Subgroups — #16020965

Order 3328.

498 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127840	C₂	1	1	yes	maximal, characteristic, complemented
2	#13127826	C₂	1	1	yes	maximal, characteristic
3	#13117104	C₂	1	2	yes	maximal
4	#13116900	C₂	1	2	yes	maximal
5	#13116884	C₂	1	2	yes	maximal, complemented
6	#13116883	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116878	C₂	1	1	yes	maximal, characteristic
8	#13116744	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116707	C₂	1	2	yes	maximal, complemented
10	C₄×#88040	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719437	—	2	1	no	—
13	#11719426	—	2	1	no	—
14	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
15	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
16	C₁₃×#494	C₂²	1	1	yes	characteristic
17	C₁₃×#494	C₂²	1	1	yes	characteristic
18	#11718852	C₂²	1	2	yes	—
19	#11718846	—	2	2	no	—
20	Q₁₆×Dic₁₃	C₂²	1	2	yes	—
21	Q₁₆×Dic₁₃	C₂²	1	2	yes	—
22	#11718843	—	2	2	no	—
23	#11718842	C₂²	1	2	yes	—
24	C₂×#88069	C₂²	1	2	yes	—
25	C₂×#88068	C₂²	1	2	yes	—
26	#11718751	—	2	1	no	—
27	#11718744	—	2	1	no	—
28	C₂×#88040	C₂²	1	1	yes	characteristic
29	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
30	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
31	C₈×Dic₂₆	C₂²	1	1	yes	characteristic
32	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
33	#11718156	C₂²	1	2	yes	—
34	#11718156	C₂²	1	2	yes	—
35	#11718101	C₂²	1	1	yes	characteristic
36	#11718097	C₂²	1	1	yes	characteristic
37	#11718095	C₂²	1	1	yes	characteristic
38	#11718094	C₂²	1	2	yes	—
39	#11718093	C₂²	1	2	yes	—
40	C₄⋊#6090	C₂²	1	1	yes	characteristic
41	C₂×C₁₃×Q₁₆	—	2	1	no	—
42	C₂×C₁₃×Q₁₆	—	2	1	no	—
43	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
44	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
45	C₁₃×C₄⋊Q₈	—	2	1	no	—
46	C₁₃×C₄⋊Q₈	—	2	1	no	—
47	#88201	—	2	2	no	—
48	Q₈×Dic₁₃	—	2	2	no	—
49	Q₈×Dic₁₃	—	2	2	no	—
50	#88198	—	2	2	no	—
51	C₂×#6107	—	2	2	no	—
52	C₂×#6106	—	2	2	no	—
53	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
54	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
55	#88152	—	4	2	no	—
56	#88151	—	4	1	no	—
57	#88146	—	4	1	no	—
58	#88144	—	4	1	no	—
59	#88143	—	4	2	no	—
60	#88141	—	4	1	no	—
61	C₄×D₅₂	—	2	1	no	—
62	C₄×Dic₂₆	—	2	1	no	—
63	C₄×Dic₂₆	D₄	1	1	yes	characteristic
64	C₁₃×Q₃₂	—	2	4	no	—
65	C₁₃×Q₃₂	—	4	2	no	—
66	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	#88075	—	2	2	no	—
70	#88075	—	2	2	no	—
71	#88069	—	4	2	no	—
72	#88068	—	4	2	no	—
73	#88061	—	2	1	no	—
74	#88058	C₂³	1	1	yes	characteristic
75	#88058	C₂³	1	1	yes	characteristic
76	#88056	—	2	1	no	—
77	#88054	D₄	1	2	yes	—
78	C₈×Dic₁₃	C₂³	1	2	yes	—
79	C₂×#6090	C₂³	1	2	yes	—
80	C₄⋊#1055	D₄	1	1	yes	characteristic
81	#88040	—	2	2	no	—
82	#33965	—	13	1	no	maximal, Sylow
83	C₂×Q₈×C₁₃	—	2	2	no	—
84	C₂×Q₈×C₁₃	—	2	2	no	—
85	C₂×D₅₂	—	2	1	no	—
86	C₂×C₄×D₁₃	—	4	1	no	—
87	C₂×Dic₂₆	—	2	1	no	—
88	C₂×Dic₂₆	D₈	1	1	yes	characteristic
89	C₁₃×Q₁₆	—	2	2	no	—
90	C₁₃×Q₁₆	—	2	2	no	—
91	C₁₃×Q₁₆	—	2	2	no	—
92	C₁₃×Q₁₆	—	2	2	no	—
93	C₁₃×Q₁₆	—	8	1	no	—
94	C₁₃×Q₁₆	—	8	1	no	—
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	4	1	no	—
98	C₁₃×C₄⋊C₄	—	4	1	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₄⋊C₄	—	4	1	no	—
101	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
102	#6107	—	8	2	no	—
103	#6106	—	8	2	no	—
104	#6103	—	4	1	no	—
105	#6103	—	4	2	no	—
106	#6103	—	4	2	no	—
107	C₄⋊Dic₁₃	—	2	1	no	—
108	C₄⋊Dic₁₃	—	2	1	no	—
109	C₄⋊Dic₁₃	—	4	2	no	—
110	C₄⋊Dic₁₃	—	4	2	no	—
111	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
112	#6101	—	4	1	no	—
113	#6101	—	4	2	no	—
114	#6101	—	4	2	no	—
115	#6101	D₈	1	2	yes	—
116	C₄×Dic₁₃	—	4	1	no	—
117	C₄×Dic₁₃	—	4	2	no	—
118	C₄×Dic₁₃	—	4	2	no	—
119	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
120	C₂×#1055	C₂×D₄	1	2	yes	—
121	Dic₅₂	—	2	1	no	—
122	Dic₅₂	—	2	1	no	—
123	D₁₀₄	—	2	1	no	—
124	D₁₀₄	—	2	1	no	—
125	C₁₃×C₁₆	—	2	1	no	—
126	C₁₃×C₁₆	#41	1	2	yes	—
127	#6090	—	2	2	no	—
128	#3395	—	13	1	no	—
129	#3393	—	13	1	no	—
130	#2251	—	13	2	no	—
131	#2247	D₁₃	1	1	yes	characteristic, complemented
132	#2211	—	13	2	no	—
133	#2209	—	13	2	no	—
134	#2209	—	13	1	no	—
135	#2208	—	13	1	no	—
136	#2183	—	13	1	no	—
137	#2176	—	13	2	no	—
138	C₄×SD₃₂	—	13	1	no	—
139	C₂²×D₁₃	—	4	1	no	—
140	Q₈×C₁₃	—	4	2	no	—
141	Q₈×C₁₃	—	4	2	no	—
142	Q₈×C₁₃	—	4	2	no	—
143	Q₈×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×Dic₁₃	—	4	1	no	—
150	C₂×Dic₁₃	—	4	1	no	—
151	C₂×Dic₁₃	—	4	1	no	—
152	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
153	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
154	D₅₂	—	4	1	no	—
155	D₅₂	—	4	1	no	—
156	C₄×D₁₃	—	8	2	no	—
157	Dic₂₆	—	2	2	no	—
158	Dic₂₆	—	4	1	no	—
159	Dic₂₆	—	4	1	no	—
160	C₈×C₁₃	—	2	1	no	—
161	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
162	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
163	#1055	—	2	2	no	—
164	#557	—	26	1	no	—
165	#556	—	26	1	no	—
166	C₂×Q₃₂	—	13	2	no	—
167	C₂×Q₃₂	C₂×D₁₃	1	2	yes	—
168	C₂×Q₃₂	C₂×D₁₃	1	2	yes	—
169	C₂×SD₃₂	—	13	2	no	—
170	C₂×SD₃₂	—	13	1	no	—
171	#495	—	13	2	no	—
172	#494	—	13	2	no	—
173	#494	C₂×D₁₃	1	1	yes	characteristic
174	#494	C₂×D₁₃	1	1	yes	characteristic
175	#464	—	26	2	no	—
176	#462	—	26	1	no	—
177	#462	—	26	2	no	—
178	#461	—	26	1	no	—
179	C₈×Q₈	—	13	1	no	—
180	C₄×Q₁₆	—	13	2	no	—
181	C₄×Q₁₆	—	13	2	no	—
182	C₄×Q₁₆	—	13	1	no	—
183	C₄×D₈	—	13	1	no	—
184	#367	—	13	1	no	—
185	C₄⋊C₁₆	—	13	2	no	—
186	C₄⋊C₁₆	—	13	1	no	—
187	#358	—	13	2	no	—
188	#358	—	13	2	no	—
189	#358	—	13	1	no	—
190	#357	—	13	1	no	—
191	#346	—	13	2	no	—
192	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
193	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
194	C₂×D₁₃	—	8	1	no	—
195	C₂×D₁₃	—	8	1	no	—
196	C₂×D₁₃	—	8	1	no	—
197	C₄×C₁₃	—	2	1	no	—
198	C₄×C₁₃	—	8	2	no	—
199	C₄×C₁₃	—	8	2	no	—
200	C₄×C₁₃	#510	1	2	yes	—
201	C₄×C₁₃	#550	1	1	yes	characteristic
202	C₄×C₁₃	#576	1	1	yes	characteristic
203	Dic₁₃	—	2	2	no	—
204	Dic₁₃	—	2	2	no	—
205	Dic₁₃	—	8	1	no	—
206	Dic₁₃	—	8	1	no	—
207	Dic₁₃	—	8	1	no	—
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	—	2	1	no	—
210	C₂×Q₁₆	—	13	1	no	—
211	C₂×Q₁₆	—	26	2	no	—
212	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
213	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
214	C₂×SD₁₆	—	26	2	no	—
215	C₂×D₈	—	13	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	26	2	no	—
219	#126	—	52	2	no	—
220	#125	—	52	2	no	—
221	#125	—	52	1	no	—
222	#124	—	26	2	no	—
223	C₂²⋊Q₈	—	52	1	no	—
224	C₄×Q₈	—	13	1	no	—
225	C₄×Q₈	—	26	2	no	—
226	C₄×Q₈	—	26	2	no	—
227	C₄×Q₈	—	26	1	no	—
228	C₄×Q₈	—	52	1	no	—
229	C₄×D₄	—	26	1	no	—
230	#117	—	52	1	no	—
231	Q₃₂	—	2	4	no	—
232	Q₃₂	—	4	2	no	—
233	Q₃₂	—	52	2	no	—
234	SD₃₂	—	26	2	no	—
235	SD₃₂	—	52	2	no	—
236	C₂×C₁₆	—	13	2	no	—
237	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
238	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
239	#107	—	13	1	no	—
240	#107	—	13	1	no	—
241	C₄⋊C₈	—	13	2	no	—
242	C₄⋊C₈	—	13	1	no	—
243	#103	—	26	2	no	—
244	#103	—	26	2	no	—
245	#103	—	26	1	no	—
246	#102	—	26	1	no	—
247	C₄×C₈	—	13	2	no	—
248	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
249	C₂×C₁₃	#3391	1	1	yes	characteristic
250	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
251	C₂×C₁₃	#3417	1	1	yes	characteristic
252	D₁₃	—	16	1	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	13	1	no	—
256	C₂×Q₈	—	26	1	no	—
257	C₂×D₄	—	26	1	no	—
258	C₂²×C₄	—	52	1	no	—
259	Q₁₆	—	2	2	no	—
260	Q₁₆	—	2	2	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	—	2	2	no	—
263	Q₁₆	—	8	1	no	—
264	Q₁₆	—	8	1	no	—
265	Q₁₆	—	26	1	no	—
266	Q₁₆	—	26	1	no	—
267	Q₁₆	—	104	2	no	—
268	SD₁₆	—	104	2	no	—
269	D₈	—	26	1	no	—
270	D₈	—	26	1	no	—
271	C₂×C₈	—	13	2	no	—
272	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
273	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	13	1	no	—
279	C₄⋊C₄	—	13	2	no	—
280	C₄⋊C₄	—	26	1	no	—
281	C₄⋊C₄	—	26	1	no	—
282	C₄⋊C₄	—	52	2	no	—
283	C₄⋊C₄	—	52	2	no	—
284	C₄⋊C₄	—	52	2	no	—
285	C₄⋊C₄	—	52	2	no	—
286	C₄⋊C₄	—	52	1	no	—
287	C₂²⋊C₄	—	52	2	no	—
288	C₂²⋊C₄	—	52	2	no	—
289	C₂²⋊C₄	—	52	1	no	—
290	C₄²	—	13	2	no	—
291	C₄²	—	52	2	no	—
292	C₄²	—	52	2	no	—
293	C₄²	—	52	1	no	—
294	C₄²	D₄×D₁₃	1	1	yes	characteristic
295	C₁₆	—	2	1	no	—
296	C₁₆	—	26	2	no	—
297	C₁₆	#6129	1	2	yes	—
298	C₁₃	#33965	1	1	yes	characteristic, complemented, Sylow
299	C₂³	—	52	1	no	—
300	Q₈	—	4	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	26	2	no	—
305	Q₈	—	52	1	no	—
306	Q₈	—	52	1	no	—
307	D₄	—	52	1	no	—
308	D₄	—	52	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	13	2	no	—
312	C₂×C₄	—	13	2	no	—
313	C₂×C₄	—	52	1	no	—
314	C₂×C₄	—	52	1	no	—
315	C₂×C₄	—	52	1	no	—
316	C₂×C₄	—	104	2	no	—
317	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
318	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
320	C₈	—	2	1	no	—
321	C₈	—	26	2	no	—
322	C₈	C₂×#6129	1	1	yes	characteristic
323	C₈	#88251	1	1	yes	characteristic
324	C₂²	—	104	1	no	—
325	C₂²	—	104	1	no	—
326	C₂²	—	104	1	no	—
327	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	26	2	no	—
332	C₄	—	26	2	no	—
333	C₄	—	104	1	no	—
334	C₄	—	104	1	no	—
335	C₄	—	104	1	no	—
336	C₄	#11718579	1	2	yes	—
337	C₄	#11719423	1	1	yes	characteristic
338	C₄	#11719462	1	1	yes	characteristic
339	C₂	—	208	1	no	—
340	C₂	#13127809	1	1	yes	characteristic
341	C₂	C₂×#11718579	1	1	yes	characteristic
342	C₂	#13127985	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×#2247	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×D₁₀₄	C₂³
hypercenter	#2247	D₁₃
hypocenter	C₁₃	#33965
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

Complements

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

N	H ≅ G/N	type
1. #13127840	C₂	semidirect
5. #13116884	C₂	semidirect
6. #13116883	C₂	semidirect
8. #13116744	C₂	semidirect
9. #13116707	C₂	semidirect
11. C₁₃×#2247	C₂	semidirect
131. #2247	D₁₃	semidirect
298. C₁₃	#33965	semidirect

Maximal subgroups

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

subgroup	index
#13127840	2
#13127826	2
#13117104	2
#13116900	2
#13116884	2
#13116883	2
#13116878	2
#13116744	2
#13116707	2
C₄×#88040	2
C₁₃×#2247	2
#33965	13