Subgroups — #16016099

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718562	C₂	1	1	yes	maximal, characteristic, complemented
2	#13123530	C₂	1	1	yes	maximal, characteristic
3	#13116666	C₂	1	4	yes	maximal, complemented
4	#13116659	C₂	1	4	yes	maximal, complemented
5	#13116594	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116449	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116447	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116432	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	#11719496	C₄	1	1	yes	characteristic
12	C₂×#88155	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×M₃₂	—	2	4	no	—
14	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
15	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
16	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
18	#11718868	C₄	1	1	yes	characteristic
19	#11718772	C₄	1	4	yes	—
20	#11718771	C₄	1	1	yes	characteristic
21	C₈×C₂²⋊D₁₃	C₂²	1	4	yes	—
22	#11718763	C₄	1	1	yes	characteristic
23	#11718762	C₂²	1	1	yes	characteristic
24	C₂×#88052	C₂²	1	1	yes	characteristic
25	C₂×#88052	C₂²	1	1	yes	characteristic
26	#11718562	—	2	4	no	—
27	C₂×#88038	C₂²	1	2	yes	complemented
28	C₂×C₁₆×D₁₃	C₂²	1	2	yes	complemented
29	C₂²⋊#6090	C₂²	1	4	yes	—
30	#11718099	C₂²	1	4	yes	complemented
31	#11718094	C₂²	1	2	yes	complemented
32	#11718093	C₂²	1	4	yes	complemented
33	C₁₆×Dic₁₃	C₂²	1	2	yes	complemented
34	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×#6127	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	—	2	2	no	—
37	C₂²×C₈×C₁₃	—	2	2	no	—
38	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	#88207	C₈	1	2	yes	—
43	#88184	C₈	1	2	yes	—
44	#88183	C₈	1	2	yes	—
45	C₄×C₂²⋊D₁₃	C₂×C₄	1	4	yes	—
46	#88180	C₂×C₄	1	1	yes	characteristic
47	#88178	C₈	1	2	yes	—
48	C₂×#6099	C₂×C₄	1	1	yes	characteristic
49	C₂×#6099	C₂×C₄	1	1	yes	characteristic
50	#88155	—	2	2	no	—
51	#88155	—	2	2	no	—
52	C₂×#6094	C₂×C₄	1	2	yes	—
53	C₂×C₈×D₁₃	C₂³	1	2	yes	—
54	C₁₃×M₃₂	—	2	4	no	—
55	C₁₃×M₃₂	—	2	4	no	—
56	C₂×C₁₃×C₁₆	—	2	2	no	—
57	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
58	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
61	C₂²⋊#1055	C₂×C₄	1	4	yes	—
62	#88060	C₂×C₄	1	4	yes	—
63	#88055	C₂×C₄	1	2	yes	—
64	#88054	C₂×C₄	1	4	yes	—
65	C₈×Dic₁₃	C₂³	1	2	yes	—
66	#88052	—	2	2	no	—
67	#88052	—	2	2	no	—
68	C₂×#6090	C₂³	1	2	yes	—
69	C₂×#6090	C₂³	1	2	yes	—
70	#88038	—	2	4	no	—
71	C₁₆×D₁₃	—	2	4	no	—
72	#33433	—	13	1	no	maximal, Sylow
73	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₃	—	2	2	no	—
76	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
81	C₂×C₂²⋊D₁₃	C₂×C₈	1	4	yes	—
82	#6127	—	2	2	no	—
83	#6127	—	2	2	no	—
84	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
86	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	4	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₁₃	#41	1	2	yes	—
95	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₂²⋊Dic₁₃	C₂×C₈	1	4	yes	—
97	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
98	#6103	C₂×C₈	1	4	yes	—
99	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
100	#6101	C₂×C₈	1	4	yes	—
101	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
102	#6099	—	2	2	no	—
103	#6099	—	2	2	no	—
104	C₂×#1055	C₂²×C₄	1	2	yes	—
105	C₂×#1055	C₂²×C₄	1	2	yes	—
106	#6094	—	2	4	no	—
107	C₈×D₁₃	—	2	2	no	—
108	C₈×D₁₃	—	2	2	no	—
109	C₁₃×C₁₆	—	2	2	no	—
110	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
111	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
112	#6090	—	2	2	no	—
113	#6090	—	2	2	no	—
114	C₂×#504	—	13	1	no	—
115	C₂²×M₃₂	D₁₃	1	1	yes	characteristic, complemented
116	#2929	—	13	1	no	—
117	#2169	—	13	4	no	—
118	#2168	—	13	4	no	—
119	#2151	—	13	1	no	—
120	#2113	—	13	1	no	—
121	#2112	—	13	1	no	—
122	#2108	—	13	1	no	—
123	C₂³×C₁₃	—	2	2	no	—
124	C₂³×C₁₃	—	4	2	no	—
125	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
126	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
127	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	4	no	—
132	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#131	1	2	yes	—
136	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
137	C₂²⋊D₁₃	—	2	4	no	—
138	C₂²⋊D₁₃	—	4	4	no	—
139	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
140	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
141	D₅₂	—	2	2	no	—
142	C₄×D₁₃	—	2	2	no	—
143	C₄×D₁₃	—	2	2	no	—
144	Dic₂₆	—	2	2	no	—
145	C₈×C₁₃	—	2	2	no	—
146	C₈×C₁₃	C₄×D₄	1	2	yes	—
147	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
149	#1055	—	2	2	no	—
150	#1055	—	2	2	no	—
151	C₂×#131	—	13	1	no	—
152	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
153	#525	—	13	1	no	—
154	#504	—	26	4	no	—
155	C₂×M₃₂	—	2	4	no	—
156	C₂×M₃₂	—	13	1	no	—
157	C₂×M₃₂	—	13	1	no	—
158	C₂×M₃₂	—	13	2	no	—
159	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
160	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
161	C₂²×C₁₆	—	13	2	no	—
162	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
163	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
164	#435	—	13	4	no	—
165	C₈×D₄	—	13	4	no	—
166	#424	—	13	1	no	—
167	#408	—	13	1	no	—
168	#407	—	13	1	no	—
169	#405	—	13	1	no	—
170	C₄⋊C₁₆	—	13	4	no	—
171	C₂²⋊C₁₆	—	13	4	no	—
172	C₂²⋊C₁₆	—	13	4	no	—
173	#346	—	13	2	no	—
174	C₄×C₁₆	—	13	2	no	—
175	C₂²×C₁₃	—	2	2	no	—
176	C₂²×C₁₃	—	2	2	no	—
177	C₂²×C₁₃	—	2	2	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	—	4	4	no	—
180	C₂²×C₁₃	C₈×D₄	1	2	yes	—
181	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
182	C₂×D₁₃	—	2	4	no	—
183	C₂×D₁₃	—	2	2	no	—
184	C₄×C₁₃	—	2	2	no	—
185	C₄×C₁₃	C₈×D₄	1	2	yes	—
186	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
187	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
188	Dic₁₃	—	2	4	no	—
189	Dic₁₃	—	2	2	no	—
190	C₂×#41	—	13	1	no	—
191	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
192	#131	—	26	2	no	—
193	#131	—	26	2	no	—
194	C₂×M₁₆	—	13	1	no	—
195	C₂×M₁₆	—	13	1	no	—
196	C₂×M₁₆	—	13	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	13	2	no	—
200	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
201	C₂²×C₈	C₄×D₁₃	1	2	yes	—
202	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
203	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
204	#123	—	13	2	no	—
205	C₂²⋊Q₈	—	13	2	no	—
206	#121	—	13	2	no	—
207	#120	—	13	2	no	—
208	C₄×D₄	—	13	4	no	—
209	#117	—	13	1	no	—
210	M₃₂	—	2	4	no	—
211	M₃₂	—	2	4	no	—
212	M₃₂	—	26	2	no	—
213	M₃₂	—	26	2	no	—
214	M₃₂	—	26	4	no	—
215	C₂×C₁₆	—	2	2	no	—
216	C₂×C₁₆	—	13	2	no	—
217	C₂×C₁₆	—	13	2	no	—
218	C₂×C₁₆	—	26	4	no	—
219	C₂×C₁₆	C₂²⋊D₁₃	1	4	yes	complemented
220	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
221	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
222	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
223	C₄⋊C₈	—	13	4	no	—
224	C₂²⋊C₈	—	13	4	no	—
225	C₂²⋊C₈	—	13	4	no	—
226	#97	—	13	2	no	—
227	C₄×C₈	—	13	2	no	—
228	C₂×C₁₃	—	2	2	no	—
229	C₂×C₁₃	—	4	2	no	—
230	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
231	C₂×C₁₃	#3407	1	2	yes	—
232	D₁₃	—	4	2	no	—
233	C₂⁴	C₈×D₁₃	1	1	yes	characteristic
234	#41	—	26	2	no	—
235	#41	—	26	2	no	—
236	C₂×Q₈	—	13	1	no	—
237	C₂×D₄	—	13	2	no	—
238	C₂×D₄	—	13	4	no	—
239	C₂×D₄	—	13	1	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	13	2	no	—
243	C₂²×C₄	C₈×D₁₃	1	2	yes	—
244	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
246	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
247	M₁₆	—	26	2	no	—
248	M₁₆	—	26	2	no	—
249	M₁₆	—	26	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	13	2	no	—
255	C₂×C₈	—	13	2	no	—
256	C₂×C₈	—	26	2	no	—
257	C₂×C₈	—	26	2	no	—
258	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
259	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
260	C₂×C₈	#6127	1	2	yes	—
261	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
262	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
263	C₄⋊C₄	—	13	2	no	—
264	C₄⋊C₄	—	13	4	no	—
265	C₂²⋊C₄	—	13	2	no	—
266	C₂²⋊C₄	—	13	4	no	—
267	C₂²⋊C₄	—	13	4	no	—
268	C₄²	—	13	2	no	—
269	C₁₆	—	2	2	no	—
270	C₁₆	—	26	2	no	—
271	C₁₆	—	26	2	no	—
272	C₁₆	C₂×C₂²⋊D₁₃	1	2	yes	—
273	C₁₆	C₂×C₂²⋊D₁₃	1	2	yes	—
274	C₁₃	#33433	1	1	yes	characteristic, complemented, Sylow
275	C₂³	—	2	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	13	2	no	—
278	C₂³	C₂×C₈×D₁₃	1	2	yes	—
279	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
280	Q₈	—	26	2	no	—
281	D₄	—	26	4	no	—
282	D₄	—	26	2	no	—
283	D₄	—	52	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	13	4	no	—
289	C₂×C₄	—	13	2	no	—
290	C₂×C₄	—	26	2	no	—
291	C₂×C₄	—	26	2	no	—
292	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
293	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
294	C₂×C₄	#88155	1	2	yes	—
295	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
296	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	26	2	no	—
299	C₈	—	26	2	no	—
300	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
301	C₈	C₂×#6127	1	1	yes	characteristic
302	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	4	4	no	—
308	C₂²	—	26	4	no	—
309	C₂²	—	26	2	no	—
310	C₂²	C₈×C₂²⋊D₁₃	1	2	yes	—
311	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	26	4	no	—
314	C₄	—	26	2	no	—
315	C₄	C₈×C₂²⋊D₁₃	1	2	yes	—
316	C₄	C₂×#88155	1	1	yes	characteristic
317	C₄	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	2	no	—
320	C₂	—	52	2	no	—
321	C₂	C₂×C₈×C₂²⋊D₁₃	1	1	yes	characteristic
322	C₂	#13127938	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₂²×C₁₃×M₃₂	C₂
prime radical	#11719496	C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂²×M₃₂	D₁₃
hypocenter	C₁₃	#33433
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#11718562	C₂	semidirect
3. #13116666	C₂	semidirect
4. #13116659	C₂	semidirect
5. #13116594	C₂	semidirect
6. #13116449	C₂	semidirect
7. #13116447	C₂	semidirect
8. #13116432	C₂	semidirect
9. C₂²×C₁₃×M₃₂	C₂	semidirect
27. C₂×#88038	C₂²	semidirect
28. C₂×C₁₆×D₁₃	C₂²	semidirect
30. #11718099	C₂²	semidirect
31. #11718094	C₂²	semidirect
32. #11718093	C₂²	semidirect
33. C₁₆×Dic₁₃	C₂²	semidirect
57. C₂×C₁₃×C₁₆	D₄	semidirect
115. C₂²×M₃₂	D₁₃	semidirect
219. C₂×C₁₆	C₂²⋊D₁₃	semidirect
274. C₁₃	#33433	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718562	2
#13123530	2
#13116666	2
#13116659	2
#13116594	2
#13116449	2
#13116447	2
#13116432	2
C₂²×C₁₃×M₃₂	2
#33433	13