Subgroups — #15683018

Order 2944.

548 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625120	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#67267	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625101	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092668	C₂	1	2	yes	maximal, complemented
5	#13092665	C₂	1	2	yes	maximal, complemented
6	#13092651	C₂	1	2	yes	maximal, complemented
7	#13092648	C₂	1	2	yes	maximal, complemented
8	#13092460	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092421	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092418	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#525	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₂₃	—	2	1	no	—
15	C₂²×D₉₂	C₂²	1	1	yes	characteristic
16	C₂₃×#123	—	2	1	no	—
17	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₂₃×#121	C₂²	1	1	yes	characteristic, complemented
20	C₂₃×#121	C₂²	1	1	yes	characteristic, complemented
21	C₂₃×#120	—	2	1	no	—
22	C₄×D₄×C₂₃	—	2	1	no	—
23	C₄×D₄×C₂₃	—	2	1	no	—
24	C₂₃×#117	C₂²	1	1	yes	characteristic
25	#10625120	—	2	4	no	—
26	C₂×#67274	C₂²	1	2	yes	complemented
27	C₂×#67272	C₂²	1	2	yes	complemented
28	#10625101	—	2	1	no	—
29	#10625101	—	2	1	no	—
30	#10625101	C₂²	1	2	yes	—
31	#10625101	C₂²	1	2	yes	—
32	C₂×(C₂×D₂₃)⋊C₄	—	2	1	no	—
33	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
34	C₂×#67267	—	2	4	no	—
35	C₂×#67267	C₂²	1	1	yes	characteristic
36	C₂×#67267	C₂²	1	1	yes	characteristic
37	C₂×#67267	C₂²	1	2	yes	complemented
38	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	complemented
39	#10625017	C₂²	1	2	yes	complemented
40	#10625014	C₂²	1	2	yes	complemented
41	#10624991	C₂²	1	2	yes	complemented
42	#10624991	C₂²	1	2	yes	complemented
43	#10624990	C₂²	1	2	yes	complemented
44	#10624989	C₂²	1	2	yes	complemented
45	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
46	C₂³×D₂₃	—	2	1	no	—
47	#41×C₂₃	—	2	2	no	—
48	#41×C₂₃	—	2	2	no	—
49	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₂₃	—	2	1	no	—
51	C₂×D₄×C₂₃	—	2	1	no	—
52	C₂×D₄×C₂₃	—	2	1	no	—
53	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	—	2	2	no	—
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	—	2	1	no	—
57	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₂₃	—	2	2	no	—
64	C₂×C₂²⋊D₂₃	—	2	2	no	—
65	C₂×C₂²⋊D₂₃	—	4	4	no	—
66	C₂²×Dic₂₃	—	2	1	no	—
67	C₂×D₉₂	—	2	2	no	—
68	C₂×D₉₂	—	2	2	no	—
69	C₂×D₉₂	—	2	1	no	—
70	C₂×D₉₂	C₂³	1	2	yes	—
71	C₄⋊C₄×C₂₃	—	2	1	no	—
72	C₄⋊C₄×C₂₃	—	2	1	no	—
73	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
75	C₂²⋊C₄×C₂₃	—	2	1	no	—
76	C₂²⋊C₄×C₂₃	—	2	1	no	—
77	C₂²⋊C₄×C₂₃	—	2	1	no	—
78	C₂²⋊C₄×C₂₃	—	2	1	no	—
79	C₂²⋊C₄×C₂₃	—	2	1	no	—
80	C₄²×C₂₃	—	2	1	no	—
81	#67274	—	2	4	no	—
82	#67272	—	2	4	no	—
83	(C₂×D₂₃)⋊C₄	—	2	2	no	—
84	(C₂×D₂₃)⋊C₄	—	2	2	no	—
85	C₄⋊Dic₂₃	—	2	1	no	—
86	C₄⋊Dic₂₃	C₂³	1	2	yes	—
87	#67267	—	2	2	no	—
88	#67267	—	2	4	no	—
89	#67267	D₄	1	4	yes	complemented
90	C₂×C₂₃⋊C₈	—	2	4	no	—
91	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
92	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
93	C₂³×C₂₃	—	2	2	no	—
94	C₂³×C₂₃	—	2	2	no	—
95	C₂³×C₂₃	—	2	1	no	—
96	C₂³×C₂₃	—	4	1	no	—
97	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×D₂₃	—	2	2	no	—
101	C₂²×D₂₃	—	4	2	no	—
102	C₂²×D₂₃	—	4	2	no	—
103	C₂²×D₂₃	—	4	2	no	—
104	Q₈×C₂₃	—	2	1	no	—
105	Q₈×C₂₃	—	2	1	no	—
106	D₄×C₂₃	—	2	1	no	—
107	D₄×C₂₃	—	2	1	no	—
108	D₄×C₂₃	—	4	2	no	—
109	D₄×C₂₃	—	4	2	no	—
110	D₄×C₂₃	—	4	2	no	—
111	C₂×C₄×C₂₃	—	2	2	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	2	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	—	4	2	no	—
122	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
123	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₂₃	#41	1	2	yes	—
127	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₂₃	—	4	4	no	—
129	C₂²⋊D₂₃	—	4	4	no	—
130	C₂×Dic₂₃	—	2	2	no	—
131	C₂×Dic₂₃	—	4	2	no	—
132	D₉₂	—	2	2	no	—
133	D₉₂	—	2	2	no	—
134	D₉₂	—	4	2	no	—
135	C₂₃⋊C₈	—	2	2	no	—
136	C₂₃⋊C₈	C₂×D₄	1	4	yes	complemented
137	#3055	—	23	1	no	maximal, Sylow
138	C₂²×C₂₃	—	2	1	no	—
139	C₂²×C₂₃	—	2	1	no	—
140	C₂²×C₂₃	—	2	1	no	—
141	C₂²×C₂₃	—	2	1	no	—
142	C₂²×C₂₃	—	2	1	no	—
143	C₂²×C₂₃	—	2	1	no	—
144	C₂²×C₂₃	—	2	1	no	—
145	C₂²×C₂₃	—	4	4	no	—
146	C₂²×C₂₃	—	4	1	no	—
147	C₂²×C₂₃	—	4	1	no	—
148	C₂²×C₂₃	—	4	1	no	—
149	C₂²×C₂₃	#121	1	2	yes	—
150	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	complemented
151	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₂₃	—	4	2	no	—
153	C₂×D₂₃	—	4	2	no	—
154	C₂×D₂₃	—	4	2	no	—
155	C₂×D₂₃	—	8	4	no	—
156	C₄×C₂₃	—	2	1	no	—
157	C₄×C₂₃	—	2	1	no	—
158	C₄×C₂₃	—	4	1	no	—
159	C₄×C₂₃	—	4	1	no	—
160	C₄×C₂₃	—	4	1	no	—
161	C₄×C₂₃	#121	1	2	yes	—
162	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
163	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
164	Dic₂₃	—	4	2	no	—
165	C₂×AGL(1,ℤ/8)	—	23	1	no	—
166	C₂²×M₁₆	—	23	1	no	—
167	#525	D₂₃	1	1	yes	characteristic, complemented
168	C₂×#121	—	23	1	no	—
169	#469	—	23	2	no	—
170	#468	—	23	2	no	—
171	#466	—	23	2	no	—
172	#465	—	23	2	no	—
173	#428	—	23	1	no	—
174	#418	—	23	1	no	—
175	#417	—	23	1	no	—
176	C₂×C₂₃	—	2	2	no	—
177	C₂×C₂₃	—	2	2	no	—
178	C₂×C₂₃	—	4	1	no	—
179	C₂×C₂₃	—	8	1	no	—
180	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
181	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂×C₂₃	#575	1	1	yes	characteristic
183	D₂₃	—	8	2	no	—
184	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
185	C₂²×D₄	—	23	1	no	—
186	C₂²×D₄	—	46	1	no	—
187	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
188	AGL(1,ℤ/8)	—	46	4	no	—
189	C₂×SD₁₆	—	23	2	no	—
190	C₂×D₈	—	23	2	no	—
191	C₂×M₁₆	—	23	1	no	—
192	C₂×M₁₆	—	23	1	no	—
193	C₂×M₁₆	—	23	2	no	—
194	C₂×M₁₆	—	46	4	no	—
195	C₂²×C₈	—	23	2	no	—
196	#123	—	2	1	no	—
197	C₂²⋊Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
198	C₂²⋊Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
199	#121	—	23	2	no	—
200	#121	—	23	2	no	—
201	#121	—	46	1	no	—
202	#121	—	46	1	no	—
203	#121	C₂×D₂₃	1	1	yes	characteristic, complemented
204	#121	C₂×D₂₃	1	1	yes	characteristic, complemented
205	#120	—	2	1	no	—
206	C₄×D₄	—	2	1	no	—
207	C₄×D₄	—	2	1	no	—
208	#117	C₂×D₂₃	1	1	yes	characteristic
209	C₂×C₄⋊C₄	—	23	1	no	—
210	C₂×C₂²⋊C₄	—	46	1	no	—
211	#107	—	23	2	no	—
212	#106	—	23	2	no	—
213	#103	—	23	2	no	—
214	#102	—	23	2	no	—
215	#102	—	23	2	no	—
216	#102	—	23	2	no	—
217	C₂₃	#3055	1	1	yes	characteristic, complemented, Sylow
218	C₂⁴	—	46	1	no	—
219	C₂⁴	C₂²⋊D₂₃	1	1	yes	characteristic
220	#41	—	2	2	no	—
221	#41	—	2	2	no	—
222	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
223	C₂×D₄	—	2	1	no	—
224	C₂×D₄	—	2	1	no	—
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	—	23	2	no	—
227	C₂×D₄	—	46	2	no	—
228	C₂×D₄	—	46	2	no	—
229	C₂×D₄	—	46	2	no	—
230	C₂×D₄	—	46	2	no	—
231	C₂×D₄	—	46	1	no	—
232	C₂×D₄	—	92	4	no	—
233	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	46	1	no	—
238	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
241	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
243	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
244	SD₁₆	—	46	4	no	—
245	D₈	—	46	4	no	—
246	M₁₆	—	23	4	no	—
247	M₁₆	—	46	4	no	—
248	M₁₆	—	46	2	no	—
249	C₂×C₈	—	23	2	no	—
250	C₂×C₈	—	23	2	no	—
251	C₂×C₈	—	46	4	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	23	2	no	—
255	C₄⋊C₄	—	46	1	no	—
256	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	46	2	no	—
264	C₂²⋊C₄	—	46	2	no	—
265	C₄²	—	2	1	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	4	1	no	—
270	C₂³	—	46	2	no	—
271	C₂³	—	92	2	no	—
272	C₂³	—	92	2	no	—
273	C₂³	—	92	2	no	—
274	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
275	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
276	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	46	2	no	—
285	D₄	—	46	2	no	—
286	D₄	—	92	4	no	—
287	D₄	—	92	4	no	—
288	D₄	—	92	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	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	—	4	2	no	—
300	C₂×C₄	—	46	2	no	—
301	C₂×C₄	—	92	2	no	—
302	C₂×C₄	D₄×D₂₃	1	2	yes	—
303	C₂×C₄	#67290	1	2	yes	—
304	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
305	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
306	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
308	C₈	—	23	4	no	—
309	C₈	—	46	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	92	2	no	—
322	C₂²	—	92	2	no	—
323	C₂²	—	92	2	no	—
324	C₂²	—	184	4	no	—
325	C₂²	#10625110	1	2	yes	—
326	C₂²	#10625120	1	2	yes	complemented
327	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	92	2	no	—
334	C₄	#10625110	1	2	yes	—
335	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
336	C₄	C₂×#67290	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	—	2	2	no	—
339	C₂	—	4	1	no	—
340	C₂	—	8	1	no	—
341	C₂	—	184	2	no	—
342	C₂	C₂×#10625110	1	1	yes	characteristic
343	C₂	C₂×#10625120	1	1	yes	characteristic
344	C₂	#13093215	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#525	C₂
socle	C₂²×C₂₃	C₂²×D₄
hypercenter	#525	D₂₃
hypocenter	C₂₃	#3055
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#10625120	C₂	semidirect
2. C₂²×#67267	C₂	semidirect
3. C₂×#10625101	C₂	semidirect
4. #13092668	C₂	semidirect
5. #13092665	C₂	semidirect
6. #13092651	C₂	semidirect
7. #13092648	C₂	semidirect
8. #13092460	C₂	semidirect
9. #13092421	C₂	semidirect
10. #13092418	C₂	semidirect
11. C₂₃×#525	C₂	semidirect
17. C₂₃×C₂²⋊Q₈	C₂²	semidirect
18. C₂₃×C₂²⋊Q₈	C₂²	semidirect
19. C₂₃×#121	C₂²	semidirect
20. C₂₃×#121	C₂²	semidirect
26. C₂×#67274	C₂²	semidirect
27. C₂×#67272	C₂²	semidirect
37. C₂×#67267	C₂²	semidirect
38. C₂²×C₂₃⋊C₈	C₂²	semidirect
39. #10625017	C₂²	semidirect
40. #10625014	C₂²	semidirect
41. #10624991	C₂²	semidirect
42. #10624991	C₂²	semidirect
43. #10624990	C₂²	semidirect
44. #10624989	C₂²	semidirect
89. #67267	D₄	semidirect
136. C₂₃⋊C₈	C₂×D₄	semidirect
150. C₂²×C₂₃	AGL(1,ℤ/8)	semidirect
167. #525	D₂₃	semidirect
197. C₂²⋊Q₈	C₂×D₂₃	semidirect
198. C₂²⋊Q₈	C₂×D₂₃	semidirect
203. #121	C₂×D₂₃	semidirect
204. #121	C₂×D₂₃	semidirect
217. C₂₃	#3055	semidirect
326. C₂²	#10625120	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625120	2
C₂²×#67267	2
C₂×#10625101	2
#13092668	2
#13092665	2
#13092651	2
#13092648	2
#13092460	2
#13092421	2
#13092418	2
C₂₃×#525	2
#3055	23