Subgroups — #15676454

Order 2944.

552 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625115	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic
4	#13092322	C₂	1	1	yes	maximal, characteristic
5	#13092314	C₂	1	2	yes	maximal
6	#13092304	C₂	1	2	yes	maximal
7	#13092299	C₂	1	2	yes	maximal
8	C₄⋊C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
9	C₄×Dic₂₃⋊C₄	C₂	1	1	yes	maximal, characteristic
10	#13092268	C₂	1	2	yes	maximal
11	C₂₃×#391	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
16	Q₈×Dic₂₃	—	2	4	no	—
17	#10625115	C₄	1	8	yes	complemented
18	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
21	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
23	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
24	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
26	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
27	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
29	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
30	C₄×Dic₄₆	—	2	4	no	—
31	C₂₃×#95	C₂²	1	2	yes	—
32	C₂₃×#95	C₂²	1	2	yes	—
33	C₂₃⋊#95	C₂²	1	2	yes	—
34	C₂₃⋊#95	C₂²	1	1	yes	characteristic
35	C₂₃⋊#95	C₂²	1	1	yes	characteristic
36	C₂₃⋊#95	C₂²	1	2	yes	—
37	C₂₃⋊#95	C₂²	1	2	yes	—
38	C₂×Q₈×C₂₃	—	2	2	no	—
39	C₂×Q₈×C₂₃	—	2	2	no	—
40	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
47	C₂²×Dic₂₃	C₂³	1	2	yes	—
48	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
49	C₂²×Dic₂₃	C₂³	1	2	yes	—
50	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
51	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
52	C₂×Dic₄₆	—	2	2	no	—
53	C₂×Dic₄₆	—	2	2	no	—
54	C₂×Dic₄₆	C₂×C₄	1	4	yes	—
55	C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₄²×C₂₃	—	2	2	no	—
57	C₄⋊Dic₂₃	—	2	4	no	—
58	C₄⋊Dic₂₃	D₄	1	4	yes	—
59	Dic₂₃⋊C₄	—	2	4	no	—
60	Dic₂₃⋊C₄	—	2	2	no	—
61	Dic₂₃⋊C₄	C₂×C₄	1	8	yes	—
62	Dic₂₃⋊C₄	C₂×C₄	1	8	yes	—
63	Dic₂₃⋊C₄	Q₈	1	4	yes	complemented
64	C₄×Dic₂₃	—	2	2	no	—
65	C₄×Dic₂₃	—	2	4	no	—
66	C₄×Dic₂₃	—	2	4	no	—
67	C₄×Dic₂₃	—	2	4	no	—
68	C₄×Dic₂₃	C₂×C₄	1	4	yes	—
69	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
70	Q₈×C₂₃	—	2	8	no	—
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	1	no	—
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	2	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	2	no	—
81	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
85	C₂×C₄×C₂₃	#41	1	2	yes	—
86	C₂×Dic₂₃	—	2	1	no	—
87	C₂×Dic₂₃	—	2	2	no	—
88	C₂×Dic₂₃	—	2	2	no	—
89	C₂×Dic₂₃	—	2	2	no	—
90	C₂×Dic₂₃	—	2	1	no	—
91	C₂×Dic₂₃	—	2	2	no	—
92	C₂×Dic₂₃	—	2	1	no	—
93	C₂×Dic₂₃	—	2	1	no	—
94	C₂×Dic₂₃	—	2	1	no	—
95	C₂×Dic₂₃	—	2	1	no	—
96	C₂×Dic₂₃	—	2	1	no	—
97	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
98	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
99	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
100	C₂×Dic₂₃	C₂×D₄	1	4	yes	—
101	C₂×Dic₂₃	C₂×Q₈	1	2	yes	—
102	C₂×Dic₂₃	C₂×Q₈	1	2	yes	—
103	C₂×Dic₂₃	#41	1	2	yes	—
104	Dic₄₆	—	2	8	no	—
105	#2365	—	23	1	no	maximal, Sylow
106	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
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	C₂×#41	1	1	yes	characteristic
111	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
112	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
113	C₄×C₂₃	—	2	4	no	—
114	C₄×C₂₃	—	2	2	no	—
115	C₄×C₂₃	—	4	1	no	—
116	C₄×C₂₃	—	4	2	no	—
117	C₄×C₂₃	C₄×D₄	1	4	yes	—
118	Dic₂₃	—	2	4	no	—
119	Dic₂₃	—	2	2	no	—
120	Dic₂₃	—	2	4	no	—
121	Dic₂₃	—	2	2	no	—
122	Dic₂₃	—	4	1	no	—
123	Dic₂₃	C₄×Q₈	1	4	yes	complemented
124	C₂×C₄⋊Q₈	—	23	1	no	—
125	C₂×C₄×Q₈	—	23	1	no	—
126	C₂×C₄×Q₈	—	23	1	no	—
127	#391	—	23	1	no	—
128	#391	D₂₃	1	1	yes	characteristic
129	#389	—	23	2	no	—
130	#387	—	23	2	no	—
131	#387	—	23	2	no	—
132	C₄×C₄⋊C₄	—	23	1	no	—
133	C₄×C₄⋊C₄	—	23	1	no	—
134	C₄×C₄⋊C₄	—	23	2	no	—
135	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
137	C₂×C₂₃	#519	1	1	yes	characteristic
138	C₂×C₂₃	#547	1	1	yes	characteristic
139	C₂×C₂₃	D₄×Q₈	1	1	yes	characteristic
140	C₂×C₂₃	#556	1	1	yes	characteristic
141	C₂×C₂₃	#557	1	1	yes	characteristic
142	C₂²×Q₈	—	23	1	no	—
143	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
144	C₄⋊Q₈	—	23	8	no	—
145	C₄×Q₈	—	46	4	no	—
146	C₄×Q₈	—	46	4	no	—
147	C₂×C₄⋊C₄	—	23	1	no	—
148	C₂×C₄⋊C₄	—	23	2	no	—
149	C₂×C₄⋊C₄	—	23	2	no	—
150	C₂×C₄⋊C₄	—	23	1	no	—
151	C₂×C₄⋊C₄	—	23	2	no	—
152	C₂×C₄⋊C₄	—	23	2	no	—
153	C₂×C₄⋊C₄	—	23	1	no	—
154	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
155	C₂×C₄²	—	23	1	no	—
156	C₂×C₄²	—	23	2	no	—
157	C₂×C₄²	—	23	2	no	—
158	C₂×C₄²	—	23	1	no	—
159	C₂×C₄²	—	23	2	no	—
160	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
161	#95	—	23	1	no	—
162	#95	—	23	2	no	—
163	#95	—	23	1	no	—
164	#95	—	23	2	no	—
165	#95	—	23	2	no	—
166	#95	C₂×D₂₃	1	2	yes	—
167	#95	C₂×D₂₃	1	2	yes	—
168	C₂₃	#2365	1	1	yes	characteristic, complemented, Sylow
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	23	4	no	—
172	C₂×Q₈	—	46	2	no	—
173	C₂×Q₈	—	46	2	no	—
174	C₂×Q₈	C₄×D₂₃	1	4	yes	—
175	C₂²×C₄	—	23	1	no	—
176	C₂²×C₄	—	23	1	no	—
177	C₂²×C₄	—	23	1	no	—
178	C₂²×C₄	—	23	2	no	—
179	C₂²×C₄	—	23	2	no	—
180	C₂²×C₄	—	23	1	no	—
181	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
182	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
183	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
184	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	23	8	no	—
188	C₄⋊C₄	—	23	8	no	—
189	C₄⋊C₄	—	23	4	no	—
190	C₄⋊C₄	—	23	4	no	—
191	C₄⋊C₄	—	46	4	no	—
192	C₄⋊C₄	—	46	4	no	—
193	C₄⋊C₄	—	46	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	23	4	no	—
196	C₄²	—	46	4	no	—
197	C₄²	—	46	4	no	—
198	C₄²	—	46	4	no	—
199	C₄²	—	46	2	no	—
200	C₂³	C₂³×D₂₃	1	1	yes	characteristic
201	Q₈	—	2	8	no	—
202	Q₈	—	46	8	no	—
203	C₂×C₄	—	2	1	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	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	23	2	no	—
214	C₂×C₄	—	23	2	no	—
215	C₂×C₄	—	23	2	no	—
216	C₂×C₄	—	23	2	no	—
217	C₂×C₄	—	23	4	no	—
218	C₂×C₄	—	23	4	no	—
219	C₂×C₄	—	23	2	no	—
220	C₂×C₄	—	46	1	no	—
221	C₂×C₄	—	46	2	no	—
222	C₂×C₄	—	46	2	no	—
223	C₂×C₄	—	46	1	no	—
224	C₂×C₄	—	46	2	no	—
225	C₂×C₄	—	46	2	no	—
226	C₂×C₄	—	46	1	no	—
227	C₂×C₄	—	46	1	no	—
228	C₂×C₄	—	46	1	no	—
229	C₂×C₄	—	46	1	no	—
230	C₂×C₄	—	46	1	no	—
231	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
232	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
233	C₂×C₄	D₄×D₂₃	1	2	yes	—
234	C₂×C₄	#67290	1	2	yes	—
235	C₂×C₄	Q₈×D₂₃	1	2	yes	—
236	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
237	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
238	C₂²	C₂×#67290	1	1	yes	characteristic
239	C₂²	C₂×Q₈×D₂₃	1	1	yes	characteristic
240	C₂²	#10625158	1	1	yes	characteristic
241	C₂²	#41×D₂₃	1	1	yes	characteristic
242	C₂²	#10625161	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	2	2	no	—
245	C₄	—	4	2	no	—
246	C₄	—	4	1	no	—
247	C₄	—	23	4	no	—
248	C₄	—	46	4	no	—
249	C₄	—	46	2	no	—
250	C₄	—	46	4	no	—
251	C₄	—	46	2	no	—
252	C₄	—	92	1	no	—
253	C₄	(C₂²⋊D₂₃)⋊C₄	1	4	yes	—
254	C₂	C₂×(C₂²⋊D₂₃)⋊C₄	1	1	yes	characteristic
255	C₂	C₄×Q₈×D₂₃	1	1	yes	characteristic
256	C₂	#13092893	1	1	yes	characteristic
257	C₂	#13093069	1	1	yes	characteristic
258	C₂	#13093080	1	1	yes	characteristic
259	C₂	#13093136	1	1	yes	characteristic
260	C₂	#13093142	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#391	C₂
socle, prime radical, cosocle	C₂³×C₂₃	C₂⁴
hypercenter	#391	D₂₃
hypocenter	C₂₃	#2365
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
17. #10625115	C₄	semidirect
63. Dic₂₃⋊C₄	Q₈	semidirect
123. Dic₂₃	C₄×Q₈	semidirect
168. C₂₃	#2365	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625115	2
C₂×Q₈×Dic₂₃	2
C₂×C₄×Dic₄₆	2
#13092322	2
#13092314	2
#13092304	2
#13092299	2
C₄⋊C₄×Dic₂₃	2
C₄×Dic₂₃⋊C₄	2
#13092268	2
C₂₃×#391	2
#2365	23