Subgroups — C₂²×C₂₃×#555 #19210333

Order 5888.

8710 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#555	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂₃×#124	C₂	1	4	yes	maximal, complemented
4	C₂²×C₂₃×#121	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×D₄×C₂₃	C₂	1	2	yes	maximal, complemented
6	C₂²×C₂₃×#117	C₂	1	4	yes	maximal, complemented
7	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
8	C₂³×D₄×C₂₃	C₂²	1	2	yes	—
9	C₂³×D₄×C₂₃	C₂²	1	4	yes	—
10	C₂⁴×C₄×C₂₃	C₂²	1	4	yes	—
11	C₂₃×#555	C₂²	1	256	yes	direct product factor
12	C₂×C₂₃×C₄⋊Q₈	C₂²	1	24	yes	complemented
13	C₂×C₂₃×#124	C₂²	1	96	yes	complemented
14	C₂×C₂₃×#121	C₂²	1	96	yes	complemented
15	C₂×C₄×D₄×C₂₃	C₂²	1	48	yes	complemented
16	C₂×C₂₃×#117	C₂²	1	96	yes	complemented
17	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
19	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	—
20	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
21	C₂²×C₄²×C₂₃	C₂²	1	4	yes	—
22	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	C₂⁵×C₂₃	C₂³	1	4	yes	—
24	C₂²×Q₈×C₂₃	—	4	8	no	—
25	C₂²×Q₈×C₂₃	C₂³	1	24	yes	—
26	C₂²×D₄×C₂₃	—	2	16	no	—
27	C₂²×D₄×C₂₃	—	4	16	no	—
28	C₂²×D₄×C₂₃	C₂³	1	24	yes	—
29	C₂²×D₄×C₂₃	C₂³	1	48	yes	—
30	C₂³×C₄×C₂₃	—	2	32	no	—
31	C₂³×C₄×C₂₃	C₂³	1	48	yes	—
32	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
33	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
34	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
36	C₂₃×C₄⋊Q₈	C₂³	1	64	yes	complemented
37	C₂₃×#124	C₂³	1	256	yes	complemented
38	C₂₃×#121	C₂³	1	256	yes	complemented
39	C₄×D₄×C₂₃	C₂³	1	128	yes	complemented
40	C₂₃×#117	C₂³	1	256	yes	complemented
41	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
42	C₂×C₄⋊C₄×C₂₃	C₂³	1	48	yes	—
43	C₂×C₂²⋊C₄×C₂₃	C₂³	1	96	yes	—
44	C₂×C₂²⋊C₄×C₂₃	C₂³	1	48	yes	—
45	C₂×C₄²×C₂₃	C₂³	1	48	yes	—
46	C₂×C₄²×C₂₃	C₂³	1	12	yes	—
47	C₂⁴×C₂₃	—	2	16	no	—
48	C₂⁴×C₂₃	—	2	16	no	—
49	C₂⁴×C₂₃	—	2	16	no	—
50	C₂⁴×C₂₃	C₂⁴	1	24	yes	—
51	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
52	C₂×Q₈×C₂₃	—	4	48	no	—
53	C₂×Q₈×C₂₃	C₂⁴	1	32	yes	—
54	C₂×D₄×C₂₃	—	2	96	no	—
55	C₂×D₄×C₂₃	—	4	96	no	—
56	C₂×D₄×C₂₃	C₂⁴	1	32	yes	—
57	C₂×D₄×C₂₃	C₂⁴	1	64	yes	—
58	C₂²×C₄×C₂₃	—	2	16	no	—
59	C₂²×C₄×C₂₃	—	2	16	no	—
60	C₂²×C₄×C₂₃	—	2	4	no	—
61	C₂²×C₄×C₂₃	—	2	192	no	—
62	C₂²×C₄×C₂₃	#41	1	16	yes	—
63	C₂²×C₄×C₂₃	C₂⁴	1	24	yes	—
64	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
65	C₂²×C₄×C₂₃	C₂⁴	1	24	yes	—
66	C₂²×C₄×C₂₃	C₂⁴	1	64	yes	—
67	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
68	C₄⋊C₄×C₂₃	C₂⁴	1	32	yes	—
69	C₄⋊C₄×C₂₃	C₂⁴	1	64	yes	—
70	C₂²⋊C₄×C₂₃	C₂⁴	1	128	yes	—
71	C₂²⋊C₄×C₂₃	C₂⁴	1	64	yes	—
72	C₄²×C₂₃	C₂⁴	1	64	yes	—
73	C₄²×C₂₃	C₂⁴	1	16	yes	—
74	C₂²×#555	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₂₃	—	2	48	no	—
76	C₂³×C₂₃	—	2	48	no	—
77	C₂³×C₂₃	—	2	48	no	—
78	C₂³×C₂₃	—	4	64	no	—
79	C₂³×C₂₃	C₂×#41	1	8	yes	—
80	C₂³×C₂₃	H(5,𝔽₂)	1	4	yes	—
81	C₂³×C₂₃	C₂⁵	1	16	yes	—
82	C₂³×C₂₃	C₂⁵	1	3	yes	—
83	Q₈×C₂₃	—	4	32	no	—
84	D₄×C₂₃	—	2	64	no	—
85	D₄×C₂₃	—	4	64	no	—
86	C₂×C₄×C₂₃	—	2	48	no	—
87	C₂×C₄×C₂₃	—	2	48	no	—
88	C₂×C₄×C₂₃	—	2	12	no	—
89	C₂×C₄×C₂₃	—	2	128	no	—
90	C₂×C₄×C₂₃	C₂×#41	1	48	yes	—
91	C₂×C₄×C₂₃	C₂⁵	1	16	yes	—
92	C₂×C₄×C₂₃	C₂⁵	1	4	yes	—
93	C₂×C₄×C₂₃	C₂⁵	1	16	yes	—
94	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
95	C₂×#555	C₂×C₂₃	1	48	yes	direct product factor
96	C₂²×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
97	C₂²×#124	C₂×C₂₃	1	4	yes	complemented
98	C₂²×#121	C₂×C₂₃	1	4	yes	complemented
99	C₂²×C₄×D₄	C₂×C₂₃	1	2	yes	complemented
100	C₂²×#117	C₂×C₂₃	1	4	yes	complemented
101	C₂²×C₂₃	—	2	16	no	—
102	C₂²×C₂₃	—	2	16	no	—
103	C₂²×C₂₃	—	2	16	no	—
104	C₂²×C₂₃	—	4	96	no	—
105	C₂²×C₂₃	#555	1	16	yes	direct product factor
106	C₂²×C₂₃	C₂²×#41	1	12	yes	—
107	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	6	yes	—
108	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
109	C₄×C₂₃	—	2	16	no	—
110	C₄×C₂₃	—	2	16	no	—
111	C₄×C₂₃	—	2	4	no	—
112	C₄×C₂₃	C₂²×#41	1	16	yes	—
113	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
114	C₂³×D₄	C₂²×C₂₃	1	2	yes	—
115	C₂³×D₄	C₂²×C₂₃	1	4	yes	—
116	C₂⁴×C₄	C₂²×C₂₃	1	4	yes	—
117	#555	C₂²×C₂₃	1	256	yes	direct product factor
118	C₂×C₄⋊Q₈	C₂²×C₂₃	1	24	yes	complemented
119	C₂×#124	C₂²×C₂₃	1	96	yes	complemented
120	C₂×#121	C₂²×C₂₃	1	96	yes	complemented
121	C₂×C₄×D₄	C₂²×C₂₃	1	48	yes	complemented
122	C₂×#117	C₂²×C₂₃	1	96	yes	complemented
123	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
124	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
125	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	—
126	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
127	C₂²×C₄²	C₂²×C₂₃	1	4	yes	—
128	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
129	C₂×C₂₃	—	4	16	no	—
130	C₂×C₂₃	C₂×#555	1	12	yes	direct product factor
131	C₂×C₂₃	C₂³×#41	1	2	yes	—
132	C₂×C₂₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
133	C₂⁵	C₂³×C₂₃	1	4	yes	—
134	C₂²×Q₈	—	4	8	no	—
135	C₂²×Q₈	C₂³×C₂₃	1	24	yes	—
136	C₂²×D₄	—	2	16	no	—
137	C₂²×D₄	—	4	16	no	—
138	C₂²×D₄	C₂³×C₂₃	1	24	yes	—
139	C₂²×D₄	C₂³×C₂₃	1	48	yes	—
140	C₂³×C₄	—	2	32	no	—
141	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
142	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
143	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
144	C₂³×C₄	C₂³×C₂₃	1	48	yes	—
145	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
146	C₄⋊Q₈	C₂³×C₂₃	1	64	yes	complemented
147	#124	C₂³×C₂₃	1	256	yes	complemented
148	#121	C₂³×C₂₃	1	256	yes	complemented
149	C₄×D₄	C₂³×C₂₃	1	128	yes	complemented
150	#117	C₂³×C₂₃	1	256	yes	complemented
151	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
152	C₂×C₄⋊C₄	C₂³×C₂₃	1	48	yes	—
153	C₂×C₂²⋊C₄	C₂³×C₂₃	1	96	yes	—
154	C₂×C₂²⋊C₄	C₂³×C₂₃	1	48	yes	—
155	C₂×C₄²	C₂³×C₂₃	1	48	yes	—
156	C₂×C₄²	C₂³×C₂₃	1	12	yes	—
157	C₂₃	C₂²×#555	1	1	yes	characteristic, direct product factor, Sylow
158	C₂⁴	—	2	16	no	—
159	C₂⁴	—	2	16	no	—
160	C₂⁴	—	2	16	no	—
161	C₂⁴	C₂⁴×C₂₃	1	24	yes	—
162	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
163	C₂×Q₈	—	4	48	no	—
164	C₂×Q₈	C₂⁴×C₂₃	1	32	yes	—
165	C₂×D₄	—	2	96	no	—
166	C₂×D₄	—	4	96	no	—
167	C₂×D₄	C₂⁴×C₂₃	1	32	yes	—
168	C₂×D₄	C₂⁴×C₂₃	1	64	yes	—
169	C₂²×C₄	—	2	16	no	—
170	C₂²×C₄	—	2	16	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	192	no	—
173	C₂²×C₄	#41×C₂₃	1	16	yes	—
174	C₂²×C₄	C₂⁴×C₂₃	1	24	yes	—
175	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
176	C₂²×C₄	C₂⁴×C₂₃	1	24	yes	—
177	C₂²×C₄	C₂⁴×C₂₃	1	64	yes	—
178	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
179	C₄⋊C₄	C₂⁴×C₂₃	1	32	yes	—
180	C₄⋊C₄	C₂⁴×C₂₃	1	64	yes	—
181	C₂²⋊C₄	C₂⁴×C₂₃	1	128	yes	—
182	C₂²⋊C₄	C₂⁴×C₂₃	1	64	yes	—
183	C₄²	C₂⁴×C₂₃	1	64	yes	—
184	C₄²	C₂⁴×C₂₃	1	16	yes	—
185	C₂³	—	2	48	no	—
186	C₂³	—	2	48	no	—
187	C₂³	—	2	48	no	—
188	C₂³	—	4	64	no	—
189	C₂³	C₂×#41×C₂₃	1	8	yes	—
190	C₂³	C₂₃×H(5,𝔽₂)	1	4	yes	—
191	C₂³	C₂⁵×C₂₃	1	16	yes	—
192	C₂³	C₂⁵×C₂₃	1	3	yes	—
193	Q₈	—	4	32	no	—
194	D₄	—	2	64	no	—
195	D₄	—	4	64	no	—
196	C₂×C₄	—	2	48	no	—
197	C₂×C₄	—	2	48	no	—
198	C₂×C₄	—	2	12	no	—
199	C₂×C₄	—	2	128	no	—
200	C₂×C₄	C₂×#41×C₂₃	1	48	yes	—
201	C₂×C₄	C₂⁵×C₂₃	1	16	yes	—
202	C₂×C₄	C₂⁵×C₂₃	1	4	yes	—
203	C₂×C₄	C₂⁵×C₂₃	1	16	yes	—
204	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
205	C₂²	—	2	16	no	—
206	C₂²	—	2	16	no	—
207	C₂²	—	2	16	no	—
208	C₂²	—	4	96	no	—
209	C₂²	C₂₃×#555	1	16	yes	direct product factor
210	C₂²	C₂²×#41×C₂₃	1	12	yes	—
211	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	6	yes	—
212	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
213	C₄	—	2	16	no	—
214	C₄	—	2	16	no	—
215	C₄	—	2	4	no	—
216	C₄	C₂²×#41×C₂₃	1	16	yes	—
217	C₂	—	4	16	no	—
218	C₂	C₂×C₂₃×#555	1	12	yes	direct product factor
219	C₂	C₂³×#41×C₂₃	1	2	yes	—
220	C₂	C₂²×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#555	C₂	direct (N × H)
2. C₂²×C₂₃×C₄⋊Q₈	C₂	semidirect
3. C₂²×C₂₃×#124	C₂	semidirect
4. C₂²×C₂₃×#121	C₂	semidirect
5. C₂²×C₄×D₄×C₂₃	C₂	semidirect
6. C₂²×C₂₃×#117	C₂	semidirect
11. C₂₃×#555	C₂²	direct (N × H)
12. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
13. C₂×C₂₃×#124	C₂²	semidirect
14. C₂×C₂₃×#121	C₂²	semidirect
15. C₂×C₄×D₄×C₂₃	C₂²	semidirect
16. C₂×C₂₃×#117	C₂²	semidirect
36. C₂₃×C₄⋊Q₈	C₂³	semidirect
37. C₂₃×#124	C₂³	semidirect
38. C₂₃×#121	C₂³	semidirect
39. C₄×D₄×C₂₃	C₂³	semidirect
40. C₂₃×#117	C₂³	semidirect
74. C₂²×#555	C₂₃	direct (N × H)
95. C₂×#555	C₂×C₂₃	direct (N × H)
96. C₂²×C₄⋊Q₈	C₂×C₂₃	semidirect
97. C₂²×#124	C₂×C₂₃	semidirect
98. C₂²×#121	C₂×C₂₃	semidirect
99. C₂²×C₄×D₄	C₂×C₂₃	semidirect
100. C₂²×#117	C₂×C₂₃	semidirect
105. C₂²×C₂₃	#555	direct (N × H)
117. #555	C₂²×C₂₃	direct (N × H)
118. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
119. C₂×#124	C₂²×C₂₃	semidirect
120. C₂×#121	C₂²×C₂₃	semidirect
121. C₂×C₄×D₄	C₂²×C₂₃	semidirect
122. C₂×#117	C₂²×C₂₃	semidirect
130. C₂×C₂₃	C₂×#555	direct (N × H)
146. C₄⋊Q₈	C₂³×C₂₃	semidirect
147. #124	C₂³×C₂₃	semidirect
148. #121	C₂³×C₂₃	semidirect
149. C₄×D₄	C₂³×C₂₃	semidirect
150. #117	C₂³×C₂₃	semidirect
157. C₂₃	C₂²×#555	direct (N × H)
209. C₂²	C₂₃×#555	direct (N × H)
218. C₂	C₂×C₂₃×#555	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×C₂₃×#555	2
C₂²×C₂₃×C₄⋊Q₈	2
C₂²×C₂₃×#124	2
C₂²×C₂₃×#121	2
C₂²×C₄×D₄×C₂₃	2
C₂²×C₂₃×#117	2
C₂²×#555	23

Subgroups — C22×C23×#555 #19210333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#555 #19210333