Subgroups — #15683866

Order 2944.

564 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625086	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10625085	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625087	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×SD₁₆×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092992	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092980	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092809	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092808	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092804	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092803	C₂	1	1	yes	maximal, characteristic, complemented
11	#13092801	C₂	1	1	yes	maximal, characteristic, complemented
12	#13092798	C₂	1	1	yes	maximal, characteristic, complemented
13	#13092788	C₂	1	1	yes	maximal, characteristic, complemented
14	#13092371	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#492	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×#67290	C₂²	1	1	yes	characteristic
19	C₂×D₄×D₂₃	C₂²	1	1	yes	characteristic
20	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
23	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
24	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
25	C₂₃×#127	C₂²	1	1	yes	characteristic
26	#10625118	C₂²	1	1	yes	characteristic, complemented
27	#10625115	C₂²	1	1	yes	characteristic, complemented
28	C₂×#67275	C₂²	1	1	yes	characteristic, complemented
29	C₂×#67275	C₂²	1	1	yes	characteristic, complemented
30	C₂×#67274	C₂²	1	1	yes	characteristic
31	C₂×#67274	C₂²	1	1	yes	characteristic
32	#10625111	C₂²	1	1	yes	characteristic
33	#10625110	—	2	2	no	—
34	#10625108	—	2	2	no	—
35	#10625107	C₂²	1	1	yes	characteristic
36	C₂×#67273	C₂²	1	1	yes	characteristic
37	C₂×#67273	C₂²	1	1	yes	characteristic, complemented
38	C₂×#67272	C₂²	1	1	yes	characteristic
39	C₂×#67272	C₂²	1	1	yes	characteristic, complemented
40	#10625087	—	2	4	no	—
41	#10625086	—	2	4	no	—
42	#10625085	—	2	4	no	—
43	SD₁₆×D₂₃	—	2	4	no	—
44	C₂×Dic₉₂	C₂²	1	1	yes	characteristic, complemented
45	C₂×D₁₈₄	C₂²	1	1	yes	characteristic
46	C₂×#67263	C₂²	1	1	yes	characteristic
47	C₂×#67263	C₂²	1	1	yes	characteristic, complemented
48	C₂×#67262	C₂²	1	1	yes	characteristic
49	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
50	#10625067	—	2	2	no	—
51	#10625065	—	2	2	no	—
52	#10625046	C₂²	1	1	yes	characteristic
53	#10625046	C₂²	1	1	yes	characteristic, complemented
54	#10625043	C₂²	1	1	yes	characteristic
55	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
56	#10624997	C₂²	1	1	yes	characteristic, complemented
57	C₈×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
58	#10624985	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₂₃	—	2	1	no	—
60	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₂₃	—	2	2	no	—
63	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₂₃	—	2	1	no	—
66	C₂×C₂²⋊D₂₃	—	2	1	no	—
67	C₂×C₂²⋊D₂₃	—	2	2	no	—
68	C₂²×Dic₂₃	—	2	1	no	—
69	#67292	—	2	2	no	—
70	#67292	—	2	2	no	—
71	Q₈×D₂₃	—	2	2	no	—
72	Q₈×D₂₃	—	2	2	no	—
73	#67290	—	2	2	no	—
74	#67290	—	2	2	no	—
75	D₄×D₂₃	—	2	2	no	—
76	D₄×D₂₃	—	2	2	no	—
77	C₂×D₉₂	—	2	1	no	—
78	C₂×D₉₂	C₂³	1	1	yes	characteristic
79	C₂×D₉₂	C₂³	1	1	yes	characteristic
80	C₂×C₄×D₂₃	—	2	1	no	—
81	C₂×C₄×D₂₃	—	2	1	no	—
82	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
83	C₂×Dic₄₆	—	2	1	no	—
84	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
85	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
86	SD₁₆×C₂₃	—	2	2	no	—
87	SD₁₆×C₂₃	—	2	2	no	—
88	SD₁₆×C₂₃	—	2	2	no	—
89	SD₁₆×C₂₃	—	2	2	no	—
90	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
91	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₂₃	—	2	2	no	—
93	C₄²×C₂₃	C₂³	1	1	yes	characteristic
94	C₂²⋊Dic₂₃	—	2	2	no	—
95	#67275	—	2	2	no	—
96	#67275	—	2	2	no	—
97	#67274	—	2	2	no	—
98	#67274	—	2	2	no	—
99	#67273	—	2	2	no	—
100	#67273	—	2	2	no	—
101	#67272	—	2	2	no	—
102	#67272	—	2	2	no	—
103	(C₂×D₂₃)⋊C₄	—	2	2	no	—
104	(C₂×D₂₃)⋊C₄	—	2	2	no	—
105	(C₂×D₂₃)⋊C₄	—	2	2	no	—
106	(C₂×D₂₃)⋊C₄	D₄	1	2	yes	—
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	D₁₈₄	—	2	2	no	—
114	#67263	—	2	2	no	—
115	#67263	—	2	2	no	—
116	#67262	D₄	1	4	yes	complemented
117	C₈×D₂₃	D₄	1	4	yes	complemented
118	C₂³×C₂₃	—	2	1	no	—
119	C₂³×C₂₃	—	2	1	no	—
120	C₂²×D₂₃	—	2	1	no	—
121	C₂²×D₂₃	—	2	1	no	—
122	C₂²×D₂₃	—	4	2	no	—
123	C₂²×D₂₃	—	8	2	no	—
124	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
125	Q₈×C₂₃	—	2	1	no	—
126	Q₈×C₂₃	—	2	1	no	—
127	Q₈×C₂₃	—	2	1	no	—
128	Q₈×C₂₃	—	2	1	no	—
129	D₄×C₂₃	—	2	1	no	—
130	D₄×C₂₃	—	2	1	no	—
131	D₄×C₂₃	—	2	1	no	—
132	D₄×C₂₃	—	2	1	no	—
133	D₄×C₂₃	—	8	2	no	—
134	C₂×C₄×C₂₃	—	2	1	no	—
135	C₂×C₄×C₂₃	—	2	1	no	—
136	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
137	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
138	C₂²⋊D₂₃	—	4	2	no	—
139	C₂²⋊D₂₃	—	4	2	no	—
140	C₂²⋊D₂₃	—	8	2	no	—
141	C₂×Dic₂₃	—	2	1	no	—
142	C₂×Dic₂₃	—	2	1	no	—
143	C₂×Dic₂₃	—	4	2	no	—
144	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
145	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
146	D₉₂	—	2	1	no	—
147	D₉₂	—	2	1	no	—
148	D₉₂	—	2	1	no	—
149	D₉₂	—	2	1	no	—
150	D₉₂	—	4	2	no	—
151	C₄×D₂₃	—	4	2	no	—
152	C₄×D₂₃	—	4	2	no	—
153	C₄×D₂₃	C₂×D₄	1	2	yes	—
154	C₄×D₂₃	C₂×D₄	1	2	yes	—
155	Dic₄₆	—	2	1	no	—
156	Dic₄₆	—	2	1	no	—
157	Dic₄₆	—	2	1	no	—
158	Dic₄₆	—	2	1	no	—
159	Dic₄₆	—	4	2	no	—
160	C₈×C₂₃	C₂×D₄	1	2	yes	—
161	C₈×C₂₃	C₂×D₄	1	2	yes	—
162	C₂₃⋊C₈	C₂×D₄	1	2	yes	—
163	C₂₃⋊C₈	C₂×D₄	1	2	yes	—
164	#3154	—	23	1	no	maximal, Sylow
165	C₂²×C₂₃	—	4	1	no	—
166	C₂²×C₂₃	—	4	1	no	—
167	C₂²×C₂₃	—	4	2	no	—
168	C₂²×C₂₃	—	4	2	no	—
169	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
170	C₂×D₂₃	—	2	2	no	—
171	C₂×D₂₃	—	4	1	no	—
172	C₂×D₂₃	—	4	1	no	—
173	C₂×D₂₃	—	4	2	no	—
174	C₂×D₂₃	—	4	2	no	—
175	C₂×D₂₃	—	8	4	no	—
176	C₂×D₂₃	#127	1	2	yes	—
177	C₄×C₂₃	—	2	2	no	—
178	C₄×C₂₃	—	4	1	no	—
179	C₄×C₂₃	—	4	1	no	—
180	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
181	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
182	Dic₂₃	—	2	2	no	—
183	Dic₂₃	—	4	1	no	—
184	Dic₂₃	—	4	1	no	—
185	Dic₂₃	#127	1	2	yes	—
186	C₂×#137	—	23	1	no	—
187	C₂×AGL(1,ℤ/8)	—	23	1	no	—
188	C₂×#135	—	23	1	no	—
189	C₂²×SD₁₆	—	23	1	no	—
190	#536	—	23	1	no	—
191	#535	—	23	1	no	—
192	#497	—	23	1	no	—
193	#497	—	23	1	no	—
194	#496	—	23	1	no	—
195	#496	—	23	1	no	—
196	#495	—	23	1	no	—
197	#495	—	23	1	no	—
198	#492	—	23	1	no	—
199	#492	D₂₃	1	1	yes	characteristic, complemented
200	#405	—	23	1	no	—
201	C₂×C₂₃	—	8	1	no	—
202	C₂×C₂₃	—	8	1	no	—
203	C₂×C₂₃	C₂×#127	1	1	yes	characteristic
204	C₂×C₂₃	#577	1	2	yes	—
205	D₂₃	—	2	2	no	—
206	D₂₃	—	8	1	no	—
207	D₂₃	—	8	1	no	—
208	C₂×#41	—	23	1	no	—
209	C₂×#41	—	23	1	no	—
210	C₂²×Q₈	—	23	1	no	—
211	C₂²×D₄	—	23	1	no	—
212	#137	—	46	4	no	—
213	AGL(1,ℤ/8)	—	46	4	no	—
214	#135	—	46	4	no	—
215	C₂×Q₁₆	—	23	1	no	—
216	C₂×Q₁₆	—	23	1	no	—
217	C₂×Q₁₆	—	23	1	no	—
218	C₂×SD₁₆	—	23	1	no	—
219	C₂×SD₁₆	—	23	1	no	—
220	C₂×SD₁₆	—	23	1	no	—
221	C₂×SD₁₆	—	23	1	no	—
222	C₂×SD₁₆	—	23	1	no	—
223	C₂×SD₁₆	—	23	1	no	—
224	C₂×SD₁₆	—	46	4	no	—
225	C₂×SD₁₆	C₂×D₂₃	1	1	yes	characteristic
226	C₂×SD₁₆	C₂×D₂₃	1	1	yes	characteristic
227	C₂×SD₁₆	C₂×D₂₃	1	1	yes	characteristic, complemented
228	C₂×SD₁₆	C₂×D₂₃	1	1	yes	characteristic, complemented
229	C₂×D₈	—	23	1	no	—
230	C₂×D₈	—	23	1	no	—
231	C₂×D₈	—	23	1	no	—
232	C₂×M₁₆	—	23	1	no	—
233	C₂²×C₈	—	23	1	no	—
234	C₄⋊Q₈	—	23	1	no	—
235	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
236	#127	—	23	1	no	—
237	#127	C₂×D₂₃	1	1	yes	characteristic
238	#124	—	23	1	no	—
239	#124	—	23	1	no	—
240	#124	—	23	1	no	—
241	#124	—	23	1	no	—
242	#123	—	46	2	no	—
243	C₂²⋊Q₈	—	46	2	no	—
244	#121	—	46	2	no	—
245	#120	—	46	2	no	—
246	#117	—	23	1	no	—
247	#97	—	23	1	no	—
248	#97	—	23	1	no	—
249	C₄×C₈	—	23	1	no	—
250	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic, complemented
251	C₂₃	#3154	1	1	yes	characteristic, complemented, Sylow
252	C₂⁴	—	46	1	no	—
253	#41	—	46	2	no	—
254	#41	—	46	2	no	—
255	#41	—	46	2	no	—
256	#41	—	46	2	no	—
257	C₂×Q₈	—	23	1	no	—
258	C₂×Q₈	—	23	1	no	—
259	C₂×Q₈	—	46	1	no	—
260	C₂×Q₈	—	46	2	no	—
261	C₂×Q₈	—	46	2	no	—
262	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
263	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	23	1	no	—
266	C₂×D₄	—	23	1	no	—
267	C₂×D₄	—	46	1	no	—
268	C₂×D₄	—	46	2	no	—
269	C₂×D₄	—	46	2	no	—
270	C₂×D₄	—	46	1	no	—
271	C₂×D₄	—	46	2	no	—
272	C₂×D₄	—	46	1	no	—
273	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
274	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
275	C₂²×C₄	—	23	1	no	—
276	C₂²×C₄	—	46	1	no	—
277	C₂²×C₄	—	46	1	no	—
278	C₂²×C₄	—	46	1	no	—
279	Q₁₆	—	46	2	no	—
280	Q₁₆	—	46	2	no	—
281	Q₁₆	—	46	2	no	—
282	SD₁₆	—	2	2	no	—
283	SD₁₆	—	2	2	no	—
284	SD₁₆	—	2	2	no	—
285	SD₁₆	—	2	2	no	—
286	SD₁₆	—	46	2	no	—
287	SD₁₆	—	46	2	no	—
288	SD₁₆	—	46	2	no	—
289	SD₁₆	—	46	2	no	—
290	SD₁₆	—	46	2	no	—
291	SD₁₆	—	46	2	no	—
292	D₈	—	46	2	no	—
293	D₈	—	46	2	no	—
294	D₈	—	46	2	no	—
295	M₁₆	—	23	4	no	—
296	C₂×C₈	—	23	1	no	—
297	C₂×C₈	—	23	1	no	—
298	C₂×C₈	—	23	4	no	—
299	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
300	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	23	2	no	—
303	C₄⋊C₄	—	46	2	no	—
304	C₂²⋊C₄	—	23	2	no	—
305	C₂²⋊C₄	—	46	2	no	—
306	C₂²⋊C₄	—	46	2	no	—
307	C₂²⋊C₄	—	46	2	no	—
308	C₂²⋊C₄	—	46	2	no	—
309	C₄²	—	23	1	no	—
310	C₄²	C₂²×D₂₃	1	1	yes	characteristic
311	C₂³	—	2	1	no	—
312	C₂³	—	2	1	no	—
313	C₂³	—	23	1	no	—
314	C₂³	—	46	1	no	—
315	C₂³	—	46	1	no	—
316	C₂³	—	92	2	no	—
317	C₂³	—	184	2	no	—
318	Q₈	—	2	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	2	1	no	—
321	Q₈	—	2	1	no	—
322	Q₈	—	46	1	no	—
323	Q₈	—	46	1	no	—
324	Q₈	—	46	1	no	—
325	Q₈	—	46	1	no	—
326	Q₈	—	92	2	no	—
327	D₄	—	2	1	no	—
328	D₄	—	2	1	no	—
329	D₄	—	2	1	no	—
330	D₄	—	2	1	no	—
331	D₄	—	8	2	no	—
332	D₄	—	46	1	no	—
333	D₄	—	46	1	no	—
334	D₄	—	46	1	no	—
335	D₄	—	46	1	no	—
336	D₄	—	92	2	no	—
337	D₄	—	92	2	no	—
338	D₄	—	92	2	no	—
339	D₄	—	184	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	23	1	no	—
343	C₂×C₄	—	23	2	no	—
344	C₂×C₄	—	23	2	no	—
345	C₂×C₄	—	23	2	no	—
346	C₂×C₄	—	46	1	no	—
347	C₂×C₄	—	46	1	no	—
348	C₂×C₄	—	92	2	no	—
349	C₂×C₄	—	92	2	no	—
350	C₂×C₄	—	92	2	no	—
351	C₂×C₄	D₄×D₂₃	1	2	yes	—
352	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
353	C₈	—	23	2	no	—
354	C₈	—	23	2	no	—
355	C₈	D₄×D₂₃	1	2	yes	—
356	C₈	D₄×D₂₃	1	2	yes	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	23	2	no	—
362	C₂²	—	46	2	no	—
363	C₂²	—	92	1	no	—
364	C₂²	—	92	1	no	—
365	C₂²	—	92	2	no	—
366	C₂²	—	92	2	no	—
367	C₂²	—	184	4	no	—
368	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
369	C₄	—	2	2	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	23	2	no	—
373	C₄	—	46	2	no	—
374	C₄	—	92	1	no	—
375	C₄	—	92	1	no	—
376	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
377	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
378	C₂	—	8	1	no	—
379	C₂	—	8	1	no	—
380	C₂	—	46	2	no	—
381	C₂	—	184	1	no	—
382	C₂	—	184	1	no	—
383	C₂	#127×D₂₃	1	1	yes	characteristic
384	C₂	#13093223	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#492	C₂
socle	C₂²×C₂₃	C₂²×D₄
hypercenter	#492	D₂₃
hypocenter	C₂₃	#3154
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

Complements

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

N	H ≅ G/N	type
1. C₂×#10625086	C₂	semidirect
2. C₂×#10625085	C₂	semidirect
3. C₂×#10625087	C₂	semidirect
4. C₂×SD₁₆×D₂₃	C₂	semidirect
5. #13092992	C₂	semidirect
6. #13092980	C₂	semidirect
7. #13092809	C₂	semidirect
8. #13092808	C₂	semidirect
9. #13092804	C₂	semidirect
10. #13092803	C₂	semidirect
11. #13092801	C₂	semidirect
12. #13092798	C₂	semidirect
13. #13092788	C₂	semidirect
14. #13092371	C₂	semidirect
15. C₂₃×#492	C₂	semidirect
22. C₂×SD₁₆×C₂₃	C₂²	semidirect
23. C₂×SD₁₆×C₂₃	C₂²	semidirect
24. C₂₃×C₄⋊Q₈	C₂²	semidirect
26. #10625118	C₂²	semidirect
27. #10625115	C₂²	semidirect
28. C₂×#67275	C₂²	semidirect
29. C₂×#67275	C₂²	semidirect
37. C₂×#67273	C₂²	semidirect
39. C₂×#67272	C₂²	semidirect
44. C₂×Dic₉₂	C₂²	semidirect
47. C₂×#67263	C₂²	semidirect
53. #10625046	C₂²	semidirect
55. C₄×C₈×C₂₃	C₂²	semidirect
56. #10624997	C₂²	semidirect
57. C₈×Dic₂₃	C₂²	semidirect
58. #10624985	C₂²	semidirect
116. #67262	D₄	semidirect
117. C₈×D₂₃	D₄	semidirect
199. #492	D₂₃	semidirect
227. C₂×SD₁₆	C₂×D₂₃	semidirect
228. C₂×SD₁₆	C₂×D₂₃	semidirect
235. C₄⋊Q₈	C₂×D₂₃	semidirect
250. C₄×C₈	C₂×D₂₃	semidirect
251. C₂₃	#3154	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625086	2
C₂×#10625085	2
C₂×#10625087	2
C₂×SD₁₆×D₂₃	2
#13092992	2
#13092980	2
#13092809	2
#13092808	2
#13092804	2
#13092803	2
#13092801	2
#13092798	2
#13092788	2
#13092371	2
C₂₃×#492	2
#3154	23