Subgroups — C₂₃×#3108 #15670727

Order 2944.

390 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#529	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#483	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#480	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#479	C₂	1	2	yes	maximal, complemented
6	C₂₃×#464	C₂	1	2	yes	maximal, complemented
7	C₂₃×#451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#448	C₂	1	1	yes	maximal, characteristic
9	C₄×Q₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
10	C₄×SD₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
11	C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂₃×#126	—	2	1	no	—
16	C₂₃×#125	C₂²	1	1	yes	characteristic
17	C₂₃×#124	C₂²	1	1	yes	characteristic
18	C₂₃×#123	—	2	1	no	—
19	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂₃×#121	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₂₃	—	2	1	no	—
22	C₄×Q₈×C₂₃	—	2	1	no	—
23	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
24	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
26	C₄×D₄×C₂₃	—	2	1	no	—
27	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#117	—	2	1	no	—
29	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₂₃	—	2	1	no	—
32	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#107	C₂²	1	2	yes	—
34	C₂₃×#106	C₂²	1	2	yes	—
35	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
36	C₂₃×#103	C₂²	1	2	yes	—
37	C₂₃×#103	C₂²	1	2	yes	—
38	C₂₃×#103	C₂²	1	2	yes	—
39	C₂₃×#102	C₂²	1	2	yes	—
40	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₂₃	C₂²	1	2	yes	—
43	C₂×Q₈×C₂₃	—	2	4	no	—
44	C₂×Q₈×C₂₃	—	2	1	no	—
45	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
47	C₂×D₄×C₂₃	—	2	1	no	—
48	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	—	2	1	no	—
50	C₂²×C₄×C₂₃	—	2	1	no	—
51	C₂²×C₄×C₂₃	—	2	1	no	—
52	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
55	Q₁₆×C₂₃	—	2	4	no	—
56	SD₁₆×C₂₃	—	2	4	no	—
57	C₂×C₈×C₂₃	C₂³	1	2	yes	—
58	C₂×C₈×C₂₃	C₂³	1	2	yes	—
59	C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₄⋊C₄×C₂₃	—	2	1	no	—
62	C₄⋊C₄×C₂₃	—	2	1	no	—
63	C₄⋊C₄×C₂₃	—	2	1	no	—
64	C₄⋊C₄×C₂₃	—	2	1	no	—
65	C₄⋊C₄×C₂₃	—	2	1	no	—
66	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
67	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄×C₂₃	—	2	1	no	—
70	C₂²⋊C₄×C₂₃	—	2	1	no	—
71	C₂²⋊C₄×C₂₃	—	2	1	no	—
72	C₂²⋊C₄×C₂₃	—	2	1	no	—
73	C₂²⋊C₄×C₂₃	—	2	1	no	—
74	C₄²×C₂₃	—	2	2	no	—
75	C₄²×C₂₃	—	2	2	no	—
76	C₄²×C₂₃	—	2	1	no	—
77	C₄²×C₂₃	D₄	1	1	yes	characteristic
78	C₄²×C₂₃	D₄	1	1	yes	characteristic
79	C₄²×C₂₃	C₂³	1	1	yes	characteristic
80	C₄²×C₂₃	C₂³	1	1	yes	characteristic
81	C₂³×C₂₃	—	2	1	no	—
82	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₂₃	—	2	4	no	—
84	Q₈×C₂₃	—	2	2	no	—
85	Q₈×C₂₃	—	2	2	no	—
86	Q₈×C₂₃	#41	1	4	yes	—
87	D₄×C₂₃	—	2	2	no	—
88	D₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	—	2	4	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	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
106	C₈×C₂₃	—	2	2	no	—
107	C₈×C₂₃	—	2	2	no	—
108	#3108	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂²×C₂₃	—	2	1	no	—
110	C₂²×C₂₃	—	2	1	no	—
111	C₂²×C₂₃	—	4	1	no	—
112	C₂²×C₂₃	—	4	1	no	—
113	C₂²×C₂₃	—	4	1	no	—
114	C₂²×C₂₃	#135	1	2	yes	—
115	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
116	C₄×C₂₃	—	2	4	no	—
117	C₄×C₂₃	—	2	1	no	—
118	C₄×C₂₃	—	2	1	no	—
119	C₄×C₂₃	—	2	1	no	—
120	C₄×C₂₃	—	2	1	no	—
121	C₄×C₂₃	—	4	1	no	—
122	C₄×C₂₃	—	4	2	no	—
123	C₄×C₂₃	—	4	1	no	—
124	C₄×C₂₃	—	4	1	no	—
125	C₄×C₂₃	#137	1	2	yes	—
126	C₄×C₂₃	C₂×#41	1	2	yes	—
127	#529	C₂×C₂₃	1	1	yes	characteristic
128	C₂×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
129	#483	C₂×C₂₃	1	1	yes	characteristic, complemented
130	#480	C₂×C₂₃	1	1	yes	characteristic
131	#479	C₂×C₂₃	1	2	yes	complemented
132	#464	C₂×C₂₃	1	2	yes	complemented
133	#451	C₂×C₂₃	1	1	yes	characteristic, complemented
134	#448	C₂×C₂₃	1	1	yes	characteristic
135	C₄×Q₁₆	C₂×C₂₃	1	2	yes	complemented
136	C₄×SD₁₆	C₂×C₂₃	1	2	yes	complemented
137	#431	C₂×C₂₃	1	1	yes	characteristic, complemented
138	C₂×C₂₃	—	2	2	no	—
139	C₂×C₂₃	—	8	1	no	—
140	C₂×C₂₃	#525	1	1	yes	characteristic
141	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
142	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
143	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
145	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
146	#126	—	2	1	no	—
147	#125	C₂²×C₂₃	1	1	yes	characteristic
148	#124	C₂²×C₂₃	1	1	yes	characteristic
149	#123	—	2	1	no	—
150	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
151	#121	C₂²×C₂₃	1	1	yes	characteristic
152	C₄×Q₈	—	2	1	no	—
153	C₄×Q₈	—	2	1	no	—
154	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
155	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
156	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
157	C₄×D₄	—	2	1	no	—
158	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
159	#117	—	2	1	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
164	#107	C₂²×C₂₃	1	2	yes	—
165	#106	C₂²×C₂₃	1	2	yes	—
166	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
167	#103	C₂²×C₂₃	1	2	yes	—
168	#103	C₂²×C₂₃	1	2	yes	—
169	#103	C₂²×C₂₃	1	2	yes	—
170	#102	C₂²×C₂₃	1	2	yes	—
171	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
172	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
173	C₄×C₈	C₂²×C₂₃	1	2	yes	—
174	C₂₃	#3108	1	1	yes	characteristic, direct product factor, Sylow
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	2	1	no	—
177	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
178	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
185	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
186	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
187	Q₁₆	—	2	4	no	—
188	SD₁₆	—	2	4	no	—
189	C₂×C₈	C₂³×C₂₃	1	2	yes	—
190	C₂×C₈	C₂³×C₂₃	1	2	yes	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
199	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
200	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
201	C₂²⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	D₄×C₂₃	1	1	yes	characteristic
210	C₄²	D₄×C₂₃	1	1	yes	characteristic
211	C₄²	C₂³×C₂₃	1	1	yes	characteristic
212	C₄²	C₂³×C₂₃	1	1	yes	characteristic
213	C₂³	—	2	1	no	—
214	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	Q₈	—	2	2	no	—
217	Q₈	—	2	2	no	—
218	Q₈	#41×C₂₃	1	4	yes	—
219	D₄	—	2	2	no	—
220	D₄	—	4	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
233	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
234	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
235	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
237	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	C₂₃×#135	1	2	yes	—
246	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	4	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	1	no	—
256	C₄	C₂₃×#137	1	2	yes	—
257	C₄	C₂×#41×C₂₃	1	2	yes	—
258	C₂	—	2	2	no	—
259	C₂	—	8	1	no	—
260	C₂	C₂₃×#525	1	1	yes	characteristic
261	C₂	C₂×C₂₃×#135	1	1	yes	characteristic
262	C₂	C₂×C₂₃×#137	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂₃×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₈×C₂₃	C₂	semidirect
3. C₂₃×#483	C₂	semidirect
5. C₂₃×#479	C₂	semidirect
6. C₂₃×#464	C₂	semidirect
7. C₂₃×#451	C₂	semidirect
9. C₄×Q₁₆×C₂₃	C₂	semidirect
10. C₄×SD₁₆×C₂₃	C₂	semidirect
11. C₂₃×#431	C₂	semidirect
108. #3108	C₂₃	direct (N × H)
128. C₂×C₄×Q₈	C₂×C₂₃	semidirect
129. #483	C₂×C₂₃	semidirect
131. #479	C₂×C₂₃	semidirect
132. #464	C₂×C₂₃	semidirect
133. #451	C₂×C₂₃	semidirect
135. C₄×Q₁₆	C₂×C₂₃	semidirect
136. C₄×SD₁₆	C₂×C₂₃	semidirect
137. #431	C₂×C₂₃	semidirect
174. C₂₃	#3108	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×#529	2
C₂×C₄×Q₈×C₂₃	2
C₂₃×#483	2
C₂₃×#480	2
C₂₃×#479	2
C₂₃×#464	2
C₂₃×#451	2
C₂₃×#448	2
C₄×Q₁₆×C₂₃	2
C₄×SD₁₆×C₂₃	2
C₂₃×#431	2
#3108	23

Subgroups — C23×#3108 #15670727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3108 #15670727