Subgroups — C₂²×#104×C₂₂₃ #43251914

Order 28544.

856 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₄×C₂₂₃	D₄
commutator	C₄	C₂³×C₄×C₂₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₃
socle	C₂³×C₂₂₃	C₂²⋊C₄
prime radical	C₂²×#41×C₂₂₃	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₂₃ ◁ C₂³×C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×M₁₆×C₂₂₃	C₂	semidirect
3. C₂²×C₄²×C₂₂₃	C₂	semidirect
4. C₂×#104×C₂₂₃	C₂	direct (N × H)
7. C₂×#41×C₂₂₃	C₄	semidirect
9. C₂²×Q₈×C₂₂₃	C₄	semidirect
11. C₂²×D₄×C₂₂₃	C₄	semidirect
16. C₂×M₁₆×C₂₂₃	C₂²	semidirect
19. C₂×C₄²×C₂₂₃	C₂²	semidirect
20. #104×C₂₂₃	C₂²	direct (N × H)
26. #41×C₂₂₃	C₂×C₄	semidirect
29. C₂×Q₈×C₂₂₃	C₂×C₄	semidirect
33. C₂×D₄×C₂₂₃	C₂×C₄	semidirect
46. M₁₆×C₂₂₃	C₂³	semidirect
49. C₄²×C₂₂₃	C₂³	semidirect
57. Q₈×C₂₂₃	C₂²×C₄	semidirect
61. D₄×C₂₂₃	C₂²×C₄	semidirect
81. C₂²×C₂₂₃	#104	direct (N × H)
94. C₂×C₂₂₃	C₂×#104	direct (N × H)
96. C₂₂₃	C₂²×#104	direct (N × H)
97. C₂²×#104	C₂₂₃	direct (N × H)
99. C₂²×M₁₆	C₂×C₂₂₃	semidirect
100. C₂²×C₄²	C₂×C₂₂₃	semidirect
101. C₂×#104	C₂×C₂₂₃	direct (N × H)
104. C₂×#41	C₄×C₂₂₃	semidirect
106. C₂²×Q₈	C₄×C₂₂₃	semidirect
108. C₂²×D₄	C₄×C₂₂₃	semidirect
113. C₂×M₁₆	C₂²×C₂₂₃	semidirect
116. C₂×C₄²	C₂²×C₂₂₃	semidirect
117. #104	C₂²×C₂₂₃	direct (N × H)
123. #41	C₂×C₄×C₂₂₃	semidirect
126. C₂×Q₈	C₂×C₄×C₂₂₃	semidirect
130. C₂×D₄	C₂×C₄×C₂₂₃	semidirect
143. M₁₆	C₂³×C₂₂₃	semidirect
146. C₄²	C₂³×C₂₂₃	semidirect
154. Q₈	C₂²×C₄×C₂₂₃	semidirect
158. D₄	C₂²×C₄×C₂₂₃	semidirect
178. C₂²	#104×C₂₂₃	direct (N × H)
191. C₂	C₂×#104×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×#41×C₂₂₃	2
C₂²×M₁₆×C₂₂₃	2
C₂²×C₄²×C₂₂₃	2
C₂×#104×C₂₂₃	2
C₂²×#104	223

Subgroups — C22×#104×C223 #43251914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#104×C₂₂₃ #43251914