Subgroups — C₂₃×#3522 #15671141

Order 2944.

1104 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#559	C₂	1	3	yes	maximal, complemented
2	Q₈²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#554	C₂	1	3	yes	maximal, complemented
4	C₂₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#548	C₂	1	2	yes	maximal, complemented
6	C₂₃×#547	C₂	1	6	yes	maximal, complemented
7	C₂₃×#542	C₂	1	6	yes	maximal, complemented
8	C₂₃×#532	C₂	1	3	yes	maximal, complemented
9	C₂₃×#529	C₂	1	3	yes	maximal, complemented
10	C₂₃×#520	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	2	yes	—
13	C₂×#41×C₂₃	C₂²	1	6	yes	—
14	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×C₄⋊Q₈	C₂²	1	6	yes	—
16	C₂₃×C₄⋊Q₈	C₂²	1	3	yes	—
17	C₂₃×#127	C₂²	1	3	yes	—
18	C₂₃×#126	C₂²	1	6	yes	—
19	C₂₃×#125	C₂²	1	3	yes	—
20	C₂₃×#124	C₂²	1	6	yes	—
21	C₂₃×#124	C₂²	1	3	yes	—
22	C₂₃×#124	C₂²	1	6	yes	—
23	C₂₃×#123	C₂²	1	6	yes	—
24	C₂₃×#123	C₂²	1	12	yes	—
25	C₂₃×C₂²⋊Q₈	C₂²	1	12	yes	—
26	C₂₃×C₂²⋊Q₈	C₂²	1	6	yes	—
27	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
28	C₂₃×#121	C₂²	1	6	yes	—
29	C₂₃×#121	C₂²	1	6	yes	—
30	C₂₃×#121	C₂²	1	3	yes	—
31	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
32	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
33	C₄×Q₈×C₂₃	C₂²	1	3	yes	—
34	C₄×Q₈×C₂₃	C₂²	1	3	yes	—
35	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₂₃	C₂²	1	3	yes	—
37	C₄×D₄×C₂₃	C₂²	1	6	yes	—
38	C₄×D₄×C₂₃	C₂²	1	6	yes	—
39	C₄×D₄×C₂₃	C₂²	1	3	yes	—
40	C₄×D₄×C₂₃	C₂²	1	6	yes	—
41	C₄×D₄×C₂₃	C₂²	1	6	yes	—
42	C₂₃×#117	C₂²	1	6	yes	—
43	C₂×C₄⋊C₄×C₂₃	C₂²	1	3	yes	—
44	C₂×C₄⋊C₄×C₂₃	C₂²	1	6	yes	—
45	C₂×C₄²×C₂₃	C₂²	1	3	yes	—
46	#41×C₂₃	—	2	8	no	—
47	#41×C₂₃	—	4	12	no	—
48	C₂×Q₈×C₂₃	—	2	4	no	—
49	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×Q₈×C₂₃	C₂³	1	3	yes	—
51	C₂×Q₈×C₂₃	C₂³	1	6	yes	—
52	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₂₃	C₂³	1	3	yes	—
54	C₂×D₄×C₂₃	C₂³	1	6	yes	—
55	C₂×D₄×C₂₃	C₂³	1	6	yes	—
56	C₂×D₄×C₂₃	C₂³	1	6	yes	—
57	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂³	1	3	yes	—
59	C₂²×C₄×C₂₃	C₂³	1	3	yes	—
60	C₂²×C₄×C₂₃	C₂³	1	6	yes	—
61	C₂²×C₄×C₂₃	C₂³	1	12	yes	—
62	C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
63	C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
64	C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
65	C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
66	C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
67	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
68	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
70	C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
71	C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
72	C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
73	C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
74	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
75	C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
76	C₂²⋊C₄×C₂₃	C₂³	1	12	yes	—
77	C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
78	C₄²×C₂₃	C₂³	1	3	yes	—
79	C₄²×C₂₃	C₂³	1	3	yes	—
80	C₄²×C₂₃	C₂³	1	3	yes	—
81	C₄²×C₂₃	C₂³	1	3	yes	—
82	C₄²×C₂₃	C₂³	1	6	yes	—
83	C₂³×C₂₃	C₂⁴	1	6	yes	—
84	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
85	Q₈×C₂₃	—	2	6	no	—
86	Q₈×C₂₃	—	2	2	no	—
87	Q₈×C₂₃	—	4	6	no	—
88	Q₈×C₂₃	#41	1	4	yes	complemented
89	D₄×C₂₃	—	2	12	no	—
90	D₄×C₂₃	—	2	12	no	—
91	D₄×C₂₃	—	2	6	no	—
92	D₄×C₂₃	—	4	6	no	—
93	C₂×C₄×C₂₃	—	2	6	no	—
94	C₂×C₄×C₂₃	—	2	6	no	—
95	C₂×C₄×C₂₃	—	2	12	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	4	12	no	—
98	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
100	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
101	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
102	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
103	C₂×C₄×C₂₃	C₂⁴	1	6	yes	—
104	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
105	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
106	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
107	#3522	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₂₃	—	2	1	no	—
109	C₂²×C₂₃	—	2	1	no	—
110	C₂²×C₂₃	—	2	6	no	—
111	C₂²×C₂₃	—	2	6	no	—
112	C₂²×C₂₃	—	2	6	no	—
113	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
114	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
115	C₄×C₂₃	—	2	2	no	—
116	C₄×C₂₃	—	2	3	no	—
117	C₄×C₂₃	—	2	3	no	—
118	C₄×C₂₃	—	2	3	no	—
119	C₄×C₂₃	—	2	6	no	—
120	C₄×C₂₃	—	2	1	no	—
121	C₄×C₂₃	—	2	2	no	—
122	C₄×C₂₃	C₂×#41	1	6	yes	—
123	C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
124	#559	C₂×C₂₃	1	3	yes	complemented
125	Q₈²	C₂×C₂₃	1	1	yes	characteristic, complemented
126	#554	C₂×C₂₃	1	3	yes	complemented
127	#550	C₂×C₂₃	1	1	yes	characteristic, complemented
128	#548	C₂×C₂₃	1	2	yes	complemented
129	#547	C₂×C₂₃	1	6	yes	complemented
130	#542	C₂×C₂₃	1	6	yes	complemented
131	#532	C₂×C₂₃	1	3	yes	complemented
132	#529	C₂×C₂₃	1	3	yes	complemented
133	#520	C₂×C₂₃	1	2	yes	complemented
134	C₂×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
135	C₂×C₂₃	—	2	2	no	—
136	C₂×C₂₃	—	4	6	no	—
137	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
138	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
140	C₂×#41	C₂²×C₂₃	1	2	yes	—
141	C₂×#41	C₂²×C₂₃	1	6	yes	—
142	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
143	C₄⋊Q₈	C₂²×C₂₃	1	3	yes	—
144	C₄⋊Q₈	C₂²×C₂₃	1	6	yes	—
145	#127	C₂²×C₂₃	1	3	yes	—
146	#126	C₂²×C₂₃	1	6	yes	—
147	#125	C₂²×C₂₃	1	3	yes	—
148	#124	C₂²×C₂₃	1	3	yes	—
149	#124	C₂²×C₂₃	1	6	yes	—
150	#124	C₂²×C₂₃	1	6	yes	—
151	#123	C₂²×C₂₃	1	6	yes	—
152	#123	C₂²×C₂₃	1	12	yes	—
153	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
154	C₂²⋊Q₈	C₂²×C₂₃	1	12	yes	—
155	C₂²⋊Q₈	C₂²×C₂₃	1	6	yes	—
156	#121	C₂²×C₂₃	1	3	yes	—
157	#121	C₂²×C₂₃	1	6	yes	—
158	#121	C₂²×C₂₃	1	6	yes	—
159	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
160	C₄×Q₈	C₂²×C₂₃	1	3	yes	—
161	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
162	C₄×Q₈	C₂²×C₂₃	1	3	yes	—
163	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
164	C₄×Q₈	C₂²×C₂₃	1	3	yes	—
165	C₄×D₄	C₂²×C₂₃	1	6	yes	—
166	C₄×D₄	C₂²×C₂₃	1	6	yes	—
167	C₄×D₄	C₂²×C₂₃	1	6	yes	—
168	C₄×D₄	C₂²×C₂₃	1	3	yes	—
169	C₄×D₄	C₂²×C₂₃	1	6	yes	—
170	#117	C₂²×C₂₃	1	6	yes	—
171	C₂×C₄⋊C₄	C₂²×C₂₃	1	3	yes	—
172	C₂×C₄⋊C₄	C₂²×C₂₃	1	6	yes	—
173	C₂×C₄²	C₂²×C₂₃	1	3	yes	—
174	C₂₃	#3522	1	1	yes	characteristic, direct product factor, Sylow
175	#41	—	2	8	no	—
176	#41	—	4	12	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
179	C₂×Q₈	C₂³×C₂₃	1	3	yes	—
180	C₂×Q₈	C₂³×C₂₃	1	6	yes	—
181	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
182	C₂×D₄	C₂³×C₂₃	1	3	yes	—
183	C₂×D₄	C₂³×C₂₃	1	6	yes	—
184	C₂×D₄	C₂³×C₂₃	1	6	yes	—
185	C₂×D₄	C₂³×C₂₃	1	6	yes	—
186	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
187	C₂²×C₄	C₂³×C₂₃	1	3	yes	—
188	C₂²×C₄	C₂³×C₂₃	1	3	yes	—
189	C₂²×C₄	C₂³×C₂₃	1	6	yes	—
190	C₂²×C₄	C₂³×C₂₃	1	12	yes	—
191	C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
192	C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
193	C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
194	C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
195	C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
196	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
197	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
198	C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
199	C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
200	C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
201	C₂²⋊C₄	C₂³×C₂₃	1	6	yes	—
202	C₂²⋊C₄	C₂³×C₂₃	1	6	yes	—
203	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
204	C₂²⋊C₄	C₂³×C₂₃	1	6	yes	—
205	C₂²⋊C₄	C₂³×C₂₃	1	12	yes	—
206	C₂²⋊C₄	C₂³×C₂₃	1	6	yes	—
207	C₄²	C₂³×C₂₃	1	3	yes	—
208	C₄²	C₂³×C₂₃	1	3	yes	—
209	C₄²	C₂³×C₂₃	1	3	yes	—
210	C₄²	C₂³×C₂₃	1	3	yes	—
211	C₄²	C₂³×C₂₃	1	6	yes	—
212	C₂³	C₂⁴×C₂₃	1	6	yes	—
213	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
214	Q₈	—	2	6	no	—
215	Q₈	—	2	2	no	—
216	Q₈	—	4	6	no	—
217	Q₈	#41×C₂₃	1	4	yes	complemented
218	D₄	—	2	12	no	—
219	D₄	—	2	12	no	—
220	D₄	—	2	6	no	—
221	D₄	—	4	6	no	—
222	C₂×C₄	—	2	6	no	—
223	C₂×C₄	—	2	6	no	—
224	C₂×C₄	—	2	12	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	12	no	—
227	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
228	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
229	C₂×C₄	C₂⁴×C₂₃	1	3	yes	—
230	C₂×C₄	C₂⁴×C₂₃	1	3	yes	—
231	C₂×C₄	C₂⁴×C₂₃	1	3	yes	—
232	C₂×C₄	C₂⁴×C₂₃	1	6	yes	—
233	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
234	C₂×C₄	C₂⁴×C₂₃	1	3	yes	—
235	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	6	no	—
239	C₂²	—	2	6	no	—
240	C₂²	—	2	6	no	—
241	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
242	C₂²	C₂⁵×C₂₃	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	3	no	—
245	C₄	—	2	3	no	—
246	C₄	—	2	3	no	—
247	C₄	—	2	6	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	2	no	—
250	C₄	C₂×#41×C₂₃	1	6	yes	—
251	C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
252	C₂	—	2	2	no	—
253	C₂	—	4	6	no	—
254	C₂	C₂²×#41×C₂₃	1	1	yes	characteristic
255	C₂	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
256	C₂	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#559	C₂	semidirect
2. Q₈²×C₂₃	C₂	semidirect
3. C₂₃×#554	C₂	semidirect
4. C₂₃×#550	C₂	semidirect
5. C₂₃×#548	C₂	semidirect
6. C₂₃×#547	C₂	semidirect
7. C₂₃×#542	C₂	semidirect
8. C₂₃×#532	C₂	semidirect
9. C₂₃×#529	C₂	semidirect
10. C₂₃×#520	C₂	semidirect
11. C₂×C₄×Q₈×C₂₃	C₂	semidirect
88. Q₈×C₂₃	#41	semidirect
107. #3522	C₂₃	direct (N × H)
124. #559	C₂×C₂₃	semidirect
125. Q₈²	C₂×C₂₃	semidirect
126. #554	C₂×C₂₃	semidirect
127. #550	C₂×C₂₃	semidirect
128. #548	C₂×C₂₃	semidirect
129. #547	C₂×C₂₃	semidirect
130. #542	C₂×C₂₃	semidirect
131. #532	C₂×C₂₃	semidirect
132. #529	C₂×C₂₃	semidirect
133. #520	C₂×C₂₃	semidirect
134. C₂×C₄×Q₈	C₂×C₂₃	semidirect
174. C₂₃	#3522	direct (N × H)
217. Q₈	#41×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#559	2
Q₈²×C₂₃	2
C₂₃×#554	2
C₂₃×#550	2
C₂₃×#548	2
C₂₃×#547	2
C₂₃×#542	2
C₂₃×#532	2
C₂₃×#529	2
C₂₃×#520	2
C₂×C₄×Q₈×C₂₃	2
#3522	23

Subgroups — C23×#3522 #15671141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3522 #15671141