Subgroups — C₂₃×#11930 #19159556

Order 5888.

418 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1362	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1361	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#135	—	2	1	no	—
7	C₂×C₂₃×#135	C₄	1	2	yes	—
8	C₂₃×#532	—	2	1	no	—
9	C₂×C₂₃×M₃₂	—	2	1	no	—
10	C₂₃×#495	—	2	1	no	—
11	C₂₃×#495	C₄	1	2	yes	—
12	C₂₃×#494	C₂²	1	1	yes	characteristic
13	C₂₃×#493	C₂²	1	1	yes	characteristic
14	C₂₃×#492	—	2	1	no	—
15	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×C₄⋊C₁₆	—	2	1	no	—
18	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₂₃×#335	—	2	1	no	—
20	C₂₃×#335	C₂²	1	2	yes	—
21	C₂×#41×C₂₃	—	2	2	no	—
22	C₂₃×#135	—	4	4	no	—
23	C₂₃×#135	—	4	2	no	—
24	C₂₃×#135	—	4	2	no	—
25	C₂×Q₁₆×C₂₃	—	2	1	no	—
26	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
27	C₂×SD₁₆×C₂₃	—	2	2	no	—
28	C₂×SD₁₆×C₂₃	—	2	2	no	—
29	C₂×D₈×C₂₃	—	2	1	no	—
30	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₂₃	—	4	1	no	—
32	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
34	C₂₃×C₄⋊Q₈	—	2	1	no	—
35	C₂₃×#127	—	2	1	no	—
36	C₂₃×#124	—	2	2	no	—
37	C₂₃×#121	—	4	1	no	—
38	C₂₃×#121	—	4	1	no	—
39	C₄×D₄×C₂₃	—	4	1	no	—
40	C₄×D₄×C₂₃	—	4	1	no	—
41	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
42	C₂₃×M₃₂	—	4	2	no	—
43	C₂×C₁₆×C₂₃	—	2	2	no	—
44	C₂₃×C₄⋊C₈	—	2	2	no	—
45	C₂₃×C₄⋊C₈	—	4	1	no	—
46	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
50	#41×C₂₃	—	4	4	no	—
51	#41×C₂₃	—	4	4	no	—
52	C₂×Q₈×C₂₃	—	2	2	no	—
53	C₂×D₄×C₂₃	—	2	2	no	—
54	C₂×D₄×C₂₃	—	4	2	no	—
55	C₂×D₄×C₂₃	—	4	1	no	—
56	C₂×D₄×C₂₃	—	4	1	no	—
57	C₂²×C₄×C₂₃	—	4	2	no	—
58	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
61	Q₁₆×C₂₃	—	4	2	no	—
62	Q₁₆×C₂₃	—	4	1	no	—
63	Q₁₆×C₂₃	—	4	1	no	—
64	SD₁₆×C₂₃	—	4	4	no	—
65	SD₁₆×C₂₃	—	4	4	no	—
66	D₈×C₂₃	—	4	2	no	—
67	D₈×C₂₃	—	4	1	no	—
68	D₈×C₂₃	—	4	1	no	—
69	C₂×C₈×C₂₃	—	2	2	no	—
70	C₂×C₈×C₂₃	—	4	2	no	—
71	C₂×C₈×C₂₃	—	8	2	no	—
72	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₂₃	D₈	1	2	yes	—
75	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
76	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₄⋊C₄×C₂₃	—	4	1	no	—
79	C₄⋊C₄×C₂₃	—	4	1	no	—
80	C₂²⋊C₄×C₂₃	—	4	2	no	—
81	C₂²⋊C₄×C₂₃	—	4	2	no	—
82	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₂₃	D₈	1	1	yes	characteristic
84	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
85	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₁₆×C₂₃	—	4	2	no	—
87	#11930	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₂₃	—	4	2	no	—
89	C₂³×C₂₃	#102	1	1	yes	characteristic
90	Q₈×C₂₃	—	4	2	no	—
91	Q₈×C₂₃	—	4	2	no	—
92	D₄×C₂₃	—	4	2	no	—
93	D₄×C₂₃	—	4	2	no	—
94	D₄×C₂₃	—	8	2	no	—
95	D₄×C₂₃	—	8	4	no	—
96	D₄×C₂₃	—	8	1	no	—
97	D₄×C₂₃	—	8	1	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	4	1	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	—	8	4	no	—
103	C₂×C₄×C₂₃	#100	1	2	yes	—
104	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
105	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
106	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
107	C₂×C₄×C₂₃	#102	1	2	yes	—
108	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₈×C₂₃	—	2	1	no	—
112	C₈×C₂₃	—	2	1	no	—
113	C₈×C₂₃	—	8	2	no	—
114	C₈×C₂₃	#102	1	2	yes	—
115	C₈×C₂₃	C₂×D₈	1	1	yes	characteristic
116	C₈×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
117	#3147	C₂×C₂₃	1	1	yes	characteristic
118	#2150	C₂×C₂₃	1	1	yes	characteristic, complemented
119	C₂×#335	C₂×C₂₃	1	1	yes	characteristic, complemented
120	#1362	C₂×C₂₃	1	2	yes	complemented
121	#1361	C₂×C₂₃	1	2	yes	complemented
122	C₂²×C₂₃	—	2	1	no	—
123	C₂²×C₂₃	—	2	1	no	—
124	C₂²×C₂₃	—	8	2	no	—
125	C₂²×C₂₃	—	8	2	no	—
126	C₂²×C₂₃	—	8	2	no	—
127	C₂²×C₂₃	#331	1	2	yes	—
128	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
129	C₄×C₂₃	—	2	1	no	—
130	C₄×C₂₃	—	2	1	no	—
131	C₄×C₂₃	—	2	1	no	—
132	C₄×C₂₃	—	8	2	no	—
133	C₄×C₂₃	#331	1	2	yes	—
134	C₄×C₂₃	C₂×#100	1	1	yes	characteristic
135	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
136	C₄×C₂₃	#509	1	1	yes	characteristic
137	C₄×C₂₃	#510	1	1	yes	characteristic
138	C₂×#135	—	2	1	no	—
139	C₂×#135	C₄×C₂₃	1	2	yes	—
140	#532	—	2	1	no	—
141	C₂×M₃₂	—	2	1	no	—
142	#495	—	2	1	no	—
143	#495	C₄×C₂₃	1	2	yes	—
144	#494	C₂²×C₂₃	1	1	yes	characteristic
145	#493	C₂²×C₂₃	1	1	yes	characteristic
146	#492	—	2	1	no	—
147	C₂×C₄⋊C₈	—	2	1	no	—
148	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
149	C₄⋊C₁₆	—	2	1	no	—
150	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
151	#335	—	2	1	no	—
152	#335	C₂²×C₂₃	1	2	yes	—
153	C₂×C₂₃	—	2	2	no	—
154	C₂×C₂₃	—	16	2	no	—
155	C₂×C₂₃	C₂×#331	1	1	yes	characteristic
156	C₂×C₂₃	#2139	1	1	yes	characteristic
157	C₂×C₂₃	#2148	1	1	yes	characteristic
158	C₂×#41	—	2	2	no	—
159	#135	—	4	4	no	—
160	#135	—	4	2	no	—
161	#135	—	4	2	no	—
162	C₂×Q₁₆	—	2	1	no	—
163	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
164	C₂×SD₁₆	—	2	2	no	—
165	C₂×SD₁₆	—	2	2	no	—
166	C₂×D₈	—	2	1	no	—
167	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
168	C₂²×C₈	—	4	1	no	—
169	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
170	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
171	C₄⋊Q₈	—	2	1	no	—
172	#127	—	2	1	no	—
173	#124	—	2	2	no	—
174	#121	—	4	1	no	—
175	#121	—	4	1	no	—
176	C₄×D₄	—	4	1	no	—
177	C₄×D₄	—	4	1	no	—
178	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
179	M₃₂	—	4	2	no	—
180	C₂×C₁₆	—	2	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	4	1	no	—
183	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
184	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
185	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
186	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
187	C₂₃	#11930	1	1	yes	characteristic, direct product factor, Sylow
188	#41	—	4	4	no	—
189	#41	—	4	4	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
198	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
199	Q₁₆	—	4	2	no	—
200	Q₁₆	—	4	1	no	—
201	Q₁₆	—	4	1	no	—
202	SD₁₆	—	4	4	no	—
203	SD₁₆	—	4	4	no	—
204	D₈	—	4	2	no	—
205	D₈	—	4	1	no	—
206	D₈	—	4	1	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	8	2	no	—
210	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
212	C₂×C₈	D₈×C₂₃	1	2	yes	—
213	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
214	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
215	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₄⋊C₄	—	4	1	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₂²⋊C₄	—	4	2	no	—
220	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
221	C₄²	D₈×C₂₃	1	1	yes	characteristic
222	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
223	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
224	C₁₆	—	4	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	C₂₃×#102	1	1	yes	characteristic
227	Q₈	—	4	2	no	—
228	Q₈	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	8	2	no	—
232	D₄	—	8	4	no	—
233	D₄	—	8	1	no	—
234	D₄	—	8	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	4	no	—
240	C₂×C₄	C₂₃×#100	1	2	yes	—
241	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
242	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
243	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
244	C₂×C₄	C₂₃×#102	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
246	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
247	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	—	8	2	no	—
251	C₈	C₂₃×#102	1	2	yes	—
252	C₈	C₂×D₈×C₂₃	1	1	yes	characteristic
253	C₈	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	C₂₃×#331	1	2	yes	—
260	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	8	2	no	—
265	C₄	C₂₃×#331	1	2	yes	—
266	C₄	C₂×C₂₃×#100	1	1	yes	characteristic
267	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
268	C₄	C₂₃×#509	1	1	yes	characteristic
269	C₄	C₂₃×#510	1	1	yes	characteristic
270	C₂	—	2	2	no	—
271	C₂	—	16	2	no	—
272	C₂	C₂×C₂₃×#331	1	1	yes	characteristic
273	C₂	C₂₃×#2139	1	1	yes	characteristic
274	C₂	C₂₃×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
prime radical	C₂₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2150	C₂	semidirect
3. C₂×C₂₃×#335	C₂	semidirect
4. C₂₃×#1362	C₂	semidirect
5. C₂₃×#1361	C₂	semidirect
87. #11930	C₂₃	direct (N × H)
118. #2150	C₂×C₂₃	semidirect
119. C₂×#335	C₂×C₂₃	semidirect
120. #1362	C₂×C₂₃	semidirect
121. #1361	C₂×C₂₃	semidirect
187. C₂₃	#11930	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3147	2
C₂₃×#2150	2
C₂×C₂₃×#335	2
C₂₃×#1362	2
C₂₃×#1361	2
#11930	23

Subgroups — C23×#11930 #19159556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11930 #19159556