Subgroups — C₂×C₃×C₇×#104 #13056931

Order 1344.

340 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₇×#104	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₇×#104	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₃×C₇×#41	—	2	1	no	—
7	C₃×C₇×#41	—	2	1	no	—
8	C₃×C₇×#41	C₄	1	2	yes	complemented
9	C₃×C₇×#41	C₂²	1	2	yes	—
10	C₂×C₃×C₇×Q₈	C₄	1	1	yes	characteristic, complemented
11	C₂×C₃×C₇×D₄	—	2	1	no	—
12	C₂×C₃×C₇×D₄	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₃×C₄×C₇	—	2	1	no	—
14	C₂²×C₃×C₄×C₇	—	2	1	no	—
15	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
16	C₃×C₇×M₁₆	—	2	1	no	—
17	C₃×C₇×M₁₆	C₂²	1	2	yes	complemented
18	C₂×C₃×C₇×C₈	—	2	1	no	—
19	C₃×C₄²×C₇	—	2	1	no	—
20	C₃×C₄²×C₇	C₂²	1	2	yes	complemented
21	C₂×C₇×#41	C₂×C₃	1	1	yes	characteristic
22	C₂×C₇×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×C₄²×C₇	C₂×C₃	1	1	yes	characteristic, complemented
24	C₇×#104	C₂×C₃	1	4	yes	direct product factor
25	C₂×C₃×#104	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₂³×C₃×C₇	—	2	1	no	—
27	C₂³×C₃×C₇	D₄	1	1	yes	characteristic
28	C₃×C₇×Q₈	—	2	1	no	—
29	C₃×C₇×Q₈	C₂×C₄	1	2	yes	complemented
30	C₃×C₇×D₄	—	2	2	no	—
31	C₃×C₇×D₄	—	2	2	no	—
32	C₃×C₇×D₄	—	2	1	no	—
33	C₃×C₇×D₄	C₂×C₄	1	2	yes	complemented
34	C₂×C₃×C₄×C₇	—	2	4	no	—
35	C₂×C₃×C₄×C₇	—	2	1	no	—
36	C₂×C₃×C₄×C₇	—	2	2	no	—
37	C₂×C₃×C₄×C₇	—	2	2	no	—
38	C₂×C₃×C₄×C₇	—	2	2	no	—
39	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
42	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
43	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
44	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
45	C₃×C₇×C₈	—	2	2	no	—
46	C₇×#41	—	2	1	no	—
47	C₇×#41	—	2	1	no	—
48	C₇×#41	C₃×C₄	1	2	yes	complemented
49	C₇×#41	C₂²×C₃	1	2	yes	—
50	C₂×C₇×Q₈	C₃×C₄	1	1	yes	characteristic, complemented
51	C₂×C₇×D₄	—	2	1	no	—
52	C₂×C₇×D₄	C₃×C₄	1	1	yes	characteristic, complemented
53	C₂²×C₄×C₇	—	2	1	no	—
54	C₂²×C₄×C₇	—	2	1	no	—
55	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
56	C₇×M₁₆	—	2	1	no	—
57	C₇×M₁₆	C₂²×C₃	1	2	yes	complemented
58	C₂×C₇×C₈	—	2	1	no	—
59	C₄²×C₇	—	2	1	no	—
60	C₄²×C₇	C₂²×C₃	1	2	yes	complemented
61	C₂×C₃×#41	C₂×C₇	1	1	yes	characteristic
62	C₂×C₃×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
63	C₂×C₃×C₄²	C₂×C₇	1	1	yes	characteristic, complemented
64	C₃×#104	C₂×C₇	1	4	yes	direct product factor
65	C₂²×C₃×C₇	—	2	2	no	—
66	C₂²×C₃×C₇	—	2	2	no	—
67	C₂²×C₃×C₇	—	4	2	no	—
68	C₂²×C₃×C₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₃×C₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
71	C₃×C₄×C₇	—	2	4	no	—
72	C₃×C₄×C₇	—	2	2	no	—
73	C₃×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₃×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
77	C₂×#104	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
78	C₂³×C₇	—	2	1	no	—
79	C₂³×C₇	C₃×D₄	1	1	yes	characteristic
80	C₇×Q₈	—	2	1	no	—
81	C₇×Q₈	C₂×C₃×C₄	1	2	yes	complemented
82	C₇×D₄	—	2	2	no	—
83	C₇×D₄	—	2	2	no	—
84	C₇×D₄	—	2	1	no	—
85	C₇×D₄	C₂×C₃×C₄	1	2	yes	complemented
86	C₂×C₄×C₇	—	2	4	no	—
87	C₂×C₄×C₇	—	2	1	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	2	2	no	—
91	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
93	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
94	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
95	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
97	C₇×C₈	—	2	2	no	—
98	C₃×#41	—	2	1	no	—
99	C₃×#41	—	2	1	no	—
100	C₃×#41	C₄×C₇	1	2	yes	complemented
101	C₃×#41	C₂²×C₇	1	2	yes	—
102	C₂×C₃×Q₈	C₄×C₇	1	1	yes	characteristic, complemented
103	C₂×C₃×D₄	—	2	1	no	—
104	C₂×C₃×D₄	C₄×C₇	1	1	yes	characteristic, complemented
105	C₂²×C₃×C₄	—	2	1	no	—
106	C₂²×C₃×C₄	—	2	1	no	—
107	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
108	C₃×M₁₆	—	2	1	no	—
109	C₃×M₁₆	C₂²×C₇	1	2	yes	complemented
110	C₂×C₃×C₈	—	2	1	no	—
111	C₃×C₄²	—	2	1	no	—
112	C₃×C₄²	C₂²×C₇	1	2	yes	complemented
113	C₂×C₃×C₇	—	2	1	no	—
114	C₂×C₃×C₇	—	2	1	no	—
115	C₂×C₃×C₇	—	4	2	no	—
116	C₂×C₃×C₇	#104	1	2	yes	direct product factor
117	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×#41	C₂×C₃×C₇	1	1	yes	characteristic
119	C₂×M₁₆	C₂×C₃×C₇	1	1	yes	characteristic, complemented
120	C₂×C₄²	C₂×C₃×C₇	1	1	yes	characteristic, complemented
121	#104	C₂×C₃×C₇	1	4	yes	direct product factor
122	C₂²×C₇	—	2	2	no	—
123	C₂²×C₇	—	2	2	no	—
124	C₂²×C₇	—	4	2	no	—
125	C₂²×C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
128	C₄×C₇	—	2	4	no	—
129	C₄×C₇	—	2	2	no	—
130	C₄×C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic
131	C₄×C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic
132	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
133	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
134	C₂³×C₃	—	2	1	no	—
135	C₂³×C₃	C₇×D₄	1	1	yes	characteristic
136	C₃×Q₈	—	2	1	no	—
137	C₃×Q₈	C₂×C₄×C₇	1	2	yes	complemented
138	C₃×D₄	—	2	2	no	—
139	C₃×D₄	—	2	2	no	—
140	C₃×D₄	—	2	1	no	—
141	C₃×D₄	C₂×C₄×C₇	1	2	yes	complemented
142	C₂×C₃×C₄	—	2	4	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
149	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
151	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
153	C₃×C₈	—	2	2	no	—
154	C₃×C₇	C₂×#104	1	1	yes	characteristic, direct product factor, Hall
155	#41	—	2	1	no	—
156	#41	—	2	1	no	—
157	#41	C₃×C₄×C₇	1	2	yes	complemented
158	#41	C₂²×C₃×C₇	1	2	yes	—
159	C₂×Q₈	C₃×C₄×C₇	1	1	yes	characteristic, complemented
160	C₂×D₄	—	2	1	no	—
161	C₂×D₄	C₃×C₄×C₇	1	1	yes	characteristic, complemented
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
165	M₁₆	—	2	1	no	—
166	M₁₆	C₂²×C₃×C₇	1	2	yes	complemented
167	C₂×C₈	—	2	1	no	—
168	C₄²	—	2	1	no	—
169	C₄²	C₂²×C₃×C₇	1	2	yes	complemented
170	C₂×C₇	—	2	1	no	—
171	C₂×C₇	—	2	1	no	—
172	C₂×C₇	—	4	2	no	—
173	C₂×C₇	C₃×#104	1	2	yes	direct product factor
174	C₂×C₇	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
175	C₂²×C₃	—	2	2	no	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	4	2	no	—
178	C₂²×C₃	C₇×C₂²⋊C₄	1	1	yes	characteristic
179	C₂²×C₃	C₇×C₂²⋊C₄	1	1	yes	characteristic
180	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
181	C₃×C₄	—	2	4	no	—
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
184	C₃×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
185	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
186	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
187	C₂³	—	2	1	no	—
188	C₂³	C₃×C₇×D₄	1	1	yes	characteristic
189	Q₈	—	2	1	no	—
190	Q₈	C₂×C₃×C₄×C₇	1	2	yes	complemented
191	D₄	—	2	2	no	—
192	D₄	—	2	2	no	—
193	D₄	—	2	1	no	—
194	D₄	C₂×C₃×C₄×C₇	1	2	yes	complemented
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	C₂×C₃×C₄×C₇	1	1	yes	characteristic
201	C₂×C₄	C₂×C₃×C₄×C₇	1	1	yes	characteristic
202	C₂×C₄	C₃×C₇×D₄	1	1	yes	characteristic
203	C₂×C₄	C₃×C₇×D₄	1	1	yes	characteristic
204	C₂×C₄	C₃×C₇×D₄	1	1	yes	characteristic
205	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₇	C₂×C₃×#104	1	1	yes	characteristic, direct product factor, Sylow
208	C₂×C₃	—	2	1	no	—
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	—	4	2	no	—
211	C₂×C₃	C₇×#104	1	2	yes	direct product factor
212	C₂×C₃	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	C₃×C₇×C₂²⋊C₄	1	1	yes	characteristic
217	C₂²	C₃×C₇×C₂²⋊C₄	1	1	yes	characteristic
218	C₂²	C₂×C₃×C₇×D₄	1	1	yes	characteristic
219	C₄	—	2	4	no	—
220	C₄	—	2	2	no	—
221	C₄	C₃×C₇×C₂²⋊C₄	1	1	yes	characteristic
222	C₄	C₃×C₇×C₂²⋊C₄	1	1	yes	characteristic
223	C₄	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
224	C₄	C₂×C₃×C₇×D₄	1	1	yes	characteristic
225	C₃	C₂×C₇×#104	1	1	yes	characteristic, direct product factor, Sylow
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	—	4	2	no	—
229	C₂	C₃×C₇×#104	1	2	yes	direct product factor
230	C₂	C₂×C₃×C₇×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄×C₇	D₄
commutator	C₄	C₂²×C₃×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₇
socle	C₂²×C₃×C₇	C₂²⋊C₄
prime radical	C₂×C₃×C₇×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₇ ◁ C₂²×C₃×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₇×M₁₆	C₂	semidirect
3. C₂×C₃×C₄²×C₇	C₂	semidirect
4. C₃×C₇×#104	C₂	direct (N × H)
5. C₂×C₇×#104	C₃	direct (N × H)
8. C₃×C₇×#41	C₄	semidirect
10. C₂×C₃×C₇×Q₈	C₄	semidirect
12. C₂×C₃×C₇×D₄	C₄	semidirect
17. C₃×C₇×M₁₆	C₂²	semidirect
20. C₃×C₄²×C₇	C₂²	semidirect
22. C₂×C₇×M₁₆	C₂×C₃	semidirect
23. C₂×C₄²×C₇	C₂×C₃	semidirect
24. C₇×#104	C₂×C₃	direct (N × H)
25. C₂×C₃×#104	C₇	direct (N × H)
29. C₃×C₇×Q₈	C₂×C₄	semidirect
33. C₃×C₇×D₄	C₂×C₄	semidirect
48. C₇×#41	C₃×C₄	semidirect
50. C₂×C₇×Q₈	C₃×C₄	semidirect
52. C₂×C₇×D₄	C₃×C₄	semidirect
57. C₇×M₁₆	C₂²×C₃	semidirect
60. C₄²×C₇	C₂²×C₃	semidirect
62. C₂×C₃×M₁₆	C₂×C₇	semidirect
63. C₂×C₃×C₄²	C₂×C₇	semidirect
64. C₃×#104	C₂×C₇	direct (N × H)
77. C₂×#104	C₃×C₇	direct (N × H)
81. C₇×Q₈	C₂×C₃×C₄	semidirect
85. C₇×D₄	C₂×C₃×C₄	semidirect
100. C₃×#41	C₄×C₇	semidirect
102. C₂×C₃×Q₈	C₄×C₇	semidirect
104. C₂×C₃×D₄	C₄×C₇	semidirect
109. C₃×M₁₆	C₂²×C₇	semidirect
112. C₃×C₄²	C₂²×C₇	semidirect
116. C₂×C₃×C₇	#104	direct (N × H)
119. C₂×M₁₆	C₂×C₃×C₇	semidirect
120. C₂×C₄²	C₂×C₃×C₇	semidirect
121. #104	C₂×C₃×C₇	direct (N × H)
137. C₃×Q₈	C₂×C₄×C₇	semidirect
141. C₃×D₄	C₂×C₄×C₇	semidirect
154. C₃×C₇	C₂×#104	direct (N × H)
157. #41	C₃×C₄×C₇	semidirect
159. C₂×Q₈	C₃×C₄×C₇	semidirect
161. C₂×D₄	C₃×C₄×C₇	semidirect
166. M₁₆	C₂²×C₃×C₇	semidirect
169. C₄²	C₂²×C₃×C₇	semidirect
173. C₂×C₇	C₃×#104	direct (N × H)
190. Q₈	C₂×C₃×C₄×C₇	semidirect
194. D₄	C₂×C₃×C₄×C₇	semidirect
207. C₇	C₂×C₃×#104	direct (N × H)
211. C₂×C₃	C₇×#104	direct (N × H)
225. C₃	C₂×C₇×#104	direct (N × H)
229. C₂	C₃×C₇×#104	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×C₃×C₇×#41	2
C₂×C₃×C₇×M₁₆	2
C₂×C₃×C₄²×C₇	2
C₃×C₇×#104	2
C₂×C₇×#104	3
C₂×C₃×#104	7

Subgroups — C2×C3×C7×#104 #13056931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₇×#104 #13056931