Subgroups — Q₈×C₂₃×#108 #19170358

Order 5888.

658 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×M₁₆×C₂₃	C₂	1	2	yes	maximal
2	C₂×C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1950	C₂	1	6	yes	maximal
4	C₂₃×#1848	C₂	1	3	yes	maximal
5	C₄×C₂₃×#108	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#446	—	2	6	no	—
8	C₈×Q₈×C₂₃	—	2	2	no	—
9	C₈×Q₈×C₂₃	C₄	1	6	yes	—
10	C₈×Q₈×C₂₃	C₄	1	2	yes	—
11	C₂×C₂₃×#108	C₂²	1	3	yes	—
12	C₂×C₂₃×#108	C₂²	1	6	yes	—
13	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
14	C₂₃×#423	C₂²	1	6	yes	—
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	3	yes	—
16	C₄×M₁₆×C₂₃	C₂²	1	6	yes	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	3	yes	—
18	C₂₃×#341	C₂²	1	6	yes	—
19	C₂²×Q₈×C₂₃	Q₈	1	1	yes	characteristic
20	C₂×M₁₆×C₂₃	C₂³	1	6	yes	—
21	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
22	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
23	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
24	C₄×Q₈×C₂₃	C₂×C₄	1	3	yes	—
25	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
27	C₄×Q₈×C₂₃	D₄	1	3	yes	—
28	C₂×C₄⋊C₄×C₂₃	Q₈	1	3	yes	—
29	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
30	C₂₃×#108	—	2	6	no	—
31	C₂₃×#108	—	4	6	no	—
32	C₂₃×#108	Q₈	1	4	yes	direct product factor
33	C₂₃×C₄⋊C₈	—	2	6	no	—
34	C₂₃×C₄⋊C₈	—	2	6	no	—
35	C₂₃×C₄⋊C₈	C₂×C₄	1	6	yes	—
36	C₂₃×C₄⋊C₈	C₂×C₄	1	6	yes	—
37	C₂₃×#97	—	2	6	no	—
38	C₄×C₈×C₂₃	—	2	6	no	—
39	C₄×C₈×C₂₃	C₂×C₄	1	6	yes	—
40	C₄×C₈×C₂₃	C₂×C₄	1	6	yes	—
41	C₂×Q₈×C₂₃	—	2	4	no	—
42	C₂×Q₈×C₂₃	C₄⋊C₄	1	3	yes	—
43	C₂×Q₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂×Q₈	1	3	yes	—
45	C₂²×C₄×C₂₃	C₂×Q₈	1	3	yes	—
46	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
47	M₁₆×C₂₃	—	2	2	no	—
48	M₁₆×C₂₃	—	2	6	no	—
49	M₁₆×C₂₃	—	2	6	no	—
50	M₁₆×C₂₃	C₂×Q₈	1	4	yes	—
51	C₂×C₈×C₂₃	—	2	2	no	—
52	C₂×C₈×C₂₃	—	2	3	no	—
53	C₂×C₈×C₂₃	—	2	3	no	—
54	C₂×C₈×C₂₃	—	2	6	no	—
55	C₂×C₈×C₂₃	C₂²×C₄	1	6	yes	—
56	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₂₃	#41	1	2	yes	—
59	C₄⋊C₄×C₂₃	C₄⋊C₄	1	6	yes	—
60	C₄⋊C₄×C₂₃	C₄⋊C₄	1	3	yes	—
61	C₄⋊C₄×C₂₃	C₄⋊C₄	1	3	yes	—
62	C₄²×C₂₃	C₂²×C₄	1	3	yes	—
63	C₄²×C₂₃	C₂²×C₄	1	3	yes	—
64	C₄²×C₂₃	C₂×D₄	1	3	yes	—
65	C₄²×C₂₃	C₂×D₄	1	3	yes	—
66	Q₈×#108	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₂₃	C₂²×Q₈	1	1	yes	characteristic
68	Q₈×C₂₃	—	2	6	no	—
69	Q₈×C₂₃	#108	1	4	yes	direct product factor
70	C₂×C₄×C₂₃	—	2	6	no	—
71	C₂×C₄×C₂₃	—	4	3	no	—
72	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
73	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
80	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
81	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
82	C₈×C₂₃	—	2	6	no	—
83	C₈×C₂₃	—	2	4	no	—
84	C₈×C₂₃	—	4	6	no	—
85	C₈×C₂₃	C₄×Q₈	1	4	yes	—
86	Q₈×M₁₆	C₂×C₂₃	1	2	yes	—
87	C₂×C₈×Q₈	C₂×C₂₃	1	1	yes	characteristic
88	#1950	C₂×C₂₃	1	6	yes	—
89	#1848	C₂×C₂₃	1	3	yes	—
90	C₄×#108	C₂×C₂₃	1	3	yes	—
91	C₂²×C₂₃	—	2	1	no	—
92	C₂²×C₂₃	—	2	1	no	—
93	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₂₃	#557	1	1	yes	characteristic
95	C₂²×C₂₃	Q₈²	1	1	yes	characteristic
96	C₄×C₂₃	—	2	3	no	—
97	C₄×C₂₃	—	2	3	no	—
98	C₄×C₂₃	—	2	3	no	—
99	C₄×C₂₃	C₂×#108	1	6	yes	—
100	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
101	C₄×C₂₃	#520	1	1	yes	characteristic
102	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
103	C₄×C₂₃	#550	1	1	yes	characteristic
104	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
105	#446	—	2	6	no	—
106	C₈×Q₈	—	2	2	no	—
107	C₈×Q₈	C₄×C₂₃	1	6	yes	—
108	C₈×Q₈	C₄×C₂₃	1	2	yes	—
109	C₂×#108	C₂²×C₂₃	1	3	yes	—
110	C₂×#108	C₂²×C₂₃	1	6	yes	—
111	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
112	#423	C₂²×C₂₃	1	6	yes	—
113	C₂×C₄⋊C₈	C₂²×C₂₃	1	3	yes	—
114	C₄×M₁₆	C₂²×C₂₃	1	6	yes	—
115	C₂×C₄×C₈	C₂²×C₂₃	1	3	yes	—
116	#341	C₂²×C₂₃	1	6	yes	—
117	C₂×C₂₃	—	2	1	no	—
118	C₂×C₂₃	—	2	1	no	—
119	C₂×C₂₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₂₃	C₂²×#108	1	1	yes	characteristic
121	C₂×C₂₃	#2916	1	1	yes	characteristic
122	C₂²×Q₈	Q₈×C₂₃	1	1	yes	characteristic
123	C₂×M₁₆	C₂³×C₂₃	1	6	yes	—
124	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
125	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
126	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
127	C₄×Q₈	C₂×C₄×C₂₃	1	3	yes	—
128	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
129	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
130	C₄×Q₈	D₄×C₂₃	1	3	yes	—
131	C₂×C₄⋊C₄	Q₈×C₂₃	1	3	yes	—
132	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
133	#108	—	2	6	no	—
134	#108	—	4	6	no	—
135	#108	Q₈×C₂₃	1	4	yes	direct product factor
136	C₄⋊C₈	—	2	6	no	—
137	C₄⋊C₈	—	2	6	no	—
138	C₄⋊C₈	C₂×C₄×C₂₃	1	6	yes	—
139	C₄⋊C₈	C₂×C₄×C₂₃	1	6	yes	—
140	#97	—	2	6	no	—
141	C₄×C₈	—	2	6	no	—
142	C₄×C₈	C₂×C₄×C₂₃	1	6	yes	—
143	C₄×C₈	C₂×C₄×C₂₃	1	6	yes	—
144	C₂₃	Q₈×#108	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	C₄⋊C₄×C₂₃	1	3	yes	—
147	C₂×Q₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
148	C₂²×C₄	C₂×Q₈×C₂₃	1	3	yes	—
149	C₂²×C₄	C₂×Q₈×C₂₃	1	3	yes	—
150	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
151	M₁₆	—	2	2	no	—
152	M₁₆	—	2	6	no	—
153	M₁₆	—	2	6	no	—
154	M₁₆	C₂×Q₈×C₂₃	1	4	yes	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	3	no	—
157	C₂×C₈	—	2	3	no	—
158	C₂×C₈	—	2	6	no	—
159	C₂×C₈	C₂²×C₄×C₂₃	1	6	yes	—
160	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
161	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
162	C₂×C₈	#41×C₂₃	1	2	yes	—
163	C₄⋊C₄	C₄⋊C₄×C₂₃	1	6	yes	—
164	C₄⋊C₄	C₄⋊C₄×C₂₃	1	3	yes	—
165	C₄⋊C₄	C₄⋊C₄×C₂₃	1	3	yes	—
166	C₄²	C₂²×C₄×C₂₃	1	3	yes	—
167	C₄²	C₂²×C₄×C₂₃	1	3	yes	—
168	C₄²	C₂×D₄×C₂₃	1	3	yes	—
169	C₄²	C₂×D₄×C₂₃	1	3	yes	—
170	C₂³	C₂²×Q₈×C₂₃	1	1	yes	characteristic
171	Q₈	—	2	6	no	—
172	Q₈	C₂₃×#108	1	4	yes	direct product factor
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	4	3	no	—
175	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
176	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
177	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
178	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
179	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
180	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
181	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
182	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
183	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
184	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	2	6	no	—
187	C₈	—	4	6	no	—
188	C₈	C₄×Q₈×C₂₃	1	4	yes	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
192	C₂²	C₂₃×#557	1	1	yes	characteristic
193	C₂²	Q₈²×C₂₃	1	1	yes	characteristic
194	C₄	—	2	3	no	—
195	C₄	—	2	3	no	—
196	C₄	—	2	3	no	—
197	C₄	C₂×C₂₃×#108	1	6	yes	—
198	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
199	C₄	C₂₃×#520	1	1	yes	characteristic
200	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
201	C₄	C₂₃×#550	1	1	yes	characteristic
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	Q₈×C₄⋊C₄×C₂₃	1	1	yes	characteristic
205	C₂	C₂²×C₂₃×#108	1	1	yes	characteristic
206	C₂	C₂₃×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₂₃	C₂²×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
32. C₂₃×#108	Q₈	direct (N × H)
66. Q₈×#108	C₂₃	direct (N × H)
69. Q₈×C₂₃	#108	direct (N × H)
135. #108	Q₈×C₂₃	direct (N × H)
144. C₂₃	Q₈×#108	direct (N × H)
172. Q₈	C₂₃×#108	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×M₁₆×C₂₃	2
C₂×C₈×Q₈×C₂₃	2
C₂₃×#1950	2
C₂₃×#1848	2
C₄×C₂₃×#108	2
Q₈×#108	23

Subgroups — Q8×C23×#108 #19170358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₂₃×#108 #19170358