Subgroups — C₂×H(3,ℤ/4)×C₂₂₃ #43250745

Order 28544.

404 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₂₂₃	C₂	1	2	yes	maximal
3	H(3,ℤ/4)×C₂₂₃	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
5	C₂×M₁₆×C₂₂₃	—	2	2	no	—
6	C₂×M₁₆×C₂₂₃	—	2	2	no	—
7	C₂×M₁₆×C₂₂₃	C₄	1	2	yes	complemented
8	C₂×M₁₆×C₂₂₃	C₂²	1	2	yes	—
9	C₂²×C₈×C₂₂₃	C₄	1	2	yes	complemented
10	#117×C₂₂₃	—	2	2	no	—
11	#117×C₂₂₃	—	2	2	no	—
12	#117×C₂₂₃	C₄	1	4	yes	complemented
13	#117×C₂₂₃	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₂₂₃	—	2	2	no	—
15	C₂×C₂²⋊C₄×C₂₂₃	—	2	2	no	—
16	C₂×C₄²×C₂₂₃	C₄	1	4	yes	complemented
17	C₂⁴×C₂₂₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₂₃	—	2	4	no	—
19	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₂₃	D₄	1	2	yes	—
25	C₂²×C₄×C₂₂₃	D₄	1	2	yes	—
26	C₂²×C₄×C₂₂₃	Q₈	1	1	yes	characteristic
27	C₂²×C₄×C₂₂₃	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
29	M₁₆×C₂₂₃	—	2	4	no	—
30	M₁₆×C₂₂₃	—	2	4	no	—
31	C₂×C₈×C₂₂₃	—	2	2	no	—
32	C₂×C₈×C₂₂₃	—	2	2	no	—
33	C₂×C₈×C₂₂₃	C₂×C₄	1	4	yes	complemented
34	C₄⋊C₄×C₂₂₃	—	2	4	no	—
35	C₄⋊C₄×C₂₂₃	—	2	4	no	—
36	C₂²⋊C₄×C₂₂₃	—	2	4	no	—
37	C₂²⋊C₄×C₂₂₃	—	2	4	no	—
38	C₄²×C₂₂₃	—	2	4	no	—
39	C₄²×C₂₂₃	C₂×C₄	1	8	yes	complemented
40	C₂³×C₂₂₃	—	4	2	no	—
41	C₂³×C₂₂₃	C₂²⋊C₄	1	2	yes	—
42	C₂³×C₂₂₃	C₂²⋊C₄	1	2	yes	—
43	C₂³×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂³×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂³×C₂₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₂₃	—	2	8	no	—
47	C₂×C₄×C₂₂₃	—	4	8	no	—
48	C₂×C₄×C₂₂₃	C₄²	1	2	yes	—
49	C₂×C₄×C₂₂₃	C₄²	1	1	yes	characteristic
50	C₂×C₄×C₂₂₃	C₄²	1	1	yes	characteristic
51	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
60	C₂×C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
61	C₂×C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₂₂₃	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
66	C₈×C₂₂₃	—	2	4	no	—
67	C₂²×C₂₂₃	—	2	4	no	—
68	C₂²×C₂₂₃	—	2	2	no	—
69	C₂²×C₂₂₃	—	4	2	no	—
70	C₂²×C₂₂₃	—	4	1	no	—
71	C₂²×C₂₂₃	—	4	1	no	—
72	C₂²×C₂₂₃	#95	1	1	yes	characteristic
73	C₂²×C₂₂₃	#95	1	2	yes	—
74	C₂²×C₂₂₃	#95	1	1	yes	characteristic
75	C₂²×C₂₂₃	C₂×C₂²⋊C₄	1	2	yes	—
76	C₂²×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₂₂₃	—	4	8	no	—
78	C₄×C₂₂₃	#95	1	2	yes	—
79	C₄×C₂₂₃	#95	1	1	yes	characteristic
80	C₄×C₂₂₃	#95	1	1	yes	characteristic
81	C₄×C₂₂₃	C₂×C₄²	1	1	yes	characteristic
82	C₄×C₂₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₄×C₂₂₃	C₂×C₄⋊C₄	1	2	yes	—
84	C₂×C₂₂₃	—	2	2	no	—
85	C₂×C₂₂₃	—	2	2	no	—
86	C₂×C₂₂₃	—	2	1	no	—
87	C₂×C₂₂₃	—	2	1	no	—
88	C₂×C₂₂₃	H(3,ℤ/4)	1	2	yes	direct product factor
89	C₂×C₂₂₃	C₂×#95	1	1	yes	characteristic
90	C₂₂₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic, direct product factor, Sylow
91	C₂×H(3,ℤ/4)	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂²×M₁₆	C₂×C₂₂₃	1	1	yes	characteristic
93	C₂×#117	C₂×C₂₂₃	1	2	yes	—
94	H(3,ℤ/4)	C₂×C₂₂₃	1	4	yes	direct product factor
95	C₂³×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	2	2	no	—
98	C₂×M₁₆	C₄×C₂₂₃	1	2	yes	complemented
99	C₂×M₁₆	C₂²×C₂₂₃	1	2	yes	—
100	C₂²×C₈	C₄×C₂₂₃	1	2	yes	complemented
101	#117	—	2	2	no	—
102	#117	—	2	2	no	—
103	#117	C₄×C₂₂₃	1	4	yes	complemented
104	#117	C₂²×C₂₂₃	1	4	yes	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₂²⋊C₄	—	2	2	no	—
107	C₂×C₄²	C₄×C₂₂₃	1	4	yes	complemented
108	C₂⁴	D₄×C₂₂₃	1	1	yes	characteristic
109	C₂²×C₄	—	2	4	no	—
110	C₂²×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
111	C₂²×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₄×C₂₂₃	1	2	yes	—
113	C₂²×C₄	C₂×C₄×C₂₂₃	1	2	yes	—
114	C₂²×C₄	D₄×C₂₂₃	1	1	yes	characteristic
115	C₂²×C₄	D₄×C₂₂₃	1	2	yes	—
116	C₂²×C₄	D₄×C₂₂₃	1	2	yes	—
117	C₂²×C₄	Q₈×C₂₂₃	1	1	yes	characteristic
118	C₂²×C₄	Q₈×C₂₂₃	1	1	yes	characteristic
119	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
120	M₁₆	—	2	4	no	—
121	M₁₆	—	2	4	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	C₂×C₄×C₂₂₃	1	4	yes	complemented
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	2	4	no	—
127	C₂²⋊C₄	—	2	4	no	—
128	C₂²⋊C₄	—	2	4	no	—
129	C₄²	—	2	4	no	—
130	C₄²	C₂×C₄×C₂₂₃	1	8	yes	complemented
131	C₂³	—	4	2	no	—
132	C₂³	C₂²⋊C₄×C₂₂₃	1	2	yes	—
133	C₂³	C₂²⋊C₄×C₂₂₃	1	2	yes	—
134	C₂³	C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
135	C₂³	C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
136	C₂³	C₂×D₄×C₂₂₃	1	1	yes	characteristic
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	4	8	no	—
139	C₂×C₄	C₄²×C₂₂₃	1	1	yes	characteristic
140	C₂×C₄	C₄²×C₂₂₃	1	2	yes	—
141	C₂×C₄	C₄²×C₂₂₃	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊C₄×C₂₂₃	1	2	yes	—
143	C₂×C₄	C₂²⋊C₄×C₂₂₃	1	2	yes	—
144	C₂×C₄	C₂²⋊C₄×C₂₂₃	1	1	yes	characteristic
145	C₂×C₄	C₂²⋊C₄×C₂₂₃	1	1	yes	characteristic
146	C₂×C₄	C₂²⋊C₄×C₂₂₃	1	2	yes	—
147	C₂×C₄	C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
148	C₂×C₄	C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
149	C₂×C₄	C₄⋊C₄×C₂₂₃	1	2	yes	—
150	C₂×C₄	C₄⋊C₄×C₂₂₃	1	2	yes	—
151	C₂×C₄	C₄⋊C₄×C₂₂₃	1	2	yes	—
152	C₂×C₄	C₄⋊C₄×C₂₂₃	1	2	yes	—
153	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
154	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
155	C₂×C₄	C₂×D₄×C₂₂₃	1	2	yes	—
156	C₂×C₄	C₂×Q₈×C₂₂₃	1	1	yes	characteristic
157	C₈	—	2	4	no	—
158	C₂²	—	2	4	no	—
159	C₂²	—	2	2	no	—
160	C₂²	—	4	2	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	4	1	no	—
163	C₂²	#95×C₂₂₃	1	1	yes	characteristic
164	C₂²	#95×C₂₂₃	1	2	yes	—
165	C₂²	#95×C₂₂₃	1	1	yes	characteristic
166	C₂²	C₂×C₂²⋊C₄×C₂₂₃	1	2	yes	—
167	C₂²	C₂×C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
168	C₄	—	4	8	no	—
169	C₄	#95×C₂₂₃	1	2	yes	—
170	C₄	#95×C₂₂₃	1	1	yes	characteristic
171	C₄	#95×C₂₂₃	1	1	yes	characteristic
172	C₄	C₂×C₄²×C₂₂₃	1	1	yes	characteristic
173	C₄	C₂×C₂²⋊C₄×C₂₂₃	1	1	yes	characteristic
174	C₄	C₂×C₄⋊C₄×C₂₂₃	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	—	2	2	no	—
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	H(3,ℤ/4)×C₂₂₃	1	2	yes	direct product factor
180	C₂	C₂×#95×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₂₃	C₄²
commutator	C₄	C₂×C₄²×C₂₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₂₃
socle	C₂²×C₂₂₃	#95
prime radical	C₂⁴×C₂₂₃	D₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. H(3,ℤ/4)×C₂₂₃	C₂	direct (N × H)
7. C₂×M₁₆×C₂₂₃	C₄	semidirect
9. C₂²×C₈×C₂₂₃	C₄	semidirect
12. #117×C₂₂₃	C₄	semidirect
16. C₂×C₄²×C₂₂₃	C₄	semidirect
33. C₂×C₈×C₂₂₃	C₂×C₄	semidirect
39. C₄²×C₂₂₃	C₂×C₄	semidirect
88. C₂×C₂₂₃	H(3,ℤ/4)	direct (N × H)
90. C₂₂₃	C₂×H(3,ℤ/4)	direct (N × H)
91. C₂×H(3,ℤ/4)	C₂₂₃	direct (N × H)
94. H(3,ℤ/4)	C₂×C₂₂₃	direct (N × H)
98. C₂×M₁₆	C₄×C₂₂₃	semidirect
100. C₂²×C₈	C₄×C₂₂₃	semidirect
103. #117	C₄×C₂₂₃	semidirect
107. C₂×C₄²	C₄×C₂₂₃	semidirect
124. C₂×C₈	C₂×C₄×C₂₂₃	semidirect
130. C₄²	C₂×C₄×C₂₂₃	semidirect
179. C₂	H(3,ℤ/4)×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
C₂²×M₁₆×C₂₂₃	2
C₂×#117×C₂₂₃	2
H(3,ℤ/4)×C₂₂₃	2
C₂×H(3,ℤ/4)	223

Subgroups — C2×H(3,ℤ/4)×C223 #43250745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×H(3,ℤ/4)×C₂₂₃ #43250745