Subgroups — C₂₃×#9698 #19157324

Order 5888.

622 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2858	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1842	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#332	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#331	C₂	1	2	yes	maximal
6	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×#124	—	2	2	no	—
9	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₂₃	—	2	1	no	—
13	C₂₃×#390	—	2	1	no	—
14	C₂₃×#390	—	2	1	no	—
15	C₂₃×#390	C₄	1	4	yes	—
16	C₄³×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#336	—	2	1	no	—
18	C₂₃×#332	—	2	4	no	—
19	C₂₃×#331	—	2	4	no	—
20	C₂²×Q₈×C₂₃	—	2	1	no	—
21	C₂²×D₄×C₂₃	—	2	1	no	—
22	C₂²×D₄×C₂₃	—	2	1	no	—
23	C₂²×D₄×C₂₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂²×C₈×C₂₃	—	2	2	no	—
26	C₂₃×C₄⋊Q₈	—	2	2	no	—
27	C₂₃×C₄⋊Q₈	—	2	2	no	—
28	C₂₃×#127	—	2	2	no	—
29	C₂₃×#127	—	2	2	no	—
30	C₂₃×#124	—	4	4	no	—
31	C₂₃×#124	—	4	4	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
35	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
36	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	D₄	1	2	yes	—
40	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
41	C₂₃×C₄⋊C₈	—	2	2	no	—
42	C₂₃×C₄⋊C₈	—	2	2	no	—
43	C₂₃×C₄⋊C₈	—	2	2	no	—
44	C₂₃×C₄⋊C₈	—	4	4	no	—
45	C₂₃×C₄⋊C₈	D₄	1	4	yes	—
46	C₄×C₈×C₂₃	—	4	2	no	—
47	C₂⁴×C₂₃	—	2	2	no	—
48	C₂×Q₈×C₂₃	—	4	1	no	—
49	C₂×Q₈×C₂₃	—	4	1	no	—
50	C₂×Q₈×C₂₃	—	4	2	no	—
51	C₂×Q₈×C₂₃	—	4	2	no	—
52	C₂×D₄×C₂₃	—	2	4	no	—
53	C₂×D₄×C₂₃	—	4	4	no	—
54	C₂×D₄×C₂₃	—	4	2	no	—
55	C₂×D₄×C₂₃	—	4	1	no	—
56	C₂×D₄×C₂₃	—	4	1	no	—
57	C₂×D₄×C₂₃	—	4	2	no	—
58	C₂×D₄×C₂₃	—	4	2	no	—
59	C₂×D₄×C₂₃	—	4	2	no	—
60	C₂×D₄×C₂₃	—	4	1	no	—
61	C₂×D₄×C₂₃	—	4	1	no	—
62	C₂²×C₄×C₂₃	—	2	2	no	—
63	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₂₃	—	2	4	no	—
69	C₂×C₈×C₂₃	—	4	2	no	—
70	C₂×C₈×C₂₃	—	4	2	no	—
71	C₂×C₈×C₂₃	—	4	2	no	—
72	C₂×C₈×C₂₃	—	4	4	no	—
73	C₄⋊C₄×C₂₃	—	2	4	no	—
74	C₄⋊C₄×C₂₃	—	4	2	no	—
75	C₂²⋊C₄×C₂₃	—	4	8	no	—
76	C₂²⋊C₄×C₂₃	—	4	8	no	—
77	C₄²×C₂₃	—	2	2	no	—
78	C₄²×C₂₃	—	2	1	no	—
79	C₄²×C₂₃	—	2	1	no	—
80	C₄²×C₂₃	—	4	2	no	—
81	C₄²×C₂₃	D₈	1	4	yes	—
82	C₄²×C₂₃	SD₁₆	1	4	yes	—
83	C₄²×C₂₃	C₂×D₄	1	2	yes	—
84	C₄²×C₂₃	#41	1	2	yes	—
85	#9698	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₂₃	—	4	2	no	—
87	C₂³×C₂₃	—	4	4	no	—
88	C₂³×C₂₃	—	4	4	no	—
89	C₂³×C₂₃	—	4	2	no	—
90	C₂³×C₂₃	—	4	2	no	—
91	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	Q₈×C₂₃	—	8	2	no	—
93	Q₈×C₂₃	—	8	1	no	—
94	Q₈×C₂₃	—	8	1	no	—
95	D₄×C₂₃	—	8	4	no	—
96	D₄×C₂₃	—	8	2	no	—
97	D₄×C₂₃	—	8	2	no	—
98	D₄×C₂₃	—	8	2	no	—
99	D₄×C₂₃	—	8	1	no	—
100	D₄×C₂₃	—	8	1	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	4	no	—
108	C₂×C₄×C₂₃	#102	1	4	yes	—
109	C₂×C₄×C₂₃	#102	1	4	yes	—
110	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₂₃	#121	1	2	yes	—
112	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄×C₂₃	#135	1	2	yes	—
117	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
119	C₈×C₂₃	—	4	4	no	—
120	C₈×C₂₃	—	8	2	no	—
121	#2858	C₂×C₂₃	1	1	yes	characteristic
122	#1842	C₂×C₂₃	1	1	yes	characteristic, complemented
123	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
124	C₂×#332	C₂×C₂₃	1	2	yes	complemented
125	C₂×#331	C₂×C₂₃	1	2	yes	—
126	C₂²×C₂₃	—	8	4	no	—
127	C₂²×C₂₃	—	8	4	no	—
128	C₂²×C₂₃	—	8	2	no	—
129	C₂²×C₂₃	—	8	2	no	—
130	C₂²×C₂₃	—	8	2	no	—
131	C₂²×C₂₃	#390	1	1	yes	characteristic
132	C₂²×C₂₃	#413	1	2	yes	—
133	C₂²×C₂₃	C₂×#102	1	2	yes	—
134	C₂²×C₂₃	#416	1	1	yes	characteristic
135	C₂²×C₂₃	#417	1	1	yes	characteristic
136	C₄×C₂₃	—	2	4	no	—
137	C₄×C₂₃	—	2	4	no	—
138	C₄×C₂₃	—	2	2	no	—
139	C₄×C₂₃	—	8	2	no	—
140	C₄×C₂₃	#463	1	2	yes	—
141	C₄×C₂₃	#464	1	2	yes	—
142	C₄×C₂₃	#486	1	2	yes	—
143	C₄×C₂₃	#492	1	1	yes	characteristic
144	C₄×C₂₃	#493	1	1	yes	characteristic
145	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
147	C₂×#124	—	2	2	no	—
148	C₂×C₄⋊C₈	—	2	2	no	—
149	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
150	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
151	C₂×C₄×C₈	—	2	1	no	—
152	#390	—	2	1	no	—
153	#390	—	2	1	no	—
154	#390	C₄×C₂₃	1	4	yes	—
155	C₄³	C₂²×C₂₃	1	1	yes	characteristic
156	#336	—	2	1	no	—
157	#332	—	2	4	no	—
158	#331	—	2	4	no	—
159	C₂×C₂₃	—	16	2	no	—
160	C₂×C₂₃	#1540	1	2	yes	—
161	C₂×C₂₃	#1547	1	2	yes	—
162	C₂×C₂₃	#1957	1	1	yes	characteristic
163	C₂×C₂₃	#1967	1	1	yes	characteristic
164	C₂×C₂₃	#1983	1	1	yes	characteristic
165	C₂²×Q₈	—	2	1	no	—
166	C₂²×D₄	—	2	1	no	—
167	C₂²×D₄	—	2	1	no	—
168	C₂²×D₄	C₂×C₄×C₂₃	1	2	yes	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₄⋊Q₈	—	2	2	no	—
172	C₄⋊Q₈	—	2	2	no	—
173	#127	—	2	2	no	—
174	#127	—	2	2	no	—
175	#124	—	4	4	no	—
176	#124	—	4	4	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
179	C₂×C₂²⋊C₄	—	2	4	no	—
180	C₂×C₂²⋊C₄	—	2	4	no	—
181	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
182	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
184	C₂×C₄²	D₄×C₂₃	1	2	yes	—
185	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	2	2	no	—
189	C₄⋊C₈	—	4	4	no	—
190	C₄⋊C₈	D₄×C₂₃	1	4	yes	—
191	C₄×C₈	—	4	2	no	—
192	C₂₃	#9698	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁴	—	2	2	no	—
194	C₂×Q₈	—	4	1	no	—
195	C₂×Q₈	—	4	1	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	—	4	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	4	4	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	1	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
210	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	—	4	8	no	—
222	C₂²⋊C₄	—	4	8	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	4	2	no	—
227	C₄²	D₈×C₂₃	1	4	yes	—
228	C₄²	SD₁₆×C₂₃	1	4	yes	—
229	C₄²	C₂×D₄×C₂₃	1	2	yes	—
230	C₄²	#41×C₂₃	1	2	yes	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	4	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	2	no	—
235	C₂³	—	4	2	no	—
236	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
237	Q₈	—	8	2	no	—
238	Q₈	—	8	1	no	—
239	Q₈	—	8	1	no	—
240	D₄	—	8	4	no	—
241	D₄	—	8	2	no	—
242	D₄	—	8	2	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	1	no	—
245	D₄	—	8	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	C₂₃×#102	1	4	yes	—
254	C₂×C₄	C₂₃×#102	1	4	yes	—
255	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
256	C₂×C₄	C₂₃×#121	1	2	yes	—
257	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
258	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
259	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
260	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
261	C₂×C₄	C₂₃×#135	1	2	yes	—
262	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
263	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
264	C₈	—	4	4	no	—
265	C₈	—	8	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	8	2	no	—
268	C₂²	—	8	2	no	—
269	C₂²	—	8	4	no	—
270	C₂²	—	8	4	no	—
271	C₂²	C₂₃×#390	1	1	yes	characteristic
272	C₂²	C₂₃×#413	1	2	yes	—
273	C₂²	C₂×C₂₃×#102	1	2	yes	—
274	C₂²	C₂₃×#416	1	1	yes	characteristic
275	C₂²	C₂₃×#417	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	4	no	—
279	C₄	—	8	2	no	—
280	C₄	C₂₃×#463	1	2	yes	—
281	C₄	C₂₃×#464	1	2	yes	—
282	C₄	C₂₃×#486	1	2	yes	—
283	C₄	C₂₃×#492	1	1	yes	characteristic
284	C₄	C₂₃×#493	1	1	yes	characteristic
285	C₂	—	16	2	no	—
286	C₂	C₂₃×#1540	1	2	yes	—
287	C₂	C₂₃×#1547	1	2	yes	—
288	C₂	C₂₃×#1957	1	1	yes	characteristic
289	C₂	C₂₃×#1967	1	1	yes	characteristic
290	C₂	C₂₃×#1983	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1842	C₂	semidirect
3. C₄×C₂₃×C₄⋊C₈	C₂	semidirect
4. C₂×C₂₃×#332	C₂	semidirect
85. #9698	C₂₃	direct (N × H)
122. #1842	C₂×C₂₃	semidirect
123. C₄×C₄⋊C₈	C₂×C₂₃	semidirect
124. C₂×#332	C₂×C₂₃	semidirect
192. C₂₃	#9698	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×#2858	2
C₂₃×#1842	2
C₄×C₂₃×C₄⋊C₈	2
C₂×C₂₃×#332	2
C₂×C₂₃×#331	2
#9698	23

Subgroups — C23×#9698 #19157324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9698 #19157324