Subgroups — C₂₃×#2329 #15669948

Order 2944.

528 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#391	C₂	1	2	yes	maximal, complemented
3	C₂₃×#389	C₂	1	2	yes	maximal, complemented
4	C₂₃×#387	C₂	1	2	yes	maximal, complemented
5	C₂₃×#387	C₂	1	2	yes	maximal, complemented
6	C₂₃×#385	C₂	1	2	yes	maximal
7	C₂₃×#380	C₂	1	1	yes	maximal, characteristic
8	C₄×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
9	C₂×C₂₃×#95	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×C₂²⋊Q₈	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
24	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
25	C₂₃×#95	C₂²	1	2	yes	—
26	C₂₃×#95	C₂²	1	2	yes	—
27	C₂₃×#95	C₂²	1	2	yes	—
28	C₂₃×#95	C₂²	1	2	yes	—
29	C₂₃×#95	C₂²	1	2	yes	—
30	C₂₃×#95	C₂²	1	2	yes	—
31	C₂₃×#95	C₂²	1	2	yes	—
32	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₂₃	—	4	1	no	—
34	C₂×Q₈×C₂₃	—	4	1	no	—
35	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	complemented
36	C₂²×C₄×C₂₃	—	2	2	no	—
37	C₂²×C₄×C₂₃	—	4	1	no	—
38	C₂²×C₄×C₂₃	—	4	1	no	—
39	C₂²×C₄×C₂₃	—	4	1	no	—
40	C₂²×C₄×C₂₃	—	4	1	no	—
41	C₂²×C₄×C₂₃	—	4	1	no	—
42	C₂²×C₄×C₂₃	—	4	1	no	—
43	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
44	C₂²×C₄×C₂₃	Q₈	1	4	yes	complemented
45	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
53	C₄⋊C₄×C₂₃	—	4	2	no	—
54	C₄⋊C₄×C₂₃	—	4	1	no	—
55	C₄⋊C₄×C₂₃	—	4	1	no	—
56	C₄⋊C₄×C₂₃	—	4	2	no	—
57	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	complemented
58	C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	complemented
59	C₂²⋊C₄×C₂₃	—	2	4	no	—
60	C₂²⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
61	C₄²×C₂₃	—	4	2	no	—
62	C₄²×C₂₃	—	4	2	no	—
63	C₂³×C₂₃	—	2	1	no	—
64	C₂³×C₂₃	—	2	1	no	—
65	C₂³×C₂₃	—	2	1	no	—
66	C₂³×C₂₃	—	2	1	no	—
67	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
68	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
69	C₂³×C₂₃	#41	1	2	yes	—
70	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
71	Q₈×C₂₃	—	4	4	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	2	1	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	1	no	—
80	C₂×C₄×C₂₃	—	2	1	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	4	4	no	—
92	C₂×C₄×C₂₃	—	4	4	no	—
93	C₂×C₄×C₂₃	—	4	4	no	—
94	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
95	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
96	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
99	#2329	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂²×C₂₃	—	2	2	no	—
101	C₂²×C₂₃	—	2	2	no	—
102	C₂²×C₂₃	—	2	2	no	—
103	C₂²×C₂₃	—	2	2	no	—
104	C₂²×C₂₃	—	2	2	no	—
105	C₂²×C₂₃	—	2	2	no	—
106	C₂²×C₂₃	C₄×Q₈	1	4	yes	—
107	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
108	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
110	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
114	C₄×C₂₃	—	2	2	no	—
115	C₄×C₂₃	—	2	2	no	—
116	C₄×C₂₃	—	2	4	no	—
117	C₄×C₂₃	—	2	4	no	—
118	C₄×C₂₃	—	2	4	no	—
119	C₄×C₂₃	—	4	2	no	—
120	C₄×C₂₃	—	4	2	no	—
121	C₄×C₂₃	—	4	2	no	—
122	C₂×C₂²⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
123	#391	C₂×C₂₃	1	2	yes	complemented
124	#389	C₂×C₂₃	1	2	yes	complemented
125	#387	C₂×C₂₃	1	2	yes	complemented
126	#387	C₂×C₂₃	1	2	yes	complemented
127	#385	C₂×C₂₃	1	2	yes	—
128	#380	C₂×C₂₃	1	1	yes	characteristic
129	C₄×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
130	C₂×#95	C₂×C₂₃	1	1	yes	characteristic
131	C₂×C₂₃	—	2	4	no	—
132	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
133	C₂×C₂₃	#518	1	1	yes	characteristic
134	C₂×C₂₃	#520	1	1	yes	characteristic
135	C₂×C₂₃	#538	1	1	yes	characteristic
136	C₂×C₂₃	#539	1	1	yes	characteristic
137	C₂×C₂₃	#543	1	1	yes	characteristic
138	C₂×C₂₃	#544	1	1	yes	characteristic
139	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
140	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
141	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
142	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
143	C₂²⋊Q₈	C₄×C₂₃	1	8	yes	complemented
144	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
147	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
148	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
150	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
152	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
153	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
154	#95	C₂²×C₂₃	1	2	yes	—
155	#95	C₂²×C₂₃	1	2	yes	—
156	#95	C₂²×C₂₃	1	2	yes	—
157	#95	C₂²×C₂₃	1	2	yes	—
158	#95	C₂²×C₂₃	1	2	yes	—
159	#95	C₂²×C₂₃	1	2	yes	—
160	#95	C₂²×C₂₃	1	2	yes	—
161	C₂₃	#2329	1	1	yes	characteristic, direct product factor, Sylow
162	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
163	C₂×Q₈	—	4	1	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂×Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	—	4	1	no	—
169	C₂²×C₄	—	4	1	no	—
170	C₂²×C₄	—	4	1	no	—
171	C₂²×C₄	—	4	1	no	—
172	C₂²×C₄	—	4	1	no	—
173	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	—
174	C₂²×C₄	Q₈×C₂₃	1	4	yes	complemented
175	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
176	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
177	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
178	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
180	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
181	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	complemented
188	C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	complemented
189	C₂²⋊C₄	—	2	4	no	—
190	C₂²⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
191	C₄²	—	4	2	no	—
192	C₄²	—	4	2	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
198	C₂³	C₂×Q₈×C₂₃	1	2	yes	—
199	C₂³	#41×C₂₃	1	2	yes	—
200	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
201	Q₈	—	4	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
225	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
226	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
227	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
228	C₂×C₄	C₂×Q₈×C₂₃	1	4	yes	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	C₄×Q₈×C₂₃	1	4	yes	—
236	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
237	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
238	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
239	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
240	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
241	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
242	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	4	no	—
246	C₄	—	2	4	no	—
247	C₄	—	2	4	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₂	—	2	4	no	—
252	C₂	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
253	C₂	C₂₃×#518	1	1	yes	characteristic
254	C₂	C₂₃×#520	1	1	yes	characteristic
255	C₂	C₂₃×#538	1	1	yes	characteristic
256	C₂	C₂₃×#539	1	1	yes	characteristic
257	C₂	C₂₃×#543	1	1	yes	characteristic
258	C₂	C₂₃×#544	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#391	C₂	semidirect
3. C₂₃×#389	C₂	semidirect
4. C₂₃×#387	C₂	semidirect
5. C₂₃×#387	C₂	semidirect
14. C₂₃×C₂²⋊Q₈	C₄	semidirect
35. C₂×Q₈×C₂₃	C₂×C₄	semidirect
44. C₂²×C₄×C₂₃	Q₈	semidirect
57. C₄⋊C₄×C₂₃	C₂×C₄	semidirect
58. C₄⋊C₄×C₂₃	C₂×C₄	semidirect
99. #2329	C₂₃	direct (N × H)
123. #391	C₂×C₂₃	semidirect
124. #389	C₂×C₂₃	semidirect
125. #387	C₂×C₂₃	semidirect
126. #387	C₂×C₂₃	semidirect
143. C₂²⋊Q₈	C₄×C₂₃	semidirect
161. C₂₃	#2329	direct (N × H)
165. C₂×Q₈	C₂×C₄×C₂₃	semidirect
174. C₂²×C₄	Q₈×C₂₃	semidirect
187. C₄⋊C₄	C₂×C₄×C₂₃	semidirect
188. C₄⋊C₄	C₂×C₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×C₂²⋊Q₈	2
C₂₃×#391	2
C₂₃×#389	2
C₂₃×#387	2
C₂₃×#385	2
C₂₃×#380	2
C₄×C₂²⋊C₄×C₂₃	2
C₂×C₂₃×#95	2
#2329	23

Subgroups — C23×#2329 #15669948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2329 #15669948