Subgroups — #16020541

Order 3328.

470 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	#13127266	C₂	1	1	yes	maximal, characteristic
2	#13127230	C₂	1	1	yes	maximal, characteristic
3	#13117116	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117043	C₂	1	2	yes	maximal, complemented
5	#13117035	C₂	1	2	yes	maximal, complemented
6	#13116989	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116943	C₂	1	2	yes	maximal, complemented
8	C₄×#88069	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116667	C₂	1	1	yes	maximal, characteristic
10	#13116623	C₂	1	2	yes	maximal, complemented
11	C₁₃×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719437	—	2	1	no	—
13	#11719284	—	2	1	no	—
14	C₁₃×#500	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
16	#11718849	C₂²	1	2	yes	—
17	C₂×#88069	C₂²	1	1	yes	characteristic
18	#11718755	—	2	1	no	—
19	#11718682	—	2	1	no	—
20	C₂×#88038	C₂²	1	2	yes	—
21	#11718549	—	2	2	no	—
22	#11718548	C₂²	1	2	yes	—
23	#11718541	C₂²	1	2	yes	—
24	D₁₃×#107	C₂²	1	2	yes	—
25	#11718470	—	2	2	no	—
26	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
27	C₈×D₅₂	C₂²	1	1	yes	characteristic
28	#11718343	C₂²	1	1	yes	characteristic
29	C₁₃×#367	C₂²	1	2	yes	—
30	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
31	C₁₃×#358	C₂²	1	2	yes	—
32	#11718156	C₂²	1	1	yes	characteristic
33	#11718099	C₂²	1	2	yes	—
34	#11718096	C₂²	1	2	yes	—
35	#11718095	C₂²	1	2	yes	—
36	#11718084	C₂²	1	1	yes	characteristic
37	#11718082	C₂²	1	2	yes	—
38	#11718082	C₂²	1	2	yes	—
39	#11718081	C₂²	1	1	yes	characteristic
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
42	C₁₃×C₄⋊Q₈	—	2	1	no	—
43	C₄×Q₈×C₁₃	—	2	1	no	—
44	#88201	—	4	1	no	—
45	#88200	—	2	2	no	—
46	Q₈×Dic₁₃	—	4	1	no	—
47	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₁₃	C₂³	1	2	yes	—
49	#88152	—	4	2	no	—
50	#88151	—	2	2	no	—
51	#88150	—	4	2	no	—
52	#88146	—	2	2	no	—
53	C₄⋊C₄×D₁₃	—	2	2	no	—
54	C₄×D₅₂	—	4	1	no	—
55	C₄×D₅₂	D₄	1	1	yes	characteristic
56	#88124	—	4	1	no	—
57	#88123	—	2	1	no	—
58	C₄×Dic₂₆	—	2	1	no	—
59	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
60	C₁₃×#107	—	2	1	no	—
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#107	C₂³	1	1	yes	characteristic
63	C₁₃×#103	—	2	1	no	—
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	#88075	—	2	2	no	—
66	#88069	—	2	2	no	—
67	#88060	D₄	1	2	yes	—
68	#88058	—	2	1	no	—
69	#88058	C₂³	1	1	yes	characteristic
70	#88058	C₂³	1	2	yes	—
71	#88056	—	2	1	no	—
72	C₂×#6090	C₂³	1	1	yes	characteristic
73	C₂×#6090	C₂³	1	1	yes	characteristic
74	#88050	—	2	2	no	—
75	#88048	—	2	2	no	—
76	#88047	—	2	2	no	—
77	C₄⋊#1055	D₄	1	1	yes	characteristic
78	#88038	—	2	4	no	—
79	#33923	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	—	2	1	no	—
81	C₂×Q₈×C₁₃	—	4	1	no	—
82	C₂×D₅₂	D₈	1	1	yes	characteristic
83	C₂×C₄×D₁₃	—	4	2	no	—
84	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
85	C₂×Dic₂₆	—	2	1	no	—
86	C₂×Dic₂₆	—	4	1	no	—
87	C₁₃×Q₁₆	—	2	1	no	—
88	C₁₃×Q₁₆	—	2	1	no	—
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₁₃×C₄⋊C₄	—	2	1	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	4	1	no	—
94	C₁₃×C₄⋊C₄	—	4	1	no	—
95	C₄²×C₁₃	—	4	1	no	—
96	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
97	#6103	—	4	2	no	—
98	#6103	—	4	2	no	—
99	#6103	—	4	2	no	—
100	#6103	D₈	1	2	yes	—
101	C₄⋊Dic₁₃	—	2	2	no	—
102	C₄⋊Dic₁₃	—	2	1	no	—
103	C₄⋊Dic₁₃	—	4	1	no	—
104	C₄⋊Dic₁₃	—	4	2	no	—
105	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
106	#6101	—	4	1	no	—
107	#6101	—	4	2	no	—
108	#6101	—	4	2	no	—
109	C₄×Dic₁₃	—	4	1	no	—
110	C₄×Dic₁₃	—	4	2	no	—
111	C₂×#1055	C₂×D₄	1	2	yes	—
112	Dic₅₂	—	2	1	no	—
113	Dic₅₂	—	2	1	no	—
114	C₈×D₁₃	—	2	2	no	—
115	C₈×D₁₃	—	2	2	no	—
116	C₁₃×C₁₆	—	2	2	no	—
117	#6090	—	2	1	no	—
118	#6090	#41	1	2	yes	—
119	#3342	—	13	1	no	—
120	#3338	—	13	1	no	—
121	#2254	—	13	1	no	—
122	#2237	—	13	2	no	—
123	#2235	—	13	2	no	—
124	#2227	—	13	1	no	—
125	#2225	D₁₃	1	1	yes	characteristic, complemented
126	#2219	—	13	2	no	—
127	C₄×Q₃₂	—	13	1	no	—
128	#2169	—	13	1	no	—
129	#2159	—	13	2	no	—
130	C₂²×D₁₃	C₂×D₈	1	2	yes	—
131	Q₈×C₁₃	—	4	1	no	—
132	Q₈×C₁₃	—	4	1	no	—
133	Q₈×C₁₃	—	8	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	—	4	1	no	—
137	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×Dic₁₃	—	4	1	no	—
141	C₂×Dic₁₃	—	4	1	no	—
142	C₂×Dic₁₃	—	4	2	no	—
143	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
144	D₅₂	—	2	2	no	—
145	C₄×D₁₃	—	2	2	no	—
146	C₄×D₁₃	—	2	2	no	—
147	C₄×D₁₃	—	16	2	no	—
148	Dic₂₆	—	4	1	no	—
149	Dic₂₆	—	4	1	no	—
150	Dic₂₆	—	16	1	no	—
151	C₈×C₁₃	—	2	1	no	—
152	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
153	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
154	#1055	—	2	2	no	—
155	#556	—	26	1	no	—
156	#554	—	26	1	no	—
157	C₂×Q₃₂	—	13	1	no	—
158	C₂×M₃₂	—	13	2	no	—
159	#500	—	13	1	no	—
160	#500	C₂×D₁₃	1	1	yes	characteristic
161	#485	—	26	2	no	—
162	#484	—	26	2	no	—
163	#477	—	26	1	no	—
164	#475	—	26	1	no	—
165	#467	—	13	2	no	—
166	#467	—	13	2	no	—
167	C₄×Q₁₆	—	13	1	no	—
168	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
169	C₈×D₄	—	13	1	no	—
170	#427	—	13	2	no	—
171	C₂×#107	—	13	2	no	—
172	#367	—	13	2	no	—
173	#367	—	13	2	no	—
174	#367	C₂×D₁₃	1	2	yes	—
175	#366	—	13	2	no	—
176	#366	—	13	1	no	—
177	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
178	#358	—	13	1	no	—
179	#358	—	13	2	no	—
180	#358	—	13	1	no	—
181	#358	C₂×D₁₃	1	2	yes	—
182	C₂²⋊C₁₆	—	13	2	no	—
183	C₄×C₁₆	—	13	1	no	—
184	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
185	C₂×D₁₃	—	2	2	no	—
186	C₂×D₁₃	—	2	2	no	—
187	C₂×D₁₃	—	2	2	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	8	1	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	—	8	1	no	—
192	C₄×C₁₃	#510	1	2	yes	—
193	C₄×C₁₃	#547	1	1	yes	characteristic
194	C₄×C₁₃	#578	1	1	yes	characteristic
195	Dic₁₃	—	2	2	no	—
196	Dic₁₃	—	8	1	no	—
197	Dic₁₃	—	8	2	no	—
198	Dic₁₃	—	8	1	no	—
199	C₂×Q₁₆	—	13	1	no	—
200	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
201	C₂²×C₈	—	13	2	no	—
202	C₄⋊Q₈	—	2	1	no	—
203	C₄⋊Q₈	—	26	1	no	—
204	#126	—	52	2	no	—
205	#125	—	52	1	no	—
206	#124	—	52	1	no	—
207	C₂²⋊Q₈	—	26	2	no	—
208	C₂²⋊Q₈	—	26	2	no	—
209	C₂²⋊Q₈	—	52	2	no	—
210	C₄×Q₈	—	2	1	no	—
211	C₄×Q₈	—	26	1	no	—
212	C₄×Q₈	—	52	1	no	—
213	C₄×D₄	—	13	1	no	—
214	C₄×D₄	—	52	1	no	—
215	#117	—	26	2	no	—
216	C₂×C₄⋊C₄	—	26	2	no	—
217	Q₃₂	—	26	2	no	—
218	M₃₂	—	26	4	no	—
219	C₂×C₁₆	—	13	1	no	—
220	C₂×C₁₆	—	13	1	no	—
221	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
222	#107	—	2	1	no	—
223	#107	—	13	1	no	—
224	#107	—	13	2	no	—
225	#107	—	26	2	no	—
226	#107	—	26	1	no	—
227	#107	C₂²×D₁₃	1	1	yes	characteristic
228	#107	C₂²×D₁₃	1	2	yes	—
229	#106	—	26	2	no	—
230	C₄⋊C₈	—	13	1	no	—
231	#103	—	2	1	no	—
232	#103	—	26	2	no	—
233	#103	—	26	2	no	—
234	#103	—	26	1	no	—
235	C₂²⋊C₈	—	13	2	no	—
236	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
237	C₂×C₁₃	#3334	1	1	yes	characteristic
238	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
239	C₂×C₁₃	#3416	1	1	yes	characteristic
240	D₁₃	—	4	2	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	26	1	no	—
244	C₂×Q₈	—	52	1	no	—
245	C₂×D₄	—	13	1	no	—
246	C₂²×C₄	—	13	2	no	—
247	C₂²×C₄	—	52	2	no	—
248	Q₁₆	—	2	1	no	—
249	Q₁₆	—	2	1	no	—
250	Q₁₆	—	26	1	no	—
251	Q₁₆	—	26	1	no	—
252	C₂×C₈	—	13	2	no	—
253	C₂×C₈	—	26	2	no	—
254	C₂×C₈	—	26	2	no	—
255	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
256	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	13	1	no	—
262	C₄⋊C₄	—	26	1	no	—
263	C₄⋊C₄	—	26	2	no	—
264	C₄⋊C₄	—	52	2	no	—
265	C₄⋊C₄	—	52	2	no	—
266	C₄⋊C₄	—	52	2	no	—
267	C₄⋊C₄	—	52	1	no	—
268	C₄⋊C₄	—	52	1	no	—
269	C₂²⋊C₄	—	13	2	no	—
270	C₂²⋊C₄	—	52	2	no	—
271	C₂²⋊C₄	—	52	2	no	—
272	C₂²⋊C₄	—	52	2	no	—
273	C₄²	—	4	1	no	—
274	C₄²	—	52	2	no	—
275	C₄²	—	52	1	no	—
276	C₄²	D₄×D₁₃	1	1	yes	characteristic
277	C₁₆	—	2	2	no	—
278	C₁₆	—	13	2	no	—
279	C₁₆	—	26	1	no	—
280	C₁₃	#33923	1	1	yes	characteristic, complemented, Sylow
281	C₂³	—	13	2	no	—
282	Q₈	—	4	1	no	—
283	Q₈	—	4	1	no	—
284	Q₈	—	8	2	no	—
285	Q₈	—	52	1	no	—
286	Q₈	—	52	1	no	—
287	Q₈	—	208	1	no	—
288	D₄	—	26	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	13	2	no	—
293	C₂×C₄	—	26	2	no	—
294	C₂×C₄	—	26	2	no	—
295	C₂×C₄	—	52	1	no	—
296	C₂×C₄	—	52	1	no	—
297	C₂×C₄	—	52	2	no	—
298	C₂×C₄	—	208	2	no	—
299	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
300	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	26	2	no	—
304	C₈	#88254	1	1	yes	characteristic
305	C₈	#41×D₁₃	1	1	yes	characteristic
306	C₂²	—	26	2	no	—
307	C₂²	—	26	2	no	—
308	C₂²	—	26	2	no	—
309	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	8	1	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	1	no	—
314	C₄	—	26	2	no	—
315	C₄	—	104	1	no	—
316	C₄	—	104	1	no	—
317	C₄	—	104	2	no	—
318	C₄	#11718576	1	2	yes	—
319	C₄	#11719336	1	1	yes	characteristic
320	C₄	#11719484	1	1	yes	characteristic
321	C₂	—	52	2	no	—
322	C₂	#13127182	1	1	yes	characteristic
323	C₂	C₂×#11718576	1	1	yes	characteristic
324	C₂	#13127981	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	C₂×D₅₂	D₈
hypercenter	#2225	D₁₃
hypocenter	C₁₃	#33923
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
3. #13117116	C₂	semidirect
4. #13117043	C₂	semidirect
5. #13117035	C₂	semidirect
6. #13116989	C₂	semidirect
7. #13116943	C₂	semidirect
8. C₄×#88069	C₂	semidirect
10. #13116623	C₂	semidirect
11. C₁₃×#2225	C₂	semidirect
125. #2225	D₁₃	semidirect
280. C₁₃	#33923	semidirect

Maximal subgroups

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

subgroup	index
#13127266	2
#13127230	2
#13117116	2
#13117043	2
#13117035	2
#13116989	2
#13116943	2
C₄×#88069	2
#13116667	2
#13116623	2
C₁₃×#2225	2
#33923	13