Subgroups — #15676255

Order 2944.

532 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625048	C₂	1	1	yes	maximal, characteristic
2	#13092304	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092304	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092298	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092298	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092292	C₂	1	1	yes	maximal, characteristic
7	#13092284	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092282	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092278	C₂	1	1	yes	maximal, characteristic
10	C₄⋊C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
12	C₄×C₂²⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic
13	#13092262	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₂₃⋊#95	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#385	C₂	1	1	yes	maximal, characteristic
16	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂³×Dic₂₃	C₂²	1	1	yes	characteristic
18	C₂³×Dic₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×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×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
27	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
35	#10625048	C₄	1	8	yes	complemented
36	C₂₃×#95	C₂²	1	1	yes	characteristic
37	C₂₃×#95	C₂²	1	1	yes	characteristic
38	C₂₃⋊#95	C₂²	1	1	yes	characteristic
39	C₂₃⋊#95	C₂²	1	1	yes	characteristic
40	C₂₃⋊#95	C₂²	1	1	yes	characteristic
41	C₂₃⋊#95	C₂²	1	1	yes	characteristic
42	C₂₃⋊#95	C₂²	1	1	yes	characteristic
43	C₂₃⋊#95	C₂²	1	1	yes	characteristic
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	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	—	2	2	no	—
54	C₂²×C₄×C₂₃	—	2	2	no	—
55	C₂²×C₄×C₂₃	—	4	1	no	—
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×C₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×Dic₂₃	—	2	2	no	—
63	C₂²×Dic₂₃	—	2	2	no	—
64	C₂²×Dic₂₃	—	2	2	no	—
65	C₂²×Dic₂₃	—	2	2	no	—
66	C₂²×Dic₂₃	—	4	1	no	—
67	C₂²×Dic₂₃	C₂×C₄	1	4	yes	—
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	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
73	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
74	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
75	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₂₃	—	2	2	no	—
77	C₄⋊C₄×C₂₃	—	4	1	no	—
78	C₂²⋊C₄×C₂₃	—	2	2	no	—
79	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
80	C₂²⋊Dic₂₃	—	2	2	no	—
81	C₂²⋊Dic₂₃	C₂×C₄	1	4	yes	—
82	C₄⋊Dic₂₃	—	2	2	no	—
83	C₄⋊Dic₂₃	—	4	1	no	—
84	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	complemented
85	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	complemented
86	C₄×Dic₂₃	—	2	2	no	—
87	C₄×Dic₂₃	—	2	2	no	—
88	C₄×Dic₂₃	—	4	1	no	—
89	C₄×Dic₂₃	—	4	1	no	—
90	C₄×Dic₂₃	C₂×C₄	1	4	yes	complemented
91	C₄×Dic₂₃	C₂×C₄	1	4	yes	complemented
92	C₂³×C₂₃	—	2	1	no	—
93	C₂³×C₂₃	—	2	1	no	—
94	C₂³×C₂₃	—	2	1	no	—
95	C₂³×C₂₃	—	2	1	no	—
96	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
97	C₂³×C₂₃	#41	1	2	yes	—
98	C₂³×C₂₃	#41	1	2	yes	—
99	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
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	—	4	4	no	—
116	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
117	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
118	C₂×C₄×C₂₃	#41	1	2	yes	—
119	C₂×Dic₂₃	—	2	8	no	—
120	C₂×Dic₂₃	—	2	1	no	—
121	C₂×Dic₂₃	—	2	1	no	—
122	C₂×Dic₂₃	—	2	1	no	—
123	C₂×Dic₂₃	—	2	1	no	—
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	1	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	—	4	4	no	—
140	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
141	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
142	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
143	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
144	C₂×Dic₂₃	#41	1	2	yes	—
145	#2343	—	23	1	no	maximal, Sylow
146	C₂²×C₂₃	—	2	2	no	—
147	C₂²×C₂₃	—	2	2	no	—
148	C₂²×C₂₃	—	2	2	no	—
149	C₂²×C₂₃	—	2	2	no	—
150	C₂²×C₂₃	—	2	2	no	—
151	C₂²×C₂₃	—	2	2	no	—
152	C₂²×C₂₃	#117	1	4	yes	—
153	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
154	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
155	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
156	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
157	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
158	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
160	C₄×C₂₃	—	2	2	no	—
161	C₄×C₂₃	—	2	2	no	—
162	C₄×C₂₃	—	2	2	no	—
163	C₄×C₂₃	—	4	1	no	—
164	C₄×C₂₃	—	4	1	no	—
165	C₄×C₂₃	—	4	1	no	—
166	Dic₂₃	—	2	2	no	—
167	Dic₂₃	—	2	2	no	—
168	Dic₂₃	—	2	2	no	—
169	Dic₂₃	—	2	2	no	—
170	Dic₂₃	—	2	2	no	—
171	Dic₂₃	—	4	1	no	—
172	Dic₂₃	—	4	1	no	—
173	Dic₂₃	—	4	1	no	—
174	C₂×#117	—	23	1	no	—
175	#387	—	23	1	no	—
176	#387	—	23	1	no	—
177	#387	—	23	1	no	—
178	#387	—	23	1	no	—
179	#385	—	23	1	no	—
180	#385	D₂₃	1	1	yes	characteristic
181	#383	—	23	1	no	—
182	#382	—	23	1	no	—
183	#381	—	23	1	no	—
184	C₄×C₄⋊C₄	—	23	1	no	—
185	C₄×C₂²⋊C₄	—	23	1	no	—
186	C₄×C₂²⋊C₄	—	23	1	no	—
187	#376	—	23	1	no	—
188	C₂×#95	—	23	1	no	—
189	C₂×C₂₃	—	2	4	no	—
190	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
191	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
192	C₂×C₂₃	#520	1	1	yes	characteristic
193	C₂×C₂₃	#551	1	1	yes	characteristic
194	C₂×C₂₃	#551	1	1	yes	characteristic
195	C₂×C₂₃	#552	1	1	yes	characteristic
196	C₂×C₂₃	#552	1	1	yes	characteristic
197	C₂³×C₄	—	23	1	no	—
198	C₂³×C₄	—	23	1	no	—
199	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
200	#117	—	23	8	no	—
201	C₂×C₄⋊C₄	—	23	1	no	—
202	C₂×C₄⋊C₄	—	23	1	no	—
203	C₂×C₄⋊C₄	—	23	1	no	—
204	C₂×C₄⋊C₄	—	23	1	no	—
205	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
206	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
207	C₂×C₂²⋊C₄	—	23	1	no	—
208	C₂×C₂²⋊C₄	—	23	1	no	—
209	C₂×C₂²⋊C₄	C₂×D₂₃	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	C₂×D₂₃	1	1	yes	characteristic
211	C₂×C₄²	—	23	1	no	—
212	C₂×C₄²	—	23	1	no	—
213	C₂×C₄²	—	23	1	no	—
214	C₂×C₄²	—	23	1	no	—
215	C₂×C₄²	—	23	1	no	—
216	C₂×C₄²	—	23	1	no	—
217	#95	—	23	1	no	—
218	#95	—	23	1	no	—
219	#95	—	23	1	no	—
220	#95	—	23	1	no	—
221	#95	—	23	1	no	—
222	#95	—	23	1	no	—
223	#95	—	23	1	no	—
224	#95	—	23	1	no	—
225	#95	—	23	1	no	—
226	#95	—	23	1	no	—
227	#95	—	23	1	no	—
228	#95	—	23	1	no	—
229	#95	—	23	1	no	—
230	#95	—	23	1	no	—
231	#95	C₂×D₂₃	1	1	yes	characteristic
232	#95	C₂×D₂₃	1	1	yes	characteristic
233	C₂₃	#2343	1	1	yes	characteristic, complemented, Sylow
234	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	23	4	no	—
239	C₂²×C₄	—	23	1	no	—
240	C₂²×C₄	—	23	1	no	—
241	C₂²×C₄	—	23	1	no	—
242	C₂²×C₄	—	23	1	no	—
243	C₂²×C₄	—	23	1	no	—
244	C₂²×C₄	—	23	1	no	—
245	C₂²×C₄	—	23	1	no	—
246	C₂²×C₄	—	23	1	no	—
247	C₂²×C₄	—	46	2	no	—
248	C₂²×C₄	—	46	2	no	—
249	C₂²×C₄	—	46	2	no	—
250	C₂²×C₄	—	46	2	no	—
251	C₂²×C₄	—	92	1	no	—
252	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
254	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
255	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
256	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
257	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	23	4	no	—
261	C₄⋊C₄	—	23	4	no	—
262	C₄⋊C₄	—	46	2	no	—
263	C₄⋊C₄	—	92	1	no	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₂²⋊C₄	—	23	4	no	—
266	C₂²⋊C₄	—	46	2	no	—
267	C₂²⋊C₄	C₄×D₂₃	1	4	yes	—
268	C₄²	—	23	4	no	—
269	C₄²	—	23	4	no	—
270	C₄²	—	46	2	no	—
271	C₄²	—	46	2	no	—
272	C₄²	—	92	1	no	—
273	C₄²	—	92	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	C₂×C₄×D₂₃	1	2	yes	—
279	C₂³	#67290	1	2	yes	—
280	C₂³	#67292	1	2	yes	—
281	C₂³	C₂³×D₂₃	1	1	yes	characteristic
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	23	2	no	—
299	C₂×C₄	—	23	2	no	—
300	C₂×C₄	—	23	2	no	—
301	C₂×C₄	—	23	2	no	—
302	C₂×C₄	—	23	2	no	—
303	C₂×C₄	—	46	8	no	—
304	C₂×C₄	—	46	1	no	—
305	C₂×C₄	—	46	1	no	—
306	C₂×C₄	—	46	1	no	—
307	C₂×C₄	—	46	1	no	—
308	C₂×C₄	—	46	1	no	—
309	C₂×C₄	—	46	1	no	—
310	C₂×C₄	—	46	1	no	—
311	C₂×C₄	—	46	1	no	—
312	C₂×C₄	—	46	1	no	—
313	C₂×C₄	—	46	1	no	—
314	C₂×C₄	—	46	1	no	—
315	C₂×C₄	—	46	1	no	—
316	C₂×C₄	—	46	1	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	—	92	4	no	—
324	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
325	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
326	C₂×C₄	#67290	1	2	yes	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	#10625063	1	4	yes	—
334	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
335	C₂²	C₂×#67290	1	1	yes	characteristic
336	C₂²	C₂×#67290	1	1	yes	characteristic
337	C₂²	#10625155	1	1	yes	characteristic
338	C₂²	C₂×#67292	1	1	yes	characteristic
339	C₂²	#41×D₂₃	1	1	yes	characteristic
340	C₂²	#10625161	1	1	yes	characteristic
341	C₄	—	2	2	no	—
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	46	2	no	—
348	C₄	—	46	2	no	—
349	C₄	—	46	2	no	—
350	C₄	—	46	2	no	—
351	C₄	—	46	2	no	—
352	C₄	—	92	1	no	—
353	C₄	—	92	1	no	—
354	C₄	—	92	1	no	—
355	C₂	—	2	4	no	—
356	C₂	C₂×#10625063	1	1	yes	characteristic
357	C₂	C₄×#67290	1	1	yes	characteristic
358	C₂	#13092896	1	1	yes	characteristic
359	C₂	#13093094	1	1	yes	characteristic
360	C₂	#13093098	1	1	yes	characteristic
361	C₂	#13093104	1	1	yes	characteristic
362	C₂	#13093109	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#385	C₂
socle, cosocle	C₂³×C₂₃	C₂⁴
prime radical	C₂⁴×C₂₃	C₂³
hypercenter	#385	D₂₃
hypocenter	C₂₃	#2343
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ #385

Complements

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

N	H ≅ G/N	type
2. #13092304	C₂	semidirect
3. #13092304	C₂	semidirect
4. #13092298	C₂	semidirect
5. #13092298	C₂	semidirect
7. #13092284	C₂	semidirect
8. #13092282	C₂	semidirect
10. C₄⋊C₄×Dic₂₃	C₂	semidirect
13. #13092262	C₂	semidirect
35. #10625048	C₄	semidirect
84. Dic₂₃⋊C₄	C₂×C₄	semidirect
85. Dic₂₃⋊C₄	C₂×C₄	semidirect
90. C₄×Dic₂₃	C₂×C₄	semidirect
91. C₄×Dic₂₃	C₂×C₄	semidirect
233. C₂₃	#2343	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625048	2
#13092304	2
#13092298	2
#13092292	2
#13092284	2
#13092282	2
#13092278	2
C₄⋊C₄×Dic₂₃	2
C₂²⋊C₄×Dic₂₃	2
C₄×C₂²⋊Dic₂₃	2
#13092262	2
C₂×C₂₃⋊#95	2
C₂₃×#385	2
#2343	23