Subgroups — #28407332

Order 14464.

470 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299667	C₂	1	1	yes	maximal, characteristic
2	#20299666	C₂	1	1	yes	maximal, characteristic, complemented
3	#20299638	C₂	1	2	yes	maximal, complemented
4	#20299637	C₂	1	2	yes	maximal, complemented
5	Q₁₆×#90673	C₂	1	2	yes	maximal, complemented
6	SD₁₆×#90673	C₂	1	2	yes	maximal, complemented
7	#20299609	C₂	1	1	yes	maximal, characteristic, complemented
8	#20299588	C₂	1	1	yes	maximal, characteristic
9	C₂×#16813685	C₂	1	1	yes	maximal, characteristic, complemented
10	#20299564	C₂	1	1	yes	maximal, characteristic, complemented
11	#137×D₁₁₃	C₂	1	1	yes	maximal, characteristic
12	C₂×#14268655	C₂²	1	1	yes	characteristic
13	Q₈×#90673	—	2	1	no	—
14	Q₈×#90673	—	2	1	no	—
15	Q₈×#90673	C₂²	1	2	yes	—
16	Q₈×#90673	C₂²	1	2	yes	—
17	Q₈×#90673	C₂²	1	2	yes	—
18	D₄×#90673	—	2	2	no	—
19	D₄×#90673	C₂²	1	2	yes	—
20	#16813701	—	2	1	no	—
21	#16813701	—	2	2	no	—
22	#16813701	C₂²	1	1	yes	characteristic
23	#16813701	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#90673	—	2	1	no	—
25	C₂×C₄⋊#90673	C₂²	1	1	yes	characteristic
26	#16813691	C₂²	1	2	yes	—
27	#16813690	C₂²	1	2	yes	—
28	#16813685	C₂²	1	2	yes	—
29	#16813685	C₂²	1	2	yes	—
30	#16813685	C₂²	1	1	yes	characteristic
31	#16813685	C₂²	1	1	yes	characteristic
32	#16813684	C₂²	1	1	yes	characteristic
33	#16813684	C₂²	1	1	yes	characteristic
34	#16813676	C₂²	1	2	yes	—
35	C₈×#90673	C₂²	1	2	yes	—
36	#41×D₁₁₃	C₂²	1	1	yes	characteristic
37	C₂×Q₈×D₁₁₃	C₂²	1	1	yes	characteristic
38	#16813655	C₄	1	1	yes	characteristic, complemented
39	#16813654	C₄	1	2	yes	complemented
40	#16813653	C₄	1	1	yes	characteristic, complemented
41	#16813652	C₄	1	1	yes	characteristic, complemented
42	#16813651	C₄	1	2	yes	complemented
43	Q₁₆×D₁₁₃	C₂²	1	2	yes	—
44	SD₁₆×D₁₁₃	C₂²	1	2	yes	—
45	M₁₆×D₁₁₃	C₂²	1	1	yes	characteristic
46	#137×C₁₁₃	C₄	1	1	yes	characteristic, complemented
47	C₂²×#90673	—	2	1	no	—
48	C₂²×#90673	—	2	2	no	—
49	#14268655	—	2	1	no	—
50	#14268655	C₂³	1	1	yes	characteristic
51	#14268655	C₂³	1	1	yes	characteristic
52	C₂×#11744753	—	2	1	no	—
53	C₄⋊#90673	—	2	2	no	—
54	C₄⋊#90673	—	2	2	no	—
55	C₄⋊#90673	—	2	2	no	—
56	C₄⋊#90673	—	2	1	no	—
57	C₄⋊#90673	D₄	1	1	yes	characteristic
58	C₄⋊#90673	D₄	1	1	yes	characteristic
59	C₄⋊#90673	C₂³	1	1	yes	characteristic
60	C₄⋊#90673	C₂³	1	1	yes	characteristic
61	C₄⋊#90673	C₂³	1	2	yes	—
62	C₂²⋊#90673	—	2	1	no	—
63	C₂²⋊#90673	—	2	2	no	—
64	C₂²⋊#90673	D₄	1	2	yes	—
65	C₄×#90673	—	2	2	no	—
66	C₄×#90673	—	2	2	no	—
67	C₄×#90673	—	2	2	no	—
68	C₄×#90673	—	2	1	no	—
69	C₄×#90673	C₂³	1	2	yes	—
70	#14268648	C₂×C₄	1	1	yes	characteristic
71	#14268647	—	2	1	no	—
72	#14268647	C₂×C₄	1	1	yes	characteristic
73	#14268646	—	2	1	no	—
74	#14268646	C₂×C₄	1	1	yes	characteristic
75	C₂×Dic₂₂₆	—	2	1	no	—
76	C₂×Dic₂₂₆	C₂×C₄	1	1	yes	characteristic, complemented
77	Q₈×D₁₁₃	—	2	1	no	—
78	Q₈×D₁₁₃	—	2	1	no	—
79	Q₈×D₁₁₃	C₂×C₄	1	2	yes	complemented
80	Q₈×D₁₁₃	C₂³	1	2	yes	—
81	Q₈×D₁₁₃	C₂³	1	1	yes	characteristic
82	D₄×D₁₁₃	—	2	1	no	—
83	D₄×D₁₁₃	C₂³	1	1	yes	characteristic
84	C₂×C₄×D₁₁₃	—	2	1	no	—
85	C₂×C₄×D₁₁₃	—	2	1	no	—
86	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
87	Dic₄₅₂	C₂×C₄	1	2	yes	complemented
88	#14268637	C₂×C₄	1	2	yes	complemented
89	#14268637	C₂×C₄	1	2	yes	complemented
90	#14268636	C₂×C₄	1	2	yes	complemented
91	#14268635	C₂×C₄	1	2	yes	complemented
92	#14268634	C₂×C₄	1	2	yes	complemented
93	#14268631	C₂×C₄	1	1	yes	characteristic, complemented
94	#14268630	C₂×C₄	1	2	yes	complemented
95	C₈×D₁₁₃	C₂³	1	2	yes	—
96	#41×C₁₁₃	C₂×C₄	1	1	yes	characteristic
97	C₂×Q₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
98	Q₁₆×C₁₁₃	C₂×C₄	1	2	yes	complemented
99	SD₁₆×C₁₁₃	C₂×C₄	1	2	yes	complemented
100	M₁₆×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
101	C₂²×D₁₁₃	—	2	1	no	—
102	C₂²×D₁₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×#90673	—	2	1	no	—
104	C₂×#90673	—	2	1	no	—
105	C₂×#90673	—	2	2	no	—
106	C₂×#90673	—	8	2	no	—
107	C₂×#90673	—	8	1	no	—
108	C₂×#90673	C₂×D₄	1	2	yes	—
109	C₂×#90673	C₂×D₄	1	2	yes	—
110	Q₈×C₁₁₃	—	2	1	no	—
111	Q₈×C₁₁₃	C₂²×C₄	1	2	yes	—
112	Q₈×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
113	D₄×C₁₁₃	—	2	1	no	—
114	D₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₁₃	—	2	1	no	—
116	C₂×C₄×C₁₁₃	—	2	1	no	—
117	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²⋊D₁₁₃	—	2	1	no	—
119	C₂²⋊D₁₁₃	—	2	1	no	—
120	C₂²⋊D₁₁₃	—	2	1	no	—
121	C₂×Dic₁₁₃	—	2	1	no	—
122	C₂×Dic₁₁₃	—	2	1	no	—
123	C₂×Dic₁₁₃	#41	1	1	yes	characteristic
124	D₄₅₂	—	2	1	no	—
125	D₄₅₂	C₂²×C₄	1	1	yes	characteristic
126	C₄×D₁₁₃	—	2	2	no	—
127	C₄×D₁₁₃	—	2	1	no	—
128	C₄×D₁₁₃	—	2	2	no	—
129	C₄×D₁₁₃	—	2	1	no	—
130	C₄×D₁₁₃	—	2	1	no	—
131	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
132	C₄×D₁₁₃	C₂×D₄	1	1	yes	characteristic
133	C₄×D₁₁₃	#41	1	1	yes	characteristic
134	C₄×D₁₁₃	C₂⁴	1	1	yes	characteristic
135	Dic₂₂₆	—	2	1	no	—
136	Dic₂₂₆	—	2	2	no	—
137	Dic₂₂₆	—	2	2	no	—
138	Dic₂₂₆	—	2	1	no	—
139	Dic₂₂₆	C₂²×C₄	1	2	yes	—
140	Dic₂₂₆	C₂²×C₄	1	1	yes	characteristic
141	#11744753	—	2	1	no	—
142	#11744753	—	2	1	no	—
143	C₈×C₁₁₃	C₂²×C₄	1	2	yes	—
144	#11744751	C₂²×C₄	1	2	yes	—
145	C₂²×C₁₁₃	—	2	1	no	—
146	C₂²×C₁₁₃	C₄×D₄	1	1	yes	characteristic
147	C₂×D₁₁₃	—	2	1	no	—
148	C₂×D₁₁₃	—	2	1	no	—
149	C₂×D₁₁₃	—	2	1	no	—
150	C₂×D₁₁₃	—	4	1	no	—
151	C₂×D₁₁₃	—	4	1	no	—
152	C₂×D₁₁₃	C₄×D₄	1	1	yes	characteristic
153	C₂×D₁₁₃	C₂²×D₄	1	1	yes	characteristic
154	#90673	—	2	2	no	—
155	#90673	—	2	2	no	—
156	#90673	—	4	1	no	—
157	#90673	—	4	2	no	—
158	#90673	—	4	1	no	—
159	C₄×C₁₁₃	—	2	2	no	—
160	C₄×C₁₁₃	—	2	1	no	—
161	C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
162	C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
163	Dic₁₁₃	—	2	2	no	—
164	Dic₁₁₃	—	2	1	no	—
165	Dic₁₁₃	C₄×D₄	1	1	yes	characteristic
166	Dic₁₁₃	C₂×#41	1	1	yes	characteristic
167	C₂×C₁₁₃	—	2	1	no	—
168	C₂×C₁₁₃	—	4	1	no	—
169	C₂×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
170	D₁₁₃	—	2	1	no	—
171	D₁₁₃	—	4	1	no	—
172	D₁₁₃	#577	1	1	yes	characteristic
173	D₁₁₃	#578	1	1	yes	characteristic
174	#2947	—	113	1	no	maximal, Sylow
175	C₁₁₃	#2947	1	1	yes	characteristic, complemented, Sylow
176	C₂×#137	—	113	1	no	—
177	#520	—	113	1	no	—
178	#519	—	113	1	no	—
179	#441	—	113	2	no	—
180	#440	—	113	2	no	—
181	C₄×Q₁₆	—	113	2	no	—
182	C₄×SD₁₆	—	113	2	no	—
183	#428	—	113	1	no	—
184	#416	—	113	1	no	—
185	C₂×#103	—	113	1	no	—
186	#405	—	113	1	no	—
187	C₂×#41	—	113	1	no	—
188	C₂²×Q₈	—	113	1	no	—
189	#137	—	113	1	no	—
190	#137	—	113	1	no	—
191	#137	—	113	2	no	—
192	#137	—	113	2	no	—
193	#137	—	113	1	no	—
194	#137	#90673	1	1	yes	characteristic, complemented
195	C₂×Q₁₆	—	113	2	no	—
196	C₂×SD₁₆	—	113	2	no	—
197	C₂×M₁₆	—	113	1	no	—
198	C₂²×C₈	—	113	1	no	—
199	C₄×Q₈	—	113	2	no	—
200	C₄×Q₈	—	113	2	no	—
201	C₄×Q₈	—	113	2	no	—
202	C₄×Q₈	—	226	1	no	—
203	C₄×Q₈	—	226	1	no	—
204	C₄×D₄	—	113	2	no	—
205	C₄×D₄	—	226	2	no	—
206	#117	—	113	1	no	—
207	#117	—	113	1	no	—
208	#117	—	226	2	no	—
209	#117	—	226	1	no	—
210	C₂×C₄⋊C₄	—	113	1	no	—
211	C₂×C₄⋊C₄	—	226	1	no	—
212	#107	—	113	2	no	—
213	#106	—	113	2	no	—
214	#103	—	113	2	no	—
215	#103	—	113	2	no	—
216	#103	—	113	1	no	—
217	#103	—	113	1	no	—
218	#102	—	113	1	no	—
219	#102	—	113	1	no	—
220	#97	—	113	2	no	—
221	C₄×C₈	—	113	2	no	—
222	#41	—	113	1	no	—
223	#41	—	113	1	no	—
224	#41	—	113	1	no	—
225	#41	—	226	1	no	—
226	#41	—	226	1	no	—
227	#41	C₂×#90673	1	1	yes	characteristic
228	C₂×Q₈	—	113	2	no	—
229	C₂×Q₈	—	113	1	no	—
230	C₂×Q₈	—	113	2	no	—
231	C₂×Q₈	—	113	1	no	—
232	C₂×Q₈	—	226	1	no	—
233	C₂×Q₈	—	226	1	no	—
234	C₂×Q₈	—	226	1	no	—
235	C₂×Q₈	C₂×#90673	1	1	yes	characteristic, complemented
236	C₂×D₄	—	113	1	no	—
237	C₂×D₄	—	226	1	no	—
238	C₂²×C₄	—	113	1	no	—
239	C₂²×C₄	—	226	1	no	—
240	C₂²×C₄	—	226	1	no	—
241	C₂²×C₄	—	226	2	no	—
242	C₂²×C₄	—	226	1	no	—
243	Q₁₆	—	113	2	no	—
244	Q₁₆	—	113	2	no	—
245	Q₁₆	—	113	2	no	—
246	Q₁₆	C₂×#90673	1	2	yes	complemented
247	SD₁₆	—	113	2	no	—
248	SD₁₆	—	113	2	no	—
249	SD₁₆	—	113	2	no	—
250	SD₁₆	C₂×#90673	1	2	yes	complemented
251	M₁₆	—	113	1	no	—
252	M₁₆	—	113	2	no	—
253	M₁₆	C₂×#90673	1	1	yes	characteristic, complemented
254	C₂×C₈	—	113	2	no	—
255	C₂×C₈	—	113	1	no	—
256	C₂×C₈	—	113	1	no	—
257	C₂×C₈	—	226	1	no	—
258	C₂×C₈	—	226	1	no	—
259	C₄⋊C₄	—	113	1	no	—
260	C₄⋊C₄	—	113	1	no	—
261	C₄⋊C₄	—	113	1	no	—
262	C₄⋊C₄	—	113	1	no	—
263	C₄⋊C₄	—	113	2	no	—
264	C₄⋊C₄	—	226	1	no	—
265	C₄⋊C₄	—	226	2	no	—
266	C₄⋊C₄	—	226	2	no	—
267	C₄⋊C₄	—	226	2	no	—
268	C₂²⋊C₄	—	113	2	no	—
269	C₂²⋊C₄	—	226	2	no	—
270	C₂²⋊C₄	—	226	1	no	—
271	C₄²	—	113	2	no	—
272	C₄²	—	226	1	no	—
273	C₄²	—	226	2	no	—
274	C₄²	—	226	2	no	—
275	C₄²	—	226	2	no	—
276	C₂³	—	113	1	no	—
277	C₂³	—	226	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	113	2	no	—
280	Q₈	—	113	1	no	—
281	Q₈	—	226	2	no	—
282	Q₈	—	226	2	no	—
283	Q₈	—	226	1	no	—
284	Q₈	—	226	1	no	—
285	Q₈	C₂²×#90673	1	2	yes	—
286	Q₈	C₂²×#90673	1	1	yes	characteristic
287	D₄	—	2	1	no	—
288	D₄	—	113	1	no	—
289	D₄	—	226	1	no	—
290	D₄	—	226	1	no	—
291	D₄	—	226	1	no	—
292	D₄	—	226	1	no	—
293	D₄	C₂²×#90673	1	1	yes	characteristic
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	113	1	no	—
297	C₂×C₄	—	113	1	no	—
298	C₂×C₄	—	113	1	no	—
299	C₂×C₄	—	113	1	no	—
300	C₂×C₄	—	113	1	no	—
301	C₂×C₄	—	113	2	no	—
302	C₂×C₄	—	113	2	no	—
303	C₂×C₄	—	904	1	no	—
304	C₂×C₄	—	904	2	no	—
305	C₂×C₄	—	226	1	no	—
306	C₂×C₄	—	226	1	no	—
307	C₂×C₄	—	226	2	no	—
308	C₂×C₄	—	226	2	no	—
309	C₂×C₄	—	226	1	no	—
310	C₂×C₄	—	226	1	no	—
311	C₂×C₄	—	226	1	no	—
312	C₂×C₄	—	226	2	no	—
313	C₂×C₄	—	226	1	no	—
314	C₂×C₄	—	226	1	no	—
315	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
316	C₈	—	113	2	no	—
317	C₈	—	226	1	no	—
318	C₈	—	226	1	no	—
319	C₈	C₂²×#90673	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	113	1	no	—
322	C₂²	—	113	1	no	—
323	C₂²	—	452	1	no	—
324	C₂²	—	452	1	no	—
325	C₂²	—	226	1	no	—
326	C₂²	—	226	1	no	—
327	C₂²	—	226	1	no	—
328	C₂²	D₄×#90673	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	—	113	1	no	—
332	C₄	—	113	1	no	—
333	C₄	—	452	2	no	—
334	C₄	—	452	1	no	—
335	C₄	—	452	1	no	—
336	C₄	—	226	2	no	—
337	C₄	—	226	2	no	—
338	C₄	—	226	1	no	—
339	C₄	—	226	2	no	—
340	C₄	D₄×#90673	1	1	yes	characteristic
341	C₄	C₂³×#90673	1	1	yes	characteristic
342	C₂	—	2	1	no	—
343	C₂	—	4	1	no	—
344	C₂	—	113	1	no	—
345	C₂	—	113	1	no	—
346	C₂	—	452	1	no	—
347	C₂	—	226	1	no	—
348	C₂	C₂×D₄×#90673	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#90673
commutator	C₄×C₁₁₃	C₂³×C₄
Frattini	C₄	C₂³×#90673
Fitting	#137×C₁₁₃	C₄
socle	C₂×C₁₁₃	C₂×C₄×D₄
prime radical	#41×D₁₁₃	C₂²
cosocle	C₄×D₁₁₃	C₂⁴
hypercenter	#137	#90673
hypocenter	C₁₁₃	#2947
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
2. #20299666	C₂	semidirect
3. #20299638	C₂	semidirect
4. #20299637	C₂	semidirect
5. Q₁₆×#90673	C₂	semidirect
6. SD₁₆×#90673	C₂	semidirect
7. #20299609	C₂	semidirect
9. C₂×#16813685	C₂	semidirect
10. #20299564	C₂	semidirect
38. #16813655	C₄	semidirect
39. #16813654	C₄	semidirect
40. #16813653	C₄	semidirect
41. #16813652	C₄	semidirect
42. #16813651	C₄	semidirect
46. #137×C₁₁₃	C₄	semidirect
76. C₂×Dic₂₂₆	C₂×C₄	semidirect
79. Q₈×D₁₁₃	C₂×C₄	semidirect
87. Dic₄₅₂	C₂×C₄	semidirect
88. #14268637	C₂×C₄	semidirect
89. #14268637	C₂×C₄	semidirect
90. #14268636	C₂×C₄	semidirect
91. #14268635	C₂×C₄	semidirect
92. #14268634	C₂×C₄	semidirect
93. #14268631	C₂×C₄	semidirect
94. #14268630	C₂×C₄	semidirect
97. C₂×Q₈×C₁₁₃	C₂×C₄	semidirect
98. Q₁₆×C₁₁₃	C₂×C₄	semidirect
99. SD₁₆×C₁₁₃	C₂×C₄	semidirect
100. M₁₆×C₁₁₃	C₂×C₄	semidirect
175. C₁₁₃	#2947	semidirect
194. #137	#90673	semidirect
235. C₂×Q₈	C₂×#90673	semidirect
246. Q₁₆	C₂×#90673	semidirect
250. SD₁₆	C₂×#90673	semidirect
253. M₁₆	C₂×#90673	semidirect

Maximal subgroups

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

subgroup	index
#20299667	2
#20299666	2
#20299638	2
#20299637	2
Q₁₆×#90673	2
SD₁₆×#90673	2
#20299609	2
#20299588	2
C₂×#16813685	2
#20299564	2
#137×D₁₁₃	2
#2947	113