Subgroups — C₂×C₃×C₇×#102 #13056925

Order 1344.

404 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₇×#102	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₇×#102	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂⁴×C₃×C₇	—	2	1	no	—
7	C₂×C₃×C₇×D₄	—	2	1	no	—
8	C₂×C₃×C₇×D₄	—	2	2	no	—
9	C₂×C₃×C₇×D₄	C₄	1	4	yes	complemented
10	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₇	—	2	1	no	—
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₇×C₈	—	2	2	no	—
14	C₂×C₃×C₇×C₈	C₂²	1	2	yes	complemented
15	C₃×C₇×C₄⋊C₄	—	2	1	no	—
16	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	complemented
17	C₂²×C₇×D₄	C₂×C₃	1	1	yes	characteristic
18	C₂²×C₇×C₈	C₂×C₃	1	1	yes	characteristic, complemented
19	C₂×C₇×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
20	C₇×#102	C₂×C₃	1	4	yes	direct product factor
21	C₂×C₃×#102	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂³×C₃×C₇	—	2	4	no	—
23	C₂³×C₃×C₇	—	2	2	no	—
24	C₂³×C₃×C₇	—	4	2	no	—
25	C₂³×C₃×C₇	—	4	2	no	—
26	C₂³×C₃×C₇	D₄	1	1	yes	characteristic
27	C₃×C₇×D₄	—	2	4	no	—
28	C₃×C₇×D₄	—	2	2	no	—
29	C₃×C₇×D₄	C₂×C₄	1	4	yes	complemented
30	C₂×C₃×C₄×C₇	—	2	2	no	—
31	C₂×C₃×C₄×C₇	—	4	2	no	—
32	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
35	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
36	C₃×C₇×C₈	—	2	2	no	—
37	C₂⁴×C₇	—	2	1	no	—
38	C₂×C₇×D₄	—	2	1	no	—
39	C₂×C₇×D₄	—	2	2	no	—
40	C₂×C₇×D₄	C₃×C₄	1	4	yes	complemented
41	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
42	C₂²×C₄×C₇	—	2	1	no	—
43	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₇×C₈	—	2	2	no	—
45	C₂×C₇×C₈	C₂²×C₃	1	2	yes	complemented
46	C₇×C₄⋊C₄	—	2	1	no	—
47	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
48	C₂²×C₃×D₄	C₂×C₇	1	1	yes	characteristic
49	C₂²×C₃×C₈	C₂×C₇	1	1	yes	characteristic, complemented
50	C₂×C₃×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
51	C₃×#102	C₂×C₇	1	4	yes	direct product factor
52	C₂²×C₃×C₇	—	2	4	no	—
53	C₂²×C₃×C₇	—	4	8	no	—
54	C₂²×C₃×C₇	—	4	2	no	—
55	C₂²×C₃×C₇	—	4	2	no	—
56	C₂²×C₃×C₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₃×C₇	D₈	1	2	yes	—
58	C₂²×C₃×C₇	SD₁₆	1	2	yes	—
59	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
60	C₃×C₄×C₇	—	4	2	no	—
61	C₃×C₄×C₇	C₂²⋊C₄	1	2	yes	—
62	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
63	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
64	C₂×#102	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
65	C₂³×C₇	—	2	4	no	—
66	C₂³×C₇	—	2	2	no	—
67	C₂³×C₇	—	4	2	no	—
68	C₂³×C₇	—	4	2	no	—
69	C₂³×C₇	C₃×D₄	1	1	yes	characteristic
70	C₇×D₄	—	2	4	no	—
71	C₇×D₄	—	2	2	no	—
72	C₇×D₄	C₂×C₃×C₄	1	4	yes	complemented
73	C₂×C₄×C₇	—	2	2	no	—
74	C₂×C₄×C₇	—	4	2	no	—
75	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
76	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
77	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
78	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
79	C₇×C₈	—	2	2	no	—
80	C₂⁴×C₃	—	2	1	no	—
81	C₂×C₃×D₄	—	2	1	no	—
82	C₂×C₃×D₄	—	2	2	no	—
83	C₂×C₃×D₄	C₄×C₇	1	4	yes	complemented
84	C₂×C₃×D₄	C₂²×C₇	1	2	yes	—
85	C₂²×C₃×C₄	—	2	1	no	—
86	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
87	C₂×C₃×C₈	—	2	2	no	—
88	C₂×C₃×C₈	C₂²×C₇	1	2	yes	complemented
89	C₃×C₄⋊C₄	—	2	1	no	—
90	C₃×C₄⋊C₄	C₂²×C₇	1	2	yes	complemented
91	C₂×C₃×C₇	—	4	4	no	—
92	C₂×C₃×C₇	#102	1	4	yes	direct product factor
93	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₃×C₇	C₂×D₈	1	1	yes	characteristic
95	C₂×C₃×C₇	C₂×SD₁₆	1	1	yes	characteristic
96	C₂²×D₄	C₂×C₃×C₇	1	1	yes	characteristic
97	C₂²×C₈	C₂×C₃×C₇	1	1	yes	characteristic, complemented
98	C₂×C₄⋊C₄	C₂×C₃×C₇	1	1	yes	characteristic, complemented
99	#102	C₂×C₃×C₇	1	4	yes	direct product factor
100	C₂²×C₇	—	2	4	no	—
101	C₂²×C₇	—	4	8	no	—
102	C₂²×C₇	—	4	2	no	—
103	C₂²×C₇	—	4	2	no	—
104	C₂²×C₇	C₃×C₂²⋊C₄	1	2	yes	—
105	C₂²×C₇	C₃×D₈	1	2	yes	—
106	C₂²×C₇	C₃×SD₁₆	1	2	yes	—
107	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
108	C₄×C₇	—	4	2	no	—
109	C₄×C₇	C₃×C₂²⋊C₄	1	2	yes	—
110	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
111	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
112	C₂³×C₃	—	2	4	no	—
113	C₂³×C₃	—	2	2	no	—
114	C₂³×C₃	—	4	2	no	—
115	C₂³×C₃	—	4	2	no	—
116	C₂³×C₃	C₇×D₄	1	1	yes	characteristic
117	C₃×D₄	—	2	4	no	—
118	C₃×D₄	—	2	2	no	—
119	C₃×D₄	C₂×C₄×C₇	1	4	yes	complemented
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	4	2	no	—
122	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
123	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
125	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
126	C₃×C₈	—	2	2	no	—
127	C₃×C₇	C₂×#102	1	1	yes	characteristic, direct product factor, Hall
128	C₂⁴	—	2	1	no	—
129	C₂×D₄	—	2	1	no	—
130	C₂×D₄	—	2	2	no	—
131	C₂×D₄	C₃×C₄×C₇	1	4	yes	complemented
132	C₂×D₄	C₂²×C₃×C₇	1	2	yes	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	C₂²×C₃×C₇	1	2	yes	complemented
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	C₂²×C₃×C₇	1	2	yes	complemented
139	C₂×C₇	—	4	4	no	—
140	C₂×C₇	C₃×#102	1	4	yes	direct product factor
141	C₂×C₇	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₇	C₂×C₃×D₈	1	1	yes	characteristic
143	C₂×C₇	C₂×C₃×SD₁₆	1	1	yes	characteristic
144	C₂²×C₃	—	2	4	no	—
145	C₂²×C₃	—	4	8	no	—
146	C₂²×C₃	—	4	2	no	—
147	C₂²×C₃	—	4	2	no	—
148	C₂²×C₃	C₇×C₂²⋊C₄	1	2	yes	—
149	C₂²×C₃	C₇×D₈	1	2	yes	—
150	C₂²×C₃	C₇×SD₁₆	1	2	yes	—
151	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
152	C₃×C₄	—	4	2	no	—
153	C₃×C₄	C₇×C₂²⋊C₄	1	2	yes	—
154	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
155	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
156	C₂³	—	2	4	no	—
157	C₂³	—	2	2	no	—
158	C₂³	—	4	2	no	—
159	C₂³	—	4	2	no	—
160	C₂³	C₃×C₇×D₄	1	1	yes	characteristic
161	D₄	—	2	4	no	—
162	D₄	—	2	2	no	—
163	D₄	C₂×C₃×C₄×C₇	1	4	yes	complemented
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	C₂×C₃×C₄×C₇	1	2	yes	—
167	C₂×C₄	C₃×C₇×D₄	1	1	yes	characteristic
168	C₂×C₄	C₃×C₇×D₄	1	2	yes	—
169	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₇	C₂×C₃×#102	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×C₃	—	4	4	no	—
173	C₂×C₃	C₇×#102	1	4	yes	direct product factor
174	C₂×C₃	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₃	C₂×C₇×D₈	1	1	yes	characteristic
176	C₂×C₃	C₂×C₇×SD₁₆	1	1	yes	characteristic
177	C₂²	—	2	4	no	—
178	C₂²	—	4	8	no	—
179	C₂²	—	4	2	no	—
180	C₂²	—	4	2	no	—
181	C₂²	C₃×C₇×C₂²⋊C₄	1	2	yes	—
182	C₂²	C₃×C₇×D₈	1	2	yes	—
183	C₂²	C₃×C₇×SD₁₆	1	2	yes	—
184	C₂²	C₂×C₃×C₇×D₄	1	1	yes	characteristic
185	C₄	—	4	2	no	—
186	C₄	C₃×C₇×C₂²⋊C₄	1	2	yes	—
187	C₄	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
188	C₄	C₂×C₃×C₇×D₄	1	1	yes	characteristic
189	C₃	C₂×C₇×#102	1	1	yes	characteristic, direct product factor, Sylow
190	C₂	—	4	4	no	—
191	C₂	C₃×C₇×#102	1	4	yes	direct product factor
192	C₂	C₂×C₃×C₇×C₂²⋊C₄	1	1	yes	characteristic
193	C₂	C₂×C₃×C₇×D₈	1	1	yes	characteristic
194	C₂	C₂×C₃×C₇×SD₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃×C₇	D₄
commutator	C₄	C₂²×C₃×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₇
prime radical	C₂²×C₃×C₇×D₄	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₇ ◁ C₂²×C₃×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₇×C₈	C₂	semidirect
3. C₂×C₃×C₇×C₄⋊C₄	C₂	semidirect
4. C₃×C₇×#102	C₂	direct (N × H)
5. C₂×C₇×#102	C₃	direct (N × H)
9. C₂×C₃×C₇×D₄	C₄	semidirect
14. C₂×C₃×C₇×C₈	C₂²	semidirect
16. C₃×C₇×C₄⋊C₄	C₂²	semidirect
18. C₂²×C₇×C₈	C₂×C₃	semidirect
19. C₂×C₇×C₄⋊C₄	C₂×C₃	semidirect
20. C₇×#102	C₂×C₃	direct (N × H)
21. C₂×C₃×#102	C₇	direct (N × H)
29. C₃×C₇×D₄	C₂×C₄	semidirect
40. C₂×C₇×D₄	C₃×C₄	semidirect
45. C₂×C₇×C₈	C₂²×C₃	semidirect
47. C₇×C₄⋊C₄	C₂²×C₃	semidirect
49. C₂²×C₃×C₈	C₂×C₇	semidirect
50. C₂×C₃×C₄⋊C₄	C₂×C₇	semidirect
51. C₃×#102	C₂×C₇	direct (N × H)
64. C₂×#102	C₃×C₇	direct (N × H)
72. C₇×D₄	C₂×C₃×C₄	semidirect
83. C₂×C₃×D₄	C₄×C₇	semidirect
88. C₂×C₃×C₈	C₂²×C₇	semidirect
90. C₃×C₄⋊C₄	C₂²×C₇	semidirect
92. C₂×C₃×C₇	#102	direct (N × H)
97. C₂²×C₈	C₂×C₃×C₇	semidirect
98. C₂×C₄⋊C₄	C₂×C₃×C₇	semidirect
99. #102	C₂×C₃×C₇	direct (N × H)
119. C₃×D₄	C₂×C₄×C₇	semidirect
127. C₃×C₇	C₂×#102	direct (N × H)
131. C₂×D₄	C₃×C₄×C₇	semidirect
136. C₂×C₈	C₂²×C₃×C₇	semidirect
138. C₄⋊C₄	C₂²×C₃×C₇	semidirect
140. C₂×C₇	C₃×#102	direct (N × H)
163. D₄	C₂×C₃×C₄×C₇	semidirect
171. C₇	C₂×C₃×#102	direct (N × H)
173. C₂×C₃	C₇×#102	direct (N × H)
189. C₃	C₂×C₇×#102	direct (N × H)
191. C₂	C₃×C₇×#102	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×C₃×C₇×D₄	2
C₂²×C₃×C₇×C₈	2
C₂×C₃×C₇×C₄⋊C₄	2
C₃×C₇×#102	2
C₂×C₇×#102	3
C₂×C₃×#102	7

Subgroups — C2×C3×C7×#102 #13056925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₇×#102 #13056925