
35	C₂×C₇×C₄⋊C₄	—	4	2	no	—
36	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
37	C₂×C₇×C₁₆	—	2	4	no	—
38	C₂×C₇×C₁₆	C₂³	1	4	yes	—
39	C₇×#107	—	2	2	no	—
40	C₇×#107	C₂³	1	4	yes	—
41	C₇×#106	—	2	4	no	—
42	C₄×C₇×C₈	D₄	1	2	yes	—
43	C₄×C₇×C₈	C₂³	1	2	yes	—
44	C₂×#499	C₂×C₇	1	1	yes	characteristic
45	C₂×#494	C₂×C₇	1	1	yes	characteristic
46	#2245	C₂×C₇	1	8	yes	direct product factor
47	C₂×#358	C₂×C₇	1	4	yes	—
48	C₂×#346	C₂×C₇	1	1	yes	characteristic
49	C₂×C₇×Q₈	—	2	4	no	—
50	C₂×C₇×Q₈	—	4	2	no	—
51	C₂×C₇×Q₈	—	4	4	no	—
52	C₂×C₇×Q₈	—	4	4	no	—
53	C₂²×C₄×C₇	—	4	2	no	—
54	C₂²×C₄×C₇	—	4	2	no	—
55	C₂²×C₄×C₇	D₈	1	2	yes	—
56	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
57	C₇×Q₁₆	—	2	8	no	—
58	C₇×Q₁₆	—	4	4	no	—
59	C₇×Q₁₆	—	4	8	no	—
60	C₂×C₇×C₈	SD₁₆	1	4	yes	—
61	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
62	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
63	C₂×C₇×C₈	#41	1	4	yes	—
64	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
65	C₇×C₄⋊C₄	—	2	4	no	—
66	C₇×C₄⋊C₄	—	4	2	no	—
67	C₇×C₄⋊C₄	—	4	8	no	—
68	C₇×C₄⋊C₄	—	4	8	no	—
69	C₄²×C₇	D₈	1	2	yes	—
70	C₄²×C₇	C₂×D₄	1	2	yes	—
71	C₇×C₁₆	—	2	4	no	—
72	C₂²×Q₁₆	—	2	1	no	—
73	C₂²×Q₁₆	C₂²×C₇	1	2	yes	—
74	C₂×C₄⋊Q₈	—	2	1	no	—
75	C₂×#125	—	2	1	no	—
76	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
77	#499	—	2	2	no	—
78	#499	C₂²×C₇	1	4	yes	—
79	#494	—	2	2	no	—
80	#494	C₂²×C₇	1	4	yes	—
81	C₂×#107	C₂²×C₇	1	2	yes	—
82	C₂×#106	—	2	1	no	—
83	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
84	#358	C₂²×C₇	1	16	yes	—
85	#346	C₂²×C₇	1	4	yes	—
86	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
87	C₇×Q₈	—	4	4	no	—
88	C₇×Q₈	—	4	4	no	—
89	C₇×Q₈	—	8	4	no	—
90	C₂×C₄×C₇	—	2	4	no	—
91	C₂×C₄×C₇	—	4	4	no	—
92	C₂×C₄×C₇	—	4	4	no	—
93	C₂×C₄×C₇	—	8	4	no	—
94	C₂×C₄×C₇	—	8	4	no	—
95	C₂×C₄×C₇	#124	1	2	yes	—
96	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
97	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
99	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
100	C₇×C₈	#124	1	2	yes	—
101	C₇×C₈	C₂×SD₁₆	1	4	yes	—
102	C₇×C₈	C₂×#41	1	2	yes	—
103	C₂²×Q₈	—	2	2	no	—
104	C₂×Q₁₆	—	2	4	no	—
105	C₂×Q₁₆	—	2	2	no	—
106	C₂×Q₁₆	—	2	8	no	—
107	C₂×Q₁₆	—	4	4	no	—
108	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
109	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
110	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
111	C₄⋊Q₈	—	2	4	no	—
112	C₄⋊Q₈	—	4	2	no	—
113	#125	—	2	4	no	—
114	#125	—	4	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	4	2	no	—
117	C₂×C₄⋊C₄	—	4	2	no	—
118	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
119	C₂×C₁₆	—	2	4	no	—
120	C₂×C₁₆	C₂³×C₇	1	4	yes	—
121	#107	—	2	2	no	—
122	#107	C₂³×C₇	1	4	yes	—
123	#106	—	2	4	no	—
124	C₄×C₈	C₇×D₄	1	2	yes	—
125	C₄×C₈	C₂³×C₇	1	2	yes	—
126	C₂²×C₇	#486	1	2	yes	—
127	C₂²×C₇	#510	1	4	yes	—
128	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
129	C₄×C₇	—	2	4	no	—
130	C₄×C₇	—	8	4	no	—
131	C₄×C₇	—	8	4	no	—
132	C₄×C₇	#486	1	2	yes	—
133	C₄×C₇	C₂×#124	1	1	yes	characteristic
134	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
135	C₂×Q₈	—	2	4	no	—
136	C₂×Q₈	—	4	2	no	—
137	C₂×Q₈	—	4	4	no	—
138	C₂×Q₈	—	4	4	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	C₇×D₈	1	2	yes	—
142	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
143	Q₁₆	—	2	8	no	—
144	Q₁₆	—	4	4	no	—
145	Q₁₆	—	4	8	no	—
146	C₂×C₈	C₇×SD₁₆	1	4	yes	—
147	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
148	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
149	C₂×C₈	C₇×#41	1	4	yes	—
150	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₄⋊C₄	—	4	8	no	—
155	C₄²	C₇×D₈	1	2	yes	—
156	C₄²	C₂×C₇×D₄	1	2	yes	—
157	C₁₆	—	2	4	no	—
158	C₂×C₇	#2245	1	4	yes	direct product factor
159	C₂×C₇	C₂×#486	1	1	yes	characteristic
160	C₂×C₇	C₂×#510	1	2	yes	—
161	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
162	Q₈	—	4	4	no	—
163	Q₈	—	4	4	no	—
164	Q₈	—	8	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	8	4	no	—
169	C₂×C₄	—	8	4	no	—
170	C₂×C₄	C₇×#124	1	2	yes	—
171	C₂×C₄	C₂×C₇×D₈	1	4	yes	—
172	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
173	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
174	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
175	C₈	C₇×#124	1	2	yes	—
176	C₈	C₂×C₇×SD₁₆	1	4	yes	—
177	C₈	C₂×C₇×#41	1	2	yes	—
178	C₇	C₂×#2245	1	1	yes	characteristic, direct product factor, Sylow
179	C₂²	C₇×#486	1	2	yes	—
180	C₂²	C₇×#510	1	4	yes	—
181	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	8	4	no	—
184	C₄	—	8	4	no	—
185	C₄	C₇×#486	1	2	yes	—
186	C₄	C₂×C₇×#124	1	1	yes	characteristic
187	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
188	C₂	C₇×#2245	1	4	yes	direct product factor
189	C₂	C₂×C₇×#486	1	1	yes	characteristic
190	C₂	C₂×C₇×#510	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#2245	C₂	direct (N × H)
20. C₂×#2245	C₇	direct (N × H)
46. #2245	C₂×C₇	direct (N × H)
158. C₂×C₇	#2245	direct (N × H)
178. C₇	C₂×#2245	direct (N × H)
188. C₂	C₇×#2245	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×#499	2
C₂×C₇×#494	2
C₇×#2245	2
C₂×C₇×#358	2
C₂×C₇×#346	2
C₂×#2245	7

Subgroups — C2×C7×#2245 #13210916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2245 #13210916