Subgroups — #16017354

Order 3328.

710 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#13124413	C₂	1	2	yes	maximal, complemented
3	#13116929	C₂	1	4	yes	maximal, complemented
4	#13116924	C₂	1	2	yes	maximal, complemented
5	#13116919	C₂	1	2	yes	maximal, complemented
6	#13116620	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116543	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	C₂²	1	2	yes	—
10	#11719533	—	2	2	no	—
11	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#508	—	2	4	no	—
13	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	complemented
15	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	#11718894	—	2	2	no	—
18	#11718849	C₂²	1	4	yes	—
19	#11718832	—	2	4	no	—
20	#11718818	C₂²	1	4	yes	—
21	#11718767	C₂²	1	2	yes	—
22	C₂²×#6090	C₂²	1	1	yes	characteristic
23	C₂×C₁₆×D₁₃	—	2	4	no	—
24	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
25	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
26	#11718156	C₂²	1	4	yes	complemented
27	#11718154	C₂²	1	4	yes	complemented
28	#11718097	C₂²	1	2	yes	complemented
29	#11718096	C₂²	1	2	yes	complemented
30	C₂×C₁₃×#41	—	2	2	no	—
31	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
32	C₁₃×#135	—	2	4	no	—
33	C₁₃×#135	—	2	4	no	—
34	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
35	C₂×C₁₃×SD₁₆	—	2	2	no	—
36	C₂×C₁₃×D₈	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	#88200	—	2	4	no	—
39	#88192	—	2	4	no	—
40	#88191	—	4	2	no	—
41	#88189	—	4	2	no	—
42	C₄×C₂²⋊D₁₃	—	4	4	no	—
43	#88180	—	2	2	no	—
44	C₂²×#1055	D₄	1	1	yes	characteristic
45	C₂×C₈×D₁₃	—	2	2	no	—
46	C₂×C₈×D₁₃	—	2	2	no	—
47	C₂×C₈×D₁₃	D₄	1	2	yes	—
48	C₂×C₈×D₁₃	C₂³	1	2	yes	—
49	C₁₃×Q₃₂	—	2	2	no	—
50	C₁₃×SD₃₂	—	2	4	no	—
51	C₁₃×D₁₆	—	2	2	no	—
52	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
53	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
54	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
55	#88075	—	2	4	no	—
56	#88072	—	2	4	no	—
57	#88058	C₂³	1	4	yes	—
58	#88057	—	2	2	no	—
59	C₂×#6090	—	2	2	no	—
60	C₂×#6090	C₂³	1	2	yes	—
61	C₁₆×D₁₃	—	2	4	no	—
62	C₁₆×D₁₃	—	2	4	no	—
63	#33600	—	13	1	no	maximal, Sylow
64	C₂³×D₁₃	D₈	1	1	yes	characteristic
65	C₁₃×#41	—	4	4	no	—
66	C₁₃×#41	—	4	4	no	—
67	C₂×Q₈×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	—	2	2	no	—
69	C₂×D₄×C₁₃	—	4	2	no	—
70	C₂²×C₄×C₁₃	—	4	2	no	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₂²⋊D₁₃	—	4	4	no	—
73	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
74	C₂×C₄×D₁₃	—	2	2	no	—
75	C₂×C₄×D₁₃	—	2	2	no	—
76	C₂×C₄×D₁₃	D₈	1	2	yes	—
77	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
78	C₁₃×Q₁₆	—	2	2	no	—
79	C₁₃×Q₁₆	—	2	2	no	—
80	C₁₃×SD₁₆	—	4	4	no	—
81	C₁₃×D₈	—	2	2	no	—
82	C₁₃×D₈	—	2	2	no	—
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₃	#41	1	2	yes	—
86	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₂²⋊Dic₁₃	—	4	2	no	—
88	C₂²⋊Dic₁₃	—	4	4	no	—
89	#6103	—	4	4	no	—
90	C₄⋊Dic₁₃	—	2	4	no	—
91	#6101	—	4	4	no	—
92	C₄×Dic₁₃	—	4	2	no	—
93	C₂×#1055	—	2	2	no	—
94	C₂×#1055	C₂×D₄	1	2	yes	—
95	C₈×D₁₃	—	2	4	no	—
96	C₈×D₁₃	—	2	2	no	—
97	C₈×D₁₃	—	2	2	no	—
98	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
99	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
100	#6090	—	2	2	no	—
101	C₂×#508	D₁₃	1	1	yes	characteristic, complemented
102	C₂³×C₁₆	—	13	1	no	—
103	#3050	—	13	2	no	—
104	#2217	—	13	4	no	—
105	#2216	—	13	2	no	—
106	#2215	—	13	2	no	—
107	#2158	—	13	1	no	—
108	#2138	—	13	2	no	—
109	C₂³×C₁₃	—	4	2	no	—
110	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
111	C₂²×D₁₃	—	2	2	no	—
112	C₂²×D₁₃	—	2	4	no	—
113	C₂²×D₁₃	C₂×D₈	1	2	yes	—
114	Q₈×C₁₃	—	4	2	no	—
115	Q₈×C₁₃	—	4	2	no	—
116	D₄×C₁₃	—	4	2	no	—
117	D₄×C₁₃	—	4	2	no	—
118	D₄×C₁₃	—	8	4	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	8	4	no	—
121	C₂×C₄×C₁₃	#121	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#135	1	2	yes	—
124	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₂²⋊D₁₃	—	8	8	no	—
126	C₂×Dic₁₃	—	2	2	no	—
127	C₂×Dic₁₃	—	4	4	no	—
128	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
129	C₄×D₁₃	—	2	2	no	—
130	C₄×D₁₃	—	2	2	no	—
131	C₄×D₁₃	—	2	4	no	—
132	C₈×C₁₃	#121	1	2	yes	—
133	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
135	#1055	—	2	2	no	—
136	C₂×#135	C₂×D₁₃	1	2	yes	—
137	C₂³×C₈	—	13	1	no	—
138	#525	—	26	2	no	—
139	#508	—	2	4	no	—
140	C₂×Q₃₂	C₂×D₁₃	1	1	yes	characteristic, complemented
141	C₂×SD₃₂	C₂×D₁₃	1	2	yes	complemented
142	C₂×D₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
143	C₂²×C₁₆	—	13	1	no	—
144	C₂²×C₁₆	—	13	2	no	—
145	C₂²×C₁₆	—	13	2	no	—
146	C₂²×C₁₆	—	26	4	no	—
147	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
148	#467	—	13	4	no	—
149	#466	—	13	4	no	—
150	#465	—	26	4	no	—
151	#427	—	13	2	no	—
152	#416	—	26	2	no	—
153	#367	—	13	2	no	—
154	#366	—	13	2	no	—
155	#358	—	13	4	no	—
156	#357	—	13	4	no	—
157	C₂²×C₁₃	—	2	2	no	—
158	C₂²×C₁₃	—	8	4	no	—
159	C₂²×C₁₃	—	8	2	no	—
160	C₂²×C₁₃	#466	1	2	yes	—
161	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
162	C₂×D₁₃	—	2	8	no	—
163	C₂×D₁₃	—	2	2	no	—
164	C₂×D₁₃	—	2	2	no	—
165	C₂×D₁₃	#508	1	4	yes	complemented
166	C₄×C₁₃	—	8	2	no	—
167	C₄×C₁₃	#466	1	2	yes	—
168	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
169	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
170	Dic₁₃	—	2	2	no	—
171	Dic₁₃	—	8	4	no	—
172	C₂×#41	—	2	2	no	—
173	C₂³×C₄	—	13	1	no	—
174	#135	—	2	4	no	—
175	#135	—	2	4	no	—
176	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
177	C₂×SD₁₆	—	2	2	no	—
178	C₂×D₈	C₂²×D₁₃	1	2	yes	—
179	C₂²×C₈	—	13	1	no	—
180	C₂²×C₈	—	13	2	no	—
181	C₂²×C₈	—	13	2	no	—
182	C₂²×C₈	—	26	2	no	—
183	C₂²×C₈	—	26	2	no	—
184	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
185	#123	—	52	2	no	—
186	C₂²⋊Q₈	—	26	4	no	—
187	#121	—	26	4	no	—
188	#120	—	52	2	no	—
189	C₄×D₄	—	52	4	no	—
190	#117	—	26	2	no	—
191	Q₃₂	—	2	2	no	—
192	SD₃₂	—	2	4	no	—
193	D₁₆	—	2	2	no	—
194	C₂×C₁₆	—	13	2	no	—
195	C₂×C₁₆	—	26	2	no	—
196	C₂×C₁₆	—	26	4	no	—
197	C₂×C₁₆	—	26	4	no	—
198	C₂×C₁₆	C₂²⋊D₁₃	1	4	yes	complemented
199	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
200	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
201	#107	—	13	4	no	—
202	#106	—	26	2	no	—
203	#103	—	26	4	no	—
204	#102	—	26	4	no	—
205	C₂×C₁₃	—	2	2	no	—
206	C₂×C₁₃	—	16	2	no	—
207	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
208	C₂×C₁₃	C₂×#508	1	2	yes	—
209	D₁₃	—	2	4	no	—
210	C₂⁴	—	13	1	no	—
211	#41	—	4	4	no	—
212	#41	—	4	4	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	52	4	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	13	1	no	—
219	C₂²×C₄	—	13	2	no	—
220	C₂²×C₄	—	13	2	no	—
221	C₂²×C₄	—	26	2	no	—
222	C₂²×C₄	—	26	2	no	—
223	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
224	Q₁₆	—	2	2	no	—
225	Q₁₆	—	2	2	no	—
226	SD₁₆	—	4	4	no	—
227	D₈	—	2	2	no	—
228	D₈	—	2	2	no	—
229	C₂×C₈	—	13	2	no	—
230	C₂×C₈	—	26	2	no	—
231	C₂×C₈	—	26	4	no	—
232	C₂×C₈	—	26	2	no	—
233	C₂×C₈	—	26	2	no	—
234	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
235	C₂×C₈	#6129	1	2	yes	—
236	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
237	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
238	C₄⋊C₄	—	26	4	no	—
239	C₄⋊C₄	—	52	4	no	—
240	C₂²⋊C₄	—	52	4	no	—
241	C₂²⋊C₄	—	52	4	no	—
242	C₂²⋊C₄	—	52	2	no	—
243	C₄²	—	52	2	no	—
244	C₁₆	—	26	2	no	—
245	C₁₆	C₂×C₂²⋊D₁₃	1	2	yes	—
246	C₁₆	C₂×C₂²⋊D₁₃	1	2	yes	—
247	C₁₃	#33600	1	1	yes	characteristic, complemented, Sylow
248	C₂³	—	4	2	no	—
249	C₂³	—	13	2	no	—
250	C₂³	—	26	4	no	—
251	C₂³	—	26	2	no	—
252	C₂³	D₈×D₁₃	1	1	yes	characteristic
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	D₄	—	8	4	no	—
258	D₄	—	104	8	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	—	13	2	no	—
262	C₂×C₄	—	26	2	no	—
263	C₂×C₄	—	26	2	no	—
264	C₂×C₄	—	26	2	no	—
265	C₂×C₄	—	26	4	no	—
266	C₂×C₄	—	52	4	no	—
267	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
268	C₂×C₄	#88166	1	2	yes	—
269	C₂×C₄	#88192	1	2	yes	—
270	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
271	C₈	—	26	2	no	—
272	C₈	#88192	1	2	yes	—
273	C₈	C₂×#6129	1	1	yes	characteristic
274	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	8	4	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	13	4	no	—
279	C₂²	—	26	8	no	—
280	C₂²	—	26	2	no	—
281	C₂²	—	26	2	no	—
282	C₂²	#11718818	1	2	yes	—
283	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
284	C₄	—	8	2	no	—
285	C₄	—	26	2	no	—
286	C₄	—	104	4	no	—
287	C₄	#11718818	1	2	yes	—
288	C₄	C₂×#88166	1	1	yes	characteristic
289	C₄	C₂×#88192	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	16	2	no	—
292	C₂	—	26	4	no	—
293	C₂	C₂×#11718818	1	1	yes	characteristic
294	C₂	D₁₃×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#508	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	C₂×#508	D₁₃
hypocenter	C₁₃	#33600
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₆×D₁₃	C₂	semidirect
2. #13124413	C₂	semidirect
3. #13116929	C₂	semidirect
4. #13116924	C₂	semidirect
5. #13116919	C₂	semidirect
6. #13116620	C₂	semidirect
7. #13116543	C₂	semidirect
8. C₂×C₁₃×#508	C₂	semidirect
13. C₂×C₁₃×Q₃₂	C₂²	semidirect
14. C₂×C₁₃×SD₃₂	C₂²	semidirect
15. C₂×C₁₃×D₁₆	C₂²	semidirect
26. #11718156	C₂²	semidirect
27. #11718154	C₂²	semidirect
28. #11718097	C₂²	semidirect
29. #11718096	C₂²	semidirect
52. C₂×C₁₃×C₁₆	D₄	semidirect
101. C₂×#508	D₁₃	semidirect
140. C₂×Q₃₂	C₂×D₁₃	semidirect
141. C₂×SD₃₂	C₂×D₁₃	semidirect
142. C₂×D₁₆	C₂×D₁₃	semidirect
165. C₂×D₁₃	#508	semidirect
198. C₂×C₁₆	C₂²⋊D₁₃	semidirect
247. C₁₃	#33600	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₁₃	2
#13124413	2
#13116929	2
#13116924	2
#13116919	2
#13116620	2
#13116543	2
C₂×C₁₃×#508	2
#33600	13