Subgroups — #16020654

Order 3328.

518 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127728	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117077	C₂	1	2	yes	maximal, complemented
3	#13117059	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116993	C₂	1	2	yes	maximal, complemented
5	#13116973	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116873	C₂	1	2	yes	maximal, complemented
7	#13116873	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116745	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116700	C₂	1	2	yes	maximal, complemented
10	C₄×#88067	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#3391	C₂	1	1	yes	maximal, characteristic
12	C₁₃×#550	—	2	1	no	—
13	#11719272	—	2	1	no	—
14	C₁₃×#493	C₂²	1	1	yes	characteristic
15	C₁₃×#493	C₂²	1	2	yes	—
16	C₁₃×#459	—	2	2	no	—
17	C₁₃×#459	—	2	1	no	—
18	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
20	C₄×C₁₃×D₈	C₂²	1	2	yes	—
21	C₂×#88067	C₂²	1	1	yes	characteristic
22	C₂×#88067	C₂²	1	2	yes	—
23	#11718512	—	2	1	no	—
24	#11718490	—	2	2	no	—
25	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
26	C₄×Dic₅₂	C₂²	1	2	yes	—
27	#11718344	C₂²	1	2	yes	—
28	#11718343	C₂²	1	1	yes	characteristic
29	#11718154	C₂²	1	1	yes	characteristic
30	#11718154	C₂²	1	2	yes	—
31	#11718154	C₂²	1	2	yes	—
32	#11718154	C₂²	1	2	yes	—
33	#11718154	C₂²	1	2	yes	—
34	#11718084	C₂²	1	1	yes	characteristic
35	#11718082	C₂²	1	1	yes	characteristic
36	#11718081	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	C₂×C₁₃×#41	—	4	1	no	—
42	C₂×C₁₃×D₈	—	2	2	no	—
43	C₂×C₁₃×D₈	—	2	1	no	—
44	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
45	C₂×C₁₃×D₈	C₂³	1	2	yes	—
46	C₁₃×#127	—	2	1	no	—
47	C₁₃×#127	—	2	2	no	—
48	C₁₃×#121	—	4	1	no	—
49	C₁₃×#121	—	4	2	no	—
50	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
51	C₄×D₄×C₁₃	—	2	1	no	—
52	C₄×D₄×C₁₃	—	2	2	no	—
53	#88198	—	4	1	no	—
54	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
55	#88143	—	4	2	no	—
56	#88141	—	4	1	no	—
57	#88124	—	2	2	no	—
58	#88123	—	2	1	no	—
59	C₄×Dic₂₆	—	2	2	no	—
60	C₄×Dic₂₆	—	2	1	no	—
61	C₁₃×#107	C₂³	1	1	yes	characteristic
62	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
63	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
64	C₁₃×#102	—	2	2	no	—
65	C₁₃×#102	—	2	1	no	—
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂³	1	2	yes	—
68	#88067	—	2	2	no	—
69	#88067	—	2	4	no	—
70	#88058	—	2	1	no	—
71	#88058	C₂³	1	2	yes	—
72	#88058	C₂³	1	1	yes	characteristic
73	#88057	—	2	2	no	—
74	#88056	—	2	1	no	—
75	#88056	—	2	2	no	—
76	C₂×#6090	C₂³	1	1	yes	characteristic
77	C₂×#6090	C₂³	1	1	yes	characteristic
78	C₂×#6090	C₂³	1	2	yes	—
79	#33935	—	13	1	no	maximal, Sylow
80	C₁₃×#41	—	8	4	no	—
81	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
82	C₂×D₄×C₁₃	—	2	2	no	—
83	C₂×D₄×C₁₃	—	2	1	no	—
84	C₂×D₄×C₁₃	—	4	2	no	—
85	C₂×D₄×C₁₃	—	4	1	no	—
86	C₂×D₄×C₁₃	—	4	2	no	—
87	C₂×D₄×C₁₃	—	4	1	no	—
88	C₂²×C₄×C₁₃	—	4	2	no	—
89	C₂²×C₄×C₁₃	—	4	1	no	—
90	C₂×Dic₂₆	—	2	1	no	—
91	C₂×Dic₂₆	—	4	1	no	—
92	C₁₃×D₈	—	2	1	no	—
93	C₁₃×D₈	—	2	2	no	—
94	C₁₃×D₈	—	2	1	no	—
95	C₁₃×D₈	—	2	2	no	—
96	C₁₃×D₈	—	8	2	no	—
97	C₁₃×D₈	—	8	1	no	—
98	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
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	D₈	1	1	yes	characteristic
103	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
104	C₁₃×C₂²⋊C₄	—	4	2	no	—
105	C₁₃×C₂²⋊C₄	—	4	1	no	—
106	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂×D₄	1	2	yes	—
108	C₄⋊Dic₁₃	—	2	2	no	—
109	C₄⋊Dic₁₃	—	2	1	no	—
110	C₄⋊Dic₁₃	—	4	1	no	—
111	#6101	—	4	1	no	—
112	#6101	—	4	2	no	—
113	#6101	—	4	2	no	—
114	#6101	—	4	2	no	—
115	C₄×Dic₁₃	—	4	2	no	—
116	C₄×Dic₁₃	—	4	1	no	—
117	Dic₅₂	—	2	1	no	—
118	Dic₅₂	#41	1	2	yes	complemented
119	#6090	—	2	1	no	—
120	#6090	—	2	2	no	—
121	#6090	—	2	1	no	—
122	#3391	D₁₃	1	1	yes	characteristic
123	#3383	—	13	1	no	—
124	#2244	—	13	2	no	—
125	#2240	—	13	1	no	—
126	#2228	—	13	2	no	—
127	#2224	—	13	1	no	—
128	#2207	—	13	2	no	—
129	#2207	—	13	1	no	—
130	#2183	—	13	1	no	—
131	#2175	—	13	2	no	—
132	C₄×SD₃₂	—	13	1	no	—
133	C₂³×C₁₃	—	4	2	no	—
134	C₂³×C₁₃	—	4	1	no	—
135	Q₈×C₁₃	—	2	2	no	—
136	D₄×C₁₃	—	4	1	no	—
137	D₄×C₁₃	—	4	2	no	—
138	D₄×C₁₃	—	4	1	no	—
139	D₄×C₁₃	—	4	2	no	—
140	D₄×C₁₃	—	8	4	no	—
141	D₄×C₁₃	—	8	2	no	—
142	D₄×C₁₃	—	16	2	no	—
143	D₄×C₁₃	—	16	1	no	—
144	C₂×C₄×C₁₃	—	4	1	no	—
145	C₂×C₄×C₁₃	—	8	4	no	—
146	C₂×C₄×C₁₃	—	8	2	no	—
147	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
149	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×Dic₁₃	—	4	1	no	—
153	C₂×Dic₁₃	—	4	1	no	—
154	C₂×Dic₁₃	—	4	2	no	—
155	Dic₂₆	—	2	2	no	—
156	Dic₂₆	—	4	1	no	—
157	Dic₂₆	—	8	2	no	—
158	C₈×C₁₃	—	2	2	no	—
159	C₈×C₁₃	—	2	1	no	—
160	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
161	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
162	#557	—	26	1	no	—
163	#550	—	2	1	no	—
164	C₂×SD₃₂	—	13	1	no	—
165	C₂×SD₃₂	—	13	2	no	—
166	#500	—	13	1	no	—
167	#499	—	13	2	no	—
168	#493	C₂×D₁₃	1	1	yes	characteristic
169	#493	C₂×D₁₃	1	2	yes	—
170	#479	—	26	2	no	—
171	#475	—	26	1	no	—
172	#459	—	2	1	no	—
173	#459	—	2	2	no	—
174	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
175	C₄×Q₁₆	—	13	1	no	—
176	C₄×Q₁₆	—	13	2	no	—
177	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
178	C₄×D₈	C₂×D₁₃	1	2	yes	—
179	#367	—	13	1	no	—
180	#366	—	13	2	no	—
181	C₄⋊C₁₆	—	13	1	no	—
182	C₄⋊C₁₆	—	13	2	no	—
183	#358	—	13	1	no	—
184	#357	—	13	1	no	—
185	#357	—	13	2	no	—
186	#357	—	13	2	no	—
187	#357	—	13	2	no	—
188	#357	—	13	2	no	—
189	#346	—	13	2	no	—
190	C₄×C₁₆	—	13	1	no	—
191	C₂²×C₁₃	—	8	2	no	—
192	C₂²×C₁₃	—	8	1	no	—
193	C₂²×C₁₃	—	8	2	no	—
194	C₂²×C₁₃	—	8	1	no	—
195	C₂²×C₁₃	—	8	2	no	—
196	C₂²×C₁₃	—	8	1	no	—
197	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
198	C₄×C₁₃	—	2	2	no	—
199	C₄×C₁₃	—	2	2	no	—
200	C₄×C₁₃	—	2	1	no	—
201	C₄×C₁₃	—	8	1	no	—
202	C₄×C₁₃	#509	1	2	yes	—
203	C₄×C₁₃	#548	1	1	yes	characteristic
204	C₄×C₁₃	#577	1	1	yes	characteristic
205	Dic₁₃	—	4	2	no	—
206	Dic₁₃	—	8	2	no	—
207	Dic₁₃	—	8	1	no	—
208	C₂×#41	—	4	1	no	—
209	C₂×Q₁₆	—	13	1	no	—
210	C₂×D₈	—	2	2	no	—
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
213	C₂×D₈	C₂²×D₁₃	1	2	yes	—
214	C₄⋊Q₈	—	26	1	no	—
215	C₄⋊Q₈	—	52	1	no	—
216	#127	—	2	2	no	—
217	#127	—	2	1	no	—
218	#125	—	26	2	no	—
219	#125	—	52	2	no	—
220	#121	—	4	1	no	—
221	#121	—	4	2	no	—
222	C₄×Q₈	—	26	1	no	—
223	C₄×Q₈	—	26	2	no	—
224	C₄×Q₈	—	52	1	no	—
225	C₄×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
226	C₄×D₄	—	2	2	no	—
227	C₄×D₄	—	2	1	no	—
228	SD₃₂	—	26	2	no	—
229	SD₃₂	—	26	4	no	—
230	C₂×C₁₆	—	13	1	no	—
231	C₂×C₁₆	—	13	1	no	—
232	C₂×C₁₆	—	13	2	no	—
233	#107	—	13	2	no	—
234	#107	—	13	1	no	—
235	#107	—	26	1	no	—
236	#107	C₂²×D₁₃	1	1	yes	characteristic
237	#106	—	26	2	no	—
238	C₄⋊C₈	C₂²⋊D₁₃	1	1	yes	characteristic
239	C₄⋊C₈	C₂²⋊D₁₃	1	2	yes	—
240	#103	—	26	2	no	—
241	#103	—	26	1	no	—
242	#102	—	2	2	no	—
243	#102	—	2	1	no	—
244	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₁₃	1	2	yes	—
246	C₂×C₁₃	—	16	2	no	—
247	C₂×C₁₃	—	16	1	no	—
248	C₂×C₁₃	#3358	1	1	yes	characteristic
249	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
250	C₂×C₁₃	#3416	1	1	yes	characteristic
251	#41	—	8	4	no	—
252	C₂×Q₈	—	26	1	no	—
253	C₂×Q₈	—	52	1	no	—
254	C₂×Q₈	#6104	1	1	yes	characteristic
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	4	1	no	—
263	Q₁₆	—	13	2	no	—
264	Q₁₆	—	26	1	no	—
265	D₈	—	2	1	no	—
266	D₈	—	2	2	no	—
267	D₈	—	2	1	no	—
268	D₈	—	2	2	no	—
269	D₈	—	8	2	no	—
270	D₈	—	8	1	no	—
271	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
272	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
273	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	26	1	no	—
276	C₄⋊C₄	—	26	2	no	—
277	C₄⋊C₄	—	52	1	no	—
278	C₄⋊C₄	—	52	2	no	—
279	C₄⋊C₄	—	52	2	no	—
280	C₄⋊C₄	—	52	2	no	—
281	C₄⋊C₄	—	52	1	no	—
282	C₄⋊C₄	#6104	1	1	yes	characteristic
283	C₄⋊C₄	#6104	1	2	yes	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	4	1	no	—
286	C₄²	—	52	2	no	—
287	C₄²	—	52	1	no	—
288	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
289	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
290	C₁₆	—	26	1	no	—
291	C₁₆	—	26	2	no	—
292	C₁₆	—	26	1	no	—
293	C₁₃	#33935	1	1	yes	characteristic, complemented, Sylow
294	C₂³	—	4	2	no	—
295	C₂³	—	4	1	no	—
296	Q₈	—	2	2	no	—
297	Q₈	—	26	2	no	—
298	Q₈	—	52	1	no	—
299	Q₈	—	104	2	no	—
300	D₄	—	4	1	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	1	no	—
303	D₄	—	4	2	no	—
304	D₄	—	8	4	no	—
305	D₄	—	8	2	no	—
306	D₄	—	16	2	no	—
307	D₄	—	16	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	8	4	no	—
310	C₂×C₄	—	8	2	no	—
311	C₂×C₄	—	52	1	no	—
312	C₂×C₄	—	52	1	no	—
313	C₂×C₄	—	52	2	no	—
314	C₂×C₄	C₂×#6104	1	2	yes	—
315	C₂×C₄	C₂×#6104	1	1	yes	characteristic
316	C₂×C₄	C₂×#6104	1	2	yes	—
317	C₂×C₄	C₂×#6104	1	1	yes	characteristic
318	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	2	1	no	—
321	C₈	#41×D₁₃	1	1	yes	characteristic
322	C₈	#88257	1	1	yes	characteristic
323	C₂²	—	8	2	no	—
324	C₂²	—	8	1	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	1	no	—
329	C₂²	C₂²×#6104	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	52	2	no	—
335	C₄	—	104	2	no	—
336	C₄	—	104	1	no	—
337	C₄	#11718808	1	2	yes	—
338	C₄	#11719536	1	1	yes	characteristic
339	C₄	#11719541	1	1	yes	characteristic
340	C₂	—	16	2	no	—
341	C₂	—	16	1	no	—
342	C₂	#13127467	1	1	yes	characteristic
343	C₂	C₂×#11718808	1	1	yes	characteristic
344	C₂	#13127983	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting, prime radical	C₁₃×#3391	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3391	D₁₃
hypocenter	C₁₃	#33935
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3391

Complements

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

N	H ≅ G/N	type
1. #13127728	C₂	semidirect
2. #13117077	C₂	semidirect
3. #13117059	C₂	semidirect
4. #13116993	C₂	semidirect
5. #13116973	C₂	semidirect
6. #13116873	C₂	semidirect
7. #13116873	C₂	semidirect
8. #13116745	C₂	semidirect
9. #13116700	C₂	semidirect
10. C₄×#88067	C₂	semidirect
118. Dic₅₂	#41	semidirect
293. C₁₃	#33935	semidirect

Maximal subgroups

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

subgroup	index
#13127728	2
#13117077	2
#13117059	2
#13116993	2
#13116973	2
#13116873	2
#13116745	2
#13116700	2
C₄×#88067	2
C₁₃×#3391	2
#33935	13