Subgroups — C₃×C₇×D₄×Q₈ #13061969

Order 1344.

560 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×Q₈	C₂	1	2	yes	maximal, complemented
2	C₃×C₇×C₄⋊Q₈	C₂	1	3	yes	maximal, complemented
3	C₃×C₇×C₂²⋊Q₈	C₂	1	6	yes	maximal, complemented
4	C₃×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×C₇×D₄	C₂	1	3	yes	maximal
6	C₇×D₄×Q₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₇×Q₈	—	2	8	no	—
8	C₂×C₃×C₇×Q₈	C₂²	1	6	yes	—
9	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₇	C₂²	1	6	yes	—
12	C₃×C₇×C₄⋊C₄	C₂²	1	3	yes	—
13	C₃×C₇×C₄⋊C₄	C₂²	1	3	yes	—
14	C₃×C₇×C₄⋊C₄	C₂²	1	6	yes	—
15	C₃×C₇×C₂²⋊C₄	C₂²	1	6	yes	—
16	C₃×C₄²×C₇	C₂²	1	3	yes	—
17	C₂²×C₇×Q₈	C₂×C₃	1	2	yes	complemented
18	C₇×C₄⋊Q₈	C₂×C₃	1	3	yes	complemented
19	C₇×C₂²⋊Q₈	C₂×C₃	1	6	yes	complemented
20	C₄×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
21	C₄×C₇×D₄	C₂×C₃	1	3	yes	—
22	C₃×D₄×Q₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂³×C₃×C₇	C₂³	1	2	yes	—
24	C₃×C₇×Q₈	—	2	12	no	—
25	C₃×C₇×Q₈	D₄	1	4	yes	direct product factor
26	C₃×C₇×D₄	Q₈	1	4	yes	direct product factor
27	C₂×C₃×C₄×C₇	—	2	12	no	—
28	C₂×C₃×C₄×C₇	C₂³	1	3	yes	—
29	C₂×C₃×C₄×C₇	C₂³	1	6	yes	—
30	C₂×C₃×C₄×C₇	C₂³	1	3	yes	—
31	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
32	C₂×C₇×Q₈	—	2	8	no	—
33	C₂×C₇×Q₈	C₂²×C₃	1	6	yes	—
34	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
35	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
36	C₂²×C₄×C₇	C₂²×C₃	1	6	yes	—
37	C₇×C₄⋊C₄	C₂²×C₃	1	3	yes	—
38	C₇×C₄⋊C₄	C₂²×C₃	1	3	yes	—
39	C₇×C₄⋊C₄	C₂²×C₃	1	6	yes	—
40	C₇×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
41	C₄²×C₇	C₂²×C₃	1	3	yes	—
42	C₂²×C₃×Q₈	C₂×C₇	1	2	yes	complemented
43	C₃×C₄⋊Q₈	C₂×C₇	1	3	yes	complemented
44	C₃×C₂²⋊Q₈	C₂×C₇	1	6	yes	complemented
45	C₃×C₄×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
46	C₃×C₄×D₄	C₂×C₇	1	3	yes	—
47	C₂²×C₃×C₇	—	2	2	no	—
48	C₂²×C₃×C₇	—	2	2	no	—
49	C₂²×C₃×C₇	C₂×Q₈	1	4	yes	complemented
50	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
51	C₃×C₄×C₇	—	2	6	no	—
52	C₃×C₄×C₇	—	2	3	no	—
53	C₃×C₄×C₇	C₂×D₄	1	6	yes	—
54	C₃×C₄×C₇	C₂×Q₈	1	2	yes	complemented
55	D₄×Q₈	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
56	C₂³×C₇	C₂³×C₃	1	2	yes	—
57	C₇×Q₈	—	2	12	no	—
58	C₇×Q₈	C₃×D₄	1	4	yes	direct product factor
59	C₇×D₄	C₃×Q₈	1	4	yes	direct product factor
60	C₂×C₄×C₇	—	2	12	no	—
61	C₂×C₄×C₇	C₂³×C₃	1	3	yes	—
62	C₂×C₄×C₇	C₂³×C₃	1	6	yes	—
63	C₂×C₄×C₇	C₂³×C₃	1	3	yes	—
64	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
65	C₂×C₃×Q₈	—	2	8	no	—
66	C₂×C₃×Q₈	C₂²×C₇	1	6	yes	—
67	C₂×C₃×Q₈	C₂²×C₇	1	1	yes	characteristic
68	C₂×C₃×D₄	C₂²×C₇	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂²×C₇	1	6	yes	—
70	C₃×C₄⋊C₄	C₂²×C₇	1	3	yes	—
71	C₃×C₄⋊C₄	C₂²×C₇	1	3	yes	—
72	C₃×C₄⋊C₄	C₂²×C₇	1	6	yes	—
73	C₃×C₂²⋊C₄	C₂²×C₇	1	6	yes	—
74	C₃×C₄²	C₂²×C₇	1	3	yes	—
75	C₂×C₃×C₇	—	2	4	no	—
76	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
78	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
79	C₂²×Q₈	C₂×C₃×C₇	1	2	yes	complemented
80	C₄⋊Q₈	C₂×C₃×C₇	1	3	yes	complemented
81	C₂²⋊Q₈	C₂×C₃×C₇	1	6	yes	complemented
82	C₄×Q₈	C₂×C₃×C₇	1	1	yes	characteristic, complemented
83	C₄×D₄	C₂×C₃×C₇	1	3	yes	—
84	C₂²×C₇	—	2	2	no	—
85	C₂²×C₇	—	2	2	no	—
86	C₂²×C₇	C₂×C₃×Q₈	1	4	yes	complemented
87	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
88	C₄×C₇	—	2	6	no	—
89	C₄×C₇	—	2	3	no	—
90	C₄×C₇	C₂×C₃×D₄	1	6	yes	—
91	C₄×C₇	C₂×C₃×Q₈	1	2	yes	complemented
92	C₂³×C₃	C₂³×C₇	1	2	yes	—
93	C₃×Q₈	—	2	12	no	—
94	C₃×Q₈	C₇×D₄	1	4	yes	direct product factor
95	C₃×D₄	C₇×Q₈	1	4	yes	direct product factor
96	C₂×C₃×C₄	—	2	12	no	—
97	C₂×C₃×C₄	C₂³×C₇	1	3	yes	—
98	C₂×C₃×C₄	C₂³×C₇	1	6	yes	—
99	C₂×C₃×C₄	C₂³×C₇	1	3	yes	—
100	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
101	C₃×C₇	D₄×Q₈	1	1	yes	characteristic, direct product factor, Hall
102	C₂×Q₈	—	2	8	no	—
103	C₂×Q₈	C₂²×C₃×C₇	1	6	yes	—
104	C₂×Q₈	C₂²×C₃×C₇	1	1	yes	characteristic
105	C₂×D₄	C₂²×C₃×C₇	1	1	yes	characteristic
106	C₂²×C₄	C₂²×C₃×C₇	1	6	yes	—
107	C₄⋊C₄	C₂²×C₃×C₇	1	3	yes	—
108	C₄⋊C₄	C₂²×C₃×C₇	1	3	yes	—
109	C₄⋊C₄	C₂²×C₃×C₇	1	6	yes	—
110	C₂²⋊C₄	C₂²×C₃×C₇	1	6	yes	—
111	C₄²	C₂²×C₃×C₇	1	3	yes	—
112	C₂×C₇	—	2	4	no	—
113	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
114	C₂×C₇	C₂²×C₃×Q₈	1	1	yes	characteristic
115	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
116	C₂²×C₃	—	2	2	no	—
117	C₂²×C₃	—	2	2	no	—
118	C₂²×C₃	C₂×C₇×Q₈	1	4	yes	complemented
119	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
120	C₃×C₄	—	2	6	no	—
121	C₃×C₄	—	2	3	no	—
122	C₃×C₄	C₂×C₇×D₄	1	6	yes	—
123	C₃×C₄	C₂×C₇×Q₈	1	2	yes	complemented
124	C₂³	C₂³×C₃×C₇	1	2	yes	—
125	Q₈	—	2	12	no	—
126	Q₈	C₃×C₇×D₄	1	4	yes	direct product factor
127	D₄	C₃×C₇×Q₈	1	4	yes	direct product factor
128	C₂×C₄	—	2	12	no	—
129	C₂×C₄	C₂³×C₃×C₇	1	3	yes	—
130	C₂×C₄	C₂³×C₃×C₇	1	6	yes	—
131	C₂×C₄	C₂³×C₃×C₇	1	3	yes	—
132	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
133	C₇	C₃×D₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×C₃	—	2	4	no	—
135	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
136	C₂×C₃	C₂²×C₇×Q₈	1	1	yes	characteristic
137	C₂×C₃	C₇×2₋¹⁺⁴	1	1	yes	characteristic
138	C₂²	—	2	2	no	—
139	C₂²	—	2	2	no	—
140	C₂²	C₂×C₃×C₇×Q₈	1	4	yes	complemented
141	C₂²	C₂⁴×C₃×C₇	1	1	yes	characteristic
142	C₄	—	2	6	no	—
143	C₄	—	2	3	no	—
144	C₄	C₂×C₃×C₇×D₄	1	6	yes	—
145	C₄	C₂×C₃×C₇×Q₈	1	2	yes	complemented
146	C₃	C₇×D₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
147	C₂	—	2	4	no	—
148	C₂	C₂²×C₃×C₇×D₄	1	1	yes	characteristic
149	C₂	C₂²×C₃×C₇×Q₈	1	1	yes	characteristic
150	C₂	C₃×C₇×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₇
prime radical	C₂×C₃×C₇×D₄	C₂²
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. C₂²×C₃×C₇×Q₈	C₂	semidirect
2. C₃×C₇×C₄⋊Q₈	C₂	semidirect
3. C₃×C₇×C₂²⋊Q₈	C₂	semidirect
4. C₃×C₄×C₇×Q₈	C₂	semidirect
6. C₇×D₄×Q₈	C₃	direct (N × H)
17. C₂²×C₇×Q₈	C₂×C₃	semidirect
18. C₇×C₄⋊Q₈	C₂×C₃	semidirect
19. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
20. C₄×C₇×Q₈	C₂×C₃	semidirect
22. C₃×D₄×Q₈	C₇	direct (N × H)
25. C₃×C₇×Q₈	D₄	direct (N × H)
26. C₃×C₇×D₄	Q₈	direct (N × H)
42. C₂²×C₃×Q₈	C₂×C₇	semidirect
43. C₃×C₄⋊Q₈	C₂×C₇	semidirect
44. C₃×C₂²⋊Q₈	C₂×C₇	semidirect
45. C₃×C₄×Q₈	C₂×C₇	semidirect
49. C₂²×C₃×C₇	C₂×Q₈	semidirect
54. C₃×C₄×C₇	C₂×Q₈	semidirect
55. D₄×Q₈	C₃×C₇	direct (N × H)
58. C₇×Q₈	C₃×D₄	direct (N × H)
59. C₇×D₄	C₃×Q₈	direct (N × H)
79. C₂²×Q₈	C₂×C₃×C₇	semidirect
80. C₄⋊Q₈	C₂×C₃×C₇	semidirect
81. C₂²⋊Q₈	C₂×C₃×C₇	semidirect
82. C₄×Q₈	C₂×C₃×C₇	semidirect
86. C₂²×C₇	C₂×C₃×Q₈	semidirect
91. C₄×C₇	C₂×C₃×Q₈	semidirect
94. C₃×Q₈	C₇×D₄	direct (N × H)
95. C₃×D₄	C₇×Q₈	direct (N × H)
101. C₃×C₇	D₄×Q₈	direct (N × H)
118. C₂²×C₃	C₂×C₇×Q₈	semidirect
123. C₃×C₄	C₂×C₇×Q₈	semidirect
126. Q₈	C₃×C₇×D₄	direct (N × H)
127. D₄	C₃×C₇×Q₈	direct (N × H)
133. C₇	C₃×D₄×Q₈	direct (N × H)
140. C₂²	C₂×C₃×C₇×Q₈	semidirect
145. C₄	C₂×C₃×C₇×Q₈	semidirect
146. C₃	C₇×D₄×Q₈	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×Q₈	2
C₃×C₇×C₄⋊Q₈	2
C₃×C₇×C₂²⋊Q₈	2
C₃×C₄×C₇×Q₈	2
C₃×C₄×C₇×D₄	2
C₇×D₄×Q₈	3
C₃×D₄×Q₈	7

Subgroups — C3×C7×D4×Q8 #13061969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×D₄×Q₈ #13061969