Subgroups — C₂×C₈×Q₁₆×C₂₃ #19165470

Order 5888.

734 conjugacy classes of subgroups.

The subgroup lattice has 258 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	2	yes	maximal
2	C₂×C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₈×Q₁₆×C₂₃	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×Q₈⋊C₈	C₂	1	2	yes	maximal
6	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
9	C₈×Q₈×C₂₃	—	2	4	no	—
10	C₈×Q₈×C₂₃	C₂²	1	8	yes	—
11	C₄×Q₁₆×C₂₃	C₄	1	4	yes	—
12	C₄×Q₁₆×C₂₃	C₂²	1	4	yes	—
13	C₂×C₂₃×#107	C₄	1	1	yes	characteristic
14	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₂₃×#103	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₂₃	—	2	2	no	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#335	C₂²	1	4	yes	—
22	C₂₃×Q₈⋊C₈	C₂²	1	8	yes	—
23	C₈²×C₂₃	C₂²	1	4	yes	—
24	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
25	C₂×Q₁₆×C₂₃	C₈	1	8	yes	direct product factor
26	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₂₃	—	2	1	no	—
28	C₂²×C₈×C₂₃	—	2	2	no	—
29	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
33	C₄×Q₈×C₂₃	—	2	2	no	—
34	C₄×Q₈×C₂₃	—	2	2	no	—
35	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
36	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
37	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₂₃	—	2	2	no	—
40	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
41	C₂₃×#107	C₂×C₄	1	4	yes	—
42	C₂₃×C₄⋊C₈	—	2	8	no	—
43	C₂₃×C₄⋊C₈	—	2	2	no	—
44	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
45	C₂₃×#103	C₂×C₄	1	8	yes	—
46	C₄×C₈×C₂₃	—	2	8	no	—
47	C₄×C₈×C₂₃	—	2	1	no	—
48	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₂₃	D₄	1	2	yes	—
50	C₄×C₈×C₂₃	C₂³	1	2	yes	—
51	C₄×C₈×C₂₃	C₂³	1	2	yes	—
52	C₄×C₈×C₂₃	C₂³	1	2	yes	—
53	C₂×Q₈×C₂₃	—	2	2	no	—
54	C₂×Q₈×C₂₃	—	2	4	no	—
55	C₂×Q₈×C₂₃	C₂×C₈	1	8	yes	—
56	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₂₃	—	2	2	no	—
58	C₂²×C₄×C₂₃	—	2	2	no	—
59	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
62	Q₁₆×C₂₃	C₂×C₈	1	16	yes	direct product factor
63	C₂×C₈×C₂₃	—	2	4	no	—
64	C₂×C₈×C₂₃	—	2	2	no	—
65	C₂×C₈×C₂₃	—	2	2	no	—
66	C₂×C₈×C₂₃	—	2	2	no	—
67	C₂×C₈×C₂₃	—	2	4	no	—
68	C₂×C₈×C₂₃	—	4	4	no	—
69	C₂×C₈×C₂₃	C₂×C₈	1	4	yes	—
70	C₂×C₈×C₂₃	Q₁₆	1	4	yes	direct product factor
71	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
75	C₄⋊C₄×C₂₃	—	2	4	no	—
76	C₄⋊C₄×C₂₃	—	2	2	no	—
77	C₄⋊C₄×C₂₃	—	2	4	no	—
78	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
79	C₄²×C₂₃	—	2	4	no	—
80	C₄²×C₂₃	—	2	4	no	—
81	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₂₃	#41	1	1	yes	characteristic
84	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×Q₁₆	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
87	Q₈×C₂₃	—	2	4	no	—
88	Q₈×C₂₃	—	2	8	no	—
89	Q₈×C₂₃	C₂²×C₈	1	8	yes	—
90	C₂×C₄×C₂₃	—	2	4	no	—
91	C₂×C₄×C₂₃	—	2	8	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	4	no	—
94	C₂×C₄×C₂₃	—	4	4	no	—
95	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
99	C₂×C₄×C₂₃	#131	1	2	yes	—
100	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
101	C₂×C₄×C₂₃	#135	1	2	yes	—
102	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
105	C₈×C₂₃	—	2	4	no	—
106	C₈×C₂₃	—	2	2	no	—
107	C₈×C₂₃	—	2	2	no	—
108	C₈×C₂₃	—	4	4	no	—
109	C₈×C₂₃	C₂²×C₈	1	4	yes	—
110	C₈×C₂₃	C₂×Q₁₆	1	4	yes	direct product factor
111	C₂×C₈×Q₈	C₂×C₂₃	1	2	yes	—
112	C₂×C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic
113	C₈×Q₁₆	C₂×C₂₃	1	8	yes	direct product factor
114	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
115	C₂×Q₈⋊C₈	C₂×C₂₃	1	2	yes	—
116	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic
117	C₂²×C₂₃	C₈×D₄	1	2	yes	—
118	C₂²×C₂₃	C₄×Q₁₆	1	2	yes	—
119	C₂²×C₂₃	#443	1	2	yes	—
120	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
121	C₄×C₂₃	—	2	8	no	—
122	C₄×C₂₃	—	2	1	no	—
123	C₄×C₂₃	—	2	1	no	—
124	C₄×C₂₃	—	4	4	no	—
125	C₄×C₂₃	C₈×D₄	1	2	yes	—
126	C₄×C₂₃	C₄×Q₁₆	1	2	yes	—
127	C₄×C₂₃	C₂³×C₈	1	1	yes	characteristic
128	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
129	C₄×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
130	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
131	C₂²×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
132	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
133	C₈×Q₈	—	2	4	no	—
134	C₈×Q₈	C₂²×C₂₃	1	8	yes	—
135	C₄×Q₁₆	C₄×C₂₃	1	4	yes	—
136	C₄×Q₁₆	C₂²×C₂₃	1	4	yes	—
137	C₂×#107	C₄×C₂₃	1	1	yes	characteristic
138	C₂×C₄⋊C₈	—	2	2	no	—
139	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
140	C₂×#103	C₄×C₂₃	1	2	yes	—
141	C₂×C₄×C₈	—	2	2	no	—
142	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
145	#335	C₂²×C₂₃	1	4	yes	—
146	Q₈⋊C₈	C₂²×C₂₃	1	8	yes	—
147	C₈²	C₂²×C₂₃	1	4	yes	—
148	C₂×C₂₃	C₈×Q₁₆	1	4	yes	direct product factor
149	C₂×C₂₃	C₂×C₈×D₄	1	1	yes	characteristic
150	C₂×C₂₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
151	C₂×C₂₃	C₂×#443	1	1	yes	characteristic
152	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
153	C₂×Q₁₆	C₈×C₂₃	1	8	yes	direct product factor
154	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
158	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
159	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
160	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
161	C₄×Q₈	—	2	2	no	—
162	C₄×Q₈	—	2	2	no	—
163	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
164	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
169	#107	C₂×C₄×C₂₃	1	4	yes	—
170	C₄⋊C₈	—	2	8	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
173	#103	C₂×C₄×C₂₃	1	8	yes	—
174	C₄×C₈	—	2	8	no	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
177	C₄×C₈	D₄×C₂₃	1	2	yes	—
178	C₄×C₈	C₂³×C₂₃	1	2	yes	—
179	C₄×C₈	C₂³×C₂₃	1	2	yes	—
180	C₄×C₈	C₂³×C₂₃	1	2	yes	—
181	C₂₃	C₂×C₈×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	C₂×C₈×C₂₃	1	8	yes	—
185	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
190	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
191	Q₁₆	C₂×C₈×C₂₃	1	16	yes	direct product factor
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	4	4	no	—
198	C₂×C₈	C₂×C₈×C₂₃	1	4	yes	—
199	C₂×C₈	Q₁₆×C₂₃	1	4	yes	direct product factor
200	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
201	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
202	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
208	C₄²	—	2	4	no	—
209	C₄²	—	2	4	no	—
210	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
211	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
212	C₄²	#41×C₂₃	1	1	yes	characteristic
213	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
214	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	Q₈	—	2	8	no	—
217	Q₈	C₂²×C₈×C₂₃	1	8	yes	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
224	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
225	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
226	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
227	C₂×C₄	C₂₃×#131	1	2	yes	—
228	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
229	C₂×C₄	C₂₃×#135	1	2	yes	—
230	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
231	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
232	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
233	C₈	—	2	4	no	—
234	C₈	—	2	2	no	—
235	C₈	—	2	2	no	—
236	C₈	—	4	4	no	—
237	C₈	C₂²×C₈×C₂₃	1	4	yes	—
238	C₈	C₂×Q₁₆×C₂₃	1	4	yes	direct product factor
239	C₂²	C₈×D₄×C₂₃	1	2	yes	—
240	C₂²	C₄×Q₁₆×C₂₃	1	2	yes	—
241	C₂²	C₂₃×#443	1	2	yes	—
242	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
243	C₄	—	2	8	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	4	4	no	—
247	C₄	C₈×D₄×C₂₃	1	2	yes	—
248	C₄	C₄×Q₁₆×C₂₃	1	2	yes	—
249	C₄	C₂³×C₈×C₂₃	1	1	yes	characteristic
250	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
251	C₄	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
252	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
253	C₂	C₈×Q₁₆×C₂₃	1	4	yes	direct product factor
254	C₂	C₂×C₈×D₄×C₂₃	1	1	yes	characteristic
255	C₂	C₂×C₄×Q₁₆×C₂₃	1	1	yes	characteristic
256	C₂	C₂×C₂₃×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₂₃	D₄
commutator	C₄	C₂³×C₈×C₂₃
Frattini, cosocle	C₄²	C₂⁴×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
3. C₈×Q₁₆×C₂₃	C₂	direct (N × H)
25. C₂×Q₁₆×C₂₃	C₈	direct (N × H)
62. Q₁₆×C₂₃	C₂×C₈	direct (N × H)
70. C₂×C₈×C₂₃	Q₁₆	direct (N × H)
85. C₂×C₈×Q₁₆	C₂₃	direct (N × H)
110. C₈×C₂₃	C₂×Q₁₆	direct (N × H)
113. C₈×Q₁₆	C₂×C₂₃	direct (N × H)
148. C₂×C₂₃	C₈×Q₁₆	direct (N × H)
153. C₂×Q₁₆	C₈×C₂₃	direct (N × H)
181. C₂₃	C₂×C₈×Q₁₆	direct (N × H)
191. Q₁₆	C₂×C₈×C₂₃	direct (N × H)
199. C₂×C₈	Q₁₆×C₂₃	direct (N × H)
238. C₈	C₂×Q₁₆×C₂₃	direct (N × H)
253. C₂	C₈×Q₁₆×C₂₃	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×C₈×Q₈×C₂₃	2
C₂×C₄×Q₁₆×C₂₃	2
C₈×Q₁₆×C₂₃	2
C₂×C₂₃×#335	2
C₂×C₂₃×Q₈⋊C₈	2
C₂×C₈²×C₂₃	2
C₂×C₈×Q₁₆	23

Subgroups — C2×C8×Q16×C23 #19165470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×Q₁₆×C₂₃ #19165470