Subgroups — #15675751

Order 2944.

696 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625063	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625043	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×D₂₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×Dic₂₃⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092268	C₂	1	2	yes	maximal, complemented
9	C₄×(C₂×D₂₃)⋊C₄	C₂	1	2	yes	maximal
10	#13092264	C₂	1	2	yes	maximal
11	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
13	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
24	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
27	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
28	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
29	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
31	#10625063	C₄	1	8	yes	complemented
32	C₄⋊C₄×D₂₃	C₄	1	8	yes	complemented
33	#10625043	C₄	1	8	yes	complemented
34	C₂₃×#95	C₂²	1	2	yes	—
35	C₂₃⋊#95	C₂²	1	2	yes	—
36	C₂₃⋊#95	C₂²	1	2	yes	—
37	C₂₃⋊#95	C₂²	1	2	yes	—
38	C₂³×D₂₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
46	C₂²×Dic₂₃	C₂³	1	2	yes	—
47	C₂²×Dic₂₃	C₂³	1	2	yes	—
48	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₂₃	—	4	2	no	—
50	C₂×C₄×D₂₃	—	4	2	no	—
51	C₂×C₄×D₂₃	C₂×C₄	1	8	yes	—
52	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
53	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
54	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
55	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
56	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	complemented
57	C₄²×C₂₃	—	4	2	no	—
58	C₄²×C₂₃	C₂×C₄	1	4	yes	complemented
59	(C₂×D₂₃)⋊C₄	C₂×C₄	1	4	yes	—
60	(C₂×D₂₃)⋊C₄	C₂×C₄	1	4	yes	—
61	(C₂×D₂₃)⋊C₄	C₂×C₄	1	8	yes	—
62	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
63	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	complemented
64	Dic₂₃⋊C₄	C₂×C₄	1	8	yes	—
65	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	complemented
66	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	complemented
67	C₄×Dic₂₃	—	4	2	no	—
68	C₄×Dic₂₃	—	4	2	no	—
69	C₄×Dic₂₃	—	4	2	no	—
70	C₄×Dic₂₃	C₂×C₄	1	4	yes	complemented
71	C₄×Dic₂₃	C₂×C₄	1	8	yes	complemented
72	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
73	C₂²×D₂₃	—	2	1	no	—
74	C₂²×D₂₃	—	2	1	no	—
75	C₂²×D₂₃	—	2	1	no	—
76	C₂²×D₂₃	—	2	1	no	—
77	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
78	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
79	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
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	1	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
90	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
95	C₂×Dic₂₃	—	2	1	no	—
96	C₂×Dic₂₃	—	2	1	no	—
97	C₂×Dic₂₃	—	2	2	no	—
98	C₂×Dic₂₃	—	2	2	no	—
99	C₂×Dic₂₃	—	2	2	no	—
100	C₂×Dic₂₃	—	2	1	no	—
101	C₂×Dic₂₃	—	2	2	no	—
102	C₂×Dic₂₃	—	2	1	no	—
103	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
104	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
105	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
106	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
107	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
108	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
109	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
110	C₄×D₂₃	—	4	8	no	—
111	C₄×D₂₃	C₄²	1	16	yes	complemented
112	#2274	—	23	1	no	maximal, Sylow
113	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
114	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
115	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
116	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
119	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
120	C₂×D₂₃	—	2	2	no	—
121	C₂×D₂₃	—	2	2	no	—
122	C₂×D₂₃	—	2	2	no	—
123	C₂×D₂₃	—	2	2	no	—
124	C₂×D₂₃	—	2	2	no	—
125	C₂×D₂₃	—	2	2	no	—
126	C₂×D₂₃	C₂×C₄²	1	4	yes	—
127	C₄×C₂₃	—	2	4	no	—
128	C₄×C₂₃	—	2	4	no	—
129	C₄×C₂₃	—	2	2	no	—
130	C₄×C₂₃	—	2	2	no	—
131	C₄×C₂₃	C₂×C₄²	1	4	yes	—
132	Dic₂₃	—	2	4	no	—
133	Dic₂₃	—	2	2	no	—
134	Dic₂₃	—	2	2	no	—
135	Dic₂₃	—	2	4	no	—
136	Dic₂₃	C₂×C₄²	1	4	yes	—
137	C₂×#117	—	23	1	no	—
138	C₂×#117	—	23	1	no	—
139	C₂²×C₄⋊C₄	—	23	1	no	—
140	C₄×C₄⋊C₄	—	23	1	no	—
141	C₄×C₄⋊C₄	—	23	1	no	—
142	C₄×C₄⋊C₄	—	23	1	no	—
143	C₄×C₄⋊C₄	—	23	2	no	—
144	C₄×C₄⋊C₄	—	23	2	no	—
145	C₄×C₄⋊C₄	D₂₃	1	1	yes	characteristic, complemented
146	C₄×C₂²⋊C₄	—	23	2	no	—
147	C₄×C₂²⋊C₄	—	23	2	no	—
148	C₂×C₂₃	C₂²×C₄²	1	1	yes	characteristic
149	C₂×C₂₃	#518	1	1	yes	characteristic
150	C₂×C₂₃	#519	1	1	yes	characteristic
151	C₂×C₂₃	#520	1	1	yes	characteristic
152	C₂×C₂₃	#520	1	1	yes	characteristic
153	C₂×C₂₃	#520	1	1	yes	characteristic
154	C₂×C₂₃	#520	1	1	yes	characteristic
155	D₂₃	—	2	4	no	—
156	C₂³×C₄	—	23	1	no	—
157	C₂³×C₄	—	23	2	no	—
158	#117	—	23	8	no	—
159	#117	—	23	8	no	—
160	C₂×C₄⋊C₄	—	23	2	no	—
161	C₂×C₄⋊C₄	—	23	1	no	—
162	C₂×C₄⋊C₄	—	23	1	no	—
163	C₂×C₄⋊C₄	—	23	1	no	—
164	C₂×C₄⋊C₄	—	23	1	no	—
165	C₂×C₄⋊C₄	—	23	8	no	—
166	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
167	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
168	C₂×C₂²⋊C₄	—	23	1	no	—
169	C₂×C₂²⋊C₄	—	23	2	no	—
170	C₂×C₂²⋊C₄	—	23	1	no	—
171	C₂×C₄²	—	23	2	no	—
172	C₂×C₄²	—	23	1	no	—
173	C₂×C₄²	—	23	2	no	—
174	C₂×C₄²	—	23	2	no	—
175	C₂×C₄²	—	23	2	no	—
176	C₂×C₄²	C₂×D₂₃	1	2	yes	—
177	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
178	#95	—	23	2	no	—
179	#95	—	23	2	no	—
180	#95	—	23	2	no	—
181	#95	C₂×D₂₃	1	2	yes	—
182	C₂₃	#2274	1	1	yes	characteristic, complemented, Sylow
183	C₂⁴	—	23	1	no	—
184	C₂²×C₄	—	23	1	no	—
185	C₂²×C₄	—	23	4	no	—
186	C₂²×C₄	—	23	4	no	—
187	C₂²×C₄	—	23	4	no	—
188	C₂²×C₄	—	23	1	no	—
189	C₂²×C₄	—	23	2	no	—
190	C₂²×C₄	—	23	1	no	—
191	C₂²×C₄	—	23	2	no	—
192	C₂²×C₄	—	23	8	no	—
193	C₂²×C₄	—	92	2	no	—
194	C₂²×C₄	—	92	2	no	—
195	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
197	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
198	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
199	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
200	C₄⋊C₄	—	23	8	no	—
201	C₄⋊C₄	—	23	4	no	—
202	C₄⋊C₄	—	23	4	no	—
203	C₄⋊C₄	—	23	4	no	—
204	C₄⋊C₄	—	23	4	no	—
205	C₄⋊C₄	C₄×D₂₃	1	4	yes	—
206	C₄⋊C₄	C₄×D₂₃	1	4	yes	complemented
207	C₂²⋊C₄	—	23	8	no	—
208	C₂²⋊C₄	—	23	4	no	—
209	C₂²⋊C₄	—	23	4	no	—
210	C₄²	—	4	2	no	—
211	C₄²	—	23	8	no	—
212	C₄²	—	23	4	no	—
213	C₄²	—	92	2	no	—
214	C₄²	—	92	2	no	—
215	C₄²	—	92	2	no	—
216	C₄²	C₄×D₂₃	1	4	yes	complemented
217	C₂³	—	23	2	no	—
218	C₂³	—	23	2	no	—
219	C₂³	—	23	2	no	—
220	C₂³	—	46	1	no	—
221	C₂³	—	46	1	no	—
222	C₂³	—	46	1	no	—
223	C₂³	—	46	1	no	—
224	C₂³	C₂³×D₂₃	1	1	yes	characteristic
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	23	2	no	—
234	C₂×C₄	—	23	2	no	—
235	C₂×C₄	—	23	2	no	—
236	C₂×C₄	—	23	16	no	—
237	C₂×C₄	—	23	2	no	—
238	C₂×C₄	—	23	2	no	—
239	C₂×C₄	—	23	4	no	—
240	C₂×C₄	—	23	4	no	—
241	C₂×C₄	—	46	1	no	—
242	C₂×C₄	—	46	2	no	—
243	C₂×C₄	—	46	1	no	—
244	C₂×C₄	—	46	2	no	—
245	C₂×C₄	—	46	1	no	—
246	C₂×C₄	—	46	2	no	—
247	C₂×C₄	—	46	2	no	—
248	C₂×C₄	—	46	1	no	—
249	C₂×C₄	—	92	8	no	—
250	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
251	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
252	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
253	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
254	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
255	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
256	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
257	C₂²	—	23	4	no	—
258	C₂²	—	46	2	no	—
259	C₂²	—	46	2	no	—
260	C₂²	—	46	2	no	—
261	C₂²	—	46	2	no	—
262	C₂²	—	46	2	no	—
263	C₂²	—	46	2	no	—
264	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
265	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
266	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
267	C₂²	#10625155	1	1	yes	characteristic
268	C₂²	#10625158	1	1	yes	characteristic
269	C₂²	#10625160	1	1	yes	characteristic
270	C₂²	#10625161	1	1	yes	characteristic
271	C₄	—	2	4	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	23	4	no	—
276	C₄	—	46	4	no	—
277	C₄	—	46	2	no	—
278	C₄	—	46	2	no	—
279	C₄	—	46	4	no	—
280	C₄	C₄²×D₂₃	1	4	yes	—
281	C₂	—	46	4	no	—
282	C₂	C₂×C₄²×D₂₃	1	1	yes	characteristic
283	C₂	#13092889	1	1	yes	characteristic
284	C₂	#13092893	1	1	yes	characteristic
285	C₂	#13092896	1	1	yes	characteristic
286	C₂	#13092897	1	1	yes	characteristic
287	C₂	#13092898	1	1	yes	characteristic
288	C₂	#13092899	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂×C₂₃	C₂²×C₄²
Fitting	C₄×C₄⋊C₄×C₂₃	C₂
socle, cosocle	C₂³×C₂₃	C₂⁴
prime radical	C₂³×D₂₃	C₂³
hypercenter	C₄×C₄⋊C₄	D₂₃
hypocenter	C₂₃	#2274
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₄⋊C₄×Dic₂₃	C₂	semidirect
5. C₄⋊C₄×Dic₂₃	C₂	semidirect
6. C₄×Dic₂₃⋊C₄	C₂	semidirect
7. C₄×C₄⋊Dic₂₃	C₂	semidirect
8. #13092268	C₂	semidirect
11. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
31. #10625063	C₄	semidirect
32. C₄⋊C₄×D₂₃	C₄	semidirect
33. #10625043	C₄	semidirect
56. C₄⋊C₄×C₂₃	C₂×C₄	semidirect
58. C₄²×C₂₃	C₂×C₄	semidirect
63. C₄⋊Dic₂₃	C₂×C₄	semidirect
65. Dic₂₃⋊C₄	C₂×C₄	semidirect
66. Dic₂₃⋊C₄	C₂×C₄	semidirect
70. C₄×Dic₂₃	C₂×C₄	semidirect
71. C₄×Dic₂₃	C₂×C₄	semidirect
111. C₄×D₂₃	C₄²	semidirect
145. C₄×C₄⋊C₄	D₂₃	semidirect
182. C₂₃	#2274	semidirect
206. C₄⋊C₄	C₄×D₂₃	semidirect
216. C₄²	C₄×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625063	2
C₂×#10625043	2
C₂×C₄⋊C₄×D₂₃	2
C₄⋊C₄×Dic₂₃	2
C₄×Dic₂₃⋊C₄	2
C₄×C₄⋊Dic₂₃	2
#13092268	2
C₄×(C₂×D₂₃)⋊C₄	2
#13092264	2
C₄×C₄⋊C₄×C₂₃	2
#2274	23