Subgroups — #15860944

Order 3328.

1042 conjugacy classes of subgroups.

The subgroup lattice has 294 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	#13123426	C₂	1	1	yes	maximal, characteristic
3	#13123254	C₂	1	2	yes	maximal
4	#13114720	C₂	1	4	yes	maximal, complemented
5	#13114715	C₂	1	4	yes	maximal, complemented
6	C₂×#11718115	C₂	1	2	yes	maximal
7	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88180	C₂²	1	2	yes	—
12	C₂²×#6099	C₂²	1	2	yes	—
13	C₄×C₁₃×M₁₆	C₄	1	8	yes	complemented
14	C₂×C₁₃×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
16	#11718786	C₄	1	8	yes	complemented
17	#11718783	—	2	4	no	—
18	#11718782	C₄	1	8	yes	complemented
19	M₁₆×Dic₁₃	—	2	4	no	—
20	C₂×#88059	C₂²	1	4	yes	—
21	#11718767	C₂²	1	4	yes	—
22	#11718762	C₂²	1	8	yes	—
23	#11718115	C₂²	1	4	yes	—
24	#11718115	C₂²	1	4	yes	—
25	C₂³×C₄×C₁₃	—	2	1	no	—
26	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
28	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
29	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
31	C₂×C₁₃×M₁₆	D₄	1	4	yes	—
32	C₂×C₁₃×M₁₆	Q₈	1	4	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₂×C₄²×C₁₃	—	2	1	no	—
36	C₂×C₄²×C₁₃	—	2	2	no	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
41	#88180	—	2	2	no	—
42	#88180	—	2	2	no	—
43	#88180	C₂×C₄	1	4	yes	—
44	#88180	C₂³	1	4	yes	—
45	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
46	C₂×C₄×Dic₁₃	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	—	2	4	no	—
49	C₂×#6099	C₂×C₄	1	8	yes	—
50	C₂×#6099	C₂³	1	4	yes	—
51	C₂²×#1055	—	2	2	no	—
52	C₁₃×#97	C₂×C₄	1	8	yes	complemented
53	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
54	#88059	C₂×C₄	1	16	yes	complemented
55	#88058	C₂×C₄	1	8	yes	complemented
56	#88057	C₂×C₄	1	8	yes	complemented
57	#88055	—	2	8	no	—
58	C₈×Dic₁₃	—	2	8	no	—
59	#20445	—	13	1	no	maximal, Sylow
60	C₂⁴×C₁₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	—	2	4	no	—
63	C₂²×C₄×C₁₃	—	2	1	no	—
64	C₂²×C₄×C₁₃	—	4	4	no	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	#41	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₁₃	—	2	4	no	—
74	C₁₃×M₁₆	C₄²	1	16	yes	—
75	C₁₃×M₁₆	C₄⋊C₄	1	16	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
81	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
82	C₄²×C₁₃	—	2	4	no	—
83	C₄²×C₁₃	—	2	1	no	—
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
86	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂²⋊Dic₁₃	—	2	4	no	—
88	C₂²⋊Dic₁₃	—	2	4	no	—
89	C₄⋊Dic₁₃	—	2	4	no	—
90	C₄⋊Dic₁₃	C₂²×C₄	1	8	yes	—
91	C₄×Dic₁₃	—	2	4	no	—
92	C₄×Dic₁₃	—	2	4	no	—
93	#6099	—	2	8	no	—
94	#6099	—	2	4	no	—
95	#6099	C₂²×C₄	1	8	yes	—
96	C₂×#1055	—	2	8	no	—
97	C₂×#1055	—	2	4	no	—
98	C₂×#430	—	13	1	no	—
99	#2911	—	13	1	no	—
100	#2873	—	13	2	no	—
101	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
102	#1773	—	13	4	no	—
103	#1772	—	13	4	no	—
104	C₂×#339	—	13	2	no	—
105	C₂³×C₁₃	—	4	2	no	—
106	C₂³×C₁₃	#117	1	2	yes	—
107	C₂³×C₁₃	C₄×D₄	1	2	yes	—
108	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
109	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	4	8	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×C₄×C₁₃	#117	1	2	yes	—
118	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
119	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
121	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
128	C₂×Dic₁₃	—	2	8	no	—
129	C₂×Dic₁₃	—	8	4	no	—
130	C₈×C₁₃	C₂×C₄²	1	8	yes	—
131	C₈×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
132	#1055	—	2	8	no	—
133	C₂²×M₁₆	—	13	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	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
138	#430	—	13	8	no	—
139	C₂×#108	—	13	4	no	—
140	#428	—	13	8	no	—
141	#427	—	13	4	no	—
142	#405	—	13	8	no	—
143	#405	—	26	4	no	—
144	C₄×M₁₆	—	26	4	no	—
145	C₄×M₁₆	Dic₁₃	1	8	yes	complemented
146	C₂×#97	C₂×D₁₃	1	2	yes	—
147	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
148	#339	—	13	4	no	—
149	#339	—	13	4	no	—
150	C₂²×C₁₃	—	2	4	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	4	2	no	—
153	C₂²×C₁₃	—	4	2	no	—
154	C₂²×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
155	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
158	C₄×C₁₃	—	2	4	no	—
159	C₄×C₁₃	—	4	2	no	—
160	C₄×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
161	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
162	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
163	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
164	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
165	Dic₁₃	—	4	8	no	—
166	C₂³×C₄	—	2	1	no	—
167	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
168	C₂×M₁₆	—	13	8	no	—
169	C₂×M₁₆	—	13	4	no	—
170	C₂×M₁₆	—	26	2	no	—
171	C₂×M₁₆	—	26	4	no	—
172	C₂×M₁₆	Dic₂₆	1	4	yes	—
173	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
174	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
175	C₂×M₁₆	D₅₂	1	4	yes	—
176	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
177	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
178	C₂²×C₈	—	26	2	no	—
179	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
180	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
181	#117	—	13	4	no	—
182	#117	—	13	4	no	—
183	#117	—	26	2	no	—
184	#117	—	26	2	no	—
185	C₂×C₄⋊C₄	—	13	4	no	—
186	C₂×C₂²⋊C₄	—	26	2	no	—
187	C₂×C₄²	—	2	1	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	26	2	no	—
190	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
191	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
192	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
193	#108	—	13	16	no	—
194	#107	—	13	8	no	—
195	#106	—	13	8	no	—
196	#97	—	26	8	no	—
197	#97	C₂×Dic₁₃	1	8	yes	complemented
198	C₄×C₈	—	26	8	no	—
199	C₄×C₈	C₂×Dic₁₃	1	8	yes	complemented
200	C₂×C₁₃	—	2	1	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	2	4	no	—
203	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
204	C₂×C₁₃	#2956	1	2	yes	—
205	C₂⁴	#6127	1	1	yes	characteristic
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	26	4	no	—
211	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
212	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
213	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
214	C₂²×C₄	C₂×D₅₂	1	2	yes	—
215	C₂²×C₄	#6127	1	1	yes	characteristic
216	C₂²×C₄	#6127	1	2	yes	—
217	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
218	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
219	M₁₆	—	13	8	no	—
220	M₁₆	—	26	8	no	—
221	M₁₆	—	26	4	no	—
222	M₁₆	C₄×Dic₁₃	1	16	yes	—
223	M₁₆	C₄⋊Dic₁₃	1	16	yes	—
224	C₂×C₈	—	26	8	no	—
225	C₂×C₈	—	26	4	no	—
226	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
227	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
228	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
229	C₂×C₈	C₂×D₅₂	1	4	yes	—
230	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
231	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
232	C₄⋊C₄	—	13	8	no	—
233	C₄⋊C₄	—	26	4	no	—
234	C₂²⋊C₄	—	26	4	no	—
235	C₂²⋊C₄	—	26	4	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	26	4	no	—
240	C₄²	—	26	4	no	—
241	C₄²	C₂²×Dic₁₃	1	2	yes	—
242	C₄²	C₂²×Dic₁₃	1	2	yes	—
243	C₁₃	#20445	1	1	yes	characteristic, complemented, Sylow
244	C₂³	—	4	2	no	—
245	C₂³	C₄×Dic₂₆	1	2	yes	—
246	C₂³	C₄×D₅₂	1	2	yes	—
247	C₂³	#88180	1	2	yes	—
248	C₂³	C₂×#6127	1	1	yes	characteristic
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	8	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	26	8	no	—
255	C₂×C₄	—	104	4	no	—
256	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
257	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
258	C₂×C₄	C₄×D₅₂	1	4	yes	—
259	C₂×C₄	C₄×D₅₂	1	2	yes	—
260	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
261	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
262	C₂×C₄	#88180	1	2	yes	—
263	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
267	C₂×C₄	C₂×#6127	1	1	yes	characteristic
268	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
269	C₈	—	26	8	no	—
270	C₈	C₂×C₄×Dic₁₃	1	8	yes	—
271	C₈	C₂×C₄⋊Dic₁₃	1	8	yes	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	C₄×C₄⋊Dic₁₃	1	4	yes	—
277	C₂²	C₂×C₄×Dic₂₆	1	1	yes	characteristic
278	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
279	C₂²	C₂×#88180	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	4	2	no	—
282	C₄	—	52	8	no	—
283	C₄	C₄×C₄⋊Dic₁₃	1	4	yes	—
284	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
285	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
286	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
287	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	4	no	—
291	C₂	C₂×C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
292	C₂	#13123747	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	#20445
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #13114720	C₂	semidirect
5. #13114715	C₂	semidirect
13. C₄×C₁₃×M₁₆	C₄	semidirect
16. #11718786	C₄	semidirect
18. #11718782	C₄	semidirect
52. C₁₃×#97	C₂×C₄	semidirect
53. C₄×C₈×C₁₃	C₂×C₄	semidirect
54. #88059	C₂×C₄	semidirect
55. #88058	C₂×C₄	semidirect
56. #88057	C₂×C₄	semidirect
145. C₄×M₁₆	Dic₁₃	semidirect
197. #97	C₂×Dic₁₃	semidirect
199. C₄×C₈	C₂×Dic₁₃	semidirect
243. C₁₃	#20445	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
#13123426	2
#13123254	2
#13114720	2
#13114715	2
C₂×#11718115	2
C₂×C₄×C₁₃×M₁₆	2
#20445	13