Subgroups — #16016033

Order 3328.

590 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88155	C₂	1	1	yes	maximal, characteristic
2	#13116642	C₂	1	4	yes	maximal, complemented
3	#13116638	C₂	1	2	yes	maximal, complemented
4	#13116634	C₂	1	2	yes	maximal, complemented
5	#13116592	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116449	C₂	1	2	yes	maximal
7	#13116432	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88155	C₄	1	1	yes	characteristic
10	C₂×#88155	C₂²	1	1	yes	characteristic
11	C₄×#6127	C₄	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718762	C₄	1	1	yes	characteristic
15	#11718762	C₂²	1	1	yes	characteristic
16	C₂×#88052	C₂²	1	2	yes	—
17	C₄×#6099	C₄	1	1	yes	characteristic
18	#11718350	C₄	1	2	yes	—
19	#11718350	C₂²	1	2	yes	—
20	C₄×#6094	C₄	1	2	yes	—
21	C₄×C₈×D₁₃	C₂²	1	2	yes	—
22	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
25	#11718099	C₂²	1	4	yes	—
26	#11718099	C₂²	1	4	yes	—
27	#11718094	C₂²	1	4	yes	—
28	C₁₆×Dic₁₃	C₂²	1	4	yes	—
29	C₄⋊#6090	C₂²	1	2	yes	—
30	C₄⋊#6090	C₂²	1	2	yes	—
31	C₂×#6127	C₂×C₄	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	C₈	1	2	yes	—
37	#88180	C₂×C₄	1	1	yes	characteristic
38	C₂×#6099	C₂×C₄	1	1	yes	characteristic
39	C₂×#6099	C₂×C₄	1	1	yes	characteristic
40	#88155	—	2	2	no	—
41	#88155	—	2	2	no	—
42	#88155	—	2	4	no	—
43	C₂×#6094	C₂×C₄	1	2	yes	—
44	C₂×#6094	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₁₃	C₂³	1	2	yes	—
47	C₄×D₅₂	C₈	1	1	yes	characteristic
48	C₄×D₅₂	C₈	1	1	yes	characteristic
49	#88126	C₂×C₄	1	2	yes	—
50	C₄²×D₁₃	C₂×C₄	1	2	yes	—
51	C₄×Dic₂₆	C₈	1	1	yes	characteristic
52	C₄×Dic₂₆	C₈	1	1	yes	characteristic
53	C₂×C₁₃×C₁₆	—	2	4	no	—
54	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
55	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
56	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	#88055	C₂×C₄	1	2	yes	—
61	#88055	C₂×C₄	1	2	yes	—
62	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
63	C₈×Dic₁₃	C₂³	1	2	yes	—
64	#88052	—	2	4	no	—
65	C₂×#6090	C₂³	1	4	yes	—
66	#88043	C₂×C₄	1	2	yes	—
67	C₄×#1055	C₂×C₄	1	2	yes	—
68	#33423	—	13	1	no	maximal, Sylow
69	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
73	#6127	—	2	2	no	—
74	#6127	—	2	2	no	—
75	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
77	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
78	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
79	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	#41	1	2	yes	—
85	C₂×C₈×C₁₃	#41	1	2	yes	—
86	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
88	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
89	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
92	#6103	C₂×C₈	1	2	yes	—
93	#6103	C₂×C₈	1	2	yes	—
94	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
95	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
96	#6101	C₂×C₈	1	2	yes	—
97	#6101	C₂×C₈	1	2	yes	—
98	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
99	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
100	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
101	#6099	—	2	2	no	—
102	#6099	M₁₆	1	4	yes	—
103	C₂×#1055	C₂²×C₄	1	2	yes	—
104	C₂×#1055	C₂²×C₄	1	2	yes	—
105	#6094	—	2	4	no	—
106	#6094	—	2	4	no	—
107	C₈×D₁₃	—	2	2	no	—
108	C₈×D₁₃	—	2	2	no	—
109	C₈×D₁₃	—	2	4	no	—
110	C₁₃×C₁₆	—	2	2	no	—
111	C₁₃×C₁₆	—	2	2	no	—
112	#6090	—	2	4	no	—
113	C₄×#131	—	13	1	no	—
114	#2164	—	13	4	no	—
115	#2163	—	13	2	no	—
116	#2162	—	13	2	no	—
117	#2150	—	13	1	no	—
118	C₂×C₄⋊C₁₆	D₁₃	1	1	yes	characteristic, complemented
119	#2113	—	13	2	no	—
120	#2108	—	13	2	no	—
121	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
122	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	4	1	no	—
125	C₂×C₄×C₁₃	#117	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
132	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	C₂²⋊D₁₃	—	2	4	no	—
134	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
135	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
136	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
137	D₅₂	—	2	2	no	—
138	C₄×D₁₃	—	2	2	no	—
139	C₄×D₁₃	—	2	2	no	—
140	C₄×D₁₃	—	2	4	no	—
141	C₄×D₁₃	—	4	2	no	—
142	Dic₂₆	—	2	2	no	—
143	C₈×C₁₃	—	2	1	no	—
144	C₈×C₁₃	—	2	1	no	—
145	C₈×C₁₃	#117	1	2	yes	—
146	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
147	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
148	#1055	—	2	2	no	—
149	#1055	C₂×M₁₆	1	4	yes	—
150	C₂×#131	—	13	1	no	—
151	C₂×#131	—	13	1	no	—
152	C₄×#41	—	13	1	no	—
153	C₂×M₃₂	—	13	2	no	—
154	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
155	#405	—	13	1	no	—
156	#405	—	13	1	no	—
157	#405	—	13	2	no	—
158	#405	—	13	2	no	—
159	C₄×M₁₆	—	13	1	no	—
160	C₄×M₁₆	—	13	2	no	—
161	C₂×C₄×C₈	—	13	2	no	—
162	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
163	C₄⋊C₁₆	—	13	2	no	—
164	C₄⋊C₁₆	—	13	2	no	—
165	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
166	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
167	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
168	C₂²⋊C₁₆	—	13	4	no	—
169	C₂²⋊C₁₆	—	13	4	no	—
170	#346	—	13	4	no	—
171	C₄×C₁₆	—	13	4	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	#431	1	2	yes	—
175	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
176	C₂×D₁₃	—	2	2	no	—
177	C₂×D₁₃	—	2	2	no	—
178	C₂×D₁₃	—	2	2	no	—
179	C₄×C₁₃	—	2	1	no	—
180	C₄×C₁₃	—	2	1	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	#431	1	2	yes	—
183	C₄×C₁₃	#504	1	2	yes	—
184	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
185	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
186	Dic₁₃	—	2	2	no	—
187	Dic₁₃	—	2	2	no	—
188	Dic₁₃	—	2	2	no	—
189	C₂×#41	—	13	1	no	—
190	#131	—	26	4	no	—
191	#131	—	26	2	no	—
192	#131	—	26	2	no	—
193	C₂×M₁₆	—	13	1	no	—
194	C₂×M₁₆	—	13	1	no	—
195	C₂×M₁₆	—	13	2	no	—
196	C₂×M₁₆	—	13	2	no	—
197	C₂²×C₈	—	13	2	no	—
198	C₂²×C₈	—	13	2	no	—
199	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
201	C₄×Q₈	—	13	1	no	—
202	C₄×Q₈	—	13	1	no	—
203	C₄×D₄	—	13	2	no	—
204	C₄×D₄	—	13	2	no	—
205	C₄×D₄	—	13	1	no	—
206	C₄×D₄	—	13	1	no	—
207	#117	—	13	1	no	—
208	#117	—	13	2	no	—
209	C₂×C₄²	—	13	2	no	—
210	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
211	M₃₂	—	26	4	no	—
212	C₂×C₁₆	—	2	4	no	—
213	C₂×C₁₆	—	13	4	no	—
214	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
215	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
216	#97	—	13	2	no	—
217	#97	—	13	2	no	—
218	#97	—	13	2	no	—
219	C₄×C₈	—	13	2	no	—
220	C₄×C₈	—	13	2	no	—
221	C₄×C₈	—	13	2	no	—
222	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
223	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
224	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
225	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
226	C₂×C₁₃	—	2	2	no	—
227	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
228	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
229	C₂×C₁₃	#3407	1	1	yes	characteristic
230	D₁₃	—	4	2	no	—
231	#41	—	26	2	no	—
232	#41	—	26	2	no	—
233	C₂×Q₈	—	13	1	no	—
234	C₂×D₄	—	13	2	no	—
235	C₂×D₄	—	13	1	no	—
236	C₂²×C₄	—	13	2	no	—
237	C₂²×C₄	—	13	2	no	—
238	C₂²×C₄	—	13	2	no	—
239	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
240	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
242	M₁₆	—	13	4	no	—
243	M₁₆	—	26	4	no	—
244	M₁₆	—	26	2	no	—
245	M₁₆	—	26	4	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	13	2	no	—
248	C₂×C₈	—	13	2	no	—
249	C₂×C₈	—	26	4	no	—
250	C₂×C₈	—	26	2	no	—
251	C₂×C₈	—	26	2	no	—
252	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
253	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
254	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
255	C₂×C₈	#6129	1	2	yes	—
256	C₂×C₈	#6131	1	2	yes	—
257	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
258	C₄⋊C₄	—	13	2	no	—
259	C₄⋊C₄	—	13	2	no	—
260	C₄⋊C₄	—	13	1	no	—
261	C₄⋊C₄	—	13	1	no	—
262	C₂²⋊C₄	—	13	2	no	—
263	C₂²⋊C₄	—	13	2	no	—
264	C₂²⋊C₄	—	13	2	no	—
265	C₄²	—	13	2	no	—
266	C₄²	—	13	2	no	—
267	C₄²	—	13	2	no	—
268	C₄²	C₈×D₁₃	1	1	yes	characteristic
269	C₄²	C₈×D₁₃	1	1	yes	characteristic
270	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
271	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
272	C₁₆	—	2	2	no	—
273	C₁₆	—	2	2	no	—
274	C₁₆	—	26	4	no	—
275	C₁₃	#33423	1	1	yes	characteristic, complemented, Sylow
276	C₂³	—	13	2	no	—
277	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
278	Q₈	—	26	2	no	—
279	D₄	—	26	4	no	—
280	D₄	—	26	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	13	2	no	—
284	C₂×C₄	—	13	2	no	—
285	C₂×C₄	—	13	2	no	—
286	C₂×C₄	—	26	2	no	—
287	C₂×C₄	—	26	2	no	—
288	C₂×C₄	—	26	4	no	—
289	C₂×C₄	—	52	2	no	—
290	C₂×C₄	#88146	1	2	yes	—
291	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
292	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
293	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
294	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
295	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
296	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
297	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	13	4	no	—
301	C₈	—	26	2	no	—
302	C₈	#88146	1	2	yes	—
303	C₈	C₂×#6129	1	1	yes	characteristic
304	C₈	C₂×#6131	1	1	yes	characteristic
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	26	2	no	—
308	C₂²	—	26	2	no	—
309	C₂²	—	26	2	no	—
310	C₂²	#11718494	1	2	yes	—
311	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	26	2	no	—
316	C₄	—	26	2	no	—
317	C₄	—	26	2	no	—
318	C₄	#11718494	1	2	yes	—
319	C₄	#11718562	1	2	yes	—
320	C₄	C₂×#88146	1	1	yes	characteristic
321	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	52	2	no	—
324	C₂	C₂×#11718494	1	1	yes	characteristic
325	C₂	C₂×#11718562	1	1	yes	characteristic
326	C₂	#13127938	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₂×C₁₃×C₄⋊C₁₆	C₂
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₁₃
hypocenter	C₁₃	#33423
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #13116642	C₂	semidirect
3. #13116638	C₂	semidirect
4. #13116634	C₂	semidirect
5. #13116592	C₂	semidirect
7. #13116432	C₂	semidirect
8. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
118. C₂×C₄⋊C₁₆	D₁₃	semidirect
275. C₁₃	#33423	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88155	2
#13116642	2
#13116638	2
#13116634	2
#13116592	2
#13116449	2
#13116432	2
C₂×C₁₃×C₄⋊C₁₆	2
#33423	13