Subgroups — C₂₃×#7682 #19155308

Order 5888.

448 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2862	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1272	C₂	1	4	yes	maximal
5	C₂×C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
8	C₂×C₂₃×#97	—	2	1	no	—
9	C₂×C₄×C₈×C₂₃	—	2	1	no	—
10	C₂₃×#391	—	2	1	no	—
11	C₂₃×#391	—	2	1	no	—
12	C₂₃×#391	C₄	1	2	yes	—
13	C₂₃×#391	C₂²	1	2	yes	—
14	C₂₃×#382	—	2	1	no	—
15	C₂₃×#382	—	2	1	no	—
16	C₄³×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#336	—	2	1	no	—
18	C₂₃×#336	C₂²	1	2	yes	—
19	C₂₃×#336	C₂²	1	2	yes	—
20	C₂²×Q₈×C₂₃	—	2	1	no	—
21	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₂₃	—	2	2	no	—
23	C₂²×C₈×C₂₃	—	2	2	no	—
24	C₂₃×C₄⋊Q₈	—	4	2	no	—
25	C₂₃×C₄⋊Q₈	—	4	2	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
28	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
37	C₂₃×C₄⋊C₈	—	2	4	no	—
38	C₂₃×#97	—	4	2	no	—
39	C₄×C₈×C₂₃	—	2	4	no	—
40	C₂₃×#95	—	2	2	no	—
41	C₂₃×#95	—	2	2	no	—
42	C₂₃×#95	—	2	2	no	—
43	C₂₃×#95	—	2	2	no	—
44	C₂×Q₈×C₂₃	—	2	4	no	—
45	C₂×Q₈×C₂₃	—	4	2	no	—
46	C₂×Q₈×C₂₃	—	4	2	no	—
47	C₂×Q₈×C₂₃	—	4	2	no	—
48	C₂×Q₈×C₂₃	—	4	2	no	—
49	C₂×Q₈×C₂₃	—	4	2	no	—
50	C₂²×C₄×C₂₃	—	2	2	no	—
51	C₂²×C₄×C₂₃	—	2	2	no	—
52	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₂₃	—	2	4	no	—
60	C₂×C₈×C₂₃	—	2	4	no	—
61	C₂×C₈×C₂₃	—	4	2	no	—
62	C₂×C₈×C₂₃	—	4	2	no	—
63	C₂×C₈×C₂₃	—	4	2	no	—
64	C₂×C₈×C₂₃	—	4	2	no	—
65	C₄⋊C₄×C₂₃	—	4	2	no	—
66	C₄⋊C₄×C₂₃	—	4	2	no	—
67	C₄⋊C₄×C₂₃	—	4	2	no	—
68	C₄⋊C₄×C₂₃	—	4	2	no	—
69	C₄²×C₂₃	—	2	2	no	—
70	C₄²×C₂₃	—	2	2	no	—
71	C₄²×C₂₃	—	2	2	no	—
72	C₄²×C₂₃	—	2	2	no	—
73	C₄²×C₂₃	—	2	2	no	—
74	C₄²×C₂₃	—	4	1	no	—
75	C₄²×C₂₃	SD₁₆	1	2	yes	—
76	C₄²×C₂₃	Q₁₆	1	2	yes	—
77	#7682	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	Q₈×C₂₃	—	8	4	no	—
80	Q₈×C₂₃	—	8	4	no	—
81	C₂×C₄×C₂₃	—	2	1	no	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	2	1	no	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	1	no	—
92	C₂×C₄×C₂₃	—	2	1	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	4	2	no	—
95	C₂×C₄×C₂₃	—	4	2	no	—
96	C₂×C₄×C₂₃	—	4	2	no	—
97	C₂×C₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	#100	1	2	yes	—
101	C₂×C₄×C₂₃	#103	1	2	yes	—
102	C₂×C₄×C₂₃	#103	1	2	yes	—
103	C₂×C₄×C₂₃	#104	1	2	yes	—
104	C₂×C₄×C₂₃	#104	1	2	yes	—
105	C₂×C₄×C₂₃	#131	1	2	yes	—
106	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₂₃	#137	1	2	yes	—
109	C₈×C₂₃	—	4	4	no	—
110	C₈×C₂₃	—	4	4	no	—
111	#2862	C₂×C₂₃	1	1	yes	characteristic
112	#1767	C₂×C₂₃	1	1	yes	characteristic
113	#1744	C₂×C₂₃	1	1	yes	characteristic
114	#1272	C₂×C₂₃	1	4	yes	—
115	C₂²×C₂₃	#408	1	1	yes	characteristic
116	C₂²×C₂₃	#410	1	1	yes	characteristic
117	C₂²×C₂₃	C₂×#100	1	1	yes	characteristic
118	C₂²×C₂₃	C₂×#103	1	1	yes	characteristic
119	C₂²×C₂₃	#419	1	1	yes	characteristic
120	C₂²×C₂₃	C₂×#104	1	1	yes	characteristic
121	C₂²×C₂₃	C₂×#104	1	1	yes	characteristic
122	C₄×C₂₃	—	2	2	no	—
123	C₄×C₂₃	—	2	2	no	—
124	C₄×C₂₃	—	2	2	no	—
125	C₄×C₂₃	—	2	1	no	—
126	C₄×C₂₃	—	2	1	no	—
127	C₄×C₂₃	—	2	2	no	—
128	C₄×C₂₃	—	4	1	no	—
129	C₄×C₂₃	—	8	2	no	—
130	C₄×C₂₃	—	8	2	no	—
131	C₄×C₂₃	#329	1	4	yes	—
132	C₂×C₄⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
133	C₂×C₄⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
134	C₂×C₄⋊C₈	—	2	1	no	—
135	C₂×#97	—	2	1	no	—
136	C₂×C₄×C₈	—	2	1	no	—
137	#391	—	2	1	no	—
138	#391	—	2	1	no	—
139	#391	C₄×C₂₃	1	2	yes	—
140	#391	C₂²×C₂₃	1	2	yes	—
141	#382	—	2	1	no	—
142	#382	—	2	1	no	—
143	C₄³	C₂²×C₂₃	1	1	yes	characteristic
144	#336	—	2	1	no	—
145	#336	C₂²×C₂₃	1	2	yes	—
146	#336	C₂²×C₂₃	1	2	yes	—
147	C₂×C₂₃	#1469	1	1	yes	characteristic
148	C₂×C₂₃	#1485	1	1	yes	characteristic
149	C₂×C₂₃	#1493	1	1	yes	characteristic
150	C₂×C₂₃	#1503	1	1	yes	characteristic
151	C₂×C₂₃	#1506	1	1	yes	characteristic
152	C₂×C₂₃	C₂×#329	1	1	yes	characteristic
153	C₂×C₂₃	#1548	1	1	yes	characteristic
154	C₂²×Q₈	—	2	1	no	—
155	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₄⋊Q₈	—	4	2	no	—
159	C₄⋊Q₈	—	4	2	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
164	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
165	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
166	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
167	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
168	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
169	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
170	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
171	C₄⋊C₈	—	2	4	no	—
172	#97	—	4	2	no	—
173	C₄×C₈	—	2	4	no	—
174	#95	—	2	2	no	—
175	#95	—	2	2	no	—
176	#95	—	2	2	no	—
177	#95	—	2	2	no	—
178	C₂₃	#7682	1	1	yes	characteristic, direct product factor, Sylow
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
188	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	4	1	no	—
210	C₄²	SD₁₆×C₂₃	1	2	yes	—
211	C₄²	Q₁₆×C₂₃	1	2	yes	—
212	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
213	Q₈	—	8	4	no	—
214	Q₈	—	8	4	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	C₂₃×#100	1	2	yes	—
235	C₂×C₄	C₂₃×#103	1	2	yes	—
236	C₂×C₄	C₂₃×#103	1	2	yes	—
237	C₂×C₄	C₂₃×#104	1	2	yes	—
238	C₂×C₄	C₂₃×#104	1	2	yes	—
239	C₂×C₄	C₂₃×#131	1	2	yes	—
240	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
241	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
242	C₂×C₄	C₂₃×#137	1	2	yes	—
243	C₈	—	4	4	no	—
244	C₈	—	4	4	no	—
245	C₂²	C₂₃×#408	1	1	yes	characteristic
246	C₂²	C₂₃×#410	1	1	yes	characteristic
247	C₂²	C₂×C₂₃×#100	1	1	yes	characteristic
248	C₂²	C₂×C₂₃×#103	1	1	yes	characteristic
249	C₂²	C₂₃×#419	1	1	yes	characteristic
250	C₂²	C₂×C₂₃×#104	1	1	yes	characteristic
251	C₂²	C₂×C₂₃×#104	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	8	2	no	—
260	C₄	—	8	2	no	—
261	C₄	C₂₃×#329	1	4	yes	—
262	C₂	C₂₃×#1469	1	1	yes	characteristic
263	C₂	C₂₃×#1485	1	1	yes	characteristic
264	C₂	C₂₃×#1493	1	1	yes	characteristic
265	C₂	C₂₃×#1503	1	1	yes	characteristic
266	C₂	C₂₃×#1506	1	1	yes	characteristic
267	C₂	C₂×C₂₃×#329	1	1	yes	characteristic
268	C₂	C₂₃×#1548	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
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
77. #7682	C₂₃	direct (N × H)
178. C₂₃	#7682	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×#2862	2
C₂₃×#1767	2
C₂₃×#1744	2
C₂₃×#1272	2
#7682	23

Subgroups — C23×#7682 #19155308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7682 #19155308