Subgroups — #16018477

Order 3328.

558 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125422	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117116	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117091	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117065	C₂	1	1	yes	maximal, characteristic, complemented
5	#13117059	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116930	C₂	1	4	yes	maximal, complemented
7	C₄×#88067	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic
10	C₁₃×#532	—	2	1	no	—
11	#11719175	—	2	1	no	—
12	C₁₃×#500	C₂²	1	1	yes	characteristic
13	C₁₃×#493	C₂²	1	1	yes	characteristic
14	C₁₃×#486	—	2	1	no	—
15	C₁₃×#466	—	2	1	no	—
16	C₁₃×#466	C₂²	1	2	yes	—
17	C₄×C₁₃×D₈	—	2	1	no	—
18	C₄×C₁₃×D₈	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×#88067	C₂²	1	4	yes	—
21	#11718777	—	2	1	no	—
22	#11718777	C₂²	1	2	yes	—
23	C₂²×#6090	C₂²	1	2	yes	—
24	#11718360	C₂²	1	1	yes	characteristic
25	C₄×Dic₅₂	—	2	1	no	—
26	C₄×Dic₅₂	C₂²	1	2	yes	—
27	#11718343	C₂²	1	1	yes	characteristic
28	#11718342	—	2	1	no	—
29	#11718154	C₂²	1	4	yes	—
30	#11718084	C₂²	1	4	yes	—
31	#11718082	C₂²	1	4	yes	—
32	C₄×#6090	C₂²	1	2	yes	—
33	C₄×#6090	C₂²	1	1	yes	characteristic
34	C₄×#6090	C₂²	1	1	yes	characteristic
35	C₂×C₁₃×#41	—	4	1	no	—
36	C₂×C₁₃×D₈	—	2	1	no	—
37	C₂×C₁₃×D₈	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×C₄⋊Q₈	—	2	1	no	—
41	C₁₃×#127	—	2	1	no	—
42	C₁₃×#124	—	4	1	no	—
43	C₁₃×#121	—	2	2	no	—
44	C₁₃×#121	—	4	1	no	—
45	C₄×D₄×C₁₃	—	2	2	no	—
46	C₄×D₄×C₁₃	—	4	1	no	—
47	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
48	#88180	—	4	1	no	—
49	#88178	—	2	2	no	—
50	#88178	—	4	1	no	—
51	C₂×Dic₅₂	—	2	1	no	—
52	C₂×Dic₅₂	C₂³	1	2	yes	—
53	#88124	—	4	1	no	—
54	#88123	—	2	1	no	—
55	#88123	—	2	1	no	—
56	C₄×Dic₂₆	—	2	2	no	—
57	C₄×Dic₂₆	—	4	1	no	—
58	C₁₃×#107	—	2	1	no	—
59	C₁₃×#107	C₂³	1	2	yes	—
60	C₁₃×#102	—	2	2	no	—
61	C₁₃×#102	—	2	2	no	—
62	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	#88067	—	2	8	no	—
67	#88058	—	2	1	no	—
68	#88058	C₂³	1	2	yes	—
69	#88056	—	2	2	no	—
70	#88056	—	2	2	no	—
71	C₂×#6090	—	2	4	no	—
72	C₂×#6090	C₂³	1	2	yes	—
73	C₂×#6090	C₂³	1	2	yes	—
74	#33736	—	13	1	no	maximal, Sylow
75	C₁₃×#41	—	8	4	no	—
76	C₂×Q₈×C₁₃	—	4	1	no	—
77	C₂×D₄×C₁₃	—	2	2	no	—
78	C₂×D₄×C₁₃	—	4	2	no	—
79	C₂×D₄×C₁₃	—	4	1	no	—
80	C₂×D₄×C₁₃	—	4	1	no	—
81	C₂²×C₄×C₁₃	—	4	2	no	—
82	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₂₆	—	2	2	no	—
86	C₂×Dic₂₆	—	4	1	no	—
87	C₁₃×D₈	—	2	4	no	—
88	C₁₃×D₈	—	4	2	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	4	1	no	—
97	C₁₃×C₂²⋊C₄	—	4	2	no	—
98	C₁₃×C₂²⋊C₄	—	4	2	no	—
99	C₄²×C₁₃	D₈	1	1	yes	characteristic
100	C₄²×C₁₃	D₈	1	1	yes	characteristic
101	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂²⋊Dic₁₃	—	4	2	no	—
104	C₄⋊Dic₁₃	—	2	2	no	—
105	C₄⋊Dic₁₃	—	4	1	no	—
106	C₄⋊Dic₁₃	—	4	1	no	—
107	C₄⋊Dic₁₃	—	4	1	no	—
108	#6101	—	4	2	no	—
109	#6101	—	4	2	no	—
110	C₄×Dic₁₃	—	4	2	no	—
111	Dic₅₂	—	2	4	no	—
112	Dic₅₂	—	4	2	no	—
113	#6090	—	2	2	no	—
114	#6090	#41	1	4	yes	complemented
115	#3170	—	13	1	no	—
116	#3169	D₁₃	1	1	yes	characteristic
117	#2254	—	13	1	no	—
118	#2248	—	13	1	no	—
119	#2241	—	13	1	no	—
120	#2240	—	13	1	no	—
121	#2217	—	13	4	no	—
122	C₄×SD₃₂	—	13	4	no	—
123	C₂×C₄×C₁₆	—	13	1	no	—
124	C₂³×C₁₃	—	4	2	no	—
125	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
126	Q₈×C₁₃	—	8	2	no	—
127	D₄×C₁₃	—	4	2	no	—
128	D₄×C₁₃	—	4	2	no	—
129	D₄×C₁₃	—	8	4	no	—
130	D₄×C₁₃	—	8	2	no	—
131	D₄×C₁₃	—	8	2	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	8	4	no	—
137	C₂×C₄×C₁₃	SD₃₂	1	4	yes	complemented
138	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×Dic₁₃	—	4	2	no	—
145	C₂×Dic₁₃	—	4	2	no	—
146	Dic₂₆	—	4	2	no	—
147	Dic₂₆	—	4	2	no	—
148	Dic₂₆	—	8	2	no	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	#135	1	2	yes	—
152	C₈×C₁₃	C₂×#41	1	2	yes	—
153	#533	—	26	1	no	—
154	#532	—	2	1	no	—
155	C₂×SD₃₂	—	13	4	no	—
156	C₂²×C₁₆	—	13	2	no	—
157	#500	—	13	1	no	—
158	#500	C₂×D₁₃	1	1	yes	characteristic
159	#494	—	13	1	no	—
160	#493	C₂×D₁₃	1	1	yes	characteristic
161	#487	—	26	1	no	—
162	#486	—	2	1	no	—
163	#467	—	13	2	no	—
164	#467	—	26	1	no	—
165	#466	—	2	1	no	—
166	#466	C₂×D₁₃	1	2	yes	—
167	C₄×Q₁₆	—	13	2	no	—
168	C₄×Q₁₆	—	26	1	no	—
169	C₄×D₈	—	2	1	no	—
170	C₄×D₈	C₂×D₁₃	1	2	yes	—
171	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
172	#367	—	13	4	no	—
173	#358	—	13	4	no	—
174	#357	—	13	4	no	—
175	C₄×C₁₆	—	13	2	no	—
176	C₄×C₁₆	—	13	1	no	—
177	C₄×C₁₆	—	13	1	no	—
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	8	2	no	—
181	C₂²×C₁₃	—	8	2	no	—
182	C₂²×C₁₃	—	8	2	no	—
183	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
184	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	8	2	no	—
188	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
189	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
190	C₄×C₁₃	#508	1	2	yes	—
191	C₄×C₁₃	#532	1	1	yes	characteristic
192	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
193	Dic₁₃	—	8	2	no	—
194	Dic₁₃	—	8	2	no	—
195	C₂×#41	—	4	1	no	—
196	C₂×Q₁₆	—	13	2	no	—
197	C₂×Q₁₆	—	26	1	no	—
198	C₂×D₈	—	2	1	no	—
199	C₂×D₈	C₂²×D₁₃	1	2	yes	—
200	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
201	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
202	C₄⋊Q₈	—	2	1	no	—
203	C₄⋊Q₈	—	26	1	no	—
204	C₄⋊Q₈	—	26	1	no	—
205	#127	—	2	1	no	—
206	#125	—	52	1	no	—
207	#124	—	4	1	no	—
208	C₂²⋊Q₈	—	26	2	no	—
209	C₂²⋊Q₈	—	52	1	no	—
210	#121	—	2	2	no	—
211	#121	—	4	1	no	—
212	C₄×Q₈	—	26	2	no	—
213	C₄×Q₈	—	52	1	no	—
214	C₄×D₄	—	2	2	no	—
215	C₄×D₄	—	4	1	no	—
216	#117	—	52	1	no	—
217	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
218	SD₃₂	—	26	8	no	—
219	C₂×C₁₆	—	13	2	no	—
220	C₂×C₁₆	—	13	2	no	—
221	C₂×C₁₆	—	26	4	no	—
222	#107	—	2	1	no	—
223	#107	—	13	2	no	—
224	#107	—	26	1	no	—
225	#107	C₂²×D₁₃	1	2	yes	—
226	#103	—	26	2	no	—
227	#103	—	26	2	no	—
228	#102	—	2	2	no	—
229	#102	—	2	2	no	—
230	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
231	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
233	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
234	C₂×C₁₃	—	2	2	no	—
235	C₂×C₁₃	—	16	2	no	—
236	C₂×C₁₃	#3169	1	1	yes	characteristic
237	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
238	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
239	#41	—	8	4	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	26	2	no	—
242	C₂×Q₈	—	52	1	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	#6104	1	1	yes	characteristic
249	C₂²×C₄	#6104	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
251	Q₁₆	—	26	4	no	—
252	Q₁₆	—	52	2	no	—
253	D₈	—	2	4	no	—
254	D₈	—	4	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
258	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	26	2	no	—
264	C₄⋊C₄	—	52	1	no	—
265	C₄⋊C₄	—	52	1	no	—
266	C₄⋊C₄	—	52	2	no	—
267	C₄⋊C₄	—	52	2	no	—
268	C₄⋊C₄	—	52	1	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	52	2	no	—
272	C₄²	—	52	2	no	—
273	C₄²	#6104	1	1	yes	characteristic
274	C₄²	#6104	1	1	yes	characteristic
275	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
276	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
277	C₁₆	—	13	4	no	—
278	C₁₆	—	26	2	no	—
279	C₁₃	#33736	1	1	yes	characteristic, complemented, Sylow
280	C₂³	—	4	2	no	—
281	C₂³	C₂×#6104	1	1	yes	characteristic
282	Q₈	—	8	2	no	—
283	Q₈	—	52	2	no	—
284	Q₈	—	52	2	no	—
285	Q₈	—	104	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	8	4	no	—
289	D₄	—	8	2	no	—
290	D₄	—	8	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	8	4	no	—
296	C₂×C₄	—	52	2	no	—
297	C₂×C₄	—	52	2	no	—
298	C₂×C₄	#88067	1	4	yes	complemented
299	C₂×C₄	C₂×#6104	1	1	yes	characteristic
300	C₂×C₄	C₂×#6104	1	1	yes	characteristic
301	C₂×C₄	C₂×#6104	1	1	yes	characteristic
302	C₂×C₄	C₂×#6104	1	1	yes	characteristic
303	C₂×C₄	C₂×#6104	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	#88204	1	2	yes	—
308	C₈	#41×D₁₃	1	2	yes	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	C₂×#88067	1	2	yes	—
315	C₂²	C₂²×#6104	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	104	2	no	—
320	C₄	—	104	2	no	—
321	C₄	C₂×#88067	1	2	yes	—
322	C₄	C₂×#88067	1	2	yes	—
323	C₄	#11718855	1	2	yes	—
324	C₄	C₂×#88204	1	1	yes	characteristic
325	C₄	#11719537	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	16	2	no	—
328	C₂	#13125411	1	1	yes	characteristic
329	C₂	C₂²×#88067	1	1	yes	characteristic
330	C₂	C₂×#11718855	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting, prime radical	C₁₃×#3169	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3169	D₁₃
hypocenter	C₁₃	#33736
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #13125422	C₂	semidirect
2. #13117116	C₂	semidirect
3. #13117091	C₂	semidirect
4. #13117065	C₂	semidirect
5. #13117059	C₂	semidirect
6. #13116930	C₂	semidirect
7. C₄×#88067	C₂	semidirect
8. C₂×C₄×#6090	C₂	semidirect
114. #6090	#41	semidirect
137. C₂×C₄×C₁₃	SD₃₂	semidirect
279. C₁₃	#33736	semidirect
298. C₂×C₄	#88067	semidirect

Maximal subgroups

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

subgroup	index
#13125422	2
#13117116	2
#13117091	2
#13117065	2
#13117059	2
#13116930	2
C₄×#88067	2
C₂×C₄×#6090	2
C₁₃×#3169	2
#33736	13