Subgroups — C₂₃×#2732 #15670351

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	2	yes	maximal, complemented
2	C₂₃×#399	C₂	1	2	yes	maximal, complemented
3	C₂₃×#398	C₂	1	2	yes	maximal, complemented
4	C₂₃×#396	C₂	1	2	yes	maximal, complemented
5	C₂₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#390	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#386	C₂	1	2	yes	maximal
9	C₂₃×#385	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
13	C₂₃×C₂²⋊Q₈	—	2	8	no	—
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
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	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
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	2	yes	—
33	C₂×Q₈×C₂₃	—	4	1	no	—
34	C₂×Q₈×C₂₃	—	4	1	no	—
35	C₂×Q₈×C₂₃	D₄	1	4	yes	complemented
36	C₂×D₄×C₂₃	—	4	1	no	—
37	C₂×D₄×C₂₃	—	4	1	no	—
38	C₂×D₄×C₂₃	Q₈	1	4	yes	complemented
39	C₂²×C₄×C₂₃	—	2	4	no	—
40	C₂²×C₄×C₂₃	—	4	2	no	—
41	C₂²×C₄×C₂₃	—	4	2	no	—
42	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₂₃	—	2	4	no	—
51	C₄⋊C₄×C₂₃	—	2	4	no	—
52	C₄⋊C₄×C₂₃	—	2	4	no	—
53	C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₄⋊C₄×C₂₃	—	4	1	no	—
55	C₄⋊C₄×C₂₃	—	4	1	no	—
56	C₄⋊C₄×C₂₃	—	4	1	no	—
57	C₄⋊C₄×C₂₃	—	4	2	no	—
58	C₂²⋊C₄×C₂₃	—	2	4	no	—
59	C₂²⋊C₄×C₂₃	—	2	4	no	—
60	C₂²⋊C₄×C₂₃	—	2	4	no	—
61	C₂²⋊C₄×C₂₃	—	4	2	no	—
62	C₂²⋊C₄×C₂₃	—	4	2	no	—
63	C₄²×C₂₃	—	4	1	no	—
64	C₂³×C₂₃	—	2	2	no	—
65	C₂³×C₂₃	—	2	2	no	—
66	C₂³×C₂₃	—	2	2	no	—
67	C₂³×C₂₃	—	2	2	no	—
68	C₂³×C₂₃	—	2	2	no	—
69	C₂³×C₂₃	—	2	2	no	—
70	C₂³×C₂₃	C₂×Q₈	1	4	yes	—
71	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
72	Q₈×C₂₃	—	4	4	no	—
73	D₄×C₂₃	—	4	4	no	—
74	C₂×C₄×C₂₃	—	2	1	no	—
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	2	1	no	—
78	C₂×C₄×C₂₃	—	2	1	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	1	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	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	4	8	no	—
95	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
97	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
98	C₂×C₄×C₂₃	#41	1	2	yes	—
99	#2732	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂²×C₂₃	—	2	4	no	—
101	C₂²×C₂₃	—	2	4	no	—
102	C₂²×C₂₃	—	2	4	no	—
103	C₂²×C₂₃	—	4	2	no	—
104	C₂²×C₂₃	—	4	2	no	—
105	C₂²×C₂₃	—	4	2	no	—
106	C₂²×C₂₃	—	4	2	no	—
107	C₂²×C₂₃	C₂²×D₄	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	4	no	—
115	C₄×C₂₃	—	2	2	no	—
116	C₄×C₂₃	—	2	2	no	—
117	C₄×C₂₃	—	2	2	no	—
118	C₄×C₂₃	—	4	2	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	2	yes	complemented
123	#399	C₂×C₂₃	1	2	yes	complemented
124	#398	C₂×C₂₃	1	2	yes	complemented
125	#396	C₂×C₂₃	1	2	yes	complemented
126	#391	C₂×C₂₃	1	1	yes	characteristic, complemented
127	#390	C₂×C₂₃	1	1	yes	characteristic
128	#389	C₂×C₂₃	1	1	yes	characteristic, complemented
129	#386	C₂×C₂₃	1	2	yes	—
130	#385	C₂×C₂₃	1	2	yes	complemented
131	C₂×C₂₃	—	4	4	no	—
132	C₂×C₂₃	#534	1	1	yes	characteristic
133	C₂×C₂₃	#539	1	1	yes	characteristic
134	C₂×C₂₃	#543	1	1	yes	characteristic
135	C₂×C₂₃	D₄×Q₈	1	1	yes	characteristic
136	C₂×C₂₃	#550	1	1	yes	characteristic
137	C₂×C₂₃	#554	1	1	yes	characteristic
138	C₂×C₂₃	#562	1	1	yes	characteristic
139	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
140	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
141	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
142	C₂²⋊Q₈	—	2	8	no	—
143	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
144	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
145	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
154	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
155	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
156	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
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	#2732	1	1	yes	characteristic, direct product factor, Sylow
162	C₂⁴	C₂³×C₂₃	1	2	yes	—
163	C₂×Q₈	—	4	1	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂×Q₈	D₄×C₂₃	1	4	yes	complemented
166	C₂×D₄	—	4	1	no	—
167	C₂×D₄	—	4	1	no	—
168	C₂×D₄	Q₈×C₂₃	1	4	yes	complemented
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
173	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
174	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
175	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
176	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
177	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	2	4	no	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₂²⋊C₄	—	2	4	no	—
189	C₂²⋊C₄	—	2	4	no	—
190	C₂²⋊C₄	—	2	4	no	—
191	C₂²⋊C₄	—	4	2	no	—
192	C₂²⋊C₄	—	4	2	no	—
193	C₄²	—	4	1	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	C₂×Q₈×C₂₃	1	4	yes	—
201	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
202	Q₈	—	4	4	no	—
203	D₄	—	4	4	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	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	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
226	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
227	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
228	C₂×C₄	#41×C₂₃	1	2	yes	—
229	C₂²	—	2	4	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	C₂²×D₄×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	4	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₂	—	4	4	no	—
252	C₂	C₂₃×#534	1	1	yes	characteristic
253	C₂	C₂₃×#539	1	1	yes	characteristic
254	C₂	C₂₃×#543	1	1	yes	characteristic
255	C₂	D₄×Q₈×C₂₃	1	1	yes	characteristic
256	C₂	C₂₃×#550	1	1	yes	characteristic
257	C₂	C₂₃×#554	1	1	yes	characteristic
258	C₂	C₂₃×#562	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×C₂²⋊Q₈	C₂	semidirect
2. C₂₃×#399	C₂	semidirect
3. C₂₃×#398	C₂	semidirect
4. C₂₃×#396	C₂	semidirect
5. C₂₃×#391	C₂	semidirect
7. C₂₃×#389	C₂	semidirect
9. C₂₃×#385	C₂	semidirect
35. C₂×Q₈×C₂₃	D₄	semidirect
38. C₂×D₄×C₂₃	Q₈	semidirect
99. #2732	C₂₃	direct (N × H)
122. C₂×C₂²⋊Q₈	C₂×C₂₃	semidirect
123. #399	C₂×C₂₃	semidirect
124. #398	C₂×C₂₃	semidirect
125. #396	C₂×C₂₃	semidirect
126. #391	C₂×C₂₃	semidirect
128. #389	C₂×C₂₃	semidirect
130. #385	C₂×C₂₃	semidirect
161. C₂₃	#2732	direct (N × H)
165. C₂×Q₈	D₄×C₂₃	semidirect
168. C₂×D₄	Q₈×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×#399	2
C₂₃×#398	2
C₂₃×#396	2
C₂₃×#391	2
C₂₃×#390	2
C₂₃×#389	2
C₂₃×#386	2
C₂₃×#385	2
#2732	23

Subgroups — C23×#2732 #15670351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2732 #15670351