Subgroups — C₂₃×#8594 #19156220

Order 5888.

476 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1831	C₂	1	2	yes	maximal
3	C₂×C₂₃×#333	C₂	1	2	yes	maximal
4	C₂₃×#1296	C₂	1	2	yes	maximal
5	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₂₃	C₄	1	2	yes	—
7	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
8	C₂₃×#391	C₄	1	2	yes	—
9	C₂₃×#389	—	2	2	no	—
10	C₂₃×#384	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
12	C₄³×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#336	—	2	2	no	—
14	C₂₃×#336	—	2	2	no	—
15	C₂₃×#333	—	2	4	no	—
16	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₂₃	—	2	4	no	—
18	C₂₃×C₄⋊Q₈	—	2	2	no	—
19	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
20	C₂₃×C₄⋊Q₈	Q₈	1	2	yes	—
21	C₄×Q₈×C₂₃	—	4	4	no	—
22	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₂₃	—	2	2	no	—
27	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₂₃	D₄	1	2	yes	—
30	C₂×C₄²×C₂₃	D₄	1	2	yes	—
31	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
32	C₂₃×C₄⋊C₈	—	2	4	no	—
33	C₂₃×C₄⋊C₈	—	2	4	no	—
34	C₂₃×#95	—	2	4	no	—
35	C₂×Q₈×C₂₃	—	2	2	no	—
36	C₂×Q₈×C₂₃	—	4	2	no	—
37	C₂×Q₈×C₂₃	—	4	2	no	—
38	C₂×Q₈×C₂₃	C₄⋊C₄	1	4	yes	—
39	C₂²×C₄×C₂₃	—	2	2	no	—
40	C₂²×C₄×C₂₃	—	2	2	no	—
41	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₂₃	—	4	4	no	—
47	C₂×C₈×C₂₃	—	4	4	no	—
48	C₂×C₈×C₂₃	—	4	4	no	—
49	C₄⋊C₄×C₂₃	—	2	4	no	—
50	C₄⋊C₄×C₂₃	—	2	4	no	—
51	C₄⋊C₄×C₂₃	—	4	2	no	—
52	C₄⋊C₄×C₂₃	—	4	4	no	—
53	C₄⋊C₄×C₂₃	—	4	4	no	—
54	C₄²×C₂₃	—	2	2	no	—
55	C₄²×C₂₃	—	2	4	no	—
56	C₄²×C₂₃	—	2	4	no	—
57	C₄²×C₂₃	—	4	1	no	—
58	C₄²×C₂₃	—	4	4	no	—
59	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₂₃	#41	1	2	yes	—
62	#8594	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	Q₈×C₂₃	—	4	4	no	—
65	Q₈×C₂₃	—	4	4	no	—
66	C₂×C₄×C₂₃	—	2	1	no	—
67	C₂×C₄×C₂₃	—	2	1	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	4	no	—
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	4	2	no	—
76	C₂×C₄×C₂₃	—	4	2	no	—
77	C₂×C₄×C₂₃	—	4	2	no	—
78	C₂×C₄×C₂₃	—	4	4	no	—
79	C₂×C₄×C₂₃	#100	1	2	yes	—
80	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
82	C₂×C₄×C₂₃	#121	1	2	yes	—
83	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
84	C₂×C₄×C₂₃	#135	1	4	yes	—
85	C₂×C₄×C₂₃	#137	1	4	yes	—
86	C₈×C₂₃	—	8	4	no	—
87	C₄×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
88	#1831	C₂×C₂₃	1	2	yes	—
89	C₂×#333	C₂×C₂₃	1	2	yes	—
90	#1296	C₂×C₂₃	1	2	yes	—
91	C₂²×C₂₃	#380	1	1	yes	characteristic
92	C₂²×C₂₃	C₂×#100	1	1	yes	characteristic
93	C₂²×C₂₃	#413	1	1	yes	characteristic
94	C₂²×C₂₃	#416	1	2	yes	—
95	C₂²×C₂₃	#419	1	2	yes	—
96	C₄×C₂₃	—	2	1	no	—
97	C₄×C₂₃	—	2	1	no	—
98	C₄×C₂₃	—	2	2	no	—
99	C₄×C₂₃	—	4	1	no	—
100	C₄×C₂₃	—	4	4	no	—
101	C₄×C₂₃	—	4	2	no	—
102	C₄×C₂₃	—	8	2	no	—
103	C₄×C₂₃	#464	1	4	yes	—
104	C₄×C₂₃	#479	1	4	yes	—
105	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
106	C₂×C₄×Q₈	C₄×C₂₃	1	2	yes	—
107	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
108	#391	C₄×C₂₃	1	2	yes	—
109	#389	—	2	2	no	—
110	#384	C₂²×C₂₃	1	1	yes	characteristic
111	C₄×C₄⋊C₄	—	2	2	no	—
112	C₄³	C₂²×C₂₃	1	1	yes	characteristic
113	#336	—	2	2	no	—
114	#336	—	2	2	no	—
115	#333	—	2	4	no	—
116	C₂×C₂₃	#1543	1	1	yes	characteristic
117	C₂×C₂₃	#1554	1	2	yes	—
118	C₂×C₂₃	#1558	1	1	yes	characteristic
119	C₂×C₂₃	#1800	1	1	yes	characteristic
120	C₂×C₂₃	#1805	1	2	yes	—
121	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
122	C₂²×C₈	—	2	4	no	—
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
125	C₄⋊Q₈	Q₈×C₂₃	1	2	yes	—
126	C₄×Q₈	—	4	4	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
131	C₂×C₄²	—	2	2	no	—
132	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
133	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
134	C₂×C₄²	D₄×C₂₃	1	2	yes	—
135	C₂×C₄²	D₄×C₂₃	1	2	yes	—
136	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
137	C₄⋊C₈	—	2	4	no	—
138	C₄⋊C₈	—	2	4	no	—
139	#95	—	2	4	no	—
140	C₂₃	#8594	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	C₄⋊C₄×C₂₃	1	4	yes	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
148	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
149	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
150	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
152	C₂×C₈	—	4	4	no	—
153	C₂×C₈	—	4	4	no	—
154	C₂×C₈	—	4	4	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	4	no	—
162	C₄²	—	2	4	no	—
163	C₄²	—	4	1	no	—
164	C₄²	—	4	4	no	—
165	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
166	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
167	C₄²	#41×C₂₃	1	2	yes	—
168	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
169	Q₈	—	4	4	no	—
170	Q₈	—	4	4	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	C₂₃×#100	1	2	yes	—
185	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
186	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
187	C₂×C₄	C₂₃×#121	1	2	yes	—
188	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
189	C₂×C₄	C₂₃×#135	1	4	yes	—
190	C₂×C₄	C₂₃×#137	1	4	yes	—
191	C₈	—	8	4	no	—
192	C₂²	C₂₃×#380	1	1	yes	characteristic
193	C₂²	C₂×C₂₃×#100	1	1	yes	characteristic
194	C₂²	C₂₃×#413	1	1	yes	characteristic
195	C₂²	C₂₃×#416	1	2	yes	—
196	C₂²	C₂₃×#419	1	2	yes	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	2	no	—
200	C₄	—	4	1	no	—
201	C₄	—	4	4	no	—
202	C₄	—	4	2	no	—
203	C₄	—	8	2	no	—
204	C₄	C₂₃×#464	1	4	yes	—
205	C₄	C₂₃×#479	1	4	yes	—
206	C₂	C₂₃×#1543	1	1	yes	characteristic
207	C₂	C₂₃×#1554	1	2	yes	—
208	C₂	C₂₃×#1558	1	1	yes	characteristic
209	C₂	C₂₃×#1800	1	1	yes	characteristic
210	C₂	C₂₃×#1805	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
62. #8594	C₂₃	direct (N × H)
140. C₂₃	#8594	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₂₃×C₄⋊Q₈	2
C₂₃×#1831	2
C₂×C₂₃×#333	2
C₂₃×#1296	2
#8594	23

Subgroups — C23×#8594 #19156220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8594 #19156220