Subgroups — C₂×C₃×C₇×#99 #13056920

Order 1344.

420 conjugacy classes of subgroups.

The subgroup lattice has 220 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	C₂×C₃×C₇×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
3	C₃×C₇×#99	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₇×#99	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂⁴×C₃×C₇	C₄	1	1	yes	characteristic, complemented
6	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₇×D₄	—	2	2	no	—
8	C₂×C₃×C₇×D₄	—	2	2	no	—
9	C₂×C₃×C₇×D₄	C₄	1	2	yes	complemented
10	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₇	—	2	2	no	—
12	C₂²×C₃×C₄×C₇	C₄	1	1	yes	characteristic, complemented
13	C₃×C₇×C₂²⋊C₄	—	2	2	no	—
14	C₃×C₇×C₂²⋊C₄	C₂²	1	4	yes	complemented
15	C₂²×C₇×D₄	C₂×C₃	1	1	yes	characteristic
16	C₂×C₇×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
17	C₇×#99	C₂×C₃	1	4	yes	direct product factor
18	C₂×C₃×#99	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂³×C₃×C₇	—	2	1	no	—
20	C₂³×C₃×C₇	—	2	1	no	—
21	C₂³×C₃×C₇	—	4	2	no	—
22	C₂³×C₃×C₇	—	4	2	no	—
23	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₃×C₇	C₂×C₄	1	2	yes	complemented
26	C₂³×C₃×C₇	D₄	1	2	yes	—
27	C₂³×C₃×C₇	D₄	1	2	yes	—
28	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
29	C₃×C₇×D₄	—	2	4	no	—
30	C₃×C₇×D₄	—	2	4	no	—
31	C₂×C₃×C₄×C₇	—	2	1	no	—
32	C₂×C₃×C₄×C₇	—	2	1	no	—
33	C₂×C₃×C₄×C₇	—	2	4	no	—
34	C₂×C₃×C₄×C₇	—	4	4	no	—
35	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	complemented
36	C₂⁴×C₇	C₃×C₄	1	1	yes	characteristic, complemented
37	C₂⁴×C₇	C₂²×C₃	1	1	yes	characteristic
38	C₂×C₇×D₄	—	2	2	no	—
39	C₂×C₇×D₄	—	2	2	no	—
40	C₂×C₇×D₄	C₃×C₄	1	2	yes	complemented
41	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
42	C₂²×C₄×C₇	—	2	2	no	—
43	C₂²×C₄×C₇	C₃×C₄	1	1	yes	characteristic, complemented
44	C₇×C₂²⋊C₄	—	2	2	no	—
45	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
46	C₂²×C₃×D₄	C₂×C₇	1	1	yes	characteristic
47	C₂×C₃×C₂²⋊C₄	C₂×C₇	1	2	yes	complemented
48	C₃×#99	C₂×C₇	1	4	yes	direct product factor
49	C₂²×C₃×C₇	—	2	4	no	—
50	C₂²×C₃×C₇	—	2	2	no	—
51	C₂²×C₃×C₇	—	2	2	no	—
52	C₂²×C₃×C₇	—	4	1	no	—
53	C₂²×C₃×C₇	—	4	2	no	—
54	C₂²×C₃×C₇	—	4	1	no	—
55	C₂²×C₃×C₇	—	4	2	no	—
56	C₂²×C₃×C₇	—	4	2	no	—
57	C₂²×C₃×C₇	—	4	2	no	—
58	C₂²×C₃×C₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₃×C₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₃×C₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
63	C₃×C₄×C₇	—	2	2	no	—
64	C₃×C₄×C₇	—	4	4	no	—
65	C₂×#99	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
66	C₂³×C₇	—	2	1	no	—
67	C₂³×C₇	—	2	1	no	—
68	C₂³×C₇	—	4	2	no	—
69	C₂³×C₇	—	4	2	no	—
70	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
71	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
72	C₂³×C₇	C₂×C₃×C₄	1	2	yes	complemented
73	C₂³×C₇	C₃×D₄	1	2	yes	—
74	C₂³×C₇	C₃×D₄	1	2	yes	—
75	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
76	C₇×D₄	—	2	4	no	—
77	C₇×D₄	—	2	4	no	—
78	C₂×C₄×C₇	—	2	1	no	—
79	C₂×C₄×C₇	—	2	1	no	—
80	C₂×C₄×C₇	—	2	4	no	—
81	C₂×C₄×C₇	—	4	4	no	—
82	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	complemented
83	C₂⁴×C₃	C₄×C₇	1	1	yes	characteristic, complemented
84	C₂⁴×C₃	C₂²×C₇	1	1	yes	characteristic
85	C₂×C₃×D₄	—	2	2	no	—
86	C₂×C₃×D₄	—	2	2	no	—
87	C₂×C₃×D₄	C₄×C₇	1	2	yes	complemented
88	C₂×C₃×D₄	C₂²×C₇	1	2	yes	—
89	C₂²×C₃×C₄	—	2	2	no	—
90	C₂²×C₃×C₄	C₄×C₇	1	1	yes	characteristic, complemented
91	C₃×C₂²⋊C₄	—	2	2	no	—
92	C₃×C₂²⋊C₄	C₂²×C₇	1	4	yes	complemented
93	C₂×C₃×C₇	—	2	2	no	—
94	C₂×C₃×C₇	—	2	1	no	—
95	C₂×C₃×C₇	—	2	1	no	—
96	C₂×C₃×C₇	—	2	2	no	—
97	C₂×C₃×C₇	—	4	2	no	—
98	C₂×C₃×C₇	#99	1	2	yes	direct product factor
99	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×D₄	C₂×C₃×C₇	1	1	yes	characteristic
101	C₂×C₂²⋊C₄	C₂×C₃×C₇	1	2	yes	complemented
102	#99	C₂×C₃×C₇	1	4	yes	direct product factor
103	C₂²×C₇	—	2	4	no	—
104	C₂²×C₇	—	2	2	no	—
105	C₂²×C₇	—	2	2	no	—
106	C₂²×C₇	—	4	1	no	—
107	C₂²×C₇	—	4	2	no	—
108	C₂²×C₇	—	4	1	no	—
109	C₂²×C₇	—	4	2	no	—
110	C₂²×C₇	—	4	2	no	—
111	C₂²×C₇	—	4	2	no	—
112	C₂²×C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×C₇	C₃×C₂²⋊C₄	1	2	yes	—
114	C₂²×C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
116	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
117	C₄×C₇	—	2	2	no	—
118	C₄×C₇	—	4	4	no	—
119	C₂³×C₃	—	2	1	no	—
120	C₂³×C₃	—	2	1	no	—
121	C₂³×C₃	—	4	2	no	—
122	C₂³×C₃	—	4	2	no	—
123	C₂³×C₃	C₂×C₄×C₇	1	1	yes	characteristic
124	C₂³×C₃	C₂×C₄×C₇	1	1	yes	characteristic
125	C₂³×C₃	C₂×C₄×C₇	1	2	yes	complemented
126	C₂³×C₃	C₇×D₄	1	2	yes	—
127	C₂³×C₃	C₇×D₄	1	2	yes	—
128	C₂³×C₃	C₂³×C₇	1	1	yes	characteristic
129	C₃×D₄	—	2	4	no	—
130	C₃×D₄	—	2	4	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	—	4	4	no	—
135	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	complemented
136	C₃×C₇	C₂×#99	1	1	yes	characteristic, direct product factor, Hall
137	C₂⁴	C₃×C₄×C₇	1	1	yes	characteristic, complemented
138	C₂⁴	C₂²×C₃×C₇	1	1	yes	characteristic
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	C₃×C₄×C₇	1	2	yes	complemented
142	C₂×D₄	C₂²×C₃×C₇	1	2	yes	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	C₃×C₄×C₇	1	1	yes	characteristic, complemented
145	C₂²⋊C₄	—	2	2	no	—
146	C₂²⋊C₄	C₂²×C₃×C₇	1	4	yes	complemented
147	C₂×C₇	—	2	2	no	—
148	C₂×C₇	—	2	1	no	—
149	C₂×C₇	—	2	1	no	—
150	C₂×C₇	—	2	2	no	—
151	C₂×C₇	—	4	2	no	—
152	C₂×C₇	C₃×#99	1	2	yes	direct product factor
153	C₂×C₇	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₃	—	2	4	no	—
155	C₂²×C₃	—	2	2	no	—
156	C₂²×C₃	—	2	2	no	—
157	C₂²×C₃	—	4	1	no	—
158	C₂²×C₃	—	4	2	no	—
159	C₂²×C₃	—	4	1	no	—
160	C₂²×C₃	—	4	2	no	—
161	C₂²×C₃	—	4	2	no	—
162	C₂²×C₃	—	4	2	no	—
163	C₂²×C₃	C₇×C₂²⋊C₄	1	1	yes	characteristic
164	C₂²×C₃	C₇×C₂²⋊C₄	1	2	yes	—
165	C₂²×C₃	C₇×C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
167	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
168	C₃×C₄	—	2	2	no	—
169	C₃×C₄	—	4	4	no	—
170	C₂³	—	2	1	no	—
171	C₂³	—	2	1	no	—
172	C₂³	—	4	2	no	—
173	C₂³	—	4	2	no	—
174	C₂³	C₂×C₃×C₄×C₇	1	1	yes	characteristic
175	C₂³	C₂×C₃×C₄×C₇	1	1	yes	characteristic
176	C₂³	C₂×C₃×C₄×C₇	1	2	yes	complemented
177	C₂³	C₃×C₇×D₄	1	2	yes	—
178	C₂³	C₃×C₇×D₄	1	2	yes	—
179	C₂³	C₂³×C₃×C₇	1	1	yes	characteristic
180	D₄	—	2	4	no	—
181	D₄	—	2	4	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	C₂×C₃×C₄×C₇	1	2	yes	complemented
187	C₇	C₂×C₃×#99	1	1	yes	characteristic, direct product factor, Sylow
188	C₂×C₃	—	2	2	no	—
189	C₂×C₃	—	2	1	no	—
190	C₂×C₃	—	2	1	no	—
191	C₂×C₃	—	2	2	no	—
192	C₂×C₃	—	4	2	no	—
193	C₂×C₃	C₇×#99	1	2	yes	direct product factor
194	C₂×C₃	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
195	C₂²	—	2	4	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	4	1	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	2	no	—
204	C₂²	C₃×C₇×C₂²⋊C₄	1	1	yes	characteristic
205	C₂²	C₃×C₇×C₂²⋊C₄	1	2	yes	—
206	C₂²	C₃×C₇×C₂²⋊C₄	1	1	yes	characteristic
207	C₂²	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
208	C₂²	C₂×C₃×C₇×D₄	1	2	yes	—
209	C₄	—	2	2	no	—
210	C₄	—	4	4	no	—
211	C₃	C₂×C₇×#99	1	1	yes	characteristic, direct product factor, Sylow
212	C₂	—	2	2	no	—
213	C₂	—	2	1	no	—
214	C₂	—	2	1	no	—
215	C₂	—	2	2	no	—
216	C₂	—	4	2	no	—
217	C₂	C₃×C₇×#99	1	2	yes	direct product factor
218	C₂	C₂×C₃×C₇×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₇	C₂²⋊C₄
commutator	C₂²	C₂²×C₃×C₄×C₇
Frattini, cosocle	C₂³	C₂³×C₃×C₇
prime radical	C₂²×C₃×C₇×D₄	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₇ ◁ C₂⁴×C₃×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₇×C₂²⋊C₄	C₂	semidirect
3. C₃×C₇×#99	C₂	direct (N × H)
4. C₂×C₇×#99	C₃	direct (N × H)
5. C₂⁴×C₃×C₇	C₄	semidirect
9. C₂×C₃×C₇×D₄	C₄	semidirect
12. C₂²×C₃×C₄×C₇	C₄	semidirect
14. C₃×C₇×C₂²⋊C₄	C₂²	semidirect
16. C₂×C₇×C₂²⋊C₄	C₂×C₃	semidirect
17. C₇×#99	C₂×C₃	direct (N × H)
18. C₂×C₃×#99	C₇	direct (N × H)
25. C₂³×C₃×C₇	C₂×C₄	semidirect
35. C₂×C₃×C₄×C₇	C₂×C₄	semidirect
36. C₂⁴×C₇	C₃×C₄	semidirect
40. C₂×C₇×D₄	C₃×C₄	semidirect
43. C₂²×C₄×C₇	C₃×C₄	semidirect
45. C₇×C₂²⋊C₄	C₂²×C₃	semidirect
47. C₂×C₃×C₂²⋊C₄	C₂×C₇	semidirect
48. C₃×#99	C₂×C₇	direct (N × H)
65. C₂×#99	C₃×C₇	direct (N × H)
72. C₂³×C₇	C₂×C₃×C₄	semidirect
82. C₂×C₄×C₇	C₂×C₃×C₄	semidirect
83. C₂⁴×C₃	C₄×C₇	semidirect
87. C₂×C₃×D₄	C₄×C₇	semidirect
90. C₂²×C₃×C₄	C₄×C₇	semidirect
92. C₃×C₂²⋊C₄	C₂²×C₇	semidirect
98. C₂×C₃×C₇	#99	direct (N × H)
101. C₂×C₂²⋊C₄	C₂×C₃×C₇	semidirect
102. #99	C₂×C₃×C₇	direct (N × H)
125. C₂³×C₃	C₂×C₄×C₇	semidirect
135. C₂×C₃×C₄	C₂×C₄×C₇	semidirect
136. C₃×C₇	C₂×#99	direct (N × H)
137. C₂⁴	C₃×C₄×C₇	semidirect
141. C₂×D₄	C₃×C₄×C₇	semidirect
144. C₂²×C₄	C₃×C₄×C₇	semidirect
146. C₂²⋊C₄	C₂²×C₃×C₇	semidirect
152. C₂×C₇	C₃×#99	direct (N × H)
176. C₂³	C₂×C₃×C₄×C₇	semidirect
186. C₂×C₄	C₂×C₃×C₄×C₇	semidirect
187. C₇	C₂×C₃×#99	direct (N × H)
193. C₂×C₃	C₇×#99	direct (N × H)
211. C₃	C₂×C₇×#99	direct (N × H)
217. C₂	C₃×C₇×#99	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
C₂×C₃×C₇×C₂²⋊C₄	2
C₃×C₇×#99	2
C₂×C₇×#99	3
C₂×C₃×#99	7

Subgroups — C2×C3×C7×#99 #13056920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₇×#99 #13056920