Subgroups — #15770982

Order 3328.

640 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123426	C₂	1	1	yes	maximal, characteristic
2	#13114568	C₂	1	2	yes	maximal
3	C₂×#11718149	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718149	C₂	1	2	yes	maximal
5	C₁₃×#2905	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
9	C₂×#88180	C₂²	1	2	yes	—
10	C₂×#88180	C₂²	1	2	yes	—
11	C₁₃×#424	—	2	2	no	—
12	C₁₃×#424	C₄	1	4	yes	complemented
13	C₁₃×#423	—	2	2	no	—
14	#11718782	—	2	4	no	—
15	#11718767	—	2	2	no	—
16	#11718149	—	2	2	no	—
17	#11718149	—	2	4	no	—
18	#11718119	—	2	4	no	—
19	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₁₃×M₁₆	—	2	2	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	1	no	—
24	C₂×C₁₃×M₁₆	—	2	1	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃	D₄	1	2	yes	—
30	C₁₃×#117	—	2	1	no	—
31	C₁₃×#117	—	2	1	no	—
32	C₁₃×#117	C₂×C₄	1	2	yes	—
33	C₁₃×#117	Q₈	1	2	yes	—
34	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
35	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
36	C₂×C₄²×C₁₃	—	2	1	no	—
37	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
38	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
39	#88180	—	2	2	no	—
40	#88180	—	2	2	no	—
41	#88180	—	2	2	no	—
42	#88180	—	2	2	no	—
43	#88180	—	2	2	no	—
44	#88180	—	2	2	no	—
45	#88180	—	2	2	no	—
46	#88180	D₄	1	4	yes	—
47	C₂×C₄⋊Dic₁₃	—	2	2	no	—
48	C₂×C₄⋊Dic₁₃	—	2	2	no	—
49	C₂×C₄×Dic₁₃	—	2	2	no	—
50	C₂×C₄×Dic₁₃	Q₈	1	4	yes	—
51	C₁₃×C₄⋊C₈	—	2	8	no	—
52	#88058	—	2	4	no	—
53	#88057	—	2	4	no	—
54	#11234	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	#41	1	2	yes	—
66	C₂²×C₄×C₁₃	#41	1	2	yes	—
67	C₂²×Dic₁₃	—	2	4	no	—
68	C₂²×Dic₁₃	—	4	2	no	—
69	C₁₃×M₁₆	—	2	4	no	—
70	C₁₃×M₁₆	—	2	4	no	—
71	C₁₃×M₁₆	—	4	1	no	—
72	C₁₃×M₁₆	—	4	2	no	—
73	C₁₃×M₁₆	—	4	1	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₂²⋊C₄	—	2	2	no	—
82	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
83	C₄²×C₁₃	—	2	2	no	—
84	C₄²×C₁₃	—	2	2	no	—
85	C₂²⋊Dic₁₃	—	2	4	no	—
86	C₂²⋊Dic₁₃	—	4	2	no	—
87	C₂²⋊Dic₁₃	—	4	2	no	—
88	C₂²⋊Dic₁₃	—	4	2	no	—
89	C₄⋊Dic₁₃	—	2	4	no	—
90	C₄⋊Dic₁₃	—	4	2	no	—
91	C₄⋊Dic₁₃	—	4	2	no	—
92	C₄⋊Dic₁₃	—	4	2	no	—
93	C₄×Dic₁₃	—	2	4	no	—
94	C₄×Dic₁₃	—	2	4	no	—
95	C₄×Dic₁₃	—	2	4	no	—
96	C₄×Dic₁₃	—	4	2	no	—
97	#2911	—	13	1	no	—
98	#2905	D₁₃	1	1	yes	characteristic
99	#1740	—	13	2	no	—
100	C₂×H(3,ℤ/4)	—	13	1	no	—
101	C₂×H(3,ℤ/4)	—	13	2	no	—
102	C₂³×C₁₃	—	8	1	no	—
103	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂³×C₁₃	#121	1	1	yes	characteristic
105	C₂³×C₁₃	#121	1	1	yes	characteristic
106	C₂³×C₁₃	#121	1	2	yes	—
107	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	8	2	no	—
110	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#121	1	2	yes	—
116	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#123	1	2	yes	—
122	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#125	1	2	yes	—
124	C₂×C₄×C₁₃	#125	1	2	yes	—
125	C₂×C₄×C₁₃	#126	1	2	yes	—
126	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
128	C₂×Dic₁₃	—	4	4	no	—
129	C₂×Dic₁₃	—	4	4	no	—
130	C₂×Dic₁₃	—	8	4	no	—
131	C₂×Dic₁₃	—	8	4	no	—
132	C₈×C₁₃	—	2	4	no	—
133	C₈×C₁₃	—	4	2	no	—
134	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
135	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
136	C₂×#117	—	13	2	no	—
137	C₂×#117	—	13	2	no	—
138	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
139	#428	—	26	4	no	—
140	#427	—	26	2	no	—
141	#424	—	2	2	no	—
142	#424	Dic₁₃	1	4	yes	complemented
143	#423	—	2	2	no	—
144	#342	—	26	4	no	—
145	H(3,ℤ/4)	—	26	4	no	—
146	H(3,ℤ/4)	—	26	2	no	—
147	C₂²×C₁₃	—	4	2	no	—
148	C₂²×C₁₃	—	4	2	no	—
149	C₂²×C₁₃	—	4	1	no	—
150	C₂²×C₁₃	—	4	1	no	—
151	C₂²×C₁₃	—	4	1	no	—
152	C₂²×C₁₃	#380	1	1	yes	characteristic
153	C₂²×C₁₃	#389	1	2	yes	—
154	C₂²×C₁₃	#394	1	1	yes	characteristic
155	C₂²×C₁₃	#396	1	2	yes	—
156	C₂²×C₁₃	#398	1	1	yes	characteristic
157	C₄×C₁₃	—	4	4	no	—
158	C₄×C₁₃	#386	1	1	yes	characteristic
159	C₄×C₁₃	#387	1	2	yes	—
160	C₄×C₁₃	#391	1	1	yes	characteristic
161	C₄×C₁₃	#395	1	1	yes	characteristic
162	C₄×C₁₃	#398	1	2	yes	—
163	C₄×C₁₃	#400	1	1	yes	characteristic
164	Dic₁₃	—	8	4	no	—
165	Dic₁₃	—	8	4	no	—
166	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
174	C₂×M₁₆	C₂²⋊D₁₃	1	2	yes	—
175	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
176	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
177	#117	—	2	1	no	—
178	#117	—	2	1	no	—
179	#117	—	13	4	no	—
180	#117	—	26	2	no	—
181	#117	—	26	2	no	—
182	#117	—	26	2	no	—
183	#117	—	26	2	no	—
184	#117	—	26	2	no	—
185	#117	—	26	2	no	—
186	#117	—	26	2	no	—
187	#117	Dic₂₆	1	2	yes	—
188	#117	C₂×Dic₁₃	1	2	yes	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	26	2	no	—
191	C₂×C₄⋊C₄	—	26	2	no	—
192	C₂×C₂²⋊C₄	—	26	2	no	—
193	C₂×C₂²⋊C₄	—	26	2	no	—
194	C₂×C₂²⋊C₄	D₅₂	1	2	yes	—
195	C₂×C₄²	—	2	1	no	—
196	C₂×C₄²	—	13	4	no	—
197	C₂×C₄²	—	26	2	no	—
198	#107	—	26	4	no	—
199	#106	—	26	4	no	—
200	C₄⋊C₈	—	2	8	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	2	2	no	—
203	C₂×C₁₃	—	2	1	no	—
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	—	2	1	no	—
206	C₂×C₁₃	#1441	1	1	yes	characteristic
207	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	26	4	no	—
210	C₂²×C₄	—	52	2	no	—
211	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
212	C₂²×C₄	#6127	1	1	yes	characteristic
213	C₂²×C₄	#6127	1	1	yes	characteristic
214	C₂²×C₄	D₄×D₁₃	1	2	yes	—
215	C₂²×C₄	#6129	1	2	yes	—
216	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
217	C₂²×C₄	#6131	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	4	1	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	4	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	C₂²⋊Dic₁₃	1	4	yes	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	26	4	no	—
234	C₄⋊C₄	—	52	2	no	—
235	C₄⋊C₄	—	52	2	no	—
236	C₄⋊C₄	—	52	2	no	—
237	C₂²⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	26	4	no	—
239	C₂²⋊C₄	—	52	2	no	—
240	C₂²⋊C₄	—	52	2	no	—
241	C₂²⋊C₄	—	52	2	no	—
242	C₂²⋊C₄	C₄⋊Dic₁₃	1	4	yes	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	26	4	no	—
246	C₄²	—	26	4	no	—
247	C₄²	—	26	4	no	—
248	C₄²	—	52	2	no	—
249	C₁₃	#11234	1	1	yes	characteristic, complemented, Sylow
250	C₂³	—	8	1	no	—
251	C₂³	#88149	1	2	yes	—
252	C₂³	#88151	1	2	yes	—
253	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
254	C₂³	#88184	1	1	yes	characteristic
255	C₂³	#88184	1	1	yes	characteristic
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	52	4	no	—
259	C₂×C₄	—	52	4	no	—
260	C₂×C₄	—	104	4	no	—
261	C₂×C₄	—	104	4	no	—
262	C₂×C₄	#88142	1	2	yes	—
263	C₂×C₄	#88143	1	2	yes	—
264	C₂×C₄	#88144	1	2	yes	—
265	C₂×C₄	#88148	1	2	yes	—
266	C₂×C₄	#88150	1	2	yes	—
267	C₂×C₄	#88152	1	2	yes	—
268	C₂×C₄	#88178	1	1	yes	characteristic
269	C₂×C₄	#88178	1	1	yes	characteristic
270	C₂×C₄	#88180	1	1	yes	characteristic
271	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
272	C₂×C₄	#88189	1	1	yes	characteristic
273	C₂×C₄	#88191	1	1	yes	characteristic
274	C₂×C₄	#88193	1	2	yes	—
275	C₂×C₄	#88198	1	1	yes	characteristic
276	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
277	C₂×C₄	#88200	1	2	yes	—
278	C₂×C₄	#88201	1	1	yes	characteristic
279	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
280	C₈	—	2	4	no	—
281	C₈	—	4	2	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	#11718644	1	1	yes	characteristic
288	C₂²	#11718646	1	2	yes	—
289	C₂²	#11718652	1	1	yes	characteristic
290	C₂²	#11718655	1	2	yes	—
291	C₂²	#11718873	1	1	yes	characteristic
292	C₄	—	4	4	no	—
293	C₄	—	104	4	no	—
294	C₄	—	104	4	no	—
295	C₄	#11718640	1	1	yes	characteristic
296	C₄	#11718642	1	2	yes	—
297	C₄	#11718645	1	2	yes	—
298	C₄	#11718647	1	1	yes	characteristic
299	C₄	#11718881	1	1	yes	characteristic
300	C₄	#11718891	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	2	2	no	—
304	C₂	—	2	1	no	—
305	C₂	—	2	2	no	—
306	C₂	#13113089	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11718891
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₁₃×#2905	C₂
socle	C₂×C₁₃	#1441
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	#2905	D₁₃
hypocenter	C₁₃	#11234
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #2905

Complements

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

N	H ≅ G/N	type
12. C₁₃×#424	C₄	semidirect
142. #424	Dic₁₃	semidirect
249. C₁₃	#11234	semidirect

Maximal subgroups

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

subgroup	index
#13123426	2
#13114568	2
C₂×#11718149	2
C₁₃×#2905	2
#11234	13