Subgroups — #16015559

Order 3328.

826 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88052	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718762	C₂	1	1	yes	maximal, characteristic
3	#13116435	C₂	1	4	yes	maximal, complemented
4	#13116432	C₂	1	4	yes	maximal, complemented
5	M₃₂×Dic₁₃	C₂	1	4	yes	maximal, complemented
6	C₂²×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88180	C₄	1	1	yes	characteristic
9	C₂²×#6099	C₄	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₄	1	8	yes	complemented
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
13	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
14	#11718762	C₄	1	2	yes	—
15	#11718762	C₄	1	2	yes	—
16	#11718762	C₂²	1	2	yes	—
17	#11718762	C₂²	1	2	yes	—
18	C₂×#88055	C₄	1	2	yes	—
19	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
20	C₂×#88052	C₄	1	8	yes	complemented
21	C₂×#88052	C₂²	1	2	yes	—
22	C₂×#88052	C₂²	1	2	yes	—
23	C₂²×#6090	C₂²	1	2	yes	—
24	#11718094	C₂²	1	8	yes	complemented
25	C₁₆×Dic₁₃	C₂²	1	8	yes	complemented
26	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₁₃	C₈	1	2	yes	—
36	#88180	C₈	1	2	yes	—
37	#88180	C₈	1	2	yes	—
38	#88180	C₂×C₄	1	2	yes	—
39	#88180	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊Dic₁₃	C₈	1	2	yes	—
41	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
42	C₂×#6099	C₈	1	8	yes	—
43	C₂×#6099	C₂×C₄	1	2	yes	—
44	C₂×#6099	C₂×C₄	1	2	yes	—
45	C₂²×#1055	C₂×C₄	1	2	yes	—
46	C₁₃×M₃₂	C₂×C₄	1	8	yes	—
47	C₁₃×M₃₂	C₂×C₄	1	8	yes	complemented
48	C₂×C₁₃×C₁₆	C₂×C₄	1	8	yes	complemented
49	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
50	#88055	C₂×C₄	1	4	yes	—
51	#88055	C₂×C₄	1	4	yes	—
52	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
53	C₈×Dic₁₃	C₂³	1	4	yes	—
54	#88052	C₂×C₄	1	8	yes	—
55	#88052	C₂×C₄	1	8	yes	complemented
56	C₂×#6090	C₂×C₄	1	8	yes	complemented
57	C₂×#6090	C₂³	1	4	yes	—
58	#33324	—	13	1	no	maximal, Sylow
59	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
61	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×Dic₁₃	C₂×C₈	1	4	yes	—
69	C₂×C₈×C₁₃	C₄²	1	4	yes	—
70	C₂×C₈×C₁₃	C₄²	1	4	yes	—
71	C₂×C₈×C₁₃	C₄²	1	4	yes	—
72	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂²⋊Dic₁₃	C₂×C₈	1	4	yes	—
83	C₂²⋊Dic₁₃	C₂×C₈	1	4	yes	—
84	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
85	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
86	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
87	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
88	#6099	C₂×C₈	1	8	yes	—
89	#6099	C₂×C₈	1	8	yes	—
90	C₂×#1055	C₂×C₈	1	8	yes	—
91	C₂×#1055	C₂²×C₄	1	4	yes	—
92	C₁₃×C₁₆	C₂²×C₄	1	8	yes	—
93	#6090	C₂²×C₄	1	8	yes	—
94	C₂²×M₃₂	—	13	1	no	—
95	C₂²×M₃₂	D₁₃	1	1	yes	characteristic
96	C₂×#405	—	13	1	no	—
97	#2108	—	13	4	no	—
98	#2108	—	13	4	no	—
99	C₄×M₃₂	—	13	4	no	—
100	C₂³×C₁₃	—	4	2	no	—
101	C₂³×C₁₃	C₄×C₈	1	4	yes	—
102	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
103	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
104	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
106	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
107	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
108	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×Dic₁₃	—	4	4	no	—
119	C₂×Dic₁₃	C₂²×C₈	1	8	yes	—
120	C₈×C₁₃	C₂×C₄²	1	2	yes	—
121	C₈×C₁₃	C₂×C₄²	1	2	yes	—
122	C₈×C₁₃	C₂×C₄²	1	2	yes	—
123	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
125	#1055	C₂²×C₈	1	8	yes	—
126	C₂²×M₁₆	—	13	1	no	—
127	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
128	C₂×#117	—	13	1	no	—
129	C₂×M₃₂	—	13	8	no	—
130	C₂×M₃₂	—	13	2	no	—
131	C₂×M₃₂	—	13	2	no	—
132	C₂×M₃₂	Dic₁₃	1	8	yes	complemented
133	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
134	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
135	C₂²×C₁₆	—	13	2	no	—
136	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
137	#405	—	13	2	no	—
138	#405	—	13	2	no	—
139	#405	—	13	2	no	—
140	#405	—	13	2	no	—
141	C₂×#97	—	13	2	no	—
142	C₂×C₄×C₈	—	13	2	no	—
143	#346	—	13	8	no	—
144	C₄×C₁₆	—	13	8	no	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	—	2	2	no	—
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	—	4	4	no	—
149	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
150	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
151	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
152	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
153	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
154	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
155	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
156	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
157	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
158	Dic₁₃	—	2	8	no	—
159	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
160	C₂×M₁₆	—	13	8	no	—
161	C₂×M₁₆	—	13	2	no	—
162	C₂×M₁₆	—	13	2	no	—
163	C₂²×C₈	—	13	2	no	—
164	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
165	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
166	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
167	C₂²×C₈	C₄×D₁₃	1	4	yes	—
168	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
169	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
172	#117	—	13	2	no	—
173	#117	—	13	2	no	—
174	#117	—	13	2	no	—
175	#117	—	13	2	no	—
176	C₂×C₄⋊C₄	—	13	2	no	—
177	C₂×C₂²⋊C₄	—	13	2	no	—
178	C₂×C₄²	—	13	2	no	—
179	M₃₂	—	13	8	no	—
180	M₃₂	—	13	8	no	—
181	M₃₂	C₂×Dic₁₃	1	8	yes	—
182	M₃₂	C₂×Dic₁₃	1	8	yes	complemented
183	C₂×C₁₆	—	13	8	no	—
184	C₂×C₁₆	—	13	4	no	—
185	C₂×C₁₆	C₂×Dic₁₃	1	8	yes	complemented
186	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
187	#97	—	13	4	no	—
188	#97	—	13	4	no	—
189	C₄×C₈	—	13	4	no	—
190	C₄×C₈	—	13	4	no	—
191	C₂×C₁₃	—	2	2	no	—
192	C₂×C₁₃	—	2	2	no	—
193	C₂×C₁₃	—	2	2	no	—
194	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
195	C₂×C₁₃	#3407	1	2	yes	—
196	C₂⁴	C₈×D₁₃	1	1	yes	characteristic
197	C₂²×C₄	—	13	4	no	—
198	C₂²×C₄	C₈×D₁₃	1	4	yes	—
199	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
203	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
206	M₁₆	—	13	8	no	—
207	M₁₆	—	13	8	no	—
208	C₂×C₈	—	13	8	no	—
209	C₂×C₈	—	13	4	no	—
210	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
211	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
212	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
213	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
214	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
215	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
216	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
217	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
218	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
219	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
220	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
221	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
222	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
223	C₄⋊C₄	—	13	4	no	—
224	C₄⋊C₄	—	13	4	no	—
225	C₂²⋊C₄	—	13	4	no	—
226	C₂²⋊C₄	—	13	4	no	—
227	C₄²	—	13	4	no	—
228	C₄²	—	13	4	no	—
229	C₁₆	—	13	8	no	—
230	C₁₆	C₂²×Dic₁₃	1	8	yes	—
231	C₁₃	#33324	1	1	yes	characteristic, complemented, Sylow
232	C₂³	—	4	2	no	—
233	C₂³	C₈×Dic₁₃	1	4	yes	—
234	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
235	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
236	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
237	C₂×C₄	—	13	8	no	—
238	C₂×C₄	—	52	4	no	—
239	C₂×C₄	C₈×Dic₁₃	1	4	yes	—
240	C₂×C₄	C₈×Dic₁₃	1	4	yes	—
241	C₂×C₄	C₈×Dic₁₃	1	4	yes	—
242	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
243	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
244	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
245	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
246	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
247	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
249	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
250	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
251	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
252	C₈	—	13	8	no	—
253	C₈	C₂×C₄×Dic₁₃	1	2	yes	—
254	C₈	C₂×C₄×Dic₁₃	1	2	yes	—
255	C₈	C₂×C₄×Dic₁₃	1	2	yes	—
256	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
257	C₈	C₂³×Dic₁₃	1	1	yes	characteristic
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₂×C₈×Dic₁₃	1	2	yes	—
263	C₂²	C₂×C₈×Dic₁₃	1	2	yes	—
264	C₂²	C₂×C₈×Dic₁₃	1	2	yes	—
265	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
266	C₄	—	26	8	no	—
267	C₄	C₂×C₈×Dic₁₃	1	2	yes	—
268	C₄	C₂×C₈×Dic₁₃	1	2	yes	—
269	C₄	C₂×C₈×Dic₁₃	1	2	yes	—
270	C₄	C₂²×C₈×D₁₃	1	1	yes	characteristic
271	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
272	C₂	—	2	2	no	—
273	C₂	—	2	2	no	—
274	C₂	—	2	2	no	—
275	C₂	C₂²×C₈×Dic₁₃	1	1	yes	characteristic
276	C₂	#13127938	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #13116435	C₂	semidirect
4. #13116432	C₂	semidirect
5. M₃₂×Dic₁₃	C₂	semidirect
10. C₂×C₁₃×M₃₂	C₄	semidirect
20. C₂×#88052	C₄	semidirect
24. #11718094	C₂²	semidirect
25. C₁₆×Dic₁₃	C₂²	semidirect
47. C₁₃×M₃₂	C₂×C₄	semidirect
48. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
55. #88052	C₂×C₄	semidirect
56. C₂×#6090	C₂×C₄	semidirect
132. C₂×M₃₂	Dic₁₃	semidirect
182. M₃₂	C₂×Dic₁₃	semidirect
185. C₂×C₁₆	C₂×Dic₁₃	semidirect
231. C₁₃	#33324	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88052	2
C₂×#11718762	2
#13116435	2
#13116432	2
M₃₂×Dic₁₃	2
C₂²×C₁₃×M₃₂	2
#33324	13