Subgroups — #16015558

Order 3328.

826 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃×M₃₂	C₂	1	2	yes	maximal
2	C₂×#11718350	C₂	1	1	yes	maximal, characteristic
3	#13116434	C₂	1	4	yes	maximal, complemented
4	#13116431	C₂	1	4	yes	maximal, complemented
5	C₄×#88052	C₂	1	1	yes	maximal, characteristic, complemented
6	M₃₂×Dic₁₃	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#6094	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88126	C₄	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88055	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88052	C₂²	1	2	yes	—
16	C₂×#88043	C₄	1	1	yes	characteristic
17	D₁₃×M₃₂	C₄	1	16	yes	complemented
18	C₂×#88038	C₂²	1	4	yes	—
19	C₂×C₁₆×D₁₃	C₂²	1	4	yes	—
20	#11718350	C₄	1	2	yes	—
21	#11718350	C₄	1	2	yes	—
22	#11718350	C₂²	1	2	yes	—
23	#11718350	C₂²	1	2	yes	—
24	C₁₃×#346	C₂²	1	2	yes	complemented
25	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
26	#11718094	C₂²	1	4	yes	complemented
27	C₁₆×Dic₁₃	C₂²	1	4	yes	complemented
28	#11718053	C₂²	1	2	yes	complemented
29	C₄×#6090	C₂²	1	2	yes	complemented
30	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×#6103	C₈	1	2	yes	—
35	C₂×#6101	C₈	1	2	yes	—
36	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
38	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
39	C₂×#6094	C₈	1	8	yes	—
40	C₂×#6094	C₂×C₄	1	2	yes	—
41	C₂×#6094	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₁₃	C₂×C₄	1	8	yes	—
44	C₂×C₈×D₁₃	C₂³	1	2	yes	—
45	#88126	C₈	1	4	yes	—
46	#88126	C₂×C₄	1	2	yes	—
47	#88126	C₂×C₄	1	2	yes	—
48	C₁₃×M₃₂	C₂×C₄	1	8	yes	—
49	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
50	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	#88055	C₂×C₄	1	2	yes	—
55	#88055	C₂×C₄	1	2	yes	—
56	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
57	C₈×Dic₁₃	C₂³	1	2	yes	—
58	#88052	C₂×C₄	1	8	yes	—
59	C₂×#6090	C₂³	1	4	yes	—
60	#88043	C₂×C₄	1	2	yes	—
61	#88043	C₂×C₄	1	2	yes	—
62	#88038	C₂×C₄	1	16	yes	complemented
63	C₁₆×D₁₃	C₂×C₄	1	16	yes	complemented
64	#33324	—	13	1	no	maximal, Sylow
65	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
69	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×D₁₃	C₄²	1	4	yes	—
71	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
73	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₄²×C₁₃	C₂×C₈	1	2	yes	—
81	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	#6103	C₂×C₈	1	4	yes	—
84	#6103	C₂×C₈	1	4	yes	—
85	#6101	C₂×C₈	1	4	yes	—
86	#6101	C₂×C₈	1	4	yes	—
87	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
88	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
89	C₂×#1055	C₄²	1	4	yes	—
90	C₂×#1055	C₂×C₈	1	4	yes	—
91	C₂×#1055	C₂²×C₄	1	2	yes	—
92	C₂×#1055	C₂²×C₄	1	2	yes	—
93	#6094	C₂×C₈	1	8	yes	—
94	#6094	C₂×C₈	1	8	yes	—
95	C₈×D₁₃	C₄²	1	8	yes	—
96	C₈×D₁₃	C₂²×C₄	1	4	yes	—
97	C₈×D₁₃	C₂²×C₄	1	4	yes	—
98	C₁₃×C₁₆	C₂²×C₄	1	8	yes	—
99	#6090	C₂²×C₄	1	8	yes	—
100	C₂²×M₃₂	—	13	2	no	—
101	C₂×#405	—	13	1	no	—
102	#2108	—	13	4	no	—
103	#2108	—	13	4	no	—
104	C₄×M₃₂	—	13	2	no	—
105	C₄×M₃₂	—	13	1	no	—
106	C₄×M₃₂	D₁₃	1	1	yes	characteristic, complemented
107	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
108	C₂²×D₁₃	—	4	2	no	—
109	C₂²×D₁₃	C₄×C₈	1	4	yes	—
110	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×Dic₁₃	—	4	2	no	—
119	C₂×Dic₁₃	C₄×C₈	1	4	yes	—
120	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
121	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
122	C₄×D₁₃	C₄×C₈	1	8	yes	—
123	C₄×D₁₃	C₂×C₄²	1	4	yes	—
124	C₄×D₁₃	C₂²×C₈	1	4	yes	—
125	C₈×C₁₃	C₂×C₄²	1	2	yes	—
126	C₈×C₁₃	C₂²×C₈	1	2	yes	—
127	C₈×C₁₃	C₂²×C₈	1	2	yes	—
128	C₈×C₁₃	C₂³×C₄	1	2	yes	—
129	#1055	C₂×C₄²	1	4	yes	—
130	#1055	C₂²×C₈	1	4	yes	—
131	C₂²×M₁₆	—	13	1	no	—
132	C₂³×C₈	—	13	1	no	—
133	C₂×#117	—	13	1	no	—
134	C₂×M₃₂	—	13	2	no	—
135	C₂×M₃₂	—	13	16	no	—
136	C₂×M₃₂	—	13	4	no	—
137	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
138	C₂²×C₁₆	—	13	4	no	—
139	#405	—	13	2	no	—
140	#405	—	13	2	no	—
141	#405	—	13	2	no	—
142	#405	—	13	2	no	—
143	C₂×#97	—	13	1	no	—
144	C₂×#97	—	13	1	no	—
145	C₂×C₄×C₈	—	13	1	no	—
146	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
147	#346	—	13	4	no	—
148	#346	—	13	2	no	—
149	#346	C₂×D₁₃	1	2	yes	complemented
150	C₄×C₁₆	—	13	4	no	—
151	C₄×C₁₆	—	13	2	no	—
152	C₄×C₁₆	C₂×D₁₃	1	2	yes	complemented
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
155	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
156	C₂×D₁₃	—	2	4	no	—
157	C₂×D₁₃	—	4	4	no	—
158	C₂×D₁₃	C₂×C₄×C₈	1	4	yes	—
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	—	2	2	no	—
161	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
162	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
163	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
164	Dic₁₃	—	2	4	no	—
165	Dic₁₃	C₂×C₄×C₈	1	4	yes	—
166	C₂³×C₄	—	13	1	no	—
167	C₂×M₁₆	—	13	8	no	—
168	C₂×M₁₆	—	13	2	no	—
169	C₂×M₁₆	—	13	2	no	—
170	C₂²×C₈	—	13	1	no	—
171	C₂²×C₈	—	13	1	no	—
172	C₂²×C₈	—	13	2	no	—
173	C₂²×C₈	—	13	8	no	—
174	C₂²×C₈	—	13	2	no	—
175	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	#117	—	13	4	no	—
178	#117	—	13	2	no	—
179	#117	—	13	2	no	—
180	C₂×C₄⋊C₄	—	13	2	no	—
181	C₂×C₂²⋊C₄	—	13	2	no	—
182	C₂×C₄²	—	13	1	no	—
183	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
184	M₃₂	—	13	8	no	—
185	M₃₂	—	13	16	no	—
186	M₃₂	C₄×D₁₃	1	8	yes	—
187	C₂×C₁₆	—	13	4	no	—
188	C₂×C₁₆	—	13	16	no	—
189	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
190	#97	—	13	2	no	—
191	#97	—	13	2	no	—
192	#97	—	13	2	no	—
193	#97	—	13	2	no	—
194	C₄×C₈	—	13	2	no	—
195	C₄×C₈	—	13	2	no	—
196	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
197	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
198	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
199	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
200	C₂×C₁₃	—	2	2	no	—
201	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
202	C₂×C₁₃	#3407	1	2	yes	—
203	D₁₃	—	2	4	no	—
204	C₂⁴	—	13	1	no	—
205	C₂²×C₄	—	13	1	no	—
206	C₂²×C₄	—	13	4	no	—
207	C₂²×C₄	—	13	2	no	—
208	C₂²×C₄	—	13	4	no	—
209	C₂²×C₄	—	13	2	no	—
210	C₂²×C₄	—	13	2	no	—
211	C₂²×C₄	C₈×D₁₃	1	2	yes	—
212	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
213	M₁₆	—	13	8	no	—
214	M₁₆	—	13	8	no	—
215	C₂×C₈	—	13	4	no	—
216	C₂×C₈	—	13	4	no	—
217	C₂×C₈	—	13	2	no	—
218	C₂×C₈	—	13	2	no	—
219	C₂×C₈	—	13	8	no	—
220	C₂×C₈	—	13	4	no	—
221	C₂×C₈	—	13	4	no	—
222	C₂×C₈	C₈×D₁₃	1	4	yes	—
223	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
225	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
227	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
228	C₄⋊C₄	—	13	4	no	—
229	C₄⋊C₄	—	13	4	no	—
230	C₂²⋊C₄	—	13	4	no	—
231	C₂²⋊C₄	—	13	4	no	—
232	C₄²	—	13	2	no	—
233	C₄²	—	13	2	no	—
234	C₄²	C₈×D₁₃	1	2	yes	—
235	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
236	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
237	C₁₆	—	13	8	no	—
238	C₁₆	C₂×C₄×D₁₃	1	8	yes	—
239	C₁₃	#33324	1	1	yes	characteristic, complemented, Sylow
240	C₂³	—	13	2	no	—
241	C₂³	—	13	4	no	—
242	C₂³	—	52	2	no	—
243	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	13	2	no	—
246	C₂×C₄	—	13	4	no	—
247	C₂×C₄	—	13	4	no	—
248	C₂×C₄	—	13	4	no	—
249	C₂×C₄	—	13	8	no	—
250	C₂×C₄	—	13	4	no	—
251	C₂×C₄	—	52	2	no	—
252	C₂×C₄	C₄²×D₁₃	1	2	yes	—
253	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
254	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
255	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
256	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
258	C₈	—	13	4	no	—
259	C₈	—	13	4	no	—
260	C₈	C₄²×D₁₃	1	2	yes	—
261	C₈	C₂×C₈×D₁₃	1	2	yes	—
262	C₈	C₂×C₈×D₁₃	1	2	yes	—
263	C₈	C₂²×C₄×D₁₃	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	13	4	no	—
266	C₂²	—	26	4	no	—
267	C₂²	—	52	4	no	—
268	C₂²	C₄×C₈×D₁₃	1	2	yes	—
269	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	13	4	no	—
273	C₄	—	26	4	no	—
274	C₄	C₄×C₈×D₁₃	1	2	yes	—
275	C₄	C₂×C₄²×D₁₃	1	1	yes	characteristic
276	C₄	C₂²×C₈×D₁₃	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	26	4	no	—
279	C₂	C₂×C₄×C₈×D₁₃	1	1	yes	characteristic
280	C₂	#13127938	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator	C₂×C₁₃	C₂²×C₄×C₈
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#33324
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #13116434	C₂	semidirect
4. #13116431	C₂	semidirect
5. C₄×#88052	C₂	semidirect
6. M₃₂×Dic₁₃	C₂	semidirect
7. C₄×C₁₃×M₃₂	C₂	semidirect
17. D₁₃×M₃₂	C₄	semidirect
24. C₁₃×#346	C₂²	semidirect
25. C₄×C₁₃×C₁₆	C₂²	semidirect
26. #11718094	C₂²	semidirect
27. C₁₆×Dic₁₃	C₂²	semidirect
28. #11718053	C₂²	semidirect
29. C₄×#6090	C₂²	semidirect
62. #88038	C₂×C₄	semidirect
63. C₁₆×D₁₃	C₂×C₄	semidirect
106. C₄×M₃₂	D₁₃	semidirect
149. #346	C₂×D₁₃	semidirect
152. C₄×C₁₆	C₂×D₁₃	semidirect
239. C₁₃	#33324	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₃×M₃₂	2
C₂×#11718350	2
#13116434	2
#13116431	2
C₄×#88052	2
M₃₂×Dic₁₃	2
C₄×C₁₃×M₃₂	2
#33324	13