Subgroups — C₂×#446×C₂₂₃ #43251974

Order 28544.

400 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#446×C₂₂₃	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₂₂₃	C₂	1	2	yes	maximal
5	C₂×#97×C₂₂₃	C₂	1	2	yes	maximal
6	C₂×C₄×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₂₂₃	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₂₂₃	C₂²	1	2	yes	—
10	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₄×Q₈×C₂₂₃	C₄	1	4	yes	—
12	C₄×Q₈×C₂₂₃	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₂₂₃	C₄	1	2	yes	—
14	C₂×C₄⋊C₄×C₂₂₃	C₄	1	1	yes	characteristic
15	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₂₂₃	C₂²	1	2	yes	—
17	C₄⋊C₈×C₂₂₃	C₂²	1	4	yes	—
18	C₄⋊C₈×C₂₂₃	C₂²	1	8	yes	—
19	#97×C₂₂₃	C₂²	1	8	yes	—
20	C₄×C₈×C₂₂₃	C₂²	1	4	yes	—
21	C₂×Q₈×C₂₂₃	—	2	4	no	—
22	C₂×Q₈×C₂₂₃	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₂₂₃	—	2	2	no	—
29	C₂×C₈×C₂₂₃	—	2	4	no	—
30	C₂×C₈×C₂₂₃	Q₈	1	4	yes	complemented
31	C₂×C₈×C₂₂₃	C₂³	1	2	yes	—
32	C₂×C₈×C₂₂₃	C₂³	1	4	yes	—
33	C₂×C₈×C₂₂₃	C₂³	1	2	yes	—
34	C₄⋊C₄×C₂₂₃	C₂×C₄	1	8	yes	—
35	C₄⋊C₄×C₂₂₃	C₂×C₄	1	4	yes	—
36	C₄²×C₂₂₃	C₂×C₄	1	4	yes	—
37	C₄²×C₂₂₃	C₂×C₄	1	2	yes	—
38	C₄²×C₂₂₃	C₂³	1	2	yes	—
39	C₄²×C₂₂₃	C₂³	1	4	yes	—
40	C₂³×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
41	Q₈×C₂₂₃	—	2	8	no	—
42	C₂×C₄×C₂₂₃	—	2	4	no	—
43	C₂×C₄×C₂₂₃	—	2	2	no	—
44	C₂×C₄×C₂₂₃	—	2	4	no	—
45	C₂×C₄×C₂₂₃	M₁₆	1	4	yes	—
46	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₂₂₃	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₂₂₃	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₂₂₃	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₂₂₃	#41	1	2	yes	—
53	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
54	C₈×C₂₂₃	—	2	2	no	—
55	C₈×C₂₂₃	—	2	4	no	—
56	C₈×C₂₂₃	C₂×Q₈	1	4	yes	complemented
57	C₂²×C₂₂₃	C₄×Q₈	1	2	yes	—
58	C₂²×C₂₂₃	C₂×M₁₆	1	2	yes	—
59	C₂²×C₂₂₃	#131	1	2	yes	—
60	C₂²×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₂₂₃	—	2	4	no	—
62	C₄×C₂₂₃	—	2	2	no	—
63	C₄×C₂₂₃	—	2	4	no	—
64	C₄×C₂₂₃	C₄×Q₈	1	2	yes	—
65	C₄×C₂₂₃	C₂×M₁₆	1	4	yes	—
66	C₄×C₂₂₃	C₂²×Q₈	1	1	yes	characteristic
67	C₄×C₂₂₃	C₂×#41	1	1	yes	characteristic
68	C₂×C₂₂₃	#446	1	4	yes	direct product factor
69	C₂×C₂₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
70	C₂×C₂₂₃	C₂²×M₁₆	1	1	yes	characteristic
71	C₂×C₂₂₃	C₂×#131	1	1	yes	characteristic
72	C₂₂₃	C₂×#446	1	1	yes	characteristic, direct product factor, Sylow
73	C₂×#446	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂×C₄×Q₈	C₂×C₂₂₃	1	1	yes	characteristic
75	#446	C₂×C₂₂₃	1	8	yes	direct product factor
76	C₂×C₄⋊C₈	C₂×C₂₂₃	1	1	yes	characteristic
77	C₂×C₄⋊C₈	C₂×C₂₂₃	1	2	yes	—
78	C₂×#97	C₂×C₂₂₃	1	2	yes	—
79	C₂×C₄×C₈	C₂×C₂₂₃	1	1	yes	characteristic
80	C₂²×Q₈	C₄×C₂₂₃	1	1	yes	characteristic
81	C₂²×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
82	C₂²×C₈	C₂²×C₂₂₃	1	2	yes	—
83	C₂²×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
84	C₄×Q₈	C₄×C₂₂₃	1	4	yes	—
85	C₄×Q₈	C₂²×C₂₂₃	1	4	yes	—
86	C₂×C₄⋊C₄	C₄×C₂₂₃	1	2	yes	—
87	C₂×C₄⋊C₄	C₄×C₂₂₃	1	1	yes	characteristic
88	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
89	C₂×C₄²	C₂²×C₂₂₃	1	2	yes	—
90	C₄⋊C₈	C₂²×C₂₂₃	1	4	yes	—
91	C₄⋊C₈	C₂²×C₂₂₃	1	8	yes	—
92	#97	C₂²×C₂₂₃	1	8	yes	—
93	C₄×C₈	C₂²×C₂₂₃	1	4	yes	—
94	C₂×Q₈	—	2	4	no	—
95	C₂×Q₈	C₂×C₄×C₂₂₃	1	4	yes	—
96	C₂²×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
97	C₂²×C₄	C₂×C₄×C₂₂₃	1	2	yes	—
98	C₂²×C₄	C₂×C₄×C₂₂₃	1	2	yes	—
99	C₂²×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
100	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	Q₈×C₂₂₃	1	4	yes	complemented
104	C₂×C₈	C₂³×C₂₂₃	1	2	yes	—
105	C₂×C₈	C₂³×C₂₂₃	1	4	yes	—
106	C₂×C₈	C₂³×C₂₂₃	1	2	yes	—
107	C₄⋊C₄	C₂×C₄×C₂₂₃	1	8	yes	—
108	C₄⋊C₄	C₂×C₄×C₂₂₃	1	4	yes	—
109	C₄²	C₂×C₄×C₂₂₃	1	4	yes	—
110	C₄²	C₂×C₄×C₂₂₃	1	2	yes	—
111	C₄²	C₂³×C₂₂₃	1	2	yes	—
112	C₄²	C₂³×C₂₂₃	1	4	yes	—
113	C₂³	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
114	Q₈	—	2	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	M₁₆×C₂₂₃	1	4	yes	—
119	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
120	C₂×C₄	C₂²×C₄×C₂₂₃	1	4	yes	—
121	C₂×C₄	C₂²×C₄×C₂₂₃	1	4	yes	—
122	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
123	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
124	C₂×C₄	C₂×Q₈×C₂₂₃	1	2	yes	—
125	C₂×C₄	#41×C₂₂₃	1	2	yes	—
126	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	2	4	no	—
129	C₈	C₂×Q₈×C₂₂₃	1	4	yes	complemented
130	C₂²	C₄×Q₈×C₂₂₃	1	2	yes	—
131	C₂²	C₂×M₁₆×C₂₂₃	1	2	yes	—
132	C₂²	#131×C₂₂₃	1	2	yes	—
133	C₂²	C₂³×C₄×C₂₂₃	1	1	yes	characteristic
134	C₄	—	2	4	no	—
135	C₄	—	2	2	no	—
136	C₄	—	2	4	no	—
137	C₄	C₄×Q₈×C₂₂₃	1	2	yes	—
138	C₄	C₂×M₁₆×C₂₂₃	1	4	yes	—
139	C₄	C₂²×Q₈×C₂₂₃	1	1	yes	characteristic
140	C₄	C₂×#41×C₂₂₃	1	1	yes	characteristic
141	C₂	#446×C₂₂₃	1	4	yes	direct product factor
142	C₂	C₂×C₄×Q₈×C₂₂₃	1	1	yes	characteristic
143	C₂	C₂²×M₁₆×C₂₂₃	1	1	yes	characteristic
144	C₂	C₂×#131×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₂₃	C₂³
commutator	C₂²	C₂³×C₄×C₂₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₃
socle, prime radical	C₂³×C₂₂₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. #446×C₂₂₃	C₂	direct (N × H)
30. C₂×C₈×C₂₂₃	Q₈	semidirect
56. C₈×C₂₂₃	C₂×Q₈	semidirect
68. C₂×C₂₂₃	#446	direct (N × H)
72. C₂₂₃	C₂×#446	direct (N × H)
73. C₂×#446	C₂₂₃	direct (N × H)
75. #446	C₂×C₂₂₃	direct (N × H)
103. C₂×C₈	Q₈×C₂₂₃	semidirect
129. C₈	C₂×Q₈×C₂₂₃	semidirect
141. C₂	#446×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
C₂×C₄×Q₈×C₂₂₃	2
#446×C₂₂₃	2
C₂×C₄⋊C₈×C₂₂₃	2
C₂×#97×C₂₂₃	2
C₂×C₄×C₈×C₂₂₃	2
C₂×#446	223

Subgroups — C2×#446×C223 #43251974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#446×C₂₂₃ #43251974