Subgroups — #16807284

Order 3328.

526 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131251	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#6092	C₂	1	1	yes	maximal, characteristic, complemented
3	#13131222	C₂	1	1	yes	maximal, characteristic, complemented
4	#13131219	C₂	1	2	yes	maximal, complemented
5	#13131215	C₂	1	1	yes	maximal, characteristic, complemented
6	#13131215	C₂	1	2	yes	maximal, complemented
7	#13131204	C₂	1	2	yes	maximal, complemented
8	#13131200	C₂	1	1	yes	maximal, characteristic, complemented
9	#13131046	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	#13123810	C₂	1	1	yes	maximal, characteristic
12	#11719285	C₄	1	1	yes	characteristic
13	C₂×#88098	C₂²	1	2	yes	—
14	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂×D₁₃⋊C₁₆	C₂²	1	2	yes	—
17	C₁₃×#446	C₄	1	1	yes	characteristic
18	Q₈×#1055	C₂²	1	1	yes	characteristic
19	#11718501	C₄	1	2	yes	—
20	#11718496	C₄	1	1	yes	characteristic
21	#11718495	C₂²	1	2	yes	—
22	#11718495	C₂²	1	1	yes	characteristic
23	#11718369	C₂²	1	2	yes	—
24	#11718367	C₄	1	2	yes	—
25	#11718349	C₄	1	1	yes	characteristic
26	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
27	#11718259	C₂²	1	2	yes	—
28	#11718259	C₂²	1	2	yes	—
29	#11718259	C₂²	1	2	yes	—
30	#11718258	C₂²	1	2	yes	—
31	#11718257	C₂²	1	1	yes	characteristic
32	#11718257	C₂²	1	1	yes	characteristic
33	#11718231	C₂²	1	2	yes	—
34	C₄⋊#6092	C₂²	1	1	yes	characteristic
35	C₄⋊#6092	C₂²	1	1	yes	characteristic
36	C₄⋊#6092	C₂²	1	2	yes	—
37	C₄⋊#6092	C₂²	1	2	yes	—
38	C₄×#6092	C₂²	1	1	yes	characteristic
39	C₄×#6092	C₂²	1	1	yes	characteristic
40	C₄×#6092	C₂²	1	2	yes	—
41	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	#88201	C₈	1	1	yes	characteristic
43	#88200	C₈	1	1	yes	characteristic
44	C₂×#6094	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₈×D₁₃	C₂³	1	2	yes	—
48	#88152	C₈	1	2	yes	—
49	#88149	C₈	1	1	yes	characteristic
50	#88148	C₈	1	2	yes	—
51	#88144	C₈	1	1	yes	characteristic
52	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
53	C₄×D₅₂	C₂×C₄	1	2	yes	—
54	#88126	C₂×C₄	1	2	yes	—
55	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
56	C₂×#6092	C₂³	1	1	yes	characteristic
57	C₂×#6092	C₂³	1	1	yes	characteristic
58	C₂×#6092	C₂³	1	2	yes	—
59	C₂×#6092	C₂³	1	2	yes	—
60	C₂×#6092	C₂³	1	1	yes	characteristic
61	C₂×#6092	C₂³	1	1	yes	characteristic
62	#88098	—	2	4	no	—
63	D₁₃⋊C₁₆	—	2	2	no	—
64	D₁₃⋊C₁₆	—	2	4	no	—
65	D₁₃⋊C₁₆	—	2	2	no	—
66	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
67	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
68	C₁₃×#97	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
70	#88060	C₂×C₄	1	2	yes	—
71	#88060	C₂×C₄	1	2	yes	—
72	#88060	C₂×C₄	1	2	yes	—
73	#88055	C₂×C₄	1	2	yes	—
74	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
75	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
76	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
77	C₄⋊#1055	C₂×C₄	1	2	yes	—
78	C₄×#1055	C₂³	1	1	yes	characteristic
79	C₄×#1055	C₂³	1	2	yes	—
80	#33480	—	13	1	no	maximal, Sylow
81	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
82	C₂×D₅₂	C₂×C₈	1	2	yes	—
83	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
92	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
93	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
95	#6103	C₂×C₈	1	2	yes	—
96	#6103	C₂×C₈	1	2	yes	—
97	#6103	C₂×C₈	1	2	yes	—
98	#6103	C₂×C₈	1	2	yes	—
99	#6103	C₂×C₈	1	2	yes	—
100	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
101	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
102	#6101	C₂×C₈	1	2	yes	—
103	#6101	C₂×C₈	1	2	yes	—
104	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
106	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
108	C₂×#1055	C₂²×C₄	1	2	yes	—
109	C₂×#1055	C₂⁴	1	1	yes	characteristic
110	#6094	—	2	4	no	—
111	C₈×D₁₃	—	2	2	no	—
112	C₈×D₁₃	—	2	1	no	—
113	C₈×D₁₃	—	2	1	no	—
114	C₈×D₁₃	—	2	2	no	—
115	C₈×D₁₃	—	2	2	no	—
116	#6092	—	2	1	no	—
117	#6092	—	2	1	no	—
118	#6092	—	2	2	no	—
119	#6092	—	2	2	no	—
120	#6092	#41	1	2	yes	—
121	#6092	#41	1	2	yes	—
122	#2966	—	13	1	no	—
123	#2183	—	13	1	no	—
124	Q₈×C₁₆	—	13	1	no	—
125	#2169	—	13	1	no	—
126	#2168	—	13	2	no	—
127	D₄×C₁₆	—	13	1	no	—
128	D₄×C₁₆	—	13	2	no	—
129	#2164	—	13	2	no	—
130	#2163	—	13	1	no	—
131	#2108	—	13	2	no	—
132	C₂×C₄×C₁₆	—	13	1	no	—
133	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
134	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
135	Q₈×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
141	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
142	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
143	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
144	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
145	D₅₂	—	2	4	no	—
146	D₅₂	—	2	2	no	—
147	C₄×D₁₃	—	2	2	no	—
148	C₄×D₁₃	—	2	1	no	—
149	C₄×D₁₃	—	2	2	no	—
150	C₄×D₁₃	—	2	1	no	—
151	C₄×D₁₃	—	2	2	no	—
152	C₄×D₁₃	—	2	2	no	—
153	C₈×C₁₃	—	2	1	no	—
154	C₈×C₁₃	—	2	2	no	—
155	C₈×C₁₃	#131	1	2	yes	—
156	#1055	—	2	1	no	—
157	#1055	—	2	2	no	—
158	#1055	C₂×#41	1	1	yes	characteristic
159	#1055	C₂×#41	1	1	yes	characteristic
160	#529	—	13	1	no	—
161	C₂×M₃₂	—	13	2	no	—
162	C₂²×C₁₆	—	13	1	no	—
163	C₂²×C₁₆	—	13	1	no	—
164	C₂²×C₁₆	—	13	2	no	—
165	#446	#261	1	1	yes	characteristic
166	C₈×Q₈	—	13	1	no	—
167	#436	—	13	1	no	—
168	#435	—	13	2	no	—
169	C₈×D₄	—	13	2	no	—
170	C₈×D₄	—	13	1	no	—
171	#433	—	13	2	no	—
172	#432	—	13	1	no	—
173	#405	—	13	2	no	—
174	C₂×C₄×C₈	—	13	1	no	—
175	C₄⋊C₁₆	—	13	1	no	—
176	C₄⋊C₁₆	—	13	1	no	—
177	C₄⋊C₁₆	—	13	2	no	—
178	C₄⋊C₁₆	—	13	2	no	—
179	C₂²⋊C₁₆	—	13	2	no	—
180	C₂²⋊C₁₆	—	13	2	no	—
181	C₂²⋊C₁₆	—	13	2	no	—
182	#346	—	13	2	no	—
183	#346	—	13	2	no	—
184	C₄×C₁₆	—	13	1	no	—
185	C₄×C₁₆	—	13	1	no	—
186	C₄×C₁₆	—	13	1	no	—
187	C₄×C₁₆	—	13	1	no	—
188	C₄×C₁₆	—	13	2	no	—
189	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
190	C₂×D₁₃	—	2	1	no	—
191	C₂×D₁₃	—	2	1	no	—
192	C₂×D₁₃	—	2	2	no	—
193	C₂×D₁₃	—	2	2	no	—
194	C₂×D₁₃	—	2	2	no	—
195	C₂×D₁₃	#504	1	2	yes	—
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	2	no	—
198	C₄×C₁₃	—	2	2	no	—
199	C₄×C₁₃	#504	1	2	yes	—
200	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
201	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
202	Dic₁₃	—	2	1	no	—
203	Dic₁₃	—	2	1	no	—
204	Dic₁₃	—	2	1	no	—
205	Dic₁₃	—	2	2	no	—
206	C₂×M₁₆	—	13	2	no	—
207	C₂²×C₈	—	13	1	no	—
208	C₂²×C₈	—	13	1	no	—
209	C₂²×C₈	—	13	2	no	—
210	#126	—	13	2	no	—
211	#125	—	13	1	no	—
212	#124	—	13	1	no	—
213	#123	—	13	2	no	—
214	C₂²⋊Q₈	—	13	1	no	—
215	#121	—	13	1	no	—
216	C₄×Q₈	C₂×#261	1	1	yes	characteristic
217	C₄×D₄	—	13	2	no	—
218	C₄×D₄	—	13	1	no	—
219	#117	—	13	2	no	—
220	C₂×C₄²	—	13	1	no	—
221	M₃₂	—	26	4	no	—
222	C₂×C₁₆	—	13	1	no	—
223	C₂×C₁₆	—	13	1	no	—
224	C₂×C₁₆	—	13	1	no	—
225	C₂×C₁₆	—	13	1	no	—
226	C₂×C₁₆	—	13	2	no	—
227	C₂×C₁₆	—	13	2	no	—
228	C₂×C₁₆	—	26	2	no	—
229	C₂×C₁₆	—	26	4	no	—
230	C₂×C₁₆	—	26	2	no	—
231	C₄⋊C₈	—	13	1	no	—
232	C₄⋊C₈	—	13	2	no	—
233	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
234	C₄⋊C₈	C₂×#261	1	2	yes	—
235	C₂²⋊C₈	—	13	2	no	—
236	C₂²⋊C₈	—	13	2	no	—
237	C₂²⋊C₈	—	13	2	no	—
238	#97	—	13	2	no	—
239	#97	C₂×#261	1	2	yes	—
240	C₄×C₈	—	13	1	no	—
241	C₄×C₈	—	13	1	no	—
242	C₄×C₈	—	13	1	no	—
243	C₄×C₈	—	13	2	no	—
244	C₄×C₈	C₂×#261	1	1	yes	characteristic
245	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
246	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
247	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
248	D₁₃	—	2	2	no	—
249	D₁₃	—	4	2	no	—
250	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
251	C₂×D₄	—	13	2	no	—
252	C₂×D₄	—	13	1	no	—
253	C₂²×C₄	—	13	1	no	—
254	C₂²×C₄	—	13	1	no	—
255	C₂²×C₄	—	13	1	no	—
256	C₂²×C₄	—	13	2	no	—
257	M₁₆	—	26	4	no	—
258	C₂×C₈	—	13	1	no	—
259	C₂×C₈	—	13	1	no	—
260	C₂×C₈	—	13	2	no	—
261	C₂×C₈	—	26	1	no	—
262	C₂×C₈	—	26	1	no	—
263	C₂×C₈	—	26	2	no	—
264	C₂×C₈	—	26	2	no	—
265	C₂×C₈	—	26	2	no	—
266	C₂×C₈	C₂²×#261	1	1	yes	characteristic
267	C₂×C₈	C₂²×#261	1	2	yes	—
268	C₂×C₈	C₂²×#261	1	1	yes	characteristic
269	C₄⋊C₄	—	13	2	no	—
270	C₄⋊C₄	—	13	2	no	—
271	C₄⋊C₄	—	13	1	no	—
272	C₄⋊C₄	—	13	2	no	—
273	C₄⋊C₄	D₁₃⋊C₈	1	1	yes	characteristic
274	C₄⋊C₄	D₁₃⋊C₈	1	2	yes	—
275	C₂²⋊C₄	—	13	2	no	—
276	C₂²⋊C₄	—	13	2	no	—
277	C₂²⋊C₄	—	13	2	no	—
278	C₂²⋊C₄	—	13	2	no	—
279	C₂²⋊C₄	—	13	2	no	—
280	C₄²	—	13	1	no	—
281	C₄²	—	13	1	no	—
282	C₄²	—	13	1	no	—
283	C₄²	C₂²×#261	1	1	yes	characteristic
284	C₄²	C₂²×#261	1	2	yes	—
285	C₁₆	—	13	2	no	—
286	C₁₆	—	13	2	no	—
287	C₁₆	—	26	1	no	—
288	C₁₆	—	26	1	no	—
289	C₁₆	—	26	2	no	—
290	C₁₆	—	26	2	no	—
291	C₁₃	#33480	1	1	yes	characteristic, complemented, Sylow
292	C₂³	—	13	1	no	—
293	C₂³	—	13	2	no	—
294	Q₈	—	2	2	no	—
295	D₄	—	26	4	no	—
296	D₄	—	26	2	no	—
297	C₂×C₄	—	13	1	no	—
298	C₂×C₄	—	13	1	no	—
299	C₂×C₄	—	13	1	no	—
300	C₂×C₄	—	13	2	no	—
301	C₂×C₄	—	26	2	no	—
302	C₂×C₄	—	26	2	no	—
303	C₂×C₄	—	26	1	no	—
304	C₂×C₄	—	26	2	no	—
305	C₂×C₄	—	26	1	no	—
306	C₂×C₄	—	26	2	no	—
307	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
308	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
309	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
310	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
311	C₂×C₄	C₂³×#261	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	2	no	—
314	C₈	—	13	1	no	—
315	C₈	—	13	1	no	—
316	C₈	—	26	1	no	—
317	C₈	—	26	2	no	—
318	C₈	#88240	1	2	yes	—
319	C₂²	—	13	2	no	—
320	C₂²	—	26	1	no	—
321	C₂²	—	26	1	no	—
322	C₂²	—	26	2	no	—
323	C₂²	—	26	2	no	—
324	C₂²	—	26	2	no	—
325	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	26	1	no	—
330	C₄	—	26	1	no	—
331	C₄	—	26	1	no	—
332	C₄	—	26	2	no	—
333	C₄	#11719096	1	2	yes	—
334	C₄	C₂×#88240	1	1	yes	characteristic
335	C₄	#41×#261	1	1	yes	characteristic
336	C₂	—	26	2	no	—
337	C₂	—	52	2	no	—
338	C₂	#13131642	1	1	yes	characteristic
339	C₂	D₁₃⋊#504	1	1	yes	characteristic
340	C₂	C₂×#11719096	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#446	C₄
prime radical	#88149	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	#446	#261
hypocenter	C₁₃	#33480
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #13131251	C₂	semidirect
2. Q₈×#6092	C₂	semidirect
3. #13131222	C₂	semidirect
4. #13131219	C₂	semidirect
5. #13131215	C₂	semidirect
6. #13131215	C₂	semidirect
7. #13131204	C₂	semidirect
8. #13131200	C₂	semidirect
9. #13131046	C₂	semidirect
10. C₄×D₁₃⋊C₁₆	C₂	semidirect
291. C₁₃	#33480	semidirect

Maximal subgroups

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

subgroup	index
#13131251	2
Q₈×#6092	2
#13131222	2
#13131219	2
#13131215	2
#13131204	2
#13131200	2
#13131046	2
C₄×D₁₃⋊C₁₆	2
#13123810	2
#33480	13