Subgroups — #15683889

Order 2944.

532 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625089	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10625086	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625087	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092992	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092987	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092811	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092810	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092808	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092799	C₂	1	1	yes	maximal, characteristic, complemented
11	#13092798	C₂	1	1	yes	maximal, characteristic, complemented
12	#13092796	C₂	1	1	yes	maximal, characteristic, complemented
13	#13092795	C₂	1	1	yes	maximal, characteristic, complemented
14	#13092374	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#67292	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₂₃	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₂₃	C₂²	1	1	yes	characteristic
19	C₂×#67290	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂₃×#124	C₂²	1	1	yes	characteristic
26	#10625118	C₂²	1	1	yes	characteristic
27	#10625115	C₂²	1	1	yes	characteristic
28	#10625115	C₂²	1	1	yes	characteristic
29	C₂×#67275	C₂²	1	1	yes	characteristic
30	C₂×#67275	C₂²	1	1	yes	characteristic
31	C₂×#67275	C₂²	1	1	yes	characteristic
32	C₂×#67275	C₂²	1	1	yes	characteristic
33	C₂×#67275	C₂²	1	1	yes	characteristic
34	C₂×#67274	C₂²	1	1	yes	characteristic
35	#10625107	C₂²	1	1	yes	characteristic
36	C₂×#67273	C₂²	1	1	yes	characteristic
37	C₂×#67272	C₂²	1	1	yes	characteristic
38	#10625089	—	2	4	no	—
39	Q₁₆×D₂₃	—	2	4	no	—
40	#10625087	—	2	4	no	—
41	#10625086	—	2	4	no	—
42	C₂×Dic₉₂	C₂²	1	1	yes	characteristic
43	C₂×Dic₉₂	C₂²	1	1	yes	characteristic
44	C₂×#67263	C₂²	1	1	yes	characteristic
45	C₂×#67263	C₂²	1	1	yes	characteristic
46	C₂×#67262	C₂²	1	1	yes	characteristic
47	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
48	#10625067	—	2	2	no	—
49	#10625065	—	2	2	no	—
50	#10625054	—	2	2	no	—
51	#10625049	—	2	2	no	—
52	#10625046	C₂²	1	1	yes	characteristic
53	#10625043	C₂²	1	1	yes	characteristic
54	#10625040	C₂²	1	1	yes	characteristic
55	C₂₃×#97	C₂²	1	1	yes	characteristic
56	#10624997	C₂²	1	1	yes	characteristic
57	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
58	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₂₃	—	2	1	no	—
64	C₂²×Dic₂₃	—	2	1	no	—
65	#67292	—	2	2	no	—
66	#67292	—	2	2	no	—
67	Q₈×D₂₃	—	2	2	no	—
68	Q₈×D₂₃	—	2	2	no	—
69	Q₈×D₂₃	—	2	2	no	—
70	Q₈×D₂₃	—	2	2	no	—
71	#67290	—	2	2	no	—
72	#67290	—	2	2	no	—
73	C₂×D₉₂	—	2	1	no	—
74	C₂×D₉₂	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₂₃	—	2	1	no	—
76	C₂×C₄×D₂₃	—	2	1	no	—
77	C₂×C₄×D₂₃	—	2	1	no	—
78	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
79	C₂×Dic₄₆	—	2	1	no	—
80	C₂×Dic₄₆	—	2	1	no	—
81	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
82	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
83	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
84	Q₁₆×C₂₃	—	2	2	no	—
85	Q₁₆×C₂₃	—	2	2	no	—
86	SD₁₆×C₂₃	—	2	2	no	—
87	SD₁₆×C₂₃	—	2	2	no	—
88	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
89	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
90	C₄⋊C₄×C₂₃	—	2	2	no	—
91	C₂²⋊C₄×C₂₃	—	2	2	no	—
92	C₄²×C₂₃	C₂³	1	1	yes	characteristic
93	C₂²⋊Dic₂₃	—	2	2	no	—
94	#67275	—	2	2	no	—
95	#67275	—	2	2	no	—
96	#67275	—	2	2	no	—
97	#67275	—	2	2	no	—
98	#67275	—	2	2	no	—
99	#67274	—	2	2	no	—
100	#67273	—	2	2	no	—
101	#67272	—	2	2	no	—
102	(C₂×D₂₃)⋊C₄	—	2	2	no	—
103	(C₂×D₂₃)⋊C₄	—	2	2	no	—
104	(C₂×D₂₃)⋊C₄	D₄	1	2	yes	—
105	C₄⋊Dic₂₃	—	2	2	no	—
106	Dic₂₃⋊C₄	—	2	2	no	—
107	Dic₂₃⋊C₄	—	2	2	no	—
108	Dic₂₃⋊C₄	D₄	1	2	yes	—
109	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
110	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
111	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
112	Dic₉₂	—	2	2	no	—
113	Dic₉₂	—	2	2	no	—
114	#67263	—	2	2	no	—
115	#67263	—	2	2	no	—
116	#67262	D₄	1	4	yes	—
117	C₈×D₂₃	D₄	1	4	yes	—
118	C₂³×C₂₃	—	2	1	no	—
119	C₂²×D₂₃	—	2	1	no	—
120	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
121	Q₈×C₂₃	—	2	1	no	—
122	Q₈×C₂₃	—	2	1	no	—
123	Q₈×C₂₃	—	2	1	no	—
124	Q₈×C₂₃	—	2	1	no	—
125	Q₈×C₂₃	—	2	1	no	—
126	Q₈×C₂₃	—	2	1	no	—
127	D₄×C₂₃	—	2	1	no	—
128	D₄×C₂₃	—	2	1	no	—
129	C₂×C₄×C₂₃	—	2	1	no	—
130	C₂×C₄×C₂₃	—	2	1	no	—
131	C₂×C₄×C₂₃	—	2	1	no	—
132	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
133	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
134	C₂²⋊D₂₃	—	4	2	no	—
135	C₂×Dic₂₃	—	2	1	no	—
136	C₂×Dic₂₃	—	2	1	no	—
137	C₂×Dic₂₃	—	2	1	no	—
138	C₂×Dic₂₃	—	4	2	no	—
139	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
140	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
141	D₉₂	—	2	1	no	—
142	D₉₂	—	2	1	no	—
143	D₉₂	—	4	2	no	—
144	C₄×D₂₃	—	4	2	no	—
145	C₄×D₂₃	—	4	2	no	—
146	C₄×D₂₃	—	4	2	no	—
147	C₄×D₂₃	C₂×D₄	1	2	yes	—
148	C₄×D₂₃	C₂×D₄	1	2	yes	—
149	Dic₄₆	—	2	1	no	—
150	Dic₄₆	—	2	1	no	—
151	Dic₄₆	—	2	1	no	—
152	Dic₄₆	—	2	1	no	—
153	Dic₄₆	—	2	1	no	—
154	Dic₄₆	—	2	1	no	—
155	Dic₄₆	—	4	2	no	—
156	Dic₄₆	—	4	2	no	—
157	C₈×C₂₃	C₂×D₄	1	2	yes	—
158	C₈×C₂₃	C₂×D₄	1	2	yes	—
159	C₂₃⋊C₈	C₂×D₄	1	2	yes	—
160	C₂₃⋊C₈	C₂×D₄	1	2	yes	—
161	#3156	—	23	1	no	maximal, Sylow
162	C₂²×C₂₃	—	4	1	no	—
163	C₂²×C₂₃	—	4	2	no	—
164	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
165	C₂×D₂₃	—	2	2	no	—
166	C₂×D₂₃	—	4	1	no	—
167	C₂×D₂₃	—	4	2	no	—
168	C₂×D₂₃	#127	1	2	yes	—
169	C₄×C₂₃	—	2	2	no	—
170	C₄×C₂₃	—	4	1	no	—
171	C₄×C₂₃	—	4	1	no	—
172	C₄×C₂₃	—	4	1	no	—
173	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
174	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
175	Dic₂₃	—	2	2	no	—
176	Dic₂₃	—	4	1	no	—
177	Dic₂₃	—	4	1	no	—
178	Dic₂₃	—	4	1	no	—
179	Dic₂₃	#127	1	2	yes	—
180	C₂×#137	—	23	1	no	—
181	C₂×#137	—	23	1	no	—
182	C₂×#135	—	23	1	no	—
183	C₂²×Q₁₆	—	23	1	no	—
184	#536	—	23	1	no	—
185	#536	—	23	1	no	—
186	#497	—	23	1	no	—
187	#497	—	23	1	no	—
188	#497	—	23	1	no	—
189	#497	D₂₃	1	1	yes	characteristic, complemented
190	#495	—	23	1	no	—
191	#495	—	23	1	no	—
192	#494	—	23	1	no	—
193	#494	—	23	1	no	—
194	#405	—	23	1	no	—
195	C₂×C₂₃	—	8	1	no	—
196	C₂×C₂₃	C₂×#127	1	1	yes	characteristic
197	C₂×C₂₃	#578	1	2	yes	—
198	D₂₃	—	2	2	no	—
199	D₂₃	—	8	1	no	—
200	C₂×#41	—	23	1	no	—
201	C₂×#41	—	23	1	no	—
202	C₂²×Q₈	—	23	1	no	—
203	C₂²×Q₈	—	23	1	no	—
204	#137	—	46	4	no	—
205	#137	—	46	4	no	—
206	#135	—	46	4	no	—
207	C₂×Q₁₆	—	23	1	no	—
208	C₂×Q₁₆	—	23	1	no	—
209	C₂×Q₁₆	—	23	1	no	—
210	C₂×Q₁₆	—	23	1	no	—
211	C₂×Q₁₆	—	23	1	no	—
212	C₂×Q₁₆	—	23	1	no	—
213	C₂×Q₁₆	—	23	1	no	—
214	C₂×Q₁₆	—	46	4	no	—
215	C₂×Q₁₆	C₂×D₂₃	1	1	yes	characteristic
216	C₂×Q₁₆	C₂×D₂₃	1	1	yes	characteristic
217	C₂×SD₁₆	—	23	1	no	—
218	C₂×SD₁₆	—	23	1	no	—
219	C₂×SD₁₆	—	23	1	no	—
220	C₂×SD₁₆	—	23	1	no	—
221	C₂×SD₁₆	C₂×D₂₃	1	1	yes	characteristic
222	C₂×SD₁₆	C₂×D₂₃	1	1	yes	characteristic
223	C₂×D₈	—	23	1	no	—
224	C₂×M₁₆	—	23	1	no	—
225	C₂²×C₈	—	23	1	no	—
226	C₄⋊Q₈	—	23	1	no	—
227	C₄⋊Q₈	—	23	1	no	—
228	C₄⋊Q₈	—	23	1	no	—
229	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
230	#124	—	23	1	no	—
231	#124	—	23	1	no	—
232	#124	—	23	1	no	—
233	#124	C₂×D₂₃	1	1	yes	characteristic
234	#123	—	46	2	no	—
235	#123	—	46	2	no	—
236	C₂²⋊Q₈	—	46	2	no	—
237	C₂²⋊Q₈	—	46	2	no	—
238	#117	—	23	1	no	—
239	#97	—	23	1	no	—
240	#97	C₂×D₂₃	1	1	yes	characteristic
241	C₄×C₈	—	23	1	no	—
242	C₄×C₈	—	23	1	no	—
243	C₂₃	#3156	1	1	yes	characteristic, complemented, Sylow
244	#41	—	46	2	no	—
245	#41	—	46	2	no	—
246	#41	—	46	2	no	—
247	#41	—	46	2	no	—
248	C₂×Q₈	—	23	1	no	—
249	C₂×Q₈	—	23	1	no	—
250	C₂×Q₈	—	23	1	no	—
251	C₂×Q₈	—	46	1	no	—
252	C₂×Q₈	—	46	1	no	—
253	C₂×Q₈	—	46	2	no	—
254	C₂×Q₈	—	46	2	no	—
255	C₂×Q₈	—	46	2	no	—
256	C₂×Q₈	—	46	2	no	—
257	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
258	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
259	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
260	C₂×D₄	—	23	1	no	—
261	C₂×D₄	—	46	1	no	—
262	C₂×D₄	—	46	1	no	—
263	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
264	C₂²×C₄	—	23	1	no	—
265	C₂²×C₄	—	46	1	no	—
266	C₂²×C₄	—	46	1	no	—
267	C₂²×C₄	—	46	1	no	—
268	C₂²×C₄	—	46	1	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	2	2	no	—
271	Q₁₆	—	46	2	no	—
272	Q₁₆	—	46	2	no	—
273	Q₁₆	—	46	2	no	—
274	Q₁₆	—	46	2	no	—
275	Q₁₆	—	46	2	no	—
276	Q₁₆	—	46	2	no	—
277	Q₁₆	—	46	2	no	—
278	SD₁₆	—	2	2	no	—
279	SD₁₆	—	2	2	no	—
280	SD₁₆	—	46	2	no	—
281	SD₁₆	—	46	2	no	—
282	SD₁₆	—	46	2	no	—
283	SD₁₆	—	46	2	no	—
284	D₈	—	46	2	no	—
285	M₁₆	—	23	4	no	—
286	C₂×C₈	—	23	1	no	—
287	C₂×C₈	—	23	1	no	—
288	C₂×C₈	—	23	4	no	—
289	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
290	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	23	2	no	—
293	C₄⋊C₄	—	46	2	no	—
294	C₄⋊C₄	—	46	2	no	—
295	C₄⋊C₄	—	46	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	23	2	no	—
298	C₂²⋊C₄	—	46	2	no	—
299	C₂²⋊C₄	—	46	2	no	—
300	C₂²⋊C₄	—	46	2	no	—
301	C₄²	—	23	1	no	—
302	C₄²	C₂²×D₂₃	1	1	yes	characteristic
303	C₂³	—	2	1	no	—
304	C₂³	—	23	1	no	—
305	C₂³	—	46	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	2	1	no	—
312	Q₈	—	46	1	no	—
313	Q₈	—	46	1	no	—
314	Q₈	—	46	1	no	—
315	Q₈	—	46	1	no	—
316	Q₈	—	46	1	no	—
317	Q₈	—	46	1	no	—
318	Q₈	—	92	2	no	—
319	Q₈	—	92	2	no	—
320	D₄	—	2	1	no	—
321	D₄	—	2	1	no	—
322	D₄	—	46	1	no	—
323	D₄	—	46	1	no	—
324	D₄	—	92	2	no	—
325	D₄	—	92	2	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	23	1	no	—
330	C₂×C₄	—	23	2	no	—
331	C₂×C₄	—	23	2	no	—
332	C₂×C₄	—	23	2	no	—
333	C₂×C₄	—	46	1	no	—
334	C₂×C₄	—	46	1	no	—
335	C₂×C₄	—	46	1	no	—
336	C₂×C₄	—	92	2	no	—
337	C₂×C₄	—	92	2	no	—
338	C₂×C₄	—	92	2	no	—
339	C₂×C₄	—	92	2	no	—
340	C₂×C₄	D₄×D₂₃	1	2	yes	—
341	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
342	C₈	—	23	2	no	—
343	C₈	—	23	2	no	—
344	C₈	D₄×D₂₃	1	2	yes	—
345	C₈	D₄×D₂₃	1	2	yes	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	23	2	no	—
349	C₂²	—	46	2	no	—
350	C₂²	—	92	1	no	—
351	C₂²	—	92	2	no	—
352	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	23	2	no	—
358	C₄	—	46	2	no	—
359	C₄	—	92	1	no	—
360	C₄	—	92	1	no	—
361	C₄	—	92	1	no	—
362	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
363	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
364	C₂	—	8	1	no	—
365	C₂	—	46	2	no	—
366	C₂	—	184	1	no	—
367	C₂	#127×D₂₃	1	1	yes	characteristic
368	C₂	#13093233	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#497	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#10625087	C₂
hypercenter	#497	D₂₃
hypocenter	C₂₃	#3156
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #497

Complements

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

N	H ≅ G/N	type
1. C₂×#10625089	C₂	semidirect
2. C₂×#10625086	C₂	semidirect
4. C₂×Q₁₆×D₂₃	C₂	semidirect
5. #13092992	C₂	semidirect
6. #13092987	C₂	semidirect
7. #13092811	C₂	semidirect
8. #13092810	C₂	semidirect
9. #13092808	C₂	semidirect
10. #13092799	C₂	semidirect
11. #13092798	C₂	semidirect
12. #13092796	C₂	semidirect
13. #13092795	C₂	semidirect
14. #13092374	C₂	semidirect
15. C₂₃×#497	C₂	semidirect
189. #497	D₂₃	semidirect
243. C₂₃	#3156	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625089	2
C₂×#10625086	2
C₂×#10625087	2
C₂×Q₁₆×D₂₃	2
#13092992	2
#13092987	2
#13092811	2
#13092810	2
#13092808	2
#13092799	2
#13092798	2
#13092796	2
#13092795	2
#13092374	2
C₂₃×#497	2
#3156	23