Subgroups — Q₈×C₂₃×#107 #19170352

Order 5888.

838 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	C₂×C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal
3	C₂₃×#1947	C₂	1	6	yes	maximal
4	C₂₃×#1845	C₂	1	3	yes	maximal
5	C₄×C₂₃×#107	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₂₃	—	2	2	no	—
7	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₂₃	C₄	1	8	yes	complemented
9	C₂×C₂₃×#107	C₂²	1	3	yes	—
10	C₂×C₂₃×#107	C₂²	1	6	yes	—
11	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	3	yes	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	3	yes	—
14	C₂₃×#389	—	2	6	no	—
15	C₂₃×#384	C₂²	1	6	yes	—
16	C₄×C₄⋊C₄×C₂₃	C₂²	1	6	yes	—
17	C₂₃×#340	C₂²	1	6	yes	—
18	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
20	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
21	C₄×Q₈×C₂₃	—	4	8	no	—
22	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
23	C₄×Q₈×C₂₃	Q₈	1	4	yes	—
24	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
25	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
26	C₂×C₄⋊C₄×C₂₃	D₄	1	3	yes	—
27	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
28	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
29	C₂×C₄²×C₂₃	—	2	6	no	—
30	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
31	C₂₃×#107	—	2	6	no	—
32	C₂₃×#107	—	4	6	no	—
33	C₂₃×#107	Q₈	1	4	yes	direct product factor
34	C₂₃×C₄⋊C₈	C₂×C₄	1	12	yes	—
35	C₄×C₈×C₂₃	C₂×C₄	1	12	yes	—
36	C₂₃×#95	—	2	6	no	—
37	C₂×Q₈×C₂₃	C₄⋊C₄	1	4	yes	—
38	C₂×Q₈×C₂₃	D₈	1	4	yes	—
39	C₂×Q₈×C₂₃	Q₁₆	1	4	yes	—
40	C₂²×C₄×C₂₃	—	2	2	no	—
41	C₂²×C₄×C₂₃	—	2	6	no	—
42	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
44	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
45	C₂×C₈×C₂₃	—	2	3	no	—
46	C₂×C₈×C₂₃	—	2	3	no	—
47	C₂×C₈×C₂₃	C₂²×C₄	1	6	yes	—
48	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₂₃	#41	1	2	yes	—
51	C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₄⋊C₄×C₂₃	—	2	6	no	—
53	C₄⋊C₄×C₂₃	—	2	6	no	—
54	C₄⋊C₄×C₂₃	—	4	12	no	—
55	C₄⋊C₄×C₂₃	—	8	6	no	—
56	C₄⋊C₄×C₂₃	C₄⋊C₄	1	12	yes	—
57	C₄⋊C₄×C₂₃	C₂×Q₈	1	4	yes	—
58	C₄²×C₂₃	—	4	12	no	—
59	C₄²×C₂₃	C₂²×C₄	1	6	yes	—
60	C₄²×C₂₃	C₂×Q₈	1	6	yes	—
61	Q₈×#107	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
63	Q₈×C₂₃	#107	1	16	yes	direct product factor
64	C₂×C₄×C₂₃	—	2	3	no	—
65	C₂×C₄×C₂₃	—	2	3	no	—
66	C₂×C₄×C₂₃	—	2	4	no	—
67	C₂×C₄×C₂₃	—	4	2	no	—
68	C₂×C₄×C₂₃	—	4	2	no	—
69	C₂×C₄×C₂₃	—	4	6	no	—
70	C₂×C₄×C₂₃	—	4	6	no	—
71	C₂×C₄×C₂₃	—	4	6	no	—
72	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	6	yes	—
73	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	6	yes	—
74	C₂×C₄×C₂₃	C₂×D₈	1	6	yes	—
75	C₂×C₄×C₂₃	C₂×Q₁₆	1	6	yes	—
76	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂²×Q₈	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	C₄×Q₈	1	4	yes	complemented
84	C₂×C₈×Q₈	C₂×C₂₃	1	1	yes	characteristic
85	Q₈×C₄⋊C₄	C₂×C₂₃	1	2	yes	—
86	#1947	C₂×C₂₃	1	6	yes	—
87	#1845	C₂×C₂₃	1	3	yes	—
88	C₄×#107	C₂×C₂₃	1	3	yes	—
89	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₂₃	D₄×Q₈	1	1	yes	characteristic
91	C₂²×C₂₃	#550	1	1	yes	characteristic
92	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
93	C₂²×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
94	C₂²×C₂₃	#576	1	1	yes	characteristic
95	C₂²×C₂₃	#578	1	1	yes	characteristic
96	C₄×C₂₃	—	2	6	no	—
97	C₄×C₂₃	—	4	4	no	—
98	C₄×C₂₃	—	8	6	no	—
99	C₄×C₂₃	C₂×#107	1	12	yes	—
100	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
101	C₄×C₂₃	#520	1	1	yes	characteristic
102	C₄×C₂₃	#557	1	1	yes	characteristic
103	C₄×C₂₃	Q₈²	1	1	yes	characteristic
104	C₂×C₄×Q₈	—	2	2	no	—
105	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
106	C₈×Q₈	C₄×C₂₃	1	8	yes	complemented
107	C₂×#107	C₂²×C₂₃	1	3	yes	—
108	C₂×#107	C₂²×C₂₃	1	6	yes	—
109	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
110	C₂×C₄⋊C₈	C₂²×C₂₃	1	3	yes	—
111	C₂×C₄×C₈	C₂²×C₂₃	1	3	yes	—
112	#389	—	2	6	no	—
113	#384	C₂²×C₂₃	1	6	yes	—
114	C₄×C₄⋊C₄	C₂²×C₂₃	1	6	yes	—
115	#340	C₂²×C₂₃	1	6	yes	—
116	C₂×C₂₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₂₃	C₂²×#107	1	1	yes	characteristic
118	C₂×C₂₃	#2913	1	1	yes	characteristic
119	C₂×C₂₃	Q₈×D₈	1	1	yes	characteristic
120	C₂×C₂₃	Q₈×Q₁₆	1	1	yes	characteristic
121	C₂×C₂₃	#3391	1	1	yes	characteristic
122	C₂×C₂₃	#3395	1	1	yes	characteristic
123	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
124	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
125	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
126	C₄×Q₈	—	4	8	no	—
127	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
128	C₄×Q₈	Q₈×C₂₃	1	4	yes	—
129	C₂×C₄⋊C₄	—	2	6	no	—
130	C₂×C₄⋊C₄	—	2	6	no	—
131	C₂×C₄⋊C₄	D₄×C₂₃	1	3	yes	—
132	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
133	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
134	C₂×C₄²	—	2	6	no	—
135	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
136	#107	—	2	6	no	—
137	#107	—	4	6	no	—
138	#107	Q₈×C₂₃	1	4	yes	direct product factor
139	C₄⋊C₈	C₂×C₄×C₂₃	1	12	yes	—
140	C₄×C₈	C₂×C₄×C₂₃	1	12	yes	—
141	#95	—	2	6	no	—
142	C₂₃	Q₈×#107	1	1	yes	characteristic, direct product factor, Sylow
143	C₂×Q₈	C₄⋊C₄×C₂₃	1	4	yes	—
144	C₂×Q₈	D₈×C₂₃	1	4	yes	—
145	C₂×Q₈	Q₁₆×C₂₃	1	4	yes	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	6	no	—
148	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
149	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
150	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
151	C₂×C₈	—	2	3	no	—
152	C₂×C₈	—	2	3	no	—
153	C₂×C₈	C₂²×C₄×C₂₃	1	6	yes	—
154	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
155	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
156	C₂×C₈	#41×C₂₃	1	2	yes	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	6	no	—
159	C₄⋊C₄	—	2	6	no	—
160	C₄⋊C₄	—	4	12	no	—
161	C₄⋊C₄	—	8	6	no	—
162	C₄⋊C₄	C₄⋊C₄×C₂₃	1	12	yes	—
163	C₄⋊C₄	C₂×Q₈×C₂₃	1	4	yes	—
164	C₄²	—	4	12	no	—
165	C₄²	C₂²×C₄×C₂₃	1	6	yes	—
166	C₄²	C₂×Q₈×C₂₃	1	6	yes	—
167	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
168	Q₈	C₂₃×#107	1	16	yes	direct product factor
169	C₂×C₄	—	2	3	no	—
170	C₂×C₄	—	2	3	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	6	no	—
175	C₂×C₄	—	4	6	no	—
176	C₂×C₄	—	4	6	no	—
177	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	6	yes	—
178	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	6	yes	—
179	C₂×C₄	C₂×D₈×C₂₃	1	6	yes	—
180	C₂×C₄	C₂×Q₁₆×C₂₃	1	6	yes	—
181	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
182	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
183	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
184	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
185	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
186	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
187	C₈	—	2	6	no	—
188	C₈	C₄×Q₈×C₂₃	1	4	yes	complemented
189	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
190	C₂²	D₄×Q₈×C₂₃	1	1	yes	characteristic
191	C₂²	C₂₃×#550	1	1	yes	characteristic
192	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
193	C₂²	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
194	C₂²	C₂₃×#576	1	1	yes	characteristic
195	C₂²	C₂₃×#578	1	1	yes	characteristic
196	C₄	—	2	6	no	—
197	C₄	—	4	4	no	—
198	C₄	—	8	6	no	—
199	C₄	C₂×C₂₃×#107	1	12	yes	—
200	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
201	C₄	C₂₃×#520	1	1	yes	characteristic
202	C₄	C₂₃×#557	1	1	yes	characteristic
203	C₄	Q₈²×C₂₃	1	1	yes	characteristic
204	C₂	Q₈×C₄⋊C₄×C₂₃	1	1	yes	characteristic
205	C₂	C₂²×C₂₃×#107	1	1	yes	characteristic
206	C₂	C₂₃×#2913	1	1	yes	characteristic
207	C₂	Q₈×D₈×C₂₃	1	1	yes	characteristic
208	C₂	Q₈×Q₁₆×C₂₃	1	1	yes	characteristic
209	C₂	C₂₃×#3391	1	1	yes	characteristic
210	C₂	C₂₃×#3395	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈×C₂₃	C₄	semidirect
33. C₂₃×#107	Q₈	direct (N × H)
61. Q₈×#107	C₂₃	direct (N × H)
63. Q₈×C₂₃	#107	direct (N × H)
83. C₈×C₂₃	C₄×Q₈	semidirect
106. C₈×Q₈	C₄×C₂₃	semidirect
138. #107	Q₈×C₂₃	direct (N × H)
142. C₂₃	Q₈×#107	direct (N × H)
168. Q₈	C₂₃×#107	direct (N × H)
188. C₈	C₄×Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₂₃	2
Q₈×C₄⋊C₄×C₂₃	2
C₂₃×#1947	2
C₂₃×#1845	2
C₄×C₂₃×#107	2
Q₈×#107	23

Subgroups — Q8×C23×#107 #19170352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₂₃×#107 #19170352