Subgroups — #16019791

Order 3328.

580 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127183	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₅₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117065	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117035	C₂	1	2	yes	maximal, complemented
5	#13116933	C₂	1	2	yes	maximal, complemented
6	#13116882	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116754	C₂	1	2	yes	maximal, complemented
8	C₄×#88068	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116655	C₂	1	1	yes	maximal, characteristic, complemented
10	D₁₃×#358	C₂	1	2	yes	maximal, complemented
11	C₁₃×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₁₃	C₂²	1	2	yes	—
13	#11719431	—	2	1	no	—
14	Q₈×D₅₂	—	2	1	no	—
15	C₁₃×#500	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂×#88068	C₂²	1	1	yes	characteristic
18	#11718748	—	2	1	no	—
19	#11718692	—	2	1	no	—
20	C₂×#88040	C₂²	1	2	yes	—
21	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
22	#11718548	C₂²	1	2	yes	—
23	#11718544	C₂²	1	2	yes	—
24	#11718543	—	2	2	no	—
25	D₁₃×#107	C₂²	1	2	yes	—
26	#11718469	—	2	2	no	—
27	#11718360	C₂²	1	1	yes	characteristic
28	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
29	C₈×D₅₂	C₂²	1	1	yes	characteristic
30	C₁₃×#367	C₂²	1	2	yes	—
31	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₁₃×#358	C₂²	1	2	yes	—
33	#11718156	C₂²	1	1	yes	characteristic
34	#11718099	C₂²	1	2	yes	—
35	#11718095	C₂²	1	2	yes	—
36	#11718084	C₂²	1	2	yes	—
37	#11718084	C₂²	1	2	yes	—
38	#11718083	C₂²	1	1	yes	characteristic
39	#11718082	C₂²	1	1	yes	characteristic
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₂×#6131	—	4	1	no	—
42	C₂×Q₈×D₁₃	—	2	2	no	—
43	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
44	C₁₃×C₄⋊Q₈	—	2	1	no	—
45	C₄×Q₈×C₁₃	—	2	1	no	—
46	#88200	—	4	1	no	—
47	C₂×#6107	—	2	2	no	—
48	Q₁₆×D₁₃	—	2	4	no	—
49	Q₁₆×D₁₃	—	4	2	no	—
50	C₂×Dic₅₂	—	2	1	no	—
51	C₂×Dic₅₂	C₂³	1	2	yes	—
52	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
53	C₂×C₈×D₁₃	C₂³	1	2	yes	—
54	#88151	—	2	2	no	—
55	#88151	—	4	2	no	—
56	#88149	—	2	2	no	—
57	#88148	—	4	2	no	—
58	C₄⋊C₄×D₁₃	—	2	2	no	—
59	C₄×D₅₂	—	2	1	no	—
60	C₄×D₅₂	—	4	1	no	—
61	C₄×D₅₂	D₄	1	1	yes	characteristic
62	#88123	—	2	1	no	—
63	#88123	—	4	1	no	—
64	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
65	C₁₃×#107	—	2	1	no	—
66	C₁₃×#107	C₂³	1	2	yes	—
67	C₁₃×#107	C₂³	1	1	yes	characteristic
68	C₁₃×#103	—	2	1	no	—
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	#88068	—	2	2	no	—
71	#88061	—	2	1	no	—
72	#88060	D₄	1	2	yes	—
73	#88058	C₂³	1	1	yes	characteristic
74	C₂×#6090	C₂³	1	1	yes	characteristic
75	C₂×#6090	C₂³	1	1	yes	characteristic
76	#88050	—	2	2	no	—
77	#88049	—	2	2	no	—
78	#88047	—	2	2	no	—
79	C₄⋊#1055	D₄	1	1	yes	characteristic
80	#88040	—	4	2	no	—
81	C₁₆×D₁₃	—	4	2	no	—
82	#33865	—	13	1	no	maximal, Sylow
83	C₂×Q₈×C₁₃	—	2	1	no	—
84	C₂×Q₈×C₁₃	—	4	1	no	—
85	#6131	—	8	4	no	—
86	Q₈×D₁₃	—	4	4	no	—
87	Q₈×D₁₃	—	8	2	no	—
88	C₂×D₅₂	—	2	1	no	—
89	C₂×D₅₂	—	4	2	no	—
90	C₂×D₅₂	D₈	1	1	yes	characteristic
91	C₂×C₄×D₁₃	—	4	2	no	—
92	C₂×C₄×D₁₃	—	4	2	no	—
93	C₂×C₄×D₁₃	—	4	1	no	—
94	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
95	C₂×Dic₂₆	—	2	2	no	—
96	C₂×Dic₂₆	—	4	2	no	—
97	C₁₃×Q₁₆	—	2	1	no	—
98	C₁₃×Q₁₆	#41	1	2	yes	complemented
99	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
101	C₁₃×C₄⋊C₄	—	2	1	no	—
102	C₁₃×C₄⋊C₄	—	2	2	no	—
103	C₁₃×C₄⋊C₄	—	4	1	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₄²×C₁₃	—	4	1	no	—
106	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	#6107	—	4	4	no	—
108	#6103	—	4	2	no	—
109	#6103	—	4	1	no	—
110	#6103	—	4	2	no	—
111	#6103	D₈	1	2	yes	—
112	C₄⋊Dic₁₃	—	2	1	no	—
113	C₄⋊Dic₁₃	—	4	1	no	—
114	C₄⋊Dic₁₃	—	4	1	no	—
115	C₄⋊Dic₁₃	—	4	2	no	—
116	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
117	#6101	—	4	2	no	—
118	C₂×#1055	C₂×D₄	1	2	yes	—
119	Dic₅₂	—	2	2	no	—
120	Dic₅₂	—	2	2	no	—
121	Dic₅₂	—	8	1	no	—
122	D₁₀₄	—	2	1	no	—
123	D₁₀₄	—	2	1	no	—
124	C₈×D₁₃	—	2	2	no	—
125	C₈×D₁₃	—	2	2	no	—
126	C₁₃×C₁₆	—	2	2	no	—
127	#6090	—	2	1	no	—
128	#6090	—	2	1	no	—
129	#3334	—	13	1	no	—
130	D₄×Q₁₆	—	13	1	no	—
131	#2241	—	13	1	no	—
132	#2235	—	13	2	no	—
133	#2225	D₁₃	1	1	yes	characteristic, complemented
134	#2217	—	13	2	no	—
135	#2208	—	13	1	no	—
136	#2185	—	13	2	no	—
137	C₄×SD₃₂	—	13	1	no	—
138	D₄×C₁₆	—	13	1	no	—
139	C₂×#358	—	13	2	no	—
140	C₂²×D₁₃	—	4	1	no	—
141	C₂²×D₁₃	C₂×D₈	1	2	yes	—
142	Q₈×C₁₃	—	2	2	no	—
143	Q₈×C₁₃	—	4	1	no	—
144	Q₈×C₁₃	—	8	2	no	—
145	C₂×C₄×C₁₃	—	4	1	no	—
146	C₂×C₄×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	4	1	no	—
148	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×Dic₁₃	—	4	1	no	—
152	C₂×Dic₁₃	—	4	2	no	—
153	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
154	D₅₂	—	4	1	no	—
155	D₅₂	—	4	1	no	—
156	D₅₂	—	8	4	no	—
157	D₅₂	SD₃₂	1	4	yes	complemented
158	C₄×D₁₃	—	2	2	no	—
159	C₄×D₁₃	—	2	2	no	—
160	C₄×D₁₃	—	8	4	no	—
161	C₄×D₁₃	—	8	4	no	—
162	C₄×D₁₃	—	8	2	no	—
163	Dic₂₆	—	4	2	no	—
164	Dic₂₆	—	4	2	no	—
165	Dic₂₆	—	8	4	no	—
166	C₈×C₁₃	—	2	1	no	—
167	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
168	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
169	#1055	—	2	2	no	—
170	C₂²×Q₁₆	—	13	2	no	—
171	D₄×Q₈	—	26	1	no	—
172	#547	—	26	1	no	—
173	C₂×SD₃₂	—	13	1	no	—
174	C₂×SD₃₂	—	13	2	no	—
175	C₂²×C₁₆	—	13	2	no	—
176	#500	C₂×D₁₃	1	1	yes	characteristic
177	#494	—	13	1	no	—
178	#480	—	26	2	no	—
179	#474	—	26	1	no	—
180	#467	—	13	2	no	—
181	#466	—	13	2	no	—
182	#462	—	26	1	no	—
183	#451	—	26	2	no	—
184	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
185	C₄×D₈	—	13	1	no	—
186	C₈×D₄	—	13	1	no	—
187	C₂×#107	—	13	2	no	—
188	#367	—	13	1	no	—
189	#367	C₂×D₁₃	1	2	yes	—
190	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
191	#358	—	13	1	no	—
192	#358	—	13	2	no	—
193	#358	—	13	2	no	—
194	#358	—	13	2	no	—
195	#358	C₂×D₁₃	1	2	yes	—
196	#357	—	13	1	no	—
197	C₂²⋊C₁₆	—	13	2	no	—
198	C₄×C₁₆	—	13	1	no	—
199	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
200	C₂×D₁₃	—	2	2	no	—
201	C₂×D₁₃	—	2	2	no	—
202	C₂×D₁₃	—	8	1	no	—
203	C₂×D₁₃	—	8	1	no	—
204	C₂×D₁₃	—	8	1	no	—
205	C₂×D₁₃	C₂×SD₃₂	1	4	yes	—
206	C₄×C₁₃	—	2	1	no	—
207	C₄×C₁₃	—	4	2	no	—
208	C₄×C₁₃	—	8	2	no	—
209	C₄×C₁₃	—	8	1	no	—
210	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
211	C₄×C₁₃	#546	1	1	yes	characteristic
212	C₄×C₁₃	#577	1	1	yes	characteristic
213	Dic₁₃	—	2	2	no	—
214	Dic₁₃	—	8	1	no	—
215	Dic₁₃	—	8	2	no	—
216	C₂×#41	—	52	1	no	—
217	C₂²×Q₈	—	26	2	no	—
218	C₂×Q₁₆	—	13	2	no	—
219	C₂×Q₁₆	—	26	2	no	—
220	C₂×Q₁₆	—	26	4	no	—
221	C₂×Q₁₆	—	26	1	no	—
222	C₂×Q₁₆	—	52	2	no	—
223	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
224	C₂×D₈	—	13	1	no	—
225	C₂²×C₈	—	13	2	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	26	1	no	—
228	C₄⋊Q₈	—	52	1	no	—
229	#123	—	52	2	no	—
230	C₂²⋊Q₈	—	26	2	no	—
231	C₂²⋊Q₈	—	52	2	no	—
232	C₂²⋊Q₈	—	52	1	no	—
233	#121	—	26	2	no	—
234	C₄×Q₈	—	2	1	no	—
235	C₄×D₄	—	13	1	no	—
236	C₄×D₄	—	26	1	no	—
237	C₄×D₄	—	52	1	no	—
238	C₂×C₄⋊C₄	—	26	2	no	—
239	SD₃₂	—	26	2	no	—
240	SD₃₂	—	52	2	no	—
241	C₂×C₁₆	—	13	1	no	—
242	C₂×C₁₆	—	13	1	no	—
243	C₂×C₁₆	—	52	2	no	—
244	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
245	#107	—	2	1	no	—
246	#107	—	13	1	no	—
247	#107	—	26	2	no	—
248	#107	C₂²×D₁₃	1	1	yes	characteristic
249	#107	C₂²×D₁₃	1	2	yes	—
250	C₄⋊C₈	—	13	1	no	—
251	#103	—	2	1	no	—
252	#103	—	26	2	no	—
253	#102	—	26	2	no	—
254	#102	—	26	1	no	—
255	C₂²⋊C₈	—	13	2	no	—
256	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
257	C₂×C₁₃	#3294	1	1	yes	characteristic
258	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
259	C₂×C₁₃	#3417	1	1	yes	characteristic
260	D₁₃	—	2	4	no	—
261	D₁₃	—	16	1	no	—
262	#41	—	104	4	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	26	2	no	—
266	C₂×Q₈	—	52	2	no	—
267	C₂×Q₈	—	52	4	no	—
268	C₂×Q₈	—	104	2	no	—
269	C₂×D₄	—	13	1	no	—
270	C₂×D₄	—	26	1	no	—
271	C₂×D₄	—	52	2	no	—
272	C₂²×C₄	—	13	2	no	—
273	C₂²×C₄	—	52	2	no	—
274	C₂²×C₄	—	52	2	no	—
275	C₂²×C₄	—	52	1	no	—
276	Q₁₆	—	2	1	no	—
277	Q₁₆	—	26	2	no	—
278	Q₁₆	—	26	2	no	—
279	Q₁₆	—	52	4	no	—
280	Q₁₆	—	104	1	no	—
281	Q₁₆	#6131	1	2	yes	complemented
282	D₈	—	26	1	no	—
283	D₈	—	26	1	no	—
284	C₂×C₈	—	13	2	no	—
285	C₂×C₈	—	26	2	no	—
286	C₂×C₈	—	26	2	no	—
287	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
288	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	13	1	no	—
294	C₄⋊C₄	—	26	1	no	—
295	C₄⋊C₄	—	52	2	no	—
296	C₄⋊C₄	—	52	2	no	—
297	C₄⋊C₄	—	52	1	no	—
298	C₄⋊C₄	—	52	1	no	—
299	C₂²⋊C₄	—	13	2	no	—
300	C₂²⋊C₄	—	52	2	no	—
301	C₂²⋊C₄	—	52	2	no	—
302	C₂²⋊C₄	—	52	1	no	—
303	C₄²	—	4	1	no	—
304	C₄²	D₄×D₁₃	1	1	yes	characteristic
305	C₁₆	—	2	2	no	—
306	C₁₆	—	26	1	no	—
307	C₁₆	—	26	1	no	—
308	C₁₃	#33865	1	1	yes	characteristic, complemented, Sylow
309	C₂³	—	13	2	no	—
310	C₂³	—	52	1	no	—
311	Q₈	—	2	2	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	8	2	no	—
314	Q₈	—	52	2	no	—
315	Q₈	—	52	2	no	—
316	Q₈	—	104	4	no	—
317	D₄	—	13	4	no	—
318	D₄	—	52	1	no	—
319	D₄	—	52	1	no	—
320	D₄	—	104	4	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	13	2	no	—
325	C₂×C₄	—	26	2	no	—
326	C₂×C₄	—	26	2	no	—
327	C₂×C₄	—	52	1	no	—
328	C₂×C₄	—	52	2	no	—
329	C₂×C₄	—	104	4	no	—
330	C₂×C₄	—	104	4	no	—
331	C₂×C₄	—	104	2	no	—
332	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
333	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	26	2	no	—
337	C₈	C₂×#6131	1	1	yes	characteristic
338	C₈	#88256	1	1	yes	characteristic
339	C₂²	—	13	4	no	—
340	C₂²	—	26	2	no	—
341	C₂²	—	26	2	no	—
342	C₂²	—	104	1	no	—
343	C₂²	—	104	1	no	—
344	C₂²	—	104	1	no	—
345	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	4	2	no	—
348	C₄	—	8	2	no	—
349	C₄	—	8	1	no	—
350	C₄	—	26	2	no	—
351	C₄	—	104	1	no	—
352	C₄	—	104	2	no	—
353	C₄	D₁₃×SD₃₂	1	2	yes	—
354	C₄	#11719335	1	1	yes	characteristic
355	C₄	#11719483	1	1	yes	characteristic
356	C₂	—	26	4	no	—
357	C₂	—	208	1	no	—
358	C₂	#13126646	1	1	yes	characteristic
359	C₂	C₂×D₁₃×SD₃₂	1	1	yes	characteristic
360	C₂	#13127988	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2225	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13127183	C₂
hypercenter	#2225	D₁₃
hypocenter	C₁₃	#33865
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2225

Complements

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

N	H ≅ G/N	type
2. Q₁₆×D₅₂	C₂	semidirect
3. #13117065	C₂	semidirect
4. #13117035	C₂	semidirect
5. #13116933	C₂	semidirect
6. #13116882	C₂	semidirect
7. #13116754	C₂	semidirect
8. C₄×#88068	C₂	semidirect
9. #13116655	C₂	semidirect
10. D₁₃×#358	C₂	semidirect
11. C₁₃×#2225	C₂	semidirect
98. C₁₃×Q₁₆	#41	semidirect
133. #2225	D₁₃	semidirect
157. D₅₂	SD₃₂	semidirect
281. Q₁₆	#6131	semidirect
308. C₁₃	#33865	semidirect

Maximal subgroups

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

subgroup	index
#13127183	2
Q₁₆×D₅₂	2
#13117065	2
#13117035	2
#13116933	2
#13116882	2
#13116754	2
C₄×#88068	2
#13116655	2
D₁₃×#358	2
C₁₃×#2225	2
#33865	13