Subgroups — #15676271

Order 2944.

548 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×D₂₃	C₂	1	1	yes	maximal, characteristic
2	#13092314	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092312	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092303	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092302	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092300	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092276	C₂	1	1	yes	maximal, characteristic
8	#13092276	C₂	1	1	yes	maximal, characteristic
9	#13092274	C₂	1	1	yes	maximal, characteristic
10	C₄×Dic₂₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	#13092268	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×(C₂×D₂₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
13	#13092264	C₂	1	1	yes	maximal, characteristic
14	#13092264	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#387	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₂₃	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×C₄²×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×(C₂×D₂₃)⋊C₄	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×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₂₃	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	C₄⋊C₄×D₂₃	C₄	1	8	yes	complemented
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	C₂³×D₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×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×Dic₂₃	C₂³	1	1	yes	characteristic
61	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
62	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
63	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
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	C₂×C₄×D₂₃	—	2	2	no	—
68	C₂×C₄×D₂₃	—	2	2	no	—
69	C₂×C₄×D₂₃	—	4	1	no	—
70	C₂×C₄×D₂₃	—	4	1	no	—
71	C₂×C₄×D₂₃	—	4	1	no	—
72	C₂×C₄×D₂₃	—	4	1	no	—
73	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
74	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
75	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
76	C₄⋊C₄×C₂₃	—	4	1	no	—
77	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
78	C₄²×C₂₃	—	4	1	no	—
79	(C₂×D₂₃)⋊C₄	—	2	2	no	—
80	(C₂×D₂₃)⋊C₄	—	2	2	no	—
81	(C₂×D₂₃)⋊C₄	—	2	2	no	—
82	(C₂×D₂₃)⋊C₄	Q₈	1	4	yes	—
83	C₄⋊Dic₂₃	—	4	1	no	—
84	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
85	Dic₂₃⋊C₄	—	2	2	no	—
86	Dic₂₃⋊C₄	—	2	2	no	—
87	Dic₂₃⋊C₄	—	4	1	no	—
88	Dic₂₃⋊C₄	—	4	1	no	—
89	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
90	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
91	C₄×Dic₂₃	—	2	2	no	—
92	C₄×Dic₂₃	—	2	2	no	—
93	C₄×Dic₂₃	—	4	1	no	—
94	C₄×Dic₂₃	—	4	1	no	—
95	C₄×Dic₂₃	—	4	1	no	—
96	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
97	C₂²×D₂₃	—	2	1	no	—
98	C₂²×D₂₃	—	2	1	no	—
99	C₂²×D₂₃	—	2	1	no	—
100	C₂²×D₂₃	—	2	1	no	—
101	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
102	C₂²×D₂₃	C₂×Q₈	1	2	yes	—
103	C₂²×D₂₃	#41	1	2	yes	—
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	—	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	C₂²×C₄	1	2	yes	—
124	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
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	—	2	1	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	C₂×Q₈	1	2	yes	—
144	C₂×Dic₂₃	#41	1	2	yes	—
145	C₂×Dic₂₃	#41	1	2	yes	—
146	C₄×D₂₃	—	2	8	no	—
147	C₄×D₂₃	—	4	4	no	—
148	C₄×D₂₃	—	4	4	no	—
149	#2345	—	23	1	no	maximal, Sylow
150	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
151	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
152	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
153	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
154	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
155	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂×D₂₃	—	2	2	no	—
158	C₂×D₂₃	—	2	2	no	—
159	C₂×D₂₃	—	2	2	no	—
160	C₂×D₂₃	—	2	2	no	—
161	C₂×D₂₃	—	2	2	no	—
162	C₂×D₂₃	—	2	2	no	—
163	C₂×D₂₃	C₄×Q₈	1	4	yes	—
164	C₄×C₂₃	—	2	2	no	—
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	—	4	1	no	—
175	Dic₂₃	—	4	1	no	—
176	Dic₂₃	—	4	1	no	—
177	Dic₂₃	#117	1	4	yes	—
178	C₂²×C₄⋊C₄	—	23	1	no	—
179	#389	—	23	1	no	—
180	#389	—	23	1	no	—
181	#387	—	23	1	no	—
182	#387	—	23	1	no	—
183	#387	—	23	1	no	—
184	#387	D₂₃	1	1	yes	characteristic, complemented
185	#380	—	23	1	no	—
186	#380	—	23	1	no	—
187	#380	—	23	1	no	—
188	C₄×C₄⋊C₄	—	23	1	no	—
189	C₄×C₄⋊C₄	—	23	1	no	—
190	C₄×C₂²⋊C₄	—	23	1	no	—
191	C₄×C₂²⋊C₄	—	23	1	no	—
192	C₄×C₂²⋊C₄	—	23	1	no	—
193	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
194	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
195	C₂×C₂₃	#518	1	1	yes	characteristic
196	C₂×C₂₃	#553	1	1	yes	characteristic
197	C₂×C₂₃	#553	1	1	yes	characteristic
198	C₂×C₂₃	#554	1	1	yes	characteristic
199	C₂×C₂₃	#554	1	1	yes	characteristic
200	D₂₃	—	2	4	no	—
201	C₂³×C₄	—	23	1	no	—
202	C₂³×C₄	—	23	1	no	—
203	C₂³×C₄	—	23	1	no	—
204	C₂×C₄⋊C₄	—	23	1	no	—
205	C₂×C₄⋊C₄	—	23	1	no	—
206	C₂×C₄⋊C₄	—	23	1	no	—
207	C₂×C₄⋊C₄	—	23	8	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	C₂×D₂₃	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
215	C₂×C₂²⋊C₄	—	23	1	no	—
216	C₂×C₂²⋊C₄	—	23	1	no	—
217	C₂×C₂²⋊C₄	—	23	1	no	—
218	C₂×C₂²⋊C₄	—	23	1	no	—
219	C₂×C₄²	—	23	1	no	—
220	C₂×C₄²	—	23	1	no	—
221	C₂×C₄²	—	23	1	no	—
222	C₂×C₄²	—	23	1	no	—
223	C₂×C₄²	—	23	1	no	—
224	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
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	—	23	1	no	—
232	#95	—	23	1	no	—
233	#95	C₂×D₂₃	1	1	yes	characteristic
234	#95	C₂×D₂₃	1	1	yes	characteristic
235	#95	C₂×D₂₃	1	1	yes	characteristic
236	#95	C₂×D₂₃	1	1	yes	characteristic
237	C₂₃	#2345	1	1	yes	characteristic, complemented, Sylow
238	C₂⁴	—	23	1	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	4	no	—
244	C₂²×C₄	—	23	4	no	—
245	C₂²×C₄	—	23	4	no	—
246	C₂²×C₄	—	23	1	no	—
247	C₂²×C₄	—	23	1	no	—
248	C₂²×C₄	—	23	1	no	—
249	C₂²×C₄	—	46	2	no	—
250	C₂²×C₄	—	46	2	no	—
251	C₂²×C₄	—	92	1	no	—
252	C₂²×C₄	—	92	1	no	—
253	C₂²×C₄	—	92	1	no	—
254	C₂²×C₄	—	92	1	no	—
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	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	—	92	1	no	—
269	C₄⋊C₄	—	92	1	no	—
270	C₄⋊C₄	—	92	1	no	—
271	C₄⋊C₄	C₄×D₂₃	1	4	yes	—
272	C₂²⋊C₄	—	23	4	no	—
273	C₂²⋊C₄	—	46	2	no	—
274	C₂²⋊C₄	—	46	2	no	—
275	C₂²⋊C₄	—	46	2	no	—
276	C₄²	—	4	1	no	—
277	C₄²	—	46	2	no	—
278	C₄²	—	46	2	no	—
279	C₄²	—	92	1	no	—
280	C₄²	—	92	1	no	—
281	C₄²	—	92	1	no	—
282	C₂³	—	23	2	no	—
283	C₂³	—	23	2	no	—
284	C₂³	—	23	2	no	—
285	C₂³	—	46	1	no	—
286	C₂³	—	46	1	no	—
287	C₂³	—	46	1	no	—
288	C₂³	—	46	1	no	—
289	C₂³	C₂³×D₂₃	1	1	yes	characteristic
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	—	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	—	23	2	no	—
308	C₂×C₄	—	23	2	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	—	46	8	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	—	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	—	92	4	no	—
330	C₂×C₄	—	92	4	no	—
331	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
332	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
333	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
334	C₂×C₄	Q₈×D₂₃	1	2	yes	—
335	C₂²	—	23	4	no	—
336	C₂²	—	46	2	no	—
337	C₂²	—	46	2	no	—
338	C₂²	—	46	2	no	—
339	C₂²	—	46	2	no	—
340	C₂²	—	46	2	no	—
341	C₂²	—	46	2	no	—
342	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
343	C₂²	#10625155	1	1	yes	characteristic
344	C₂²	C₂×Q₈×D₂₃	1	1	yes	characteristic
345	C₂²	#41×D₂₃	1	1	yes	characteristic
346	C₂²	#41×D₂₃	1	1	yes	characteristic
347	C₂²	#41×D₂₃	1	1	yes	characteristic
348	C₂²	#10625160	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	23	4	no	—
357	C₄	—	46	2	no	—
358	C₄	—	46	2	no	—
359	C₄	—	46	2	no	—
360	C₄	—	92	1	no	—
361	C₄	—	92	1	no	—
362	C₄	—	92	1	no	—
363	C₂	—	46	4	no	—
364	C₂	#117×D₂₃	1	1	yes	characteristic
365	C₂	C₄×Q₈×D₂₃	1	1	yes	characteristic
366	C₂	#13092889	1	1	yes	characteristic
367	C₂	#13093112	1	1	yes	characteristic
368	C₂	#13093118	1	1	yes	characteristic
369	C₂	#13093121	1	1	yes	characteristic
370	C₂	#13093123	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#387	C₂
socle, cosocle	C₂³×C₂₃	C₂⁴
prime radical	C₂³×D₂₃	C₂³
hypercenter	#387	D₂₃
hypocenter	C₂₃	#2345
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ #387

Complements

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

N	H ≅ G/N	type
2. #13092314	C₂	semidirect
3. #13092312	C₂	semidirect
4. #13092303	C₂	semidirect
5. #13092302	C₂	semidirect
6. #13092300	C₂	semidirect
10. C₄×Dic₂₃⋊C₄	C₂	semidirect
11. #13092268	C₂	semidirect
15. C₂₃×#387	C₂	semidirect
39. C₄⋊C₄×D₂₃	C₄	semidirect
184. #387	D₂₃	semidirect
237. C₂₃	#2345	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×D₂₃	2
#13092314	2
#13092312	2
#13092303	2
#13092302	2
#13092300	2
#13092276	2
#13092274	2
C₄×Dic₂₃⋊C₄	2
#13092268	2
C₄×(C₂×D₂₃)⋊C₄	2
#13092264	2
C₂₃×#387	2
#2345	23