Subgroups — C₈×C₂₃×#107 #19155907

Order 5888.

446 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×#107	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1279	C₂	1	2	yes	maximal
6	C₂×C₂₃×#107	C₄	1	2	yes	—
7	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₂₃	—	2	2	no	—
9	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#340	C₄	1	2	yes	—
14	C₂₃×#336	—	2	2	no	—
15	C₂₃×#335	C₄	1	4	yes	complemented
16	C₈²×C₂₃	C₄	1	4	yes	complemented
17	C₂²×C₈×C₂₃	—	2	1	no	—
18	C₂²×C₈×C₂₃	—	2	2	no	—
19	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₂₃	—	2	2	no	—
25	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
26	C₂₃×#107	C₈	1	8	yes	direct product factor
27	C₂₃×C₄⋊C₈	—	2	2	no	—
28	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₂₃	—	2	1	no	—
30	C₄×C₈×C₂₃	—	4	4	no	—
31	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₂₃	Q₈	1	2	yes	—
36	C₂₃×#95	—	2	2	no	—
37	C₂²×C₄×C₂₃	—	2	4	no	—
38	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
41	C₂×C₈×C₂₃	—	2	4	no	—
42	C₂×C₈×C₂₃	—	2	2	no	—
43	C₂×C₈×C₂₃	—	2	2	no	—
44	C₂×C₈×C₂₃	—	2	1	no	—
45	C₂×C₈×C₂₃	—	2	1	no	—
46	C₂×C₈×C₂₃	—	2	4	no	—
47	C₂×C₈×C₂₃	—	4	2	no	—
48	C₂×C₈×C₂₃	—	4	2	no	—
49	C₂×C₈×C₂₃	C₄²	1	2	yes	—
50	C₂×C₈×C₂₃	C₄²	1	2	yes	—
51	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₂₃	C₂×C₈	1	4	yes	—
54	C₂×C₈×C₂₃	D₈	1	2	yes	—
55	C₂×C₈×C₂₃	Q₁₆	1	2	yes	—
56	C₄⋊C₄×C₂₃	—	2	4	no	—
57	C₄⋊C₄×C₂₃	C₂×C₈	1	8	yes	—
58	C₄²×C₂₃	—	4	2	no	—
59	C₄²×C₂₃	—	4	2	no	—
60	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₂₃	#41	1	1	yes	characteristic
64	C₈×#107	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	8	no	—
68	C₂×C₄×C₂₃	—	4	4	no	—
69	C₂×C₄×C₂₃	—	4	4	no	—
70	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₂₃	#131	1	2	yes	—
78	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
81	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
82	C₈×C₂₃	—	2	4	no	—
83	C₈×C₂₃	—	2	2	no	—
84	C₈×C₂₃	—	2	2	no	—
85	C₈×C₂₃	—	4	4	no	—
86	C₈×C₂₃	C₄×C₈	1	4	yes	complemented
87	C₈×C₂₃	#107	1	4	yes	direct product factor
88	C₄×#107	C₂×C₂₃	1	1	yes	characteristic
89	C₈×C₄⋊C₄	C₂×C₂₃	1	2	yes	—
90	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
91	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic
92	#1279	C₂×C₂₃	1	2	yes	—
93	C₂²×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₂₃	C₈×D₄	1	2	yes	—
95	C₂²×C₂₃	C₄×D₈	1	1	yes	characteristic
96	C₂²×C₂₃	C₄×Q₁₆	1	1	yes	characteristic
97	C₂²×C₂₃	#443	1	2	yes	—
98	C₄×C₂₃	—	2	1	no	—
99	C₄×C₂₃	—	2	1	no	—
100	C₄×C₂₃	—	4	8	no	—
101	C₄×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
102	C₄×C₂₃	#405	1	1	yes	characteristic
103	C₄×C₂₃	C₂×#107	1	1	yes	characteristic
104	C₄×C₂₃	#427	1	1	yes	characteristic
105	C₄×C₂₃	C₄×D₈	1	1	yes	characteristic
106	C₄×C₂₃	C₄×Q₁₆	1	1	yes	characteristic
107	C₄×C₂₃	C₈×Q₈	1	2	yes	—
108	C₂×#107	C₄×C₂₃	1	2	yes	—
109	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
110	C₂×C₄×C₈	—	2	2	no	—
111	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
114	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
115	#340	C₄×C₂₃	1	2	yes	—
116	#336	—	2	2	no	—
117	#335	C₄×C₂₃	1	4	yes	complemented
118	C₈²	C₄×C₂₃	1	4	yes	complemented
119	C₂×C₂₃	C₈×D₈	1	2	yes	—
120	C₂×C₂₃	C₈×Q₁₆	1	2	yes	—
121	C₂×C₂₃	C₈×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₂₃	#1772	1	1	yes	characteristic
123	C₂×C₂₃	C₄×#107	1	1	yes	characteristic
124	C₂²×C₈	—	2	1	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
127	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
128	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
129	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
130	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
131	C₂×C₄²	—	2	2	no	—
132	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
133	#107	C₈×C₂₃	1	8	yes	direct product factor
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
136	C₄×C₈	—	2	1	no	—
137	C₄×C₈	—	4	4	no	—
138	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
139	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
140	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
141	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
142	C₄×C₈	Q₈×C₂₃	1	2	yes	—
143	#95	—	2	2	no	—
144	C₂₃	C₈×#107	1	1	yes	characteristic, direct product factor, Sylow
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
148	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	C₄²×C₂₃	1	2	yes	—
158	C₂×C₈	C₄²×C₂₃	1	2	yes	—
159	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
160	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
161	C₂×C₈	C₂×C₈×C₂₃	1	4	yes	—
162	C₂×C₈	D₈×C₂₃	1	2	yes	—
163	C₂×C₈	Q₁₆×C₂₃	1	2	yes	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	C₂×C₈×C₂₃	1	8	yes	—
166	C₄²	—	4	2	no	—
167	C₄²	—	4	2	no	—
168	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
169	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
170	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
171	C₄²	#41×C₂₃	1	1	yes	characteristic
172	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	8	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
179	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
180	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
181	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
182	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
183	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
184	C₂×C₄	C₂₃×#131	1	2	yes	—
185	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
186	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
187	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
188	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
189	C₈	—	2	4	no	—
190	C₈	—	2	2	no	—
191	C₈	—	2	2	no	—
192	C₈	—	4	4	no	—
193	C₈	C₄×C₈×C₂₃	1	4	yes	complemented
194	C₈	C₂₃×#107	1	4	yes	direct product factor
195	C₂²	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
196	C₂²	C₈×D₄×C₂₃	1	2	yes	—
197	C₂²	C₄×D₈×C₂₃	1	1	yes	characteristic
198	C₂²	C₄×Q₁₆×C₂₃	1	1	yes	characteristic
199	C₂²	C₂₃×#443	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	4	8	no	—
203	C₄	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
204	C₄	C₂₃×#405	1	1	yes	characteristic
205	C₄	C₂×C₂₃×#107	1	1	yes	characteristic
206	C₄	C₂₃×#427	1	1	yes	characteristic
207	C₄	C₄×D₈×C₂₃	1	1	yes	characteristic
208	C₄	C₄×Q₁₆×C₂₃	1	1	yes	characteristic
209	C₄	C₈×Q₈×C₂₃	1	2	yes	—
210	C₂	C₈×D₈×C₂₃	1	2	yes	—
211	C₂	C₈×Q₁₆×C₂₃	1	2	yes	—
212	C₂	C₈×C₄⋊C₄×C₂₃	1	1	yes	characteristic
213	C₂	C₂₃×#1772	1	1	yes	characteristic
214	C₂	C₄×C₂₃×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₂₃	D₄
commutator	C₄	C₂×C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₄×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₂₃×#335	C₄	semidirect
16. C₈²×C₂₃	C₄	semidirect
26. C₂₃×#107	C₈	direct (N × H)
64. C₈×#107	C₂₃	direct (N × H)
86. C₈×C₂₃	C₄×C₈	semidirect
87. C₈×C₂₃	#107	direct (N × H)
117. #335	C₄×C₂₃	semidirect
118. C₈²	C₄×C₂₃	semidirect
133. #107	C₈×C₂₃	direct (N × H)
144. C₂₃	C₈×#107	direct (N × H)
193. C₈	C₄×C₈×C₂₃	semidirect
194. C₈	C₂₃×#107	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×#107	2
C₈×C₄⋊C₄×C₂₃	2
C₂×C₂₃×#335	2
C₂×C₈²×C₂₃	2
C₂₃×#1279	2
C₈×#107	23

Subgroups — C8×C23×#107 #19155907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₂₃×#107 #19155907