Subgroups — #15676076

Order 2944.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625049	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625048	C₂	1	1	yes	maximal, characteristic
4	#13092321	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092314	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092313	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092312	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092310	C₂	1	1	yes	maximal, characteristic
9	#13092305	C₂	1	1	yes	maximal, characteristic
10	#13092305	C₂	1	1	yes	maximal, characteristic
11	#13092302	C₂	1	1	yes	maximal, characteristic, complemented
12	#13092301	C₂	1	1	yes	maximal, characteristic, complemented
13	#13092275	C₂	1	1	yes	maximal, characteristic
14	#13092268	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#388	C₂	1	1	yes	maximal, characteristic
16	C₂³×Dic₂₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊Dic₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Dic₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
28	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
33	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
35	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
41	#10625049	C₄	1	8	yes	complemented
42	#10625048	—	2	4	no	—
43	C₄×Dic₄₆	—	2	4	no	—
44	C₂₃×#95	C₂²	1	1	yes	characteristic
45	C₂₃×#95	C₂²	1	1	yes	characteristic
46	C₂₃⋊#95	C₂²	1	1	yes	characteristic
47	C₂₃⋊#95	C₂²	1	1	yes	characteristic
48	C₂₃⋊#95	C₂²	1	1	yes	characteristic
49	C₂₃⋊#95	C₂²	1	1	yes	characteristic
50	C₂₃⋊#95	C₂²	1	1	yes	characteristic
51	C₂₃⋊#95	C₂²	1	1	yes	characteristic
52	C₂₃⋊#95	C₂²	1	1	yes	characteristic
53	C₂₃⋊#95	C₂²	1	1	yes	characteristic
54	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×Dic₂₃	—	2	2	no	—
62	C₂²×Dic₂₃	—	2	2	no	—
63	C₂²×Dic₂₃	C₂×C₄	1	4	yes	—
64	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
65	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
66	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
67	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
68	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
69	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
70	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
71	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
72	C₂×Dic₄₆	—	2	2	no	—
73	C₂×Dic₄₆	—	2	2	no	—
74	C₂×Dic₄₆	C₂×C₄	1	4	yes	—
75	C₄⋊C₄×C₂₃	—	4	1	no	—
76	C₂²⋊C₄×C₂₃	—	2	2	no	—
77	C₂²⋊C₄×C₂₃	—	4	1	no	—
78	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
79	C₄²×C₂₃	—	2	2	no	—
80	C₂²⋊Dic₂₃	—	2	2	no	—
81	C₂²⋊Dic₂₃	—	4	1	no	—
82	C₂²⋊Dic₂₃	C₂×C₄	1	4	yes	—
83	C₄⋊Dic₂₃	—	2	2	no	—
84	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
85	Dic₂₃⋊C₄	—	2	2	no	—
86	Dic₂₃⋊C₄	—	2	2	no	—
87	Dic₂₃⋊C₄	—	2	2	no	—
88	Dic₂₃⋊C₄	—	2	2	no	—
89	Dic₂₃⋊C₄	—	2	2	no	—
90	Dic₂₃⋊C₄	—	4	1	no	—
91	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
92	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
93	C₄×Dic₂₃	—	2	2	no	—
94	C₄×Dic₂₃	—	2	2	no	—
95	C₄×Dic₂₃	—	4	1	no	—
96	C₄×Dic₂₃	—	4	1	no	—
97	C₄×Dic₂₃	D₄	1	4	yes	—
98	C₂³×C₂₃	—	2	1	no	—
99	C₂³×C₂₃	—	2	1	no	—
100	C₂³×C₂₃	—	2	1	no	—
101	C₂³×C₂₃	—	2	1	no	—
102	C₂³×C₂₃	—	2	1	no	—
103	C₂³×C₂₃	—	2	1	no	—
104	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
105	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
106	C₂×C₄×C₂₃	—	2	1	no	—
107	C₂×C₄×C₂₃	—	2	1	no	—
108	C₂×C₄×C₂₃	—	2	1	no	—
109	C₂×C₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	2	1	no	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	—	2	1	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	2	1	no	—
117	C₂×C₄×C₂₃	—	2	1	no	—
118	C₂×C₄×C₂₃	—	2	1	no	—
119	C₂×C₄×C₂₃	—	2	1	no	—
120	C₂×C₄×C₂₃	—	2	1	no	—
121	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
122	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
123	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
124	C₂×Dic₂₃	—	2	1	no	—
125	C₂×Dic₂₃	—	2	1	no	—
126	C₂×Dic₂₃	—	2	1	no	—
127	C₂×Dic₂₃	—	2	1	no	—
128	C₂×Dic₂₃	—	2	1	no	—
129	C₂×Dic₂₃	—	2	1	no	—
130	C₂×Dic₂₃	—	2	1	no	—
131	C₂×Dic₂₃	—	2	1	no	—
132	C₂×Dic₂₃	—	2	8	no	—
133	C₂×Dic₂₃	—	2	1	no	—
134	C₂×Dic₂₃	—	2	1	no	—
135	C₂×Dic₂₃	—	2	1	no	—
136	C₂×Dic₂₃	—	2	1	no	—
137	C₂×Dic₂₃	—	2	1	no	—
138	C₂×Dic₂₃	—	2	1	no	—
139	C₂×Dic₂₃	—	2	1	no	—
140	C₂×Dic₂₃	—	2	1	no	—
141	C₂×Dic₂₃	—	2	1	no	—
142	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
143	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
144	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
145	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
146	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
147	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
148	C₂×Dic₂₃	#41	1	2	yes	—
149	Dic₄₆	—	2	8	no	—
150	#2323	—	23	1	no	maximal, Sylow
151	C₂²×C₂₃	—	2	2	no	—
152	C₂²×C₂₃	—	2	2	no	—
153	C₂²×C₂₃	—	2	2	no	—
154	C₂²×C₂₃	—	4	1	no	—
155	C₂²×C₂₃	—	4	1	no	—
156	C₂²×C₂₃	—	4	1	no	—
157	C₂²×C₂₃	—	4	1	no	—
158	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
159	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
160	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
161	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
163	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
164	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
165	C₄×C₂₃	—	2	2	no	—
166	C₄×C₂₃	—	2	2	no	—
167	C₄×C₂₃	—	2	2	no	—
168	C₄×C₂₃	—	4	1	no	—
169	C₄×C₂₃	—	4	1	no	—
170	C₄×C₂₃	—	4	1	no	—
171	Dic₂₃	—	2	2	no	—
172	Dic₂₃	—	2	2	no	—
173	Dic₂₃	—	2	2	no	—
174	Dic₂₃	—	2	2	no	—
175	Dic₂₃	—	4	1	no	—
176	Dic₂₃	—	4	1	no	—
177	Dic₂₃	—	4	1	no	—
178	Dic₂₃	C₄×D₄	1	4	yes	—
179	C₂×C₂²⋊Q₈	—	23	1	no	—
180	C₂×C₄×Q₈	—	23	1	no	—
181	C₂×#117	—	23	1	no	—
182	#391	—	23	1	no	—
183	#389	—	23	1	no	—
184	#389	—	23	1	no	—
185	#389	—	23	1	no	—
186	#388	—	23	1	no	—
187	#388	—	23	1	no	—
188	#388	—	23	1	no	—
189	#388	D₂₃	1	1	yes	characteristic
190	#387	—	23	1	no	—
191	#387	—	23	1	no	—
192	#380	—	23	1	no	—
193	C₄×C₄⋊C₄	—	23	1	no	—
194	C₂×C₂₃	—	4	2	no	—
195	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
196	C₂×C₂₃	#519	1	1	yes	characteristic
197	C₂×C₂₃	#520	1	1	yes	characteristic
198	C₂×C₂₃	#536	1	1	yes	characteristic
199	C₂×C₂₃	#537	1	1	yes	characteristic
200	C₂×C₂₃	#541	1	1	yes	characteristic
201	C₂×C₂₃	#542	1	1	yes	characteristic
202	C₂²×Q₈	—	23	1	no	—
203	C₂³×C₄	—	23	1	no	—
204	C₂²⋊Q₈	—	23	8	no	—
205	C₄×Q₈	—	46	4	no	—
206	#117	—	46	4	no	—
207	C₂×C₄⋊C₄	—	23	1	no	—
208	C₂×C₄⋊C₄	—	23	1	no	—
209	C₂×C₄⋊C₄	—	23	1	no	—
210	C₂×C₄⋊C₄	—	23	1	no	—
211	C₂×C₄⋊C₄	—	23	1	no	—
212	C₂×C₄⋊C₄	—	23	1	no	—
213	C₂×C₄⋊C₄	—	23	1	no	—
214	C₂×C₄⋊C₄	—	23	1	no	—
215	C₂×C₄⋊C₄	—	23	1	no	—
216	C₂×C₄⋊C₄	—	23	1	no	—
217	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
218	C₂×C₂²⋊C₄	—	23	1	no	—
219	C₂×C₂²⋊C₄	—	23	1	no	—
220	C₂×C₂²⋊C₄	—	23	1	no	—
221	C₂×C₂²⋊C₄	C₂×D₂₃	1	1	yes	characteristic
222	C₂×C₂²⋊C₄	C₂×D₂₃	1	1	yes	characteristic
223	C₂×C₂²⋊C₄	C₂×D₂₃	1	1	yes	characteristic
224	C₂×C₄²	—	23	1	no	—
225	C₂×C₄²	—	23	1	no	—
226	C₂×C₄²	—	23	1	no	—
227	C₂×C₄²	—	23	1	no	—
228	C₂×C₄²	—	23	1	no	—
229	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
230	#95	—	23	1	no	—
231	#95	—	23	1	no	—
232	#95	—	23	1	no	—
233	#95	—	23	1	no	—
234	#95	—	23	1	no	—
235	#95	—	23	1	no	—
236	#95	—	23	1	no	—
237	#95	—	23	1	no	—
238	#95	C₂×D₂₃	1	1	yes	characteristic
239	#95	C₂×D₂₃	1	1	yes	characteristic
240	C₂₃	#2323	1	1	yes	characteristic, complemented, Sylow
241	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
242	C₂×Q₈	—	23	4	no	—
243	C₂×Q₈	—	46	2	no	—
244	C₂×Q₈	—	46	2	no	—
245	C₂²×C₄	—	23	1	no	—
246	C₂²×C₄	—	23	4	no	—
247	C₂²×C₄	—	23	1	no	—
248	C₂²×C₄	—	23	1	no	—
249	C₂²×C₄	—	23	1	no	—
250	C₂²×C₄	—	23	1	no	—
251	C₂²×C₄	—	23	1	no	—
252	C₂²×C₄	—	23	1	no	—
253	C₂²×C₄	—	23	1	no	—
254	C₂²×C₄	—	46	2	no	—
255	C₂²×C₄	—	46	2	no	—
256	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
257	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
258	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
259	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
260	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
261	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	23	4	no	—
264	C₄⋊C₄	—	23	4	no	—
265	C₄⋊C₄	—	23	4	no	—
266	C₄⋊C₄	—	46	2	no	—
267	C₄⋊C₄	—	46	2	no	—
268	C₄⋊C₄	—	46	2	no	—
269	C₄⋊C₄	—	46	2	no	—
270	C₄⋊C₄	—	46	2	no	—
271	C₄⋊C₄	—	46	2	no	—
272	C₄⋊C₄	—	92	1	no	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	4	1	no	—
275	C₂²⋊C₄	—	23	4	no	—
276	C₂²⋊C₄	—	46	2	no	—
277	C₂²⋊C₄	—	92	1	no	—
278	C₂²⋊C₄	C₄×D₂₃	1	4	yes	—
279	C₄²	—	2	2	no	—
280	C₄²	—	23	4	no	—
281	C₄²	—	46	2	no	—
282	C₄²	—	46	2	no	—
283	C₄²	—	92	1	no	—
284	C₄²	—	92	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	C₂×C₄×D₂₃	1	2	yes	—
292	C₂³	C₂³×D₂₃	1	1	yes	characteristic
293	Q₈	—	46	8	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	23	2	no	—
310	C₂×C₄	—	23	2	no	—
311	C₂×C₄	—	23	2	no	—
312	C₂×C₄	—	23	2	no	—
313	C₂×C₄	—	23	2	no	—
314	C₂×C₄	—	23	2	no	—
315	C₂×C₄	—	23	2	no	—
316	C₂×C₄	—	46	8	no	—
317	C₂×C₄	—	46	1	no	—
318	C₂×C₄	—	46	1	no	—
319	C₂×C₄	—	46	1	no	—
320	C₂×C₄	—	46	1	no	—
321	C₂×C₄	—	46	1	no	—
322	C₂×C₄	—	46	1	no	—
323	C₂×C₄	—	46	1	no	—
324	C₂×C₄	—	46	1	no	—
325	C₂×C₄	—	46	1	no	—
326	C₂×C₄	—	46	1	no	—
327	C₂×C₄	—	46	1	no	—
328	C₂×C₄	—	46	1	no	—
329	C₂×C₄	—	46	1	no	—
330	C₂×C₄	—	46	1	no	—
331	C₂×C₄	—	46	1	no	—
332	C₂×C₄	—	46	1	no	—
333	C₂×C₄	—	46	1	no	—
334	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
335	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
336	C₂×C₄	D₄×D₂₃	1	2	yes	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
345	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
346	C₂²	#10625155	1	1	yes	characteristic
347	C₂²	#41×D₂₃	1	1	yes	characteristic
348	C₂²	#10625161	1	1	yes	characteristic
349	C₂²	#10625161	1	1	yes	characteristic
350	C₂²	#10625161	1	1	yes	characteristic
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	23	4	no	—
358	C₄	—	46	2	no	—
359	C₄	—	46	2	no	—
360	C₄	—	46	2	no	—
361	C₄	—	46	2	no	—
362	C₄	—	92	1	no	—
363	C₄	—	92	1	no	—
364	C₄	—	92	1	no	—
365	C₂	—	4	2	no	—
366	C₂	C₄×D₄×D₂₃	1	1	yes	characteristic
367	C₂	#13092894	1	1	yes	characteristic
368	C₂	#13092896	1	1	yes	characteristic
369	C₂	#13092989	1	1	yes	characteristic
370	C₂	#13092996	1	1	yes	characteristic
371	C₂	#13093024	1	1	yes	characteristic
372	C₂	#13093031	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#388	C₂
socle, cosocle	C₂³×C₂₃	C₂⁴
prime radical	C₂⁴×C₂₃	C₂³
hypercenter	#388	D₂₃
hypocenter	C₂₃	#2323
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ #388

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₄₆	C₂	semidirect
4. #13092321	C₂	semidirect
5. #13092314	C₂	semidirect
6. #13092313	C₂	semidirect
7. #13092312	C₂	semidirect
11. #13092302	C₂	semidirect
12. #13092301	C₂	semidirect
14. #13092268	C₂	semidirect
41. #10625049	C₄	semidirect
240. C₂₃	#2323	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625049	2
C₂×C₄×Dic₄₆	2
C₂×#10625048	2
#13092321	2
#13092314	2
#13092313	2
#13092312	2
#13092310	2
#13092305	2
#13092302	2
#13092301	2
#13092275	2
#13092268	2
C₂₃×#388	2
#2323	23