Subgroups — C₂₃×#9707 #19157333

Order 5888.

610 conjugacy classes of subgroups.

The subgroup lattice has 284 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	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#333	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	C₄	1	4	yes	—
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×C₂₃×#97	—	2	1	no	—
13	C₂₃×#390	—	2	1	no	—
14	C₂₃×#390	—	2	1	no	—
15	C₂₃×#390	—	2	2	no	—
16	C₄³×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#336	—	2	1	no	—
18	C₂₃×#333	—	2	4	no	—
19	C₂₃×#331	—	2	4	no	—
20	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
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	8	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
33	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
35	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	D₄	1	2	yes	—
39	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×C₄⋊C₈	—	2	2	no	—
41	C₂₃×C₄⋊C₈	—	2	2	no	—
42	C₂₃×C₄⋊C₈	—	2	2	no	—
43	C₂₃×C₄⋊C₈	—	4	4	no	—
44	C₂₃×C₄⋊C₈	D₄	1	4	yes	—
45	C₂₃×#97	—	4	2	no	—
46	C₂⁴×C₂₃	—	2	2	no	—
47	C₂×Q₈×C₂₃	—	2	4	no	—
48	C₂×Q₈×C₂₃	—	4	4	no	—
49	C₂×D₄×C₂₃	—	2	4	no	—
50	C₂×D₄×C₂₃	—	4	4	no	—
51	C₂×D₄×C₂₃	—	4	2	no	—
52	C₂×D₄×C₂₃	—	4	1	no	—
53	C₂×D₄×C₂₃	—	4	1	no	—
54	C₂×D₄×C₂₃	—	4	2	no	—
55	C₂×D₄×C₂₃	—	4	1	no	—
56	C₂×D₄×C₂₃	—	4	2	no	—
57	C₂×D₄×C₂₃	—	4	1	no	—
58	C₂×D₄×C₂₃	—	4	2	no	—
59	C₂²×C₄×C₂₃	—	2	2	no	—
60	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₂₃	—	2	4	no	—
66	C₂×C₈×C₂₃	—	4	2	no	—
67	C₂×C₈×C₂₃	—	4	2	no	—
68	C₂×C₈×C₂₃	—	4	2	no	—
69	C₂×C₈×C₂₃	—	4	4	no	—
70	C₄⋊C₄×C₂₃	—	4	2	no	—
71	C₄⋊C₄×C₂₃	—	4	1	no	—
72	C₄⋊C₄×C₂₃	—	4	1	no	—
73	C₂²⋊C₄×C₂₃	—	4	8	no	—
74	C₂²⋊C₄×C₂₃	—	4	8	no	—
75	C₄²×C₂₃	—	2	4	no	—
76	C₄²×C₂₃	—	2	2	no	—
77	C₄²×C₂₃	—	2	2	no	—
78	C₄²×C₂₃	—	2	1	no	—
79	C₄²×C₂₃	—	2	2	no	—
80	C₄²×C₂₃	—	2	1	no	—
81	C₄²×C₂₃	C₂×D₄	1	2	yes	—
82	C₄²×C₂₃	#41	1	2	yes	—
83	#9707	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₂₃	—	4	2	no	—
85	C₂³×C₂₃	—	4	4	no	—
86	C₂³×C₂₃	—	4	2	no	—
87	C₂³×C₂₃	—	4	4	no	—
88	C₂³×C₂₃	—	4	2	no	—
89	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	Q₈×C₂₃	—	8	4	no	—
91	D₄×C₂₃	—	8	4	no	—
92	D₄×C₂₃	—	8	2	no	—
93	D₄×C₂₃	—	8	2	no	—
94	D₄×C₂₃	—	8	2	no	—
95	D₄×C₂₃	—	8	1	no	—
96	D₄×C₂₃	—	8	1	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	4	4	no	—
105	C₂×C₄×C₂₃	#100	1	4	yes	—
106	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₂₃	#121	1	2	yes	—
108	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
109	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
110	C₂×C₄×C₂₃	#135	1	2	yes	—
111	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
113	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
114	C₂×C₄×C₂₃	#137	1	2	yes	—
115	C₈×C₂₃	—	4	4	no	—
116	C₈×C₂₃	—	8	2	no	—
117	#2858	C₂×C₂₃	1	1	yes	characteristic
118	#1842	C₂×C₂₃	1	1	yes	characteristic, complemented
119	#1767	C₂×C₂₃	1	1	yes	characteristic, complemented
120	C₂×#333	C₂×C₂₃	1	2	yes	complemented
121	C₂×#331	C₂×C₂₃	1	2	yes	—
122	C₂²×C₂₃	—	8	4	no	—
123	C₂²×C₂₃	—	8	4	no	—
124	C₂²×C₂₃	—	8	2	no	—
125	C₂²×C₂₃	—	8	2	no	—
126	C₂²×C₂₃	—	8	2	no	—
127	C₂²×C₂₃	#390	1	1	yes	characteristic
128	C₂²×C₂₃	C₂×#100	1	2	yes	—
129	C₂²×C₂₃	#416	1	1	yes	characteristic
130	C₂²×C₂₃	#417	1	2	yes	—
131	C₂²×C₂₃	#417	1	1	yes	characteristic
132	C₄×C₂₃	—	2	4	no	—
133	C₄×C₂₃	—	2	2	no	—
134	C₄×C₂₃	—	4	2	no	—
135	C₄×C₂₃	—	8	2	no	—
136	C₄×C₂₃	#463	1	2	yes	—
137	C₄×C₂₃	#464	1	2	yes	—
138	C₄×C₂₃	#489	1	2	yes	—
139	C₄×C₂₃	#496	1	1	yes	characteristic
140	C₄×C₂₃	#497	1	1	yes	characteristic
141	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
143	C₂×#124	C₄×C₂₃	1	4	yes	—
144	C₂×C₄⋊C₈	—	2	2	no	—
145	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
146	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
147	C₂×#97	—	2	1	no	—
148	#390	—	2	1	no	—
149	#390	—	2	1	no	—
150	#390	—	2	2	no	—
151	C₄³	C₂²×C₂₃	1	1	yes	characteristic
152	#336	—	2	1	no	—
153	#333	—	2	4	no	—
154	#331	—	2	4	no	—
155	C₂×C₂₃	—	16	2	no	—
156	C₂×C₂₃	#1551	1	2	yes	—
157	C₂×C₂₃	#1560	1	2	yes	—
158	C₂×C₂₃	#1959	1	1	yes	characteristic
159	C₂×C₂₃	#1965	1	1	yes	characteristic
160	C₂×C₂₃	#1983	1	1	yes	characteristic
161	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
162	C₂²×D₄	—	2	1	no	—
163	C₂²×D₄	—	2	1	no	—
164	C₂²×D₄	C₂×C₄×C₂₃	1	2	yes	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₄⋊Q₈	—	2	2	no	—
168	C₄⋊Q₈	—	2	2	no	—
169	#127	—	2	2	no	—
170	#127	—	2	2	no	—
171	#124	—	4	8	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₂²⋊C₄	—	2	4	no	—
175	C₂×C₂²⋊C₄	—	2	4	no	—
176	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
177	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
178	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
179	C₂×C₄²	D₄×C₂₃	1	2	yes	—
180	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	—	2	2	no	—
184	C₄⋊C₈	—	4	4	no	—
185	C₄⋊C₈	D₄×C₂₃	1	4	yes	—
186	#97	—	4	2	no	—
187	C₂₃	#9707	1	1	yes	characteristic, direct product factor, Sylow
188	C₂⁴	—	2	2	no	—
189	C₂×Q₈	—	2	4	no	—
190	C₂×Q₈	—	4	4	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	—	4	4	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂×D₄	—	4	2	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
203	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	4	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₂²⋊C₄	—	4	8	no	—
216	C₂²⋊C₄	—	4	8	no	—
217	C₄²	—	2	4	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	1	no	—
223	C₄²	C₂×D₄×C₂₃	1	2	yes	—
224	C₄²	#41×C₂₃	1	2	yes	—
225	C₂³	—	4	2	no	—
226	C₂³	—	4	4	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	4	4	no	—
229	C₂³	—	4	2	no	—
230	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
231	Q₈	—	8	4	no	—
232	D₄	—	8	4	no	—
233	D₄	—	8	2	no	—
234	D₄	—	8	2	no	—
235	D₄	—	8	2	no	—
236	D₄	—	8	1	no	—
237	D₄	—	8	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	C₂₃×#100	1	4	yes	—
247	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
248	C₂×C₄	C₂₃×#121	1	2	yes	—
249	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
250	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
251	C₂×C₄	C₂₃×#135	1	2	yes	—
252	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
254	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
255	C₂×C₄	C₂₃×#137	1	2	yes	—
256	C₈	—	4	4	no	—
257	C₈	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	4	no	—
262	C₂²	—	8	4	no	—
263	C₂²	C₂₃×#390	1	1	yes	characteristic
264	C₂²	C₂×C₂₃×#100	1	2	yes	—
265	C₂²	C₂₃×#416	1	1	yes	characteristic
266	C₂²	C₂₃×#417	1	2	yes	—
267	C₂²	C₂₃×#417	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	4	no	—
270	C₄	—	4	2	no	—
271	C₄	—	8	2	no	—
272	C₄	C₂₃×#463	1	2	yes	—
273	C₄	C₂₃×#464	1	2	yes	—
274	C₄	C₂₃×#489	1	2	yes	—
275	C₄	C₂₃×#496	1	1	yes	characteristic
276	C₄	C₂₃×#497	1	1	yes	characteristic
277	C₂	—	16	2	no	—
278	C₂	C₂₃×#1551	1	2	yes	—
279	C₂	C₂₃×#1560	1	2	yes	—
280	C₂	C₂₃×#1959	1	1	yes	characteristic
281	C₂	C₂₃×#1965	1	1	yes	characteristic
282	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. C₂₃×#1767	C₂	semidirect
4. C₂×C₂₃×#333	C₂	semidirect
83. #9707	C₂₃	direct (N × H)
118. #1842	C₂×C₂₃	semidirect
119. #1767	C₂×C₂₃	semidirect
120. C₂×#333	C₂×C₂₃	semidirect
187. C₂₃	#9707	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
C₂₃×#1767	2
C₂×C₂₃×#333	2
C₂×C₂₃×#331	2
#9707	23

Subgroups — C23×#9707 #19157333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9707 #19157333