Subgroups — #15863097

Order 3328.

772 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718786	C₂	1	1	yes	maximal, characteristic
2	#13123254	C₂	1	1	yes	maximal, characteristic
3	#13114959	C₂	1	8	yes	maximal, complemented
4	C₂×#11718125	C₂	1	2	yes	maximal
5	C₂×#11718123	C₂	1	2	yes	maximal
6	C₁₃×#2911	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#117	C₂²	1	2	yes	—
9	C₂×#88180	C₂²	1	1	yes	characteristic
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₂²×#6099	C₂²	1	1	yes	characteristic
12	C₁₃×#428	—	2	4	no	—
13	C₁₃×#427	C₂²	1	4	yes	—
14	#11718786	—	2	4	no	—
15	#11718783	C₄	1	4	yes	complemented
16	M₁₆×Dic₁₃	C₄	1	4	yes	complemented
17	C₂×#88059	C₂²	1	4	yes	—
18	#11718762	C₂²	1	4	yes	—
19	#11718125	C₂²	1	8	yes	—
20	#11718123	C₂²	1	8	yes	—
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₃×M₁₆	D₄	1	4	yes	—
25	C₂×C₁₃×M₁₆	Q₈	1	4	yes	—
26	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
27	C₁₃×#117	—	2	2	no	—
28	C₁₃×#117	—	2	2	no	—
29	C₁₃×#117	—	2	2	no	—
30	C₁₃×#117	C₂³	1	4	yes	—
31	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
33	C₂×C₄²×C₁₃	—	2	2	no	—
34	C₂×C₂²⋊Dic₁₃	—	2	1	no	—
35	#88180	—	2	1	no	—
36	#88180	—	2	1	no	—
37	#88180	C₂×C₄	1	2	yes	—
38	#88180	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₁₃	—	2	1	no	—
40	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
41	C₂×#6099	—	2	2	no	—
42	C₂×#6099	—	2	2	no	—
43	C₂×#6099	—	2	2	no	—
44	C₂×#6099	—	2	2	no	—
45	C₂×#6099	—	2	2	no	—
46	C₂×#6099	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	C₂×C₄	1	2	yes	—
49	C₂×#6099	C₂³	1	4	yes	—
50	C₂×#6099	C₂³	1	2	yes	—
51	C₂²×#1055	—	2	2	no	—
52	C₂²×#1055	C₂×C₄	1	2	yes	—
53	C₁₃×#107	—	2	4	no	—
54	C₁₃×#106	—	2	4	no	—
55	#88059	—	2	8	no	—
56	#88055	C₂×C₄	1	8	yes	complemented
57	C₈×Dic₁₃	C₂×C₄	1	8	yes	complemented
58	#20757	—	13	1	no	maximal, Sylow
59	C₂⁴×C₁₃	#41	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	#41	1	2	yes	—
67	C₂²×C₄×C₁₃	#41	1	2	yes	—
68	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₁₃	—	2	2	no	—
72	C₁₃×M₁₆	C₄⋊C₄	1	8	yes	—
73	C₁₃×M₁₆	C₄⋊C₄	1	8	yes	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
76	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
77	C₁₃×C₄⋊C₄	—	2	4	no	—
78	C₁₃×C₄⋊C₄	—	4	2	no	—
79	C₁₃×C₂²⋊C₄	—	2	4	no	—
80	C₁₃×C₂²⋊C₄	—	4	2	no	—
81	C₄²×C₁₃	—	2	4	no	—
82	C₄²×C₁₃	—	4	2	no	—
83	C₂²⋊Dic₁₃	—	2	2	no	—
84	C₂²⋊Dic₁₃	—	2	2	no	—
85	C₄⋊Dic₁₃	—	2	2	no	—
86	C₄⋊Dic₁₃	—	2	2	no	—
87	C₄×Dic₁₃	—	2	2	no	—
88	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
89	#6099	—	2	4	no	—
90	#6099	—	2	2	no	—
91	#6099	—	2	2	no	—
92	#6099	—	2	4	no	—
93	#6099	—	2	4	no	—
94	#6099	—	4	2	no	—
95	#6099	—	4	2	no	—
96	C₂×#1055	—	2	4	no	—
97	C₂×#1055	—	2	4	no	—
98	C₂×#1055	—	4	2	no	—
99	C₂×#1055	—	4	2	no	—
100	C₂×#1055	C₂²×C₄	1	4	yes	—
101	C₂×#430	—	13	1	no	—
102	#2911	D₁₃	1	1	yes	characteristic
103	#2873	—	13	1	no	—
104	#1833	—	13	8	no	—
105	C₂×#344	—	13	2	no	—
106	C₂×H(3,ℤ/4)	—	13	2	no	—
107	C₂³×C₁₃	—	8	1	no	—
108	C₂³×C₁₃	#117	1	1	yes	characteristic
109	C₂³×C₁₃	#117	1	1	yes	characteristic
110	C₂³×C₁₃	#124	1	2	yes	—
111	C₂³×C₁₃	#125	1	2	yes	—
112	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₁₃	—	4	4	no	—
114	C₂×C₄×C₁₃	—	8	4	no	—
115	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₃	#117	1	2	yes	—
118	C₂×C₄×C₁₃	#117	1	2	yes	—
119	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#124	1	2	yes	—
122	C₂×C₄×C₁₃	#124	1	2	yes	—
123	C₂×C₄×C₁₃	#124	1	2	yes	—
124	C₂×C₄×C₁₃	#125	1	2	yes	—
125	C₂×C₄×C₁₃	#125	1	2	yes	—
126	C₂×C₄×C₁₃	#125	1	2	yes	—
127	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
133	C₂×Dic₁₃	—	2	4	no	—
134	C₂×Dic₁₃	—	8	2	no	—
135	C₈×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
136	#1055	—	2	4	no	—
137	#1055	—	4	4	no	—
138	C₂²×M₁₆	—	13	1	no	—
139	C₂²×M₁₆	—	13	2	no	—
140	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
141	C₂×#117	—	13	1	no	—
142	C₂×#117	C₂×D₁₃	1	2	yes	—
143	#430	—	26	4	no	—
144	C₂×#108	—	13	4	no	—
145	#428	—	2	4	no	—
146	#427	C₂×D₁₃	1	4	yes	—
147	#405	—	13	4	no	—
148	#405	—	13	4	no	—
149	C₄×M₁₆	—	13	4	no	—
150	#344	—	13	8	no	—
151	H(3,ℤ/4)	—	13	8	no	—
152	C₂²×C₁₃	—	4	1	no	—
153	C₂²×C₁₃	—	4	2	no	—
154	C₂²×C₁₃	—	4	1	no	—
155	C₂²×C₁₃	—	4	2	no	—
156	C₂²×C₁₃	—	4	1	no	—
157	C₂²×C₁₃	#382	1	2	yes	—
158	C₂²×C₁₃	#382	1	2	yes	—
159	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
160	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#125	1	1	yes	characteristic
162	C₄×C₁₃	—	8	4	no	—
163	C₄×C₁₃	#382	1	2	yes	—
164	C₄×C₁₃	#382	1	2	yes	—
165	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
167	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
168	C₄×C₁₃	C₂×#125	1	1	yes	characteristic
169	Dic₁₃	—	4	4	no	—
170	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
171	C₂×M₁₆	—	13	2	no	—
172	C₂×M₁₆	—	13	2	no	—
173	C₂×M₁₆	—	13	4	no	—
174	C₂×M₁₆	—	26	2	no	—
175	C₂×M₁₆	—	26	2	no	—
176	C₂×M₁₆	—	26	2	no	—
177	C₂×M₁₆	—	26	2	no	—
178	C₂×M₁₆	—	26	2	no	—
179	C₂×M₁₆	—	26	2	no	—
180	C₂×M₁₆	—	26	2	no	—
181	C₂×M₁₆	Dic₂₆	1	4	yes	—
182	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
183	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
184	C₂×M₁₆	C₂²⋊D₁₃	1	4	yes	—
185	C₂²×C₈	—	13	2	no	—
186	C₂²×C₈	—	26	2	no	—
187	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	#117	—	13	2	no	—
192	#117	—	13	2	no	—
193	#117	—	26	1	no	—
194	#117	—	26	1	no	—
195	#117	C₂²×D₁₃	1	4	yes	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	26	1	no	—
198	C₂×C₂²⋊C₄	—	2	2	no	—
199	C₂×C₂²⋊C₄	—	26	1	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	13	2	no	—
202	#108	—	26	8	no	—
203	#107	—	2	4	no	—
204	#106	—	2	4	no	—
205	#97	—	13	8	no	—
206	C₄×C₈	—	13	8	no	—
207	C₂×C₁₃	—	2	1	no	—
208	C₂×C₁₃	—	2	1	no	—
209	C₂×C₁₃	—	2	1	no	—
210	C₂×C₁₃	—	2	2	no	—
211	C₂×C₁₃	—	2	2	no	—
212	C₂×C₁₃	C₂×#382	1	1	yes	characteristic
213	C₂⁴	#6129	1	1	yes	characteristic
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	26	2	no	—
216	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
219	C₂²×C₄	#6129	1	1	yes	characteristic
220	C₂²×C₄	#6129	1	2	yes	—
221	C₂²×C₄	#6131	1	1	yes	characteristic
222	C₂²×C₄	#6131	1	1	yes	characteristic
223	C₂²×C₄	#6131	1	2	yes	—
224	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
225	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
226	M₁₆	—	26	4	no	—
227	M₁₆	—	26	2	no	—
228	M₁₆	—	26	2	no	—
229	M₁₆	—	26	4	no	—
230	M₁₆	—	26	4	no	—
231	M₁₆	—	52	2	no	—
232	M₁₆	—	52	2	no	—
233	M₁₆	#6101	1	8	yes	—
234	M₁₆	#6101	1	8	yes	—
235	C₂×C₈	—	13	4	no	—
236	C₂×C₈	—	26	4	no	—
237	C₂×C₈	—	26	4	no	—
238	C₂×C₈	—	52	2	no	—
239	C₂×C₈	—	52	2	no	—
240	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
241	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
242	C₂×C₈	C₂×C₂²⋊D₁₃	1	4	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	26	2	no	—
246	C₄⋊C₄	—	26	2	no	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₂²⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	26	2	no	—
250	C₂²⋊C₄	—	26	2	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	4	2	no	—
253	C₄²	—	13	4	no	—
254	C₄²	—	26	2	no	—
255	C₁₃	#20757	1	1	yes	characteristic, complemented, Sylow
256	C₂³	—	8	1	no	—
257	C₂³	#88144	1	2	yes	—
258	C₂³	#88146	1	1	yes	characteristic
259	C₂³	#88146	1	1	yes	characteristic
260	C₂³	#88190	1	2	yes	—
261	C₂³	C₂×#6129	1	1	yes	characteristic
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	8	4	no	—
264	C₂×C₄	—	26	4	no	—
265	C₂×C₄	—	104	2	no	—
266	C₂×C₄	#88144	1	2	yes	—
267	C₂×C₄	#88144	1	2	yes	—
268	C₂×C₄	#88144	1	2	yes	—
269	C₂×C₄	#88146	1	2	yes	—
270	C₂×C₄	#88146	1	2	yes	—
271	C₂×C₄	#88146	1	1	yes	characteristic
272	C₂×C₄	#88146	1	1	yes	characteristic
273	C₂×C₄	C₂×#6101	1	2	yes	—
274	C₂×C₄	C₂×#6101	1	2	yes	—
275	C₂×C₄	#88190	1	2	yes	—
276	C₂×C₄	#88201	1	2	yes	—
277	C₂×C₄	#88201	1	2	yes	—
278	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
280	C₂×C₄	C₂×#6129	1	1	yes	characteristic
281	C₂×C₄	C₂×#6131	1	1	yes	characteristic
282	C₂×C₄	C₂×#6131	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
284	C₈	—	26	4	no	—
285	C₈	—	52	4	no	—
286	C₈	C₂×#6101	1	8	yes	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	#11718638	1	2	yes	—
293	C₂²	#11718638	1	2	yes	—
294	C₂²	C₂×#88144	1	1	yes	characteristic
295	C₂²	C₂×#88146	1	1	yes	characteristic
296	C₂²	C₂×#88190	1	1	yes	characteristic
297	C₄	—	8	4	no	—
298	C₄	—	52	4	no	—
299	C₄	#11718638	1	2	yes	—
300	C₄	#11718638	1	2	yes	—
301	C₄	C₂×#88144	1	1	yes	characteristic
302	C₄	C₂×#88146	1	1	yes	characteristic
303	C₄	C₂²×#6101	1	1	yes	characteristic
304	C₄	C₂×#88201	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	C₂×#11718638	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#88146
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#2911	C₂
socle	C₂×C₁₃	C₂×#382
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#2911	D₁₃
hypocenter	C₁₃	#20757
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #2911

Complements

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

N	H ≅ G/N	type
3. #13114959	C₂	semidirect
15. #11718783	C₄	semidirect
16. M₁₆×Dic₁₃	C₄	semidirect
56. #88055	C₂×C₄	semidirect
57. C₈×Dic₁₃	C₂×C₄	semidirect
255. C₁₃	#20757	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718786	2
#13123254	2
#13114959	2
C₂×#11718125	2
C₂×#11718123	2
C₁₃×#2911	2
#20757	13