Subgroups — C₂×D₄×Q₈×C₂₂₃ #43252481

Order 28544.

1568 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	C₂³×Q₈×C₂₂₃	C₂	1	2	yes	maximal, complemented
2	D₄×Q₈×C₂₂₃	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄⋊Q₈×C₂₂₃	C₂	1	3	yes	maximal, complemented
4	C₂×C₂²⋊Q₈×C₂₂₃	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄×C₂₂₃	C₂	1	3	yes	maximal
7	C₂²×Q₈×C₂₂₃	—	2	16	no	—
8	C₂²×Q₈×C₂₂₃	C₂²	1	6	yes	—
9	C₂²×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×C₂₂₃	C₂²	1	16	yes	complemented
11	C₂²×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₂₃	C₂²	1	6	yes	—
13	C₄⋊Q₈×C₂₂₃	C₂²	1	24	yes	complemented
14	C₂²⋊Q₈×C₂₂₃	C₂²	1	48	yes	complemented
15	C₄×Q₈×C₂₂₃	C₂²	1	8	yes	complemented
16	C₄×D₄×C₂₂₃	C₂²	1	24	yes	—
17	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	6	yes	—
18	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	3	yes	—
19	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	3	yes	—
20	C₂×C₂²⋊C₄×C₂₂₃	C₂²	1	6	yes	—
21	C₂×C₄²×C₂₂₃	C₂²	1	3	yes	—
22	C₂⁴×C₂₂₃	C₂³	1	2	yes	—
23	C₂×Q₈×C₂₂₃	—	2	64	no	—
24	C₂×Q₈×C₂₂₃	—	2	24	no	—
25	C₂×Q₈×C₂₂₃	D₄	1	8	yes	direct product factor
26	C₂×Q₈×C₂₂₃	C₂³	1	24	yes	—
27	C₂×Q₈×C₂₂₃	C₂³	1	4	yes	—
28	C₂×D₄×C₂₂₃	Q₈	1	8	yes	direct product factor
29	C₂×D₄×C₂₂₃	C₂³	1	4	yes	—
30	C₂²×C₄×C₂₂₃	—	2	24	no	—
31	C₂²×C₄×C₂₂₃	C₂³	1	24	yes	—
32	C₂²×C₄×C₂₂₃	C₂³	1	3	yes	—
33	C₂²×C₄×C₂₂₃	C₂³	1	6	yes	—
34	C₂²×C₄×C₂₂₃	C₂³	1	3	yes	—
35	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₂₂₃	C₂³	1	12	yes	—
37	C₄⋊C₄×C₂₂₃	C₂³	1	12	yes	—
38	C₄⋊C₄×C₂₂₃	C₂³	1	24	yes	—
39	C₂²⋊C₄×C₂₂₃	C₂³	1	24	yes	—
40	C₄²×C₂₂₃	C₂³	1	12	yes	—
41	C₂³×C₂₂₃	—	2	4	no	—
42	C₂³×C₂₂₃	—	2	4	no	—
43	C₂³×C₂₂₃	C₂×Q₈	1	8	yes	complemented
44	C₂³×C₂₂₃	C₂⁴	1	4	yes	—
45	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
46	Q₈×C₂₂₃	—	2	48	no	—
47	Q₈×C₂₂₃	C₂×D₄	1	16	yes	direct product factor
48	D₄×C₂₂₃	C₂×Q₈	1	16	yes	direct product factor
49	C₂×C₄×C₂₂₃	—	2	12	no	—
50	C₂×C₄×C₂₂₃	—	2	6	no	—
51	C₂×C₄×C₂₂₃	—	2	48	no	—
52	C₂×C₄×C₂₂₃	C₂×D₄	1	12	yes	—
53	C₂×C₄×C₂₂₃	C₂×Q₈	1	4	yes	complemented
54	C₂×C₄×C₂₂₃	C₂⁴	1	6	yes	—
55	C₂×C₄×C₂₂₃	C₂⁴	1	12	yes	—
56	C₂×C₄×C₂₂₃	C₂⁴	1	6	yes	—
57	C₂×C₄×C₂₂₃	C₂⁴	1	2	yes	—
58	C₂²×C₂₂₃	—	2	16	no	—
59	C₂²×C₂₂₃	—	2	4	no	—
60	C₂²×C₂₂₃	—	2	4	no	—
61	C₂²×C₂₂₃	C₂²×D₄	1	2	yes	—
62	C₂²×C₂₂₃	C₂²×Q₈	1	2	yes	—
63	C₂²×C₂₂₃	C₂²×Q₈	1	8	yes	complemented
64	C₂²×C₂₂₃	2₋¹⁺⁴	1	2	yes	—
65	C₂²×C₂₂₃	C₂⁵	1	1	yes	characteristic
66	C₄×C₂₂₃	—	2	12	no	—
67	C₄×C₂₂₃	—	2	6	no	—
68	C₄×C₂₂₃	C₂²×D₄	1	12	yes	—
69	C₄×C₂₂₃	C₂²×Q₈	1	4	yes	complemented
70	C₂×C₂₂₃	—	2	8	no	—
71	C₂×C₂₂₃	D₄×Q₈	1	4	yes	direct product factor
72	C₂×C₂₂₃	C₂³×D₄	1	1	yes	characteristic
73	C₂×C₂₂₃	C₂³×Q₈	1	1	yes	characteristic
74	C₂×C₂₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
75	C₂₂₃	C₂×D₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
76	C₂×D₄×Q₈	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×Q₈	C₂×C₂₂₃	1	2	yes	complemented
78	D₄×Q₈	C₂×C₂₂₃	1	16	yes	direct product factor
79	C₂×C₄⋊Q₈	C₂×C₂₂₃	1	3	yes	complemented
80	C₂×C₂²⋊Q₈	C₂×C₂₂₃	1	6	yes	complemented
81	C₂×C₄×Q₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
82	C₂×C₄×D₄	C₂×C₂₂₃	1	3	yes	—
83	C₂²×Q₈	—	2	16	no	—
84	C₂²×Q₈	C₂²×C₂₂₃	1	6	yes	—
85	C₂²×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
86	C₂²×Q₈	C₂²×C₂₂₃	1	16	yes	complemented
87	C₂²×D₄	C₂²×C₂₂₃	1	1	yes	characteristic
88	C₂³×C₄	C₂²×C₂₂₃	1	6	yes	—
89	C₄⋊Q₈	C₂²×C₂₂₃	1	24	yes	complemented
90	C₂²⋊Q₈	C₂²×C₂₂₃	1	48	yes	complemented
91	C₄×Q₈	C₂²×C₂₂₃	1	8	yes	complemented
92	C₄×D₄	C₂²×C₂₂₃	1	24	yes	—
93	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	3	yes	—
94	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	3	yes	—
95	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	6	yes	—
96	C₂×C₂²⋊C₄	C₂²×C₂₂₃	1	6	yes	—
97	C₂×C₄²	C₂²×C₂₂₃	1	3	yes	—
98	C₂⁴	C₂³×C₂₂₃	1	2	yes	—
99	C₂×Q₈	—	2	64	no	—
100	C₂×Q₈	—	2	24	no	—
101	C₂×Q₈	D₄×C₂₂₃	1	8	yes	direct product factor
102	C₂×Q₈	C₂³×C₂₂₃	1	24	yes	—
103	C₂×Q₈	C₂³×C₂₂₃	1	4	yes	—
104	C₂×D₄	Q₈×C₂₂₃	1	8	yes	direct product factor
105	C₂×D₄	C₂³×C₂₂₃	1	4	yes	—
106	C₂²×C₄	—	2	24	no	—
107	C₂²×C₄	C₂³×C₂₂₃	1	3	yes	—
108	C₂²×C₄	C₂³×C₂₂₃	1	6	yes	—
109	C₂²×C₄	C₂³×C₂₂₃	1	24	yes	—
110	C₂²×C₄	C₂³×C₂₂₃	1	3	yes	—
111	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
112	C₄⋊C₄	C₂³×C₂₂₃	1	12	yes	—
113	C₄⋊C₄	C₂³×C₂₂₃	1	12	yes	—
114	C₄⋊C₄	C₂³×C₂₂₃	1	24	yes	—
115	C₂²⋊C₄	C₂³×C₂₂₃	1	24	yes	—
116	C₄²	C₂³×C₂₂₃	1	12	yes	—
117	C₂³	—	2	4	no	—
118	C₂³	—	2	4	no	—
119	C₂³	C₂×Q₈×C₂₂₃	1	8	yes	complemented
120	C₂³	C₂⁴×C₂₂₃	1	4	yes	—
121	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
122	Q₈	—	2	48	no	—
123	Q₈	C₂×D₄×C₂₂₃	1	16	yes	direct product factor
124	D₄	C₂×Q₈×C₂₂₃	1	16	yes	direct product factor
125	C₂×C₄	—	2	12	no	—
126	C₂×C₄	—	2	6	no	—
127	C₂×C₄	—	2	48	no	—
128	C₂×C₄	C₂×D₄×C₂₂₃	1	12	yes	—
129	C₂×C₄	C₂×Q₈×C₂₂₃	1	4	yes	complemented
130	C₂×C₄	C₂⁴×C₂₂₃	1	6	yes	—
131	C₂×C₄	C₂⁴×C₂₂₃	1	12	yes	—
132	C₂×C₄	C₂⁴×C₂₂₃	1	6	yes	—
133	C₂×C₄	C₂⁴×C₂₂₃	1	2	yes	—
134	C₂²	—	2	16	no	—
135	C₂²	—	2	4	no	—
136	C₂²	—	2	4	no	—
137	C₂²	C₂²×D₄×C₂₂₃	1	2	yes	—
138	C₂²	C₂²×Q₈×C₂₂₃	1	2	yes	—
139	C₂²	C₂²×Q₈×C₂₂₃	1	8	yes	complemented
140	C₂²	2₋¹⁺⁴×C₂₂₃	1	2	yes	—
141	C₂²	C₂⁵×C₂₂₃	1	1	yes	characteristic
142	C₄	—	2	12	no	—
143	C₄	—	2	6	no	—
144	C₄	C₂²×D₄×C₂₂₃	1	12	yes	—
145	C₄	C₂²×Q₈×C₂₂₃	1	4	yes	complemented
146	C₂	—	2	8	no	—
147	C₂	D₄×Q₈×C₂₂₃	1	4	yes	direct product factor
148	C₂	C₂³×D₄×C₂₂₃	1	1	yes	characteristic
149	C₂	C₂³×Q₈×C₂₂₃	1	1	yes	characteristic
150	C₂	C₂×2₋¹⁺⁴×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₂₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₂₃
prime radical	C₂²×D₄×C₂₂₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₈×C₂₂₃	C₂	semidirect
2. D₄×Q₈×C₂₂₃	C₂	direct (N × H)
3. C₂×C₄⋊Q₈×C₂₂₃	C₂	semidirect
4. C₂×C₂²⋊Q₈×C₂₂₃	C₂	semidirect
5. C₂×C₄×Q₈×C₂₂₃	C₂	semidirect
10. C₂²×Q₈×C₂₂₃	C₂²	semidirect
13. C₄⋊Q₈×C₂₂₃	C₂²	semidirect
14. C₂²⋊Q₈×C₂₂₃	C₂²	semidirect
15. C₄×Q₈×C₂₂₃	C₂²	semidirect
25. C₂×Q₈×C₂₂₃	D₄	direct (N × H)
28. C₂×D₄×C₂₂₃	Q₈	direct (N × H)
43. C₂³×C₂₂₃	C₂×Q₈	semidirect
47. Q₈×C₂₂₃	C₂×D₄	direct (N × H)
48. D₄×C₂₂₃	C₂×Q₈	direct (N × H)
53. C₂×C₄×C₂₂₃	C₂×Q₈	semidirect
63. C₂²×C₂₂₃	C₂²×Q₈	semidirect
69. C₄×C₂₂₃	C₂²×Q₈	semidirect
71. C₂×C₂₂₃	D₄×Q₈	direct (N × H)
75. C₂₂₃	C₂×D₄×Q₈	direct (N × H)
76. C₂×D₄×Q₈	C₂₂₃	direct (N × H)
77. C₂³×Q₈	C₂×C₂₂₃	semidirect
78. D₄×Q₈	C₂×C₂₂₃	direct (N × H)
79. C₂×C₄⋊Q₈	C₂×C₂₂₃	semidirect
80. C₂×C₂²⋊Q₈	C₂×C₂₂₃	semidirect
81. C₂×C₄×Q₈	C₂×C₂₂₃	semidirect
86. C₂²×Q₈	C₂²×C₂₂₃	semidirect
89. C₄⋊Q₈	C₂²×C₂₂₃	semidirect
90. C₂²⋊Q₈	C₂²×C₂₂₃	semidirect
91. C₄×Q₈	C₂²×C₂₂₃	semidirect
101. C₂×Q₈	D₄×C₂₂₃	direct (N × H)
104. C₂×D₄	Q₈×C₂₂₃	direct (N × H)
119. C₂³	C₂×Q₈×C₂₂₃	semidirect
123. Q₈	C₂×D₄×C₂₂₃	direct (N × H)
124. D₄	C₂×Q₈×C₂₂₃	direct (N × H)
129. C₂×C₄	C₂×Q₈×C₂₂₃	semidirect
139. C₂²	C₂²×Q₈×C₂₂₃	semidirect
145. C₄	C₂²×Q₈×C₂₂₃	semidirect
147. C₂	D₄×Q₈×C₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₈×C₂₂₃	2
D₄×Q₈×C₂₂₃	2
C₂×C₄⋊Q₈×C₂₂₃	2
C₂×C₂²⋊Q₈×C₂₂₃	2
C₂×C₄×Q₈×C₂₂₃	2
C₂×C₄×D₄×C₂₂₃	2
C₂×D₄×Q₈	223

Subgroups — C2×D4×Q8×C223 #43252481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×Q₈×C₂₂₃ #43252481