Subgroups — #16016435

Order 3328.

602 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#6127	C₂	1	1	yes	maximal, characteristic
2	#13116738	C₂	1	2	yes	maximal, complemented
3	#13116655	C₂	1	2	yes	maximal, complemented
4	#13116653	C₂	1	4	yes	maximal, complemented
5	#13116633	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₁₆×D₁₃	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#6127	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₈×C₂²⋊D₁₃	C₄	1	2	yes	—
12	C₈×C₂²⋊D₁₃	C₂²	1	2	yes	—
13	C₂²×#6090	C₂²	1	2	yes	—
14	#11718586	C₄	1	1	yes	characteristic
15	C₂×C₁₆×D₁₃	C₂²	1	4	yes	—
16	C₈×D₅₂	C₄	1	1	yes	characteristic
17	C₈×D₅₂	C₂²	1	1	yes	characteristic
18	#11718346	C₄	1	2	yes	—
19	C₄×C₈×D₁₃	C₂²	1	2	yes	—
20	C₈×Dic₂₆	C₄	1	1	yes	characteristic
21	C₈×Dic₂₆	C₂²	1	1	yes	characteristic
22	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
23	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
24	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
25	#11718099	C₂²	1	4	yes	—
26	#11718093	C₂²	1	4	yes	—
27	C₁₆×Dic₁₃	C₂²	1	4	yes	—
28	C₄×#6090	C₂²	1	1	yes	characteristic
29	C₄×#6090	C₂²	1	2	yes	—
30	C₄×#6090	C₂²	1	1	yes	characteristic
31	C₂×#6127	C₈	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₂²⋊D₁₃	C₈	1	2	yes	—
36	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
37	#88180	C₈	1	1	yes	characteristic
38	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₁₃	C₂³	1	2	yes	—
40	C₄×D₅₂	C₈	1	1	yes	characteristic
41	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
42	#88126	C₈	1	2	yes	—
43	C₄²×D₁₃	C₂×C₄	1	2	yes	—
44	C₄×Dic₂₆	C₈	1	1	yes	characteristic
45	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
47	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₂²⋊#1055	C₂×C₄	1	2	yes	—
52	#88060	C₂×C₄	1	2	yes	—
53	#88060	C₂×C₄	1	2	yes	—
54	#88054	C₂×C₄	1	2	yes	—
55	#88054	C₂×C₄	1	2	yes	—
56	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
57	C₈×Dic₁₃	C₂³	1	2	yes	—
58	C₂×#6090	—	2	4	no	—
59	C₂×#6090	C₂³	1	2	yes	—
60	C₂×#6090	C₂³	1	2	yes	—
61	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
62	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
63	C₄×#1055	C₂×C₄	1	2	yes	—
64	C₁₆×D₁₃	—	2	8	no	—
65	C₁₆×#41	—	13	1	no	maximal, Sylow
66	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
70	#6127	C₁₆	1	8	yes	complemented
71	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
74	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
75	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
83	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
87	#6103	C₂×C₈	1	2	yes	—
88	#6103	C₂×C₈	1	2	yes	—
89	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
90	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
91	#6101	C₂×C₈	1	2	yes	—
92	#6101	C₂×C₈	1	2	yes	—
93	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
94	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
95	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
96	C₂×#1055	C₂²×C₄	1	2	yes	—
97	C₂×#1055	C₂²×C₄	1	2	yes	—
98	C₈×D₁₃	—	2	4	no	—
99	C₈×D₁₃	—	2	4	no	—
100	C₁₃×C₁₆	—	2	4	no	—
101	#6090	—	2	2	no	—
102	#6090	#41	1	4	yes	complemented
103	C₈×#41	—	13	1	no	—
104	Q₈×C₁₆	—	13	2	no	—
105	D₄×C₁₆	—	13	2	no	—
106	D₄×C₁₆	—	13	4	no	—
107	#2162	—	13	2	no	—
108	#2162	D₁₃	1	1	yes	characteristic, complemented
109	C₂×C₄×C₁₆	—	13	1	no	—
110	C₂×C₄×C₁₆	—	13	2	no	—
111	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
112	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	complemented
117	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂²⋊D₁₃	C₂×C₁₆	1	8	yes	complemented
124	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
125	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
126	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
127	D₅₂	C₂×C₁₆	1	4	yes	complemented
128	C₄×D₁₃	—	2	2	no	—
129	C₄×D₁₃	—	2	2	no	—
130	C₄×D₁₃	—	2	4	no	—
131	C₄×D₁₃	C₂×C₁₆	1	8	yes	complemented
132	Dic₂₆	C₂×C₁₆	1	4	yes	complemented
133	C₈×C₁₃	—	2	1	no	—
134	C₈×C₁₃	—	2	1	no	—
135	C₈×C₁₃	#131	1	2	yes	—
136	C₈×C₁₃	C₂×#41	1	2	yes	—
137	#1055	—	2	2	no	—
138	#1055	—	2	2	no	—
139	C₄×#41	—	13	1	no	—
140	C₂²×C₁₆	—	13	2	no	—
141	C₂²×C₁₆	—	13	4	no	—
142	C₈×Q₈	—	13	1	no	—
143	C₈×Q₈	—	13	1	no	—
144	C₈×D₄	—	13	2	no	—
145	C₈×D₄	—	13	2	no	—
146	C₈×D₄	—	13	1	no	—
147	C₈×D₄	—	13	1	no	—
148	#431	—	13	1	no	—
149	#431	—	13	2	no	—
150	C₂×C₄×C₈	—	13	2	no	—
151	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
152	C₄⋊C₁₆	—	13	4	no	—
153	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
154	C₂²⋊C₁₆	—	13	4	no	—
155	C₂²⋊C₁₆	C₂×D₁₃	1	2	yes	—
156	C₄×C₁₆	—	13	4	no	—
157	C₄×C₁₆	—	13	2	no	—
158	C₄×C₁₆	—	13	1	no	—
159	C₄×C₁₆	—	13	1	no	—
160	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	—	2	1	no	—
163	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
164	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
165	C₂×D₁₃	—	2	2	no	—
166	C₂×D₁₃	—	2	2	no	—
167	C₂×D₁₃	C₂²×C₁₆	1	4	yes	—
168	C₄×C₁₃	—	2	1	no	—
169	C₄×C₁₃	—	2	1	no	—
170	C₄×C₁₃	C₂²×C₁₆	1	2	yes	—
171	C₄×C₁₃	C₂²×C₁₆	1	2	yes	—
172	C₄×C₁₃	#504	1	2	yes	—
173	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
174	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
175	Dic₁₃	—	2	2	no	—
176	Dic₁₃	—	2	2	no	—
177	Dic₁₃	C₂²×C₁₆	1	4	yes	—
178	C₂×#41	—	13	1	no	—
179	C₂²×C₈	—	13	2	no	—
180	C₂²×C₈	—	13	2	no	—
181	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
183	C₄×Q₈	—	13	1	no	—
184	C₄×Q₈	—	13	1	no	—
185	C₄×D₄	—	13	2	no	—
186	C₄×D₄	—	13	2	no	—
187	C₄×D₄	—	13	1	no	—
188	C₄×D₄	—	13	1	no	—
189	#117	—	13	1	no	—
190	#117	—	13	2	no	—
191	C₂×C₄²	—	13	2	no	—
192	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
193	C₂×C₁₆	—	13	2	no	—
194	C₂×C₁₆	—	13	2	no	—
195	C₂×C₁₆	—	26	8	no	—
196	C₂×C₁₆	—	26	4	no	—
197	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
198	C₄⋊C₈	—	13	2	no	—
199	C₄⋊C₈	—	13	2	no	—
200	C₄⋊C₈	—	13	1	no	—
201	C₄⋊C₈	—	13	1	no	—
202	C₂²⋊C₈	—	13	2	no	—
203	C₂²⋊C₈	—	13	2	no	—
204	C₂²⋊C₈	—	13	2	no	—
205	C₄×C₈	—	13	2	no	—
206	C₄×C₈	—	13	2	no	—
207	C₄×C₈	—	13	2	no	—
208	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
209	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
210	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
211	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
212	C₂×C₁₃	—	2	2	no	—
213	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
214	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
215	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
216	D₁₃	—	2	4	no	—
217	#41	—	13	8	no	—
218	C₂×Q₈	—	13	1	no	—
219	C₂×D₄	—	13	2	no	—
220	C₂×D₄	—	13	1	no	—
221	C₂²×C₄	—	13	2	no	—
222	C₂²×C₄	—	13	2	no	—
223	C₂²×C₄	—	13	2	no	—
224	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	13	2	no	—
230	C₂×C₈	—	13	2	no	—
231	C₂×C₈	—	26	4	no	—
232	C₂×C₈	—	26	4	no	—
233	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
235	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
236	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
237	C₄⋊C₄	—	13	2	no	—
238	C₄⋊C₄	—	13	2	no	—
239	C₄⋊C₄	—	13	1	no	—
240	C₄⋊C₄	—	13	1	no	—
241	C₂²⋊C₄	—	13	2	no	—
242	C₂²⋊C₄	—	13	2	no	—
243	C₂²⋊C₄	—	13	2	no	—
244	C₄²	—	13	2	no	—
245	C₄²	—	13	2	no	—
246	C₄²	—	13	2	no	—
247	C₄²	C₈×D₁₃	1	1	yes	characteristic
248	C₄²	C₈×D₁₃	1	1	yes	characteristic
249	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
250	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
251	C₁₆	—	2	4	no	—
252	C₁₆	—	13	4	no	—
253	C₁₆	—	26	2	no	—
254	C₁₃	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
255	C₂³	—	13	2	no	—
256	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
257	Q₈	—	13	4	no	—
258	D₄	—	13	8	no	—
259	D₄	—	13	4	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	13	2	no	—
264	C₂×C₄	—	13	8	no	—
265	C₂×C₄	—	13	2	no	—
266	C₂×C₄	—	13	2	no	—
267	C₂×C₄	—	26	2	no	—
268	C₂×C₄	—	26	2	no	—
269	C₂×C₄	—	26	4	no	—
270	C₂×C₄	C₁₆×D₁₃	1	4	yes	complemented
271	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
272	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
273	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
274	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
275	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	26	2	no	—
280	C₈	—	26	2	no	—
281	C₈	#88161	1	2	yes	—
282	C₈	#41×D₁₃	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	13	4	no	—
286	C₂²	—	26	2	no	—
287	C₂²	—	26	2	no	—
288	C₂²	C₂×C₁₆×D₁₃	1	2	yes	—
289	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	13	4	no	—
293	C₄	—	26	2	no	—
294	C₄	—	26	2	no	—
295	C₄	C₂×C₁₆×D₁₃	1	2	yes	—
296	C₄	C₂×C₁₆×D₁₃	1	2	yes	—
297	C₄	#11718568	1	2	yes	—
298	C₄	#11719227	1	1	yes	characteristic
299	C₄	C₂×#88161	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	26	4	no	—
302	C₂	#13123792	1	1	yes	characteristic
303	C₂	C₂²×C₁₆×D₁₃	1	1	yes	characteristic
304	C₂	C₂×#11718568	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²×D₁₃
commutator	C₂×C₁₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2162	C₂
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂×#6127	C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2162	D₁₃
hypocenter	C₁₃	C₁₆×#41
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. #13116738	C₂	semidirect
3. #13116655	C₂	semidirect
4. #13116653	C₂	semidirect
5. #13116633	C₂	semidirect
6. C₂×C₄×#6090	C₂	semidirect
7. C₄×C₁₆×D₁₃	C₂	semidirect
8. C₁₃×#2162	C₂	semidirect
70. #6127	C₁₆	semidirect
102. #6090	#41	semidirect
108. #2162	D₁₃	semidirect
116. C₂×C₄×C₁₃	C₂×C₁₆	semidirect
123. C₂²⋊D₁₃	C₂×C₁₆	semidirect
127. D₅₂	C₂×C₁₆	semidirect
131. C₄×D₁₃	C₂×C₁₆	semidirect
132. Dic₂₆	C₂×C₁₆	semidirect
254. C₁₃	C₁₆×#41	semidirect
270. C₂×C₄	C₁₆×D₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₈×#6127	2
#13116738	2
#13116655	2
#13116653	2
#13116633	2
C₂×C₄×#6090	2
C₄×C₁₆×D₁₃	2
C₁₃×#2162	2
C₁₆×#41	13