Subgroups — C₃×C₇×#416 #13056927

Order 1344.

316 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×C₃×C₇×#41	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₇×#103	C₂	1	2	yes	maximal, complemented
5	C₃×C₇×#102	C₂	1	2	yes	maximal, complemented
6	C₇×#416	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₇×#41	—	2	2	no	—
8	C₃×C₇×#41	C₄	1	4	yes	complemented
9	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₇×D₄	—	2	1	no	—
11	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₇	—	2	1	no	—
13	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₇×C₈	—	2	2	no	—
15	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
16	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
17	C₃×C₇×C₂²⋊C₄	—	2	1	no	—
18	C₃×C₄²×C₇	—	2	1	no	—
19	C₂×C₇×#41	C₂×C₃	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂×C₃	1	1	yes	characteristic, complemented
21	C₇×#117	C₂×C₃	1	1	yes	characteristic, complemented
22	C₇×#103	C₂×C₃	1	2	yes	complemented
23	C₇×#102	C₂×C₃	1	2	yes	complemented
24	C₃×#416	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂³×C₃×C₇	—	2	1	no	—
26	C₂³×C₃×C₇	D₄	1	1	yes	characteristic
27	C₃×C₇×Q₈	—	2	1	no	—
28	C₃×C₇×Q₈	C₂×C₄	1	2	yes	—
29	C₃×C₇×D₄	—	2	4	no	—
30	C₃×C₇×D₄	—	2	1	no	—
31	C₃×C₇×D₄	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₄×C₇	—	2	1	no	—
33	C₂×C₃×C₄×C₇	—	2	4	no	—
34	C₂×C₃×C₄×C₇	—	2	2	no	—
35	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
37	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
38	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
39	C₃×C₇×C₈	—	2	2	no	—
40	C₇×#41	—	2	2	no	—
41	C₇×#41	C₃×C₄	1	4	yes	complemented
42	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₇×D₄	—	2	1	no	—
44	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
45	C₂²×C₄×C₇	—	2	1	no	—
46	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₇×C₈	—	2	2	no	—
48	C₂×C₇×C₈	C₂²×C₃	1	2	yes	—
49	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
50	C₇×C₂²⋊C₄	—	2	1	no	—
51	C₄²×C₇	—	2	1	no	—
52	C₂×C₃×#41	C₂×C₇	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂×C₇	1	1	yes	characteristic, complemented
54	C₃×#117	C₂×C₇	1	1	yes	characteristic, complemented
55	C₃×#103	C₂×C₇	1	2	yes	complemented
56	C₃×#102	C₂×C₇	1	2	yes	complemented
57	C₂²×C₃×C₇	—	2	2	no	—
58	C₂²×C₃×C₇	—	2	2	no	—
59	C₂²×C₃×C₇	—	4	2	no	—
60	C₂²×C₃×C₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
62	C₃×C₄×C₇	—	2	2	no	—
63	C₃×C₄×C₇	—	4	2	no	—
64	C₃×C₄×C₇	C₂²⋊C₄	1	2	yes	—
65	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
66	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
67	#416	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
68	C₂³×C₇	—	2	1	no	—
69	C₂³×C₇	C₃×D₄	1	1	yes	characteristic
70	C₇×Q₈	—	2	1	no	—
71	C₇×Q₈	C₂×C₃×C₄	1	2	yes	—
72	C₇×D₄	—	2	4	no	—
73	C₇×D₄	—	2	1	no	—
74	C₇×D₄	C₂×C₃×C₄	1	2	yes	—
75	C₂×C₄×C₇	—	2	1	no	—
76	C₂×C₄×C₇	—	2	4	no	—
77	C₂×C₄×C₇	—	2	2	no	—
78	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
79	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
80	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
81	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
82	C₇×C₈	—	2	2	no	—
83	C₃×#41	—	2	2	no	—
84	C₃×#41	C₄×C₇	1	4	yes	complemented
85	C₂×C₃×Q₈	C₂²×C₇	1	1	yes	characteristic
86	C₂×C₃×D₄	—	2	1	no	—
87	C₂×C₃×D₄	C₂²×C₇	1	1	yes	characteristic
88	C₂²×C₃×C₄	—	2	1	no	—
89	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
90	C₂×C₃×C₈	—	2	2	no	—
91	C₂×C₃×C₈	C₂²×C₇	1	2	yes	—
92	C₃×C₄⋊C₄	C₂²×C₇	1	2	yes	—
93	C₃×C₂²⋊C₄	—	2	1	no	—
94	C₃×C₄²	—	2	1	no	—
95	C₂×C₃×C₇	—	2	2	no	—
96	C₂×C₃×C₇	—	4	2	no	—
97	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₃×C₇	#135	1	2	yes	—
99	C₂×#41	C₂×C₃×C₇	1	1	yes	characteristic
100	C₂²×C₈	C₂×C₃×C₇	1	1	yes	characteristic, complemented
101	#117	C₂×C₃×C₇	1	1	yes	characteristic, complemented
102	#103	C₂×C₃×C₇	1	2	yes	complemented
103	#102	C₂×C₃×C₇	1	2	yes	complemented
104	C₂²×C₇	—	2	2	no	—
105	C₂²×C₇	—	2	2	no	—
106	C₂²×C₇	—	4	2	no	—
107	C₂²×C₇	C₃×C₂²⋊C₄	1	2	yes	—
108	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
109	C₄×C₇	—	2	2	no	—
110	C₄×C₇	—	4	2	no	—
111	C₄×C₇	C₃×C₂²⋊C₄	1	2	yes	—
112	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
113	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
114	C₂³×C₃	—	2	1	no	—
115	C₂³×C₃	C₇×D₄	1	1	yes	characteristic
116	C₃×Q₈	—	2	1	no	—
117	C₃×Q₈	C₂×C₄×C₇	1	2	yes	—
118	C₃×D₄	—	2	4	no	—
119	C₃×D₄	—	2	1	no	—
120	C₃×D₄	C₂×C₄×C₇	1	2	yes	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	4	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
125	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
127	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
128	C₃×C₈	—	2	2	no	—
129	C₃×C₇	#416	1	1	yes	characteristic, direct product factor, Hall
130	#41	—	2	2	no	—
131	#41	C₃×C₄×C₇	1	4	yes	complemented
132	C₂×Q₈	C₂²×C₃×C₇	1	1	yes	characteristic
133	C₂×D₄	—	2	1	no	—
134	C₂×D₄	C₂²×C₃×C₇	1	1	yes	characteristic
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	C₂²×C₃×C₇	1	2	yes	—
139	C₄⋊C₄	C₂²×C₃×C₇	1	2	yes	—
140	C₂²⋊C₄	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₂×C₇	—	2	2	no	—
143	C₂×C₇	—	4	2	no	—
144	C₂×C₇	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₇	C₃×#135	1	2	yes	—
146	C₂²×C₃	—	2	2	no	—
147	C₂²×C₃	—	2	2	no	—
148	C₂²×C₃	—	4	2	no	—
149	C₂²×C₃	C₇×C₂²⋊C₄	1	2	yes	—
150	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
151	C₃×C₄	—	2	2	no	—
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	1	no	—
157	C₂³	C₃×C₇×D₄	1	1	yes	characteristic
158	Q₈	—	2	1	no	—
159	Q₈	C₂×C₃×C₄×C₇	1	2	yes	—
160	D₄	—	2	4	no	—
161	D₄	—	2	1	no	—
162	D₄	C₂×C₃×C₄×C₇	1	2	yes	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	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	C₃×#416	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×C₃	—	2	2	no	—
173	C₂×C₃	—	4	2	no	—
174	C₂×C₃	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₃	C₇×#135	1	2	yes	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	4	2	no	—
179	C₂²	C₃×C₇×C₂²⋊C₄	1	2	yes	—
180	C₂²	C₂×C₃×C₇×D₄	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	C₃×C₇×C₂²⋊C₄	1	2	yes	—
184	C₄	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
185	C₄	C₂×C₃×C₇×D₄	1	1	yes	characteristic
186	C₃	C₇×#416	1	1	yes	characteristic, direct product factor, Sylow
187	C₂	—	2	2	no	—
188	C₂	—	4	2	no	—
189	C₂	C₂×C₃×C₇×C₂²⋊C₄	1	1	yes	characteristic
190	C₂	C₃×C₇×#135	1	2	yes	—

	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	C₂×D₄
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. C₂²×C₃×C₇×C₈	C₂	semidirect
3. C₃×C₇×#117	C₂	semidirect
4. C₃×C₇×#103	C₂	semidirect
5. C₃×C₇×#102	C₂	semidirect
6. C₇×#416	C₃	direct (N × H)
8. C₃×C₇×#41	C₄	semidirect
20. C₂²×C₇×C₈	C₂×C₃	semidirect
21. C₇×#117	C₂×C₃	semidirect
22. C₇×#103	C₂×C₃	semidirect
23. C₇×#102	C₂×C₃	semidirect
24. C₃×#416	C₇	direct (N × H)
41. C₇×#41	C₃×C₄	semidirect
53. C₂²×C₃×C₈	C₂×C₇	semidirect
54. C₃×#117	C₂×C₇	semidirect
55. C₃×#103	C₂×C₇	semidirect
56. C₃×#102	C₂×C₇	semidirect
67. #416	C₃×C₇	direct (N × H)
84. C₃×#41	C₄×C₇	semidirect
100. C₂²×C₈	C₂×C₃×C₇	semidirect
101. #117	C₂×C₃×C₇	semidirect
102. #103	C₂×C₃×C₇	semidirect
103. #102	C₂×C₃×C₇	semidirect
129. C₃×C₇	#416	direct (N × H)
131. #41	C₃×C₄×C₇	semidirect
171. C₇	C₃×#416	direct (N × H)
186. C₃	C₇×#416	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
C₂²×C₃×C₇×C₈	2
C₃×C₇×#117	2
C₃×C₇×#103	2
C₃×C₇×#102	2
C₇×#416	3
C₃×#416	7

Subgroups — C3×C7×#416 #13056927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#416 #13056927