Subgroups — #16016408

Order 3328.

506 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123765	C₂	1	1	yes	maximal, characteristic
2	#13116745	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116743	C₂	1	2	yes	maximal, complemented
4	#13116741	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₆×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₂₆	C₂	1	2	yes	maximal, complemented
7	#13116593	C₂	1	2	yes	maximal
8	#13116590	C₂	1	1	yes	maximal, characteristic
9	#13116431	C₂	1	2	yes	maximal
10	C₄×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
11	Q₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719283	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718685	C₄	1	1	yes	characteristic
15	C₂×#88038	C₂²	1	2	yes	—
16	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
18	#11718500	C₄	1	2	yes	—
19	#11718498	C₄	1	1	yes	characteristic
20	#11718492	C₄	1	1	yes	characteristic
21	#11718489	C₄	1	2	yes	—
22	#11718350	C₂²	1	2	yes	—
23	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
24	C₈×Dic₂₆	C₂²	1	1	yes	characteristic
25	C₈×Dic₂₆	C₂²	1	2	yes	—
26	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
29	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
30	#11718094	C₂²	1	2	yes	—
31	#11718093	C₂²	1	2	yes	—
32	#11718093	C₂²	1	2	yes	—
33	#11718093	C₂²	1	2	yes	—
34	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₁₃	C₂²	1	2	yes	—
37	C₄⋊#6090	C₂²	1	1	yes	characteristic
38	C₄⋊#6090	C₂²	1	2	yes	—
39	#11718053	C₂²	1	2	yes	—
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	#88200	C₈	1	1	yes	characteristic
43	#88198	C₈	1	1	yes	characteristic
44	C₂×#6094	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
47	#88151	C₈	1	1	yes	characteristic
48	#88150	C₈	1	2	yes	—
49	#88143	C₈	1	2	yes	—
50	#88142	C₈	1	1	yes	characteristic
51	#88126	C₂×C₄	1	2	yes	—
52	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
54	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
55	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
57	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
58	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
59	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	2	yes	—
62	#88055	C₂×C₄	1	2	yes	—
63	#88054	C₂×C₄	1	2	yes	—
64	#88054	C₂×C₄	1	2	yes	—
65	#88054	C₂×C₄	1	2	yes	—
66	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
68	C₈×Dic₁₃	C₂³	1	2	yes	—
69	C₂×#6090	C₂³	1	1	yes	characteristic
70	C₂×#6090	C₂³	1	2	yes	—
71	C₂×#6090	C₂³	1	1	yes	characteristic
72	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
73	C₄⋊#1055	C₂×C₄	1	2	yes	—
74	#88043	C₂×C₄	1	2	yes	—
75	C₄×#1055	C₂×C₄	1	1	yes	characteristic
76	#88038	—	2	4	no	—
77	C₁₆×D₁₃	—	2	2	no	—
78	C₁₆×D₁₃	Q₈	1	4	yes	complemented
79	#33474	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
85	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
91	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
93	#6103	C₂×C₈	1	2	yes	—
94	#6103	C₂×C₈	1	2	yes	—
95	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
96	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
97	#6101	C₂×C₈	1	2	yes	—
98	#6101	C₂×C₈	1	2	yes	—
99	#6101	C₂×C₈	1	2	yes	—
100	#6101	C₂×C₈	1	2	yes	—
101	#6101	C₂×C₈	1	2	yes	—
102	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
103	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
104	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
106	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
107	C₂×#1055	C₂²×C₄	1	2	yes	—
108	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
109	#6094	—	2	4	no	—
110	C₈×D₁₃	—	2	2	no	—
111	C₈×D₁₃	C₂×Q₈	1	2	yes	—
112	C₈×D₁₃	#41	1	2	yes	—
113	C₁₃×C₁₆	—	2	1	no	—
114	C₁₃×C₁₆	—	2	2	no	—
115	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
116	#6090	—	2	1	no	—
117	#6090	—	2	2	no	—
118	#6090	C₂×Q₈	1	2	yes	—
119	#2960	—	13	1	no	—
120	#2183	—	13	1	no	—
121	#2183	—	13	1	no	—
122	#2183	—	13	2	no	—
123	Q₈×C₁₆	—	13	2	no	—
124	Q₈×C₁₆	—	13	1	no	—
125	Q₈×C₁₆	D₁₃	1	1	yes	characteristic, complemented
126	#2151	—	13	2	no	—
127	#2150	—	13	1	no	—
128	#2108	—	13	2	no	—
129	C₂×C₄×C₁₆	—	13	1	no	—
130	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
131	Q₈×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
137	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
138	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
139	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
140	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
141	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
142	C₄×D₁₃	—	2	2	no	—
143	C₄×D₁₃	—	2	2	no	—
144	C₄×D₁₃	C₄×Q₈	1	2	yes	—
145	C₄×D₁₃	#131	1	2	yes	—
146	Dic₂₆	—	2	2	no	—
147	Dic₂₆	—	2	4	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	2	2	no	—
150	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
151	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
152	#1055	—	2	1	no	—
153	#1055	—	2	2	no	—
154	#1055	C₄×Q₈	1	2	yes	—
155	#533	—	13	1	no	—
156	C₂×M₃₂	—	13	2	no	—
157	C₂²×C₁₆	—	13	1	no	—
158	C₂²×C₁₆	—	13	1	no	—
159	#446	—	13	1	no	—
160	#446	—	13	2	no	—
161	#446	—	13	1	no	—
162	C₈×Q₈	—	13	2	no	—
163	C₈×Q₈	—	13	1	no	—
164	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
165	#424	—	13	2	no	—
166	#423	—	13	1	no	—
167	#405	—	13	2	no	—
168	C₂×C₄×C₈	—	13	1	no	—
169	C₄⋊C₁₆	—	13	2	no	—
170	C₄⋊C₁₆	—	13	2	no	—
171	C₄⋊C₁₆	—	13	2	no	—
172	C₄⋊C₁₆	—	13	1	no	—
173	C₄⋊C₁₆	—	13	2	no	—
174	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
175	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
176	#346	—	13	2	no	—
177	#346	—	13	2	no	—
178	C₄×C₁₆	—	13	1	no	—
179	C₄×C₁₆	—	13	1	no	—
180	C₄×C₁₆	—	13	2	no	—
181	C₄×C₁₆	—	13	1	no	—
182	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
183	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
184	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
185	C₂×D₁₃	—	2	1	no	—
186	C₂×D₁₃	—	2	1	no	—
187	C₂×D₁₃	C₈×Q₈	1	2	yes	—
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	#504	1	2	yes	—
192	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
193	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
194	Dic₁₃	—	2	2	no	—
195	Dic₁₃	—	2	2	no	—
196	Dic₁₃	—	2	1	no	—
197	Dic₁₃	C₈×Q₈	1	2	yes	—
198	Dic₁₃	#504	1	2	yes	—
199	C₂×M₁₆	—	13	2	no	—
200	C₂²×C₈	—	13	1	no	—
201	C₂²×C₈	—	13	1	no	—
202	C₄⋊Q₈	—	13	1	no	—
203	C₄⋊Q₈	—	13	1	no	—
204	#125	—	13	2	no	—
205	C₂²⋊Q₈	—	13	1	no	—
206	C₂²⋊Q₈	—	13	2	no	—
207	C₂²⋊Q₈	—	13	1	no	—
208	C₄×Q₈	—	13	2	no	—
209	C₄×Q₈	—	13	1	no	—
210	C₄×Q₈	C₄×D₁₃	1	1	yes	characteristic
211	#117	—	13	2	no	—
212	C₂×C₄²	—	13	1	no	—
213	M₃₂	—	26	4	no	—
214	C₂×C₁₆	—	13	1	no	—
215	C₂×C₁₆	—	13	4	no	—
216	C₂×C₁₆	—	13	1	no	—
217	C₂×C₁₆	—	13	2	no	—
218	C₂×C₁₆	—	26	2	no	—
219	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
220	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
221	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
222	C₄⋊C₈	—	13	2	no	—
223	C₄⋊C₈	—	13	2	no	—
224	C₄⋊C₈	—	13	2	no	—
225	C₄⋊C₈	—	13	1	no	—
226	C₄⋊C₈	—	13	2	no	—
227	C₄⋊C₈	C₄×D₁₃	1	1	yes	characteristic
228	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
229	#97	—	13	2	no	—
230	#97	—	13	2	no	—
231	C₄×C₈	—	13	1	no	—
232	C₄×C₈	—	13	1	no	—
233	C₄×C₈	—	13	2	no	—
234	C₄×C₈	—	13	1	no	—
235	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₁₃	1	2	yes	—
237	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
238	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
239	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
240	D₁₃	—	2	2	no	—
241	C₂×Q₈	—	13	2	no	—
242	C₂×Q₈	—	13	1	no	—
243	C₂×Q₈	C₈×D₁₃	1	1	yes	characteristic
244	C₂²×C₄	—	13	1	no	—
245	C₂²×C₄	—	13	1	no	—
246	C₂²×C₄	—	13	1	no	—
247	M₁₆	—	26	4	no	—
248	C₂×C₈	—	13	1	no	—
249	C₂×C₈	—	13	2	no	—
250	C₂×C₈	—	13	2	no	—
251	C₂×C₈	—	13	1	no	—
252	C₂×C₈	—	13	2	no	—
253	C₂×C₈	—	26	2	no	—
254	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
255	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
256	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
257	C₄⋊C₄	—	13	2	no	—
258	C₄⋊C₄	—	13	2	no	—
259	C₄⋊C₄	—	13	2	no	—
260	C₄⋊C₄	—	13	2	no	—
261	C₄⋊C₄	—	13	2	no	—
262	C₄⋊C₄	—	13	2	no	—
263	C₄⋊C₄	—	13	1	no	—
264	C₄⋊C₄	C₈×D₁₃	1	2	yes	—
265	C₄⋊C₄	C₈×D₁₃	1	1	yes	characteristic
266	C₂²⋊C₄	—	13	2	no	—
267	C₂²⋊C₄	—	13	2	no	—
268	C₄²	—	13	1	no	—
269	C₄²	—	13	1	no	—
270	C₄²	—	13	1	no	—
271	C₄²	—	13	2	no	—
272	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₁₃	1	2	yes	—
274	C₁₆	—	2	1	no	—
275	C₁₆	—	2	2	no	—
276	C₁₆	—	13	2	no	—
277	C₁₆	—	26	1	no	—
278	C₁₆	—	26	2	no	—
279	C₁₆	Q₈×D₁₃	1	2	yes	—
280	C₁₃	#33474	1	1	yes	characteristic, complemented, Sylow
281	C₂³	—	13	1	no	—
282	Q₈	—	2	2	no	—
283	Q₈	—	26	4	no	—
284	Q₈	—	26	2	no	—
285	C₂×C₄	—	13	1	no	—
286	C₂×C₄	—	13	1	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	13	2	no	—
289	C₂×C₄	—	13	1	no	—
290	C₂×C₄	—	13	2	no	—
291	C₂×C₄	—	13	2	no	—
292	C₂×C₄	—	26	2	no	—
293	C₂×C₄	—	26	2	no	—
294	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
295	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
296	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
297	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	2	no	—
301	C₈	—	13	2	no	—
302	C₈	—	26	1	no	—
303	C₈	—	26	2	no	—
304	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
305	C₈	#41×D₁₃	1	1	yes	characteristic
306	C₂²	—	13	2	no	—
307	C₂²	—	26	1	no	—
308	C₂²	—	26	1	no	—
309	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	13	2	no	—
314	C₄	—	13	2	no	—
315	C₄	—	26	2	no	—
316	C₄	—	26	1	no	—
317	C₄	—	26	2	no	—
318	C₄	#11718562	1	2	yes	—
319	C₄	C₄×Q₈×D₁₃	1	1	yes	characteristic
320	C₄	D₁₃×#131	1	1	yes	characteristic
321	C₂	—	26	2	no	—
322	C₂	C₈×Q₈×D₁₃	1	1	yes	characteristic
323	C₂	D₁₃×#504	1	1	yes	characteristic
324	C₂	C₂×#11718562	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	Q₈×C₁₃×C₁₆	C₂
prime radical	C₂²×D₁₃	C₂²×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	Q₈×C₁₆	D₁₃
hypocenter	C₁₃	#33474
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #13116745	C₂	semidirect
3. #13116743	C₂	semidirect
4. #13116741	C₂	semidirect
5. C₁₆×Dic₂₆	C₂	semidirect
6. C₁₆×Dic₂₆	C₂	semidirect
11. Q₈×C₁₃×C₁₆	C₂	semidirect
78. C₁₆×D₁₃	Q₈	semidirect
125. Q₈×C₁₆	D₁₃	semidirect
280. C₁₃	#33474	semidirect

Maximal subgroups

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

subgroup	index
#13123765	2
#13116745	2
#13116743	2
#13116741	2
C₁₆×Dic₂₆	2
#13116593	2
#13116590	2
#13116431	2
C₄×C₁₆×D₁₃	2
Q₈×C₁₃×C₁₆	2
#33474	13