Subgroups — #16019029

Order 3328.

504 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125350	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117116	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117109	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117059	C₂	1	1	yes	maximal, characteristic, complemented
5	#13117059	C₂	1	1	yes	maximal, characteristic, complemented
6	#13117024	C₂	1	4	yes	maximal, complemented
7	#13116993	C₂	1	4	yes	maximal, complemented
8	#13116635	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic
10	C₁₃×#532	—	2	1	no	—
11	#11719175	—	2	1	no	—
12	C₁₃×#500	C₂²	1	1	yes	characteristic
13	C₁₃×#493	C₂²	1	1	yes	characteristic
14	C₁₃×#486	—	2	1	no	—
15	C₁₃×#466	—	2	1	no	—
16	C₁₃×#466	C₂²	1	2	yes	—
17	C₄×C₁₃×D₈	—	2	1	no	—
18	C₄×C₁₃×D₈	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	#11718767	—	2	1	no	—
21	#11718767	C₂²	1	2	yes	—
22	#11718344	—	2	1	no	—
23	#11718344	C₂²	1	2	yes	—
24	#11718343	C₂²	1	1	yes	characteristic
25	#11718343	C₂²	1	1	yes	characteristic
26	#11718341	—	2	1	no	—
27	#11718154	C₂²	1	4	yes	—
28	#11718154	C₂²	1	4	yes	—
29	C₂²⋊#6090	C₂²	1	2	yes	—
30	#11718082	C₂²	1	4	yes	—
31	#11718081	C₂²	1	4	yes	—
32	C₄⋊#6090	C₂²	1	2	yes	—
33	C₄×#6090	C₂²	1	1	yes	characteristic
34	C₄×#6090	C₂²	1	1	yes	characteristic
35	C₂×C₁₃×#41	—	4	1	no	—
36	C₂×C₁₃×D₈	—	2	1	no	—
37	C₂×C₁₃×D₈	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×C₄⋊Q₈	—	2	1	no	—
41	C₁₃×#127	—	2	1	no	—
42	C₁₃×#124	—	4	1	no	—
43	C₁₃×#121	—	2	2	no	—
44	C₁₃×#121	—	4	1	no	—
45	C₄×D₄×C₁₃	—	2	2	no	—
46	C₄×D₄×C₁₃	—	4	1	no	—
47	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
48	#88180	—	2	2	no	—
49	#88178	—	4	1	no	—
50	#88178	—	4	1	no	—
51	#88124	—	2	2	no	—
52	#88123	—	2	1	no	—
53	#88123	—	2	1	no	—
54	C₄×Dic₂₆	—	4	1	no	—
55	C₄×Dic₂₆	—	4	1	no	—
56	C₁₃×#107	—	2	1	no	—
57	C₁₃×#107	C₂³	1	2	yes	—
58	C₁₃×#102	—	2	2	no	—
59	C₁₃×#102	—	2	2	no	—
60	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	#88058	—	2	1	no	—
65	#88058	—	2	1	no	—
66	#88058	C₂³	1	2	yes	—
67	#88058	C₂³	1	2	yes	—
68	#88057	—	2	2	no	—
69	#88057	—	2	2	no	—
70	C₂×#6090	C₂³	1	2	yes	—
71	C₂×#6090	C₂³	1	2	yes	—
72	#33794	—	13	1	no	maximal, Sylow
73	C₁₃×#41	—	8	4	no	—
74	C₂×Q₈×C₁₃	—	4	1	no	—
75	C₂×D₄×C₁₃	—	2	2	no	—
76	C₂×D₄×C₁₃	—	4	2	no	—
77	C₂×D₄×C₁₃	—	4	1	no	—
78	C₂×D₄×C₁₃	—	4	1	no	—
79	C₂²×C₄×C₁₃	—	4	2	no	—
80	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₂₆	—	4	1	no	—
84	C₂×Dic₂₆	—	4	1	no	—
85	C₁₃×D₈	—	2	4	no	—
86	C₁₃×D₈	—	4	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	4	1	no	—
95	C₁₃×C₂²⋊C₄	—	4	2	no	—
96	C₁₃×C₂²⋊C₄	—	4	2	no	—
97	C₄²×C₁₃	D₈	1	1	yes	characteristic
98	C₄²×C₁₃	D₈	1	1	yes	characteristic
99	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂²⋊Dic₁₃	—	4	2	no	—
102	C₄⋊Dic₁₃	—	2	2	no	—
103	C₄⋊Dic₁₃	—	2	2	no	—
104	C₄⋊Dic₁₃	—	4	1	no	—
105	C₄⋊Dic₁₃	—	4	1	no	—
106	#6101	—	4	2	no	—
107	#6101	—	4	2	no	—
108	C₄×Dic₁₃	—	4	2	no	—
109	#6090	—	2	2	no	—
110	#6090	—	2	2	no	—
111	#3169	D₁₃	1	1	yes	characteristic
112	#3160	—	13	1	no	—
113	#2254	—	13	1	no	—
114	#2252	—	13	1	no	—
115	#2240	—	13	1	no	—
116	#2240	—	13	1	no	—
117	#2234	—	13	4	no	—
118	#2228	—	13	4	no	—
119	#2162	—	13	1	no	—
120	C₂³×C₁₃	—	4	2	no	—
121	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
122	Q₈×C₁₃	—	8	2	no	—
123	D₄×C₁₃	—	4	2	no	—
124	D₄×C₁₃	—	4	2	no	—
125	D₄×C₁₃	—	8	4	no	—
126	D₄×C₁₃	—	8	2	no	—
127	D₄×C₁₃	—	8	2	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	8	4	no	—
134	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
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	2	no	—
141	C₂×Dic₁₃	—	4	2	no	—
142	Dic₂₆	—	8	2	no	—
143	Dic₂₆	—	8	2	no	—
144	C₈×C₁₃	—	2	1	no	—
145	C₈×C₁₃	—	2	1	no	—
146	C₈×C₁₃	#137	1	2	yes	—
147	C₈×C₁₃	2₋¹⁺⁴	1	2	yes	—
148	#533	—	26	1	no	—
149	#532	—	2	1	no	—
150	#500	—	13	1	no	—
151	#500	—	13	1	no	—
152	#500	C₂×D₁₃	1	1	yes	characteristic
153	#499	—	13	2	no	—
154	#499	—	26	1	no	—
155	#498	—	26	1	no	—
156	#493	C₂×D₁₃	1	1	yes	characteristic
157	#486	—	2	1	no	—
158	#466	—	2	1	no	—
159	#466	C₂×D₁₃	1	2	yes	—
160	C₄×D₈	—	2	1	no	—
161	C₄×D₈	C₂×D₁₃	1	2	yes	—
162	#427	—	13	2	no	—
163	#427	—	26	1	no	—
164	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
165	#367	—	13	4	no	—
166	#366	—	13	4	no	—
167	C₄⋊C₁₆	—	13	2	no	—
168	#357	—	13	4	no	—
169	#357	—	13	4	no	—
170	C₂²⋊C₁₆	—	13	2	no	—
171	C₄×C₁₆	—	13	1	no	—
172	C₄×C₁₆	—	13	1	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	—	2	1	no	—
175	C₂²×C₁₃	—	2	1	no	—
176	C₂²×C₁₃	—	8	2	no	—
177	C₂²×C₁₃	—	8	2	no	—
178	C₂²×C₁₃	—	8	2	no	—
179	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
180	C₄×C₁₃	—	2	1	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	8	2	no	—
184	C₄×C₁₃	#508	1	2	yes	—
185	C₄×C₁₃	#509	1	2	yes	—
186	C₄×C₁₃	#536	1	1	yes	characteristic
187	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
188	Dic₁₃	—	8	2	no	—
189	Dic₁₃	—	8	2	no	—
190	C₂×#41	—	4	1	no	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	C₂²×D₁₃	1	2	yes	—
193	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
195	C₄⋊Q₈	—	2	1	no	—
196	C₄⋊Q₈	—	26	1	no	—
197	C₄⋊Q₈	—	26	1	no	—
198	#127	—	2	1	no	—
199	#125	—	26	2	no	—
200	#124	—	4	1	no	—
201	C₂²⋊Q₈	—	52	1	no	—
202	C₂²⋊Q₈	—	52	1	no	—
203	#121	—	2	2	no	—
204	#121	—	4	1	no	—
205	C₄×Q₈	—	52	1	no	—
206	C₄×Q₈	—	52	1	no	—
207	C₄×D₄	—	2	2	no	—
208	C₄×D₄	—	4	1	no	—
209	#117	—	26	2	no	—
210	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
211	C₂×C₁₆	—	13	2	no	—
212	C₂×C₁₆	—	13	2	no	—
213	#107	—	2	1	no	—
214	#107	—	13	2	no	—
215	#107	—	13	2	no	—
216	#107	—	26	1	no	—
217	#107	—	26	1	no	—
218	#107	C₂²×D₁₃	1	2	yes	—
219	#106	—	26	2	no	—
220	#106	—	26	2	no	—
221	#102	—	2	2	no	—
222	#102	—	2	2	no	—
223	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
224	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
225	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
226	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
227	C₂×C₁₃	—	4	1	no	—
228	C₂×C₁₃	—	16	2	no	—
229	C₂×C₁₃	#3226	1	1	yes	characteristic
230	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
231	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
232	#41	—	8	4	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂×Q₈	—	52	1	no	—
235	C₂×Q₈	—	52	1	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	#6104	1	1	yes	characteristic
242	C₂²×C₄	#6104	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
244	D₈	—	2	4	no	—
245	D₈	—	4	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	26	2	no	—
255	C₄⋊C₄	—	26	2	no	—
256	C₄⋊C₄	—	52	2	no	—
257	C₄⋊C₄	—	52	2	no	—
258	C₄⋊C₄	—	52	1	no	—
259	C₄⋊C₄	—	52	1	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	52	2	no	—
263	C₄²	—	52	2	no	—
264	C₄²	#6104	1	1	yes	characteristic
265	C₄²	#6104	1	1	yes	characteristic
266	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
267	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
268	C₁₆	—	26	2	no	—
269	C₁₆	—	26	2	no	—
270	C₁₃	#33794	1	1	yes	characteristic, complemented, Sylow
271	C₂³	—	4	2	no	—
272	C₂³	C₂×#6104	1	1	yes	characteristic
273	Q₈	—	8	2	no	—
274	Q₈	—	104	2	no	—
275	Q₈	—	104	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	8	4	no	—
279	D₄	—	8	2	no	—
280	D₄	—	8	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	4	no	—
287	C₂×C₄	—	52	2	no	—
288	C₂×C₄	—	52	2	no	—
289	C₂×C₄	C₂×#6104	1	1	yes	characteristic
290	C₂×C₄	C₂×#6104	1	1	yes	characteristic
291	C₂×C₄	C₂×#6104	1	1	yes	characteristic
292	C₂×C₄	C₂×#6104	1	1	yes	characteristic
293	C₂×C₄	C₂×#6104	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	#88205	1	2	yes	—
298	C₈	#88257	1	2	yes	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	C₂²×#6104	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	8	2	no	—
310	C₄	—	104	2	no	—
311	C₄	—	104	2	no	—
312	C₄	#11718808	1	2	yes	—
313	C₄	#11718855	1	2	yes	—
314	C₄	C₂×#88205	1	1	yes	characteristic
315	C₄	#11719530	1	1	yes	characteristic
316	C₂	—	4	1	no	—
317	C₂	—	16	2	no	—
318	C₂	#13125954	1	1	yes	characteristic
319	C₂	C₂×#11718855	1	1	yes	characteristic
320	C₂	C₂×#11718808	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×#3169	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3169	D₁₃
hypocenter	C₁₃	#33794
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #13125350	C₂	semidirect
2. #13117116	C₂	semidirect
3. #13117109	C₂	semidirect
4. #13117059	C₂	semidirect
5. #13117059	C₂	semidirect
6. #13117024	C₂	semidirect
7. #13116993	C₂	semidirect
8. #13116635	C₂	semidirect
270. C₁₃	#33794	semidirect

Maximal subgroups

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

subgroup	index
#13125350	2
#13117116	2
#13117109	2
#13117059	2
#13117024	2
#13116993	2
#13116635	2
C₁₃×#3169	2
#33794	13