Subgroups — C₂²×C₂₃×#441 #19207779

Order 5888.

2790 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₂₃	C₂	1	2	yes	maximal
3	C₂×C₂₃×#441	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₂₃×#106	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₃×#103	C₂	1	2	yes	maximal
6	C₂²×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
8	C₂²×Q₁₆×C₂₃	C₄	1	16	yes	complemented
9	C₂²×Q₁₆×C₂₃	C₂²	1	12	yes	—
10	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₂₃	—	2	12	no	—
13	C₂×C₄×Q₈×C₂₃	C₂²	1	24	yes	—
14	C₂²×C₄⋊C₄×C₂₃	—	2	2	no	—
15	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₂₃	—	2	2	no	—
17	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#441	C₂²	1	64	yes	direct product factor
19	C₂×C₂₃×#106	C₂²	1	12	yes	—
20	C₂×C₂₃×#103	C₂²	1	24	yes	—
21	C₂×C₂₃×#97	C₂²	1	12	yes	—
22	C₂²×Q₈×C₂₃	—	2	6	no	—
23	C₂²×Q₈×C₂₃	—	2	8	no	—
24	C₂²×Q₈×C₂₃	C₂×C₄	1	16	yes	—
25	C₂²×Q₈×C₂₃	C₂³	1	12	yes	—
26	C₂³×C₄×C₂₃	—	2	2	no	—
27	C₂³×C₄×C₂₃	—	2	2	no	—
28	C₂³×C₄×C₂₃	D₄	1	2	yes	—
29	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂×Q₁₆×C₂₃	C₂×C₄	1	96	yes	complemented
31	C₂×Q₁₆×C₂₃	C₂³	1	16	yes	—
32	C₂²×C₈×C₂₃	—	2	4	no	—
33	C₂²×C₈×C₂₃	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
35	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
36	C₄×Q₈×C₂₃	—	2	48	no	—
37	C₄×Q₈×C₂₃	C₂³	1	32	yes	—
38	C₂×C₄⋊C₄×C₂₃	—	2	24	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
40	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
41	C₂×C₄²×C₂₃	—	2	24	no	—
42	C₂×C₄²×C₂₃	D₄	1	6	yes	—
43	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
44	C₂₃×#106	C₂³	1	16	yes	—
45	C₂₃×#103	C₂³	1	32	yes	—
46	C₂₃×#97	C₂³	1	16	yes	—
47	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂×Q₈×C₂₃	—	2	48	no	—
49	C₂×Q₈×C₂₃	—	2	12	no	—
50	C₂×Q₈×C₂₃	—	2	24	no	—
51	C₂×Q₈×C₂₃	C₂²×C₄	1	48	yes	—
52	C₂×Q₈×C₂₃	C₂⁴	1	8	yes	—
53	C₂²×C₄×C₂₃	—	2	16	no	—
54	C₂²×C₄×C₂₃	—	2	8	no	—
55	C₂²×C₄×C₂₃	—	2	12	no	—
56	C₂²×C₄×C₂₃	—	2	12	no	—
57	C₂²×C₄×C₂₃	—	4	8	no	—
58	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
60	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
61	C₂²×C₄×C₂₃	#41	1	4	yes	—
62	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
63	Q₁₆×C₂₃	C₂²×C₄	1	64	yes	complemented
64	C₂×C₈×C₂₃	—	2	12	no	—
65	C₂×C₈×C₂₃	C₂²×C₄	1	24	yes	—
66	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
67	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
68	C₄⋊C₄×C₂₃	—	2	32	no	—
69	C₄⋊C₄×C₂₃	—	2	12	no	—
70	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
71	C₄²×C₂₃	—	2	32	no	—
72	C₄²×C₂₃	C₂×D₄	1	12	yes	—
73	C₄²×C₂₃	C₂⁴	1	4	yes	—
74	C₂²×#441	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₂₃	C₄×D₄	1	4	yes	—
76	C₂³×C₂₃	#137	1	8	yes	—
77	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
78	Q₈×C₂₃	—	2	48	no	—
79	Q₈×C₂₃	—	2	8	no	—
80	Q₈×C₂₃	C₂³×C₄	1	16	yes	—
81	C₂×C₄×C₂₃	—	2	48	no	—
82	C₂×C₄×C₂₃	—	2	24	no	—
83	C₂×C₄×C₂₃	—	2	8	no	—
84	C₂×C₄×C₂₃	—	2	8	no	—
85	C₂×C₄×C₂₃	—	4	24	no	—
86	C₂×C₄×C₂₃	C₄×D₄	1	12	yes	—
87	C₂×C₄×C₂₃	C₂³×C₄	1	6	yes	—
88	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
89	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
90	C₂×C₄×C₂₃	C₂×#41	1	6	yes	—
91	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
92	C₈×C₂₃	—	2	4	no	—
93	C₈×C₂₃	C₂³×C₄	1	8	yes	—
94	C₂³×Q₁₆	C₂×C₂₃	1	1	yes	characteristic
95	C₂²×C₄×Q₈	C₂×C₂₃	1	2	yes	—
96	C₂×#441	C₂×C₂₃	1	24	yes	direct product factor
97	C₂²×#106	C₂×C₂₃	1	1	yes	characteristic
98	C₂²×#103	C₂×C₂₃	1	2	yes	—
99	C₂²×#97	C₂×C₂₃	1	1	yes	characteristic
100	C₂²×C₂₃	#441	1	16	yes	direct product factor
101	C₂²×C₂₃	C₂×C₄×D₄	1	6	yes	—
102	C₂²×C₂₃	C₂×#137	1	12	yes	—
103	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
104	C₄×C₂₃	—	2	16	no	—
105	C₄×C₂₃	—	2	8	no	—
106	C₄×C₂₃	—	4	8	no	—
107	C₄×C₂₃	C₂×C₄×D₄	1	6	yes	—
108	C₄×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
109	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
110	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
111	C₂²×Q₁₆	C₄×C₂₃	1	16	yes	complemented
112	C₂²×Q₁₆	C₂²×C₂₃	1	12	yes	—
113	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
114	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
115	C₂×C₄×Q₈	—	2	12	no	—
116	C₂×C₄×Q₈	C₂²×C₂₃	1	24	yes	—
117	C₂²×C₄⋊C₄	—	2	2	no	—
118	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
119	C₂²×C₄²	—	2	2	no	—
120	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
121	#441	C₂²×C₂₃	1	64	yes	direct product factor
122	C₂×#106	C₂²×C₂₃	1	12	yes	—
123	C₂×#103	C₂²×C₂₃	1	24	yes	—
124	C₂×#97	C₂²×C₂₃	1	12	yes	—
125	C₂×C₂₃	C₂×#441	1	12	yes	direct product factor
126	C₂×C₂₃	C₂²×C₄×D₄	1	1	yes	characteristic
127	C₂×C₂₃	C₂²×#137	1	2	yes	—
128	C₂²×Q₈	—	2	6	no	—
129	C₂²×Q₈	—	2	8	no	—
130	C₂²×Q₈	C₂×C₄×C₂₃	1	16	yes	—
131	C₂²×Q₈	C₂³×C₂₃	1	12	yes	—
132	C₂³×C₄	—	2	2	no	—
133	C₂³×C₄	—	2	2	no	—
134	C₂³×C₄	D₄×C₂₃	1	2	yes	—
135	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
136	C₂×Q₁₆	C₂×C₄×C₂₃	1	96	yes	complemented
137	C₂×Q₁₆	C₂³×C₂₃	1	16	yes	—
138	C₂²×C₈	—	2	4	no	—
139	C₂²×C₈	C₂×C₄×C₂₃	1	8	yes	—
140	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
141	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
142	C₄×Q₈	—	2	48	no	—
143	C₄×Q₈	C₂³×C₂₃	1	32	yes	—
144	C₂×C₄⋊C₄	—	2	24	no	—
145	C₂×C₄⋊C₄	—	2	6	no	—
146	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
147	C₂×C₄²	—	2	24	no	—
148	C₂×C₄²	D₄×C₂₃	1	6	yes	—
149	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
150	#106	C₂³×C₂₃	1	16	yes	—
151	#103	C₂³×C₂₃	1	32	yes	—
152	#97	C₂³×C₂₃	1	16	yes	—
153	C₂₃	C₂²×#441	1	1	yes	characteristic, direct product factor, Sylow
154	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
155	C₂×Q₈	—	2	48	no	—
156	C₂×Q₈	—	2	12	no	—
157	C₂×Q₈	—	2	24	no	—
158	C₂×Q₈	C₂²×C₄×C₂₃	1	48	yes	—
159	C₂×Q₈	C₂⁴×C₂₃	1	8	yes	—
160	C₂²×C₄	—	2	16	no	—
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	2	12	no	—
163	C₂²×C₄	—	2	12	no	—
164	C₂²×C₄	—	4	8	no	—
165	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
166	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
167	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
168	C₂²×C₄	#41×C₂₃	1	4	yes	—
169	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
170	Q₁₆	C₂²×C₄×C₂₃	1	64	yes	complemented
171	C₂×C₈	—	2	12	no	—
172	C₂×C₈	C₂²×C₄×C₂₃	1	24	yes	—
173	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
174	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
175	C₄⋊C₄	—	2	32	no	—
176	C₄⋊C₄	—	2	12	no	—
177	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
178	C₄²	—	2	32	no	—
179	C₄²	C₂×D₄×C₂₃	1	12	yes	—
180	C₄²	C₂⁴×C₂₃	1	4	yes	—
181	C₂³	C₄×D₄×C₂₃	1	4	yes	—
182	C₂³	C₂₃×#137	1	8	yes	—
183	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
184	Q₈	—	2	8	no	—
185	Q₈	—	2	48	no	—
186	Q₈	C₂³×C₄×C₂₃	1	16	yes	—
187	C₂×C₄	—	2	24	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	2	48	no	—
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	—	4	24	no	—
192	C₂×C₄	C₄×D₄×C₂₃	1	12	yes	—
193	C₂×C₄	C₂³×C₄×C₂₃	1	6	yes	—
194	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
195	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
196	C₂×C₄	C₂×#41×C₂₃	1	6	yes	—
197	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
198	C₈	—	2	4	no	—
199	C₈	C₂³×C₄×C₂₃	1	8	yes	—
200	C₂²	C₂₃×#441	1	16	yes	direct product factor
201	C₂²	C₂×C₄×D₄×C₂₃	1	6	yes	—
202	C₂²	C₂×C₂₃×#137	1	12	yes	—
203	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
204	C₄	—	2	16	no	—
205	C₄	—	2	8	no	—
206	C₄	—	4	8	no	—
207	C₄	C₂×C₄×D₄×C₂₃	1	6	yes	—
208	C₄	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
209	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
210	C₂	C₂×C₂₃×#441	1	12	yes	direct product factor
211	C₂	C₂²×C₄×D₄×C₂₃	1	1	yes	characteristic
212	C₂	C₂²×C₂₃×#137	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₃	C₂×D₄
commutator	C₄	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₂₃×#441	C₂	direct (N × H)
8. C₂²×Q₁₆×C₂₃	C₄	semidirect
18. C₂₃×#441	C₂²	direct (N × H)
30. C₂×Q₁₆×C₂₃	C₂×C₄	semidirect
63. Q₁₆×C₂₃	C₂²×C₄	semidirect
74. C₂²×#441	C₂₃	direct (N × H)
96. C₂×#441	C₂×C₂₃	direct (N × H)
100. C₂²×C₂₃	#441	direct (N × H)
111. C₂²×Q₁₆	C₄×C₂₃	semidirect
121. #441	C₂²×C₂₃	direct (N × H)
125. C₂×C₂₃	C₂×#441	direct (N × H)
136. C₂×Q₁₆	C₂×C₄×C₂₃	semidirect
153. C₂₃	C₂²×#441	direct (N × H)
170. Q₁₆	C₂²×C₄×C₂₃	semidirect
200. C₂²	C₂₃×#441	direct (N × H)
210. C₂	C₂×C₂₃×#441	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×Q₁₆×C₂₃	2
C₂²×C₄×Q₈×C₂₃	2
C₂×C₂₃×#441	2
C₂²×C₂₃×#106	2
C₂²×C₂₃×#103	2
C₂²×C₂₃×#97	2
C₂²×#441	23

Subgroups — C22×C23×#441 #19207779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#441 #19207779