Subgroups — C₃×C₇×#541 #13061961

Order 1344.

384 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₇×#126	C₂	1	4	yes	maximal
3	C₃×C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₇×#125	C₂	1	4	yes	maximal, complemented
5	C₃×C₇×C₂²⋊Q₈	C₂	1	2	yes	maximal
6	C₃×C₄×C₇×Q₈	C₂	1	2	yes	maximal, complemented
7	C₃×C₇×#117	C₂	1	1	yes	maximal, characteristic
8	C₇×#541	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₇×Q₈	C₂²	1	2	yes	—
10	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₃×C₇×C₄⋊C₄	C₂²	1	4	yes	—
14	C₃×C₇×C₄⋊C₄	C₂²	1	4	yes	—
15	C₃×C₇×C₄⋊C₄	C₂²	1	4	yes	—
16	C₃×C₇×C₄⋊C₄	C₂²	1	4	yes	—
17	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₃×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
19	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
20	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
21	C₃×C₄²×C₇	C₂²	1	4	yes	—
22	C₇×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
23	C₇×#126	C₂×C₃	1	4	yes	—
24	C₇×#125	C₂×C₃	1	1	yes	characteristic, complemented
25	C₇×#125	C₂×C₃	1	4	yes	complemented
26	C₇×C₂²⋊Q₈	C₂×C₃	1	2	yes	—
27	C₄×C₇×Q₈	C₂×C₃	1	2	yes	complemented
28	C₇×#117	C₂×C₃	1	1	yes	characteristic
29	C₃×#541	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
30	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
31	C₃×C₇×Q₈	—	2	4	no	—
32	C₂×C₃×C₄×C₇	—	2	2	no	—
33	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
34	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
35	C₂×C₃×C₄×C₇	C₂³	1	4	yes	—
36	C₂×C₃×C₄×C₇	C₂³	1	4	yes	—
37	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
39	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
40	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
41	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
42	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
43	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
44	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
45	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
46	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
47	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
48	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
49	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
50	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
51	C₄²×C₇	C₂²×C₃	1	4	yes	—
52	C₃×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
53	C₃×#126	C₂×C₇	1	4	yes	—
54	C₃×#125	C₂×C₇	1	1	yes	characteristic, complemented
55	C₃×#125	C₂×C₇	1	4	yes	complemented
56	C₃×C₂²⋊Q₈	C₂×C₇	1	2	yes	—
57	C₃×C₄×Q₈	C₂×C₇	1	2	yes	complemented
58	C₃×#117	C₂×C₇	1	1	yes	characteristic
59	C₂²×C₃×C₇	—	2	1	no	—
60	C₂²×C₃×C₇	—	2	2	no	—
61	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
62	C₃×C₄×C₇	—	2	1	no	—
63	C₃×C₄×C₇	—	2	4	no	—
64	C₃×C₄×C₇	—	2	2	no	—
65	C₃×C₄×C₇	—	2	2	no	—
66	C₃×C₄×C₇	—	2	4	no	—
67	C₃×C₄×C₇	#41	1	2	yes	—
68	#541	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
69	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
70	C₇×Q₈	—	2	4	no	—
71	C₂×C₄×C₇	—	2	2	no	—
72	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
73	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
74	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
75	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
76	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
77	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
78	C₂×C₃×Q₈	C₂²×C₇	1	2	yes	—
79	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
80	C₃×C₄⋊C₄	C₂²×C₇	1	2	yes	—
81	C₃×C₄⋊C₄	C₂²×C₇	1	2	yes	—
82	C₃×C₄⋊C₄	C₂²×C₇	1	4	yes	—
83	C₃×C₄⋊C₄	C₂²×C₇	1	4	yes	—
84	C₃×C₄⋊C₄	C₂²×C₇	1	4	yes	—
85	C₃×C₄⋊C₄	C₂²×C₇	1	4	yes	—
86	C₃×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
87	C₃×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
88	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
89	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
90	C₃×C₄²	C₂²×C₇	1	4	yes	—
91	C₂×C₃×C₇	—	4	1	no	—
92	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
93	C₂×C₃×C₇	2₋¹⁺⁴	1	2	yes	—
94	C₄⋊Q₈	C₂×C₃×C₇	1	1	yes	characteristic, complemented
95	#126	C₂×C₃×C₇	1	4	yes	—
96	#125	C₂×C₃×C₇	1	1	yes	characteristic, complemented
97	#125	C₂×C₃×C₇	1	4	yes	complemented
98	C₂²⋊Q₈	C₂×C₃×C₇	1	2	yes	—
99	C₄×Q₈	C₂×C₃×C₇	1	2	yes	complemented
100	#117	C₂×C₃×C₇	1	1	yes	characteristic
101	C₂²×C₇	—	2	1	no	—
102	C₂²×C₇	—	2	2	no	—
103	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
104	C₄×C₇	—	2	1	no	—
105	C₄×C₇	—	2	4	no	—
106	C₄×C₇	—	2	2	no	—
107	C₄×C₇	—	2	2	no	—
108	C₄×C₇	—	2	4	no	—
109	C₄×C₇	C₃×#41	1	2	yes	—
110	C₂³×C₃	C₂³×C₇	1	1	yes	characteristic
111	C₃×Q₈	—	2	4	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
114	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
115	C₂×C₃×C₄	C₂³×C₇	1	4	yes	—
116	C₂×C₃×C₄	C₂³×C₇	1	4	yes	—
117	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
119	C₃×C₇	#541	1	1	yes	characteristic, direct product factor, Hall
120	C₂×Q₈	C₂²×C₃×C₇	1	2	yes	—
121	C₂²×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
122	C₄⋊C₄	C₂²×C₃×C₇	1	2	yes	—
123	C₄⋊C₄	C₂²×C₃×C₇	1	2	yes	—
124	C₄⋊C₄	C₂²×C₃×C₇	1	4	yes	—
125	C₄⋊C₄	C₂²×C₃×C₇	1	4	yes	—
126	C₄⋊C₄	C₂²×C₃×C₇	1	4	yes	—
127	C₄⋊C₄	C₂²×C₃×C₇	1	4	yes	—
128	C₂²⋊C₄	C₂²×C₃×C₇	1	2	yes	—
129	C₂²⋊C₄	C₂²×C₃×C₇	1	4	yes	—
130	C₄²	C₂²×C₃×C₇	1	1	yes	characteristic
131	C₄²	C₂²×C₃×C₇	1	1	yes	characteristic
132	C₄²	C₂²×C₃×C₇	1	4	yes	—
133	C₂×C₇	—	4	1	no	—
134	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
135	C₂×C₇	C₃×2₋¹⁺⁴	1	2	yes	—
136	C₂²×C₃	—	2	1	no	—
137	C₂²×C₃	—	2	2	no	—
138	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
139	C₃×C₄	—	2	1	no	—
140	C₃×C₄	—	2	4	no	—
141	C₃×C₄	—	2	2	no	—
142	C₃×C₄	—	2	2	no	—
143	C₃×C₄	—	2	4	no	—
144	C₃×C₄	C₇×#41	1	2	yes	—
145	C₂³	C₂³×C₃×C₇	1	1	yes	characteristic
146	Q₈	—	2	4	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	C₂³×C₃×C₇	1	2	yes	—
149	C₂×C₄	C₂³×C₃×C₇	1	2	yes	—
150	C₂×C₄	C₂³×C₃×C₇	1	4	yes	—
151	C₂×C₄	C₂³×C₃×C₇	1	4	yes	—
152	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
153	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
154	C₇	C₃×#541	1	1	yes	characteristic, direct product factor, Sylow
155	C₂×C₃	—	4	1	no	—
156	C₂×C₃	C₂×C₇×#41	1	1	yes	characteristic
157	C₂×C₃	C₇×2₋¹⁺⁴	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	C₂⁴×C₃×C₇	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	4	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	2	4	no	—
166	C₄	C₃×C₇×#41	1	2	yes	—
167	C₃	C₇×#541	1	1	yes	characteristic, direct product factor, Sylow
168	C₂	—	4	1	no	—
169	C₂	C₂×C₃×C₇×#41	1	1	yes	characteristic
170	C₂	C₃×C₇×2₋¹⁺⁴	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×C₄⋊Q₈	C₂	semidirect
3. C₃×C₇×#125	C₂	semidirect
4. C₃×C₇×#125	C₂	semidirect
6. C₃×C₄×C₇×Q₈	C₂	semidirect
8. C₇×#541	C₃	direct (N × H)
22. C₇×C₄⋊Q₈	C₂×C₃	semidirect
24. C₇×#125	C₂×C₃	semidirect
25. C₇×#125	C₂×C₃	semidirect
27. C₄×C₇×Q₈	C₂×C₃	semidirect
29. C₃×#541	C₇	direct (N × H)
52. C₃×C₄⋊Q₈	C₂×C₇	semidirect
54. C₃×#125	C₂×C₇	semidirect
55. C₃×#125	C₂×C₇	semidirect
57. C₃×C₄×Q₈	C₂×C₇	semidirect
68. #541	C₃×C₇	direct (N × H)
94. C₄⋊Q₈	C₂×C₃×C₇	semidirect
96. #125	C₂×C₃×C₇	semidirect
97. #125	C₂×C₃×C₇	semidirect
99. C₄×Q₈	C₂×C₃×C₇	semidirect
119. C₃×C₇	#541	direct (N × H)
154. C₇	C₃×#541	direct (N × H)
167. C₃	C₇×#541	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₇×C₄⋊Q₈	2
C₃×C₇×#126	2
C₃×C₇×#125	2
C₃×C₇×C₂²⋊Q₈	2
C₃×C₄×C₇×Q₈	2
C₃×C₇×#117	2
C₇×#541	3
C₃×#541	7

Subgroups — C3×C7×#541 #13061961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#541 #13061961