Subgroups — #15777965

Order 3328.

490 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718783	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718145	C₂	1	1	yes	maximal, characteristic
3	M₃₂×Dic₁₃	C₂	1	2	yes	maximal, complemented
4	#13112919	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#349	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
7	C₂×#88180	C₄	1	1	yes	characteristic
8	C₂³×#1055	C₄	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
10	#11718783	—	2	2	no	—
11	#11718783	—	2	2	no	—
12	C₂×#88055	C₄	1	1	yes	characteristic
13	C₂×#88055	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88052	C₂²	1	2	yes	—
17	C₁₃×#349	C₄	1	4	yes	complemented
18	#11718145	C₄	1	4	yes	complemented
19	#11718094	—	2	2	no	—
20	C₁₆×Dic₁₃	—	2	2	no	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊Dic₁₃	C₈	1	2	yes	—
29	#88180	—	2	2	no	—
30	#88180	—	2	2	no	—
31	C₂×C₄⋊Dic₁₃	C₈	1	2	yes	—
32	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×#1055	—	2	1	no	—
35	C₂²×#1055	—	2	1	no	—
36	C₂²×#1055	—	2	2	no	—
37	C₂²×#1055	C₈	1	4	yes	—
38	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
39	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
40	C₁₃×M₃₂	—	2	2	no	—
41	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
42	C₂×C₁₃×C₁₆	—	2	2	no	—
43	#88055	—	2	2	no	—
44	#88055	—	2	2	no	—
45	C₈×Dic₁₃	—	2	1	no	—
46	C₈×Dic₁₃	—	2	1	no	—
47	C₈×Dic₁₃	D₄	1	2	yes	—
48	C₈×Dic₁₃	D₄	1	2	yes	—
49	#88052	—	2	2	no	—
50	#88052	C₂×C₄	1	4	yes	—
51	C₂×#6090	—	2	2	no	—
52	#12602	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₃	—	2	1	no	—
58	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
63	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
64	C₁₃×M₁₆	—	2	4	no	—
65	C₁₃×M₁₆	—	2	4	no	—
66	C₂×C₈×C₁₃	—	2	1	no	—
67	C₂×C₈×C₁₃	—	2	1	no	—
68	C₂×C₈×C₁₃	C₄²	1	2	yes	—
69	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₃	#41	1	2	yes	—
73	C₂²⋊Dic₁₃	—	2	4	no	—
74	C₄⋊Dic₁₃	—	2	4	no	—
75	C₄×Dic₁₃	—	2	1	no	—
76	C₄×Dic₁₃	—	2	1	no	—
77	C₄×Dic₁₃	C₂²⋊C₄	1	2	yes	—
78	C₄×Dic₁₃	M₁₆	1	2	yes	—
79	C₂×#1055	—	2	1	no	—
80	C₂×#1055	—	2	1	no	—
81	C₂×#1055	—	2	4	no	—
82	C₂×#1055	—	2	4	no	—
83	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×#1055	C₂×C₈	1	2	yes	—
86	C₂×#1055	C₂×C₈	1	2	yes	—
87	C₂×#1055	M₁₆	1	2	yes	—
88	C₁₃×C₁₆	—	2	4	no	—
89	#6090	—	2	4	no	—
90	C₂×#405	—	13	1	no	—
91	C₂×#349	—	13	1	no	—
92	C₂×#349	D₁₃	1	1	yes	characteristic
93	C₄×M₃₂	—	13	2	no	—
94	#1377	—	13	2	no	—
95	C₂³×C₁₃	—	2	1	no	—
96	C₂³×C₁₃	—	2	1	no	—
97	C₂³×C₁₃	—	4	2	no	—
98	C₂³×C₁₃	C₄×C₈	1	2	yes	—
99	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
113	C₂×Dic₁₃	—	2	2	no	—
114	C₂×Dic₁₃	—	4	2	no	—
115	C₂×Dic₁₃	—	4	2	no	—
116	C₂×Dic₁₃	C₂²⋊C₈	1	4	yes	—
117	C₈×C₁₃	—	2	2	no	—
118	C₈×C₁₃	C₄×D₄	1	2	yes	—
119	C₈×C₁₃	C₄×D₄	1	2	yes	—
120	#1055	—	2	2	no	—
121	#1055	C₂²⋊C₈	1	2	yes	—
122	#1055	C₂²⋊C₈	1	2	yes	—
123	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
124	C₂³×C₈	—	13	1	no	—
125	C₂×#117	—	13	1	no	—
126	C₂×M₃₂	—	13	2	no	—
127	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
128	#405	—	26	2	no	—
129	#405	—	26	2	no	—
130	C₂×#97	—	13	1	no	—
131	C₂×#97	—	13	1	no	—
132	C₂×C₄×C₈	—	13	1	no	—
133	C₂×C₄×C₈	—	13	1	no	—
134	#349	—	13	4	no	—
135	#349	Dic₁₃	1	4	yes	complemented
136	#346	—	26	2	no	—
137	C₄×C₁₆	—	26	2	no	—
138	C₂²×C₁₃	—	2	2	no	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	—	4	2	no	—
141	C₂²×C₁₃	—	4	2	no	—
142	C₂²×C₁₃	—	4	2	no	—
143	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₂²×C₁₃	#431	1	1	yes	characteristic
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
148	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
149	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₄×C₁₃	#431	1	1	yes	characteristic
151	Dic₁₃	—	2	4	no	—
152	Dic₁₃	—	4	2	no	—
153	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
157	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
158	C₂²×C₈	—	13	1	no	—
159	C₂²×C₈	—	13	4	no	—
160	C₂²×C₈	—	13	1	no	—
161	C₂²×C₈	—	26	2	no	—
162	C₂²×C₈	—	26	1	no	—
163	C₂²×C₈	—	26	1	no	—
164	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
166	#117	—	26	2	no	—
167	#117	—	26	2	no	—
168	C₂×C₄⋊C₄	—	13	2	no	—
169	C₂×C₂²⋊C₄	—	13	2	no	—
170	C₂×C₄²	—	13	1	no	—
171	C₂×C₄²	—	13	1	no	—
172	M₃₂	—	2	2	no	—
173	M₃₂	—	13	4	no	—
174	M₃₂	—	26	2	no	—
175	M₃₂	C₂×Dic₁₃	1	4	yes	—
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	26	2	no	—
178	#97	—	26	2	no	—
179	#97	—	26	2	no	—
180	C₄×C₈	—	13	2	no	—
181	C₄×C₈	—	13	2	no	—
182	C₄×C₈	—	26	1	no	—
183	C₄×C₈	—	26	1	no	—
184	C₂×C₁₃	—	2	1	no	—
185	C₂×C₁₃	—	2	1	no	—
186	C₂×C₁₃	—	4	2	no	—
187	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
188	C₂×C₁₃	#2115	1	2	yes	—
189	C₂⁴	C₈×D₁₃	1	1	yes	characteristic
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	13	2	no	—
195	C₂²×C₄	—	13	2	no	—
196	C₂²×C₄	C₈×D₁₃	1	2	yes	—
197	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
198	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
199	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	13	1	no	—
205	C₂×C₈	—	13	1	no	—
206	C₂×C₈	—	13	2	no	—
207	C₂×C₈	—	13	2	no	—
208	C₂×C₈	—	13	2	no	—
209	C₂×C₈	—	26	4	no	—
210	C₂×C₈	—	26	4	no	—
211	C₂×C₈	—	26	1	no	—
212	C₂×C₈	—	26	1	no	—
213	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
214	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
215	C₂×C₈	D₄×D₁₃	1	2	yes	—
216	C₂×C₈	#6129	1	2	yes	—
217	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
218	C₄⋊C₄	—	26	4	no	—
219	C₂²⋊C₄	—	26	4	no	—
220	C₄²	—	13	2	no	—
221	C₄²	—	13	2	no	—
222	C₄²	—	26	1	no	—
223	C₄²	—	26	1	no	—
224	C₁₆	—	2	4	no	—
225	C₁₆	—	26	4	no	—
226	C₁₃	#12602	1	1	yes	characteristic, complemented, Sylow
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	4	2	no	—
230	C₂³	C₈×Dic₁₃	1	2	yes	—
231	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	13	4	no	—
239	C₂×C₄	—	26	2	no	—
240	C₂×C₄	—	52	2	no	—
241	C₂×C₄	—	52	2	no	—
242	C₂×C₄	C₈×Dic₁₃	1	2	yes	—
243	C₂×C₄	#88131	1	1	yes	characteristic
244	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
245	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
246	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
247	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	13	2	no	—
251	C₈	—	13	2	no	—
252	C₈	—	26	2	no	—
253	C₈	#88135	1	2	yes	—
254	C₈	D₄×Dic₁₃	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	#11718379	1	1	yes	characteristic
261	C₂²	D₁₃×C₂²⋊C₈	1	1	yes	characteristic
262	C₂²	C₂×C₈×Dic₁₃	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	26	4	no	—
265	C₄	—	52	2	no	—
266	C₄	#11718379	1	1	yes	characteristic
267	C₄	D₁₃×C₂²⋊C₈	1	1	yes	characteristic
268	C₄	C₂²⋊C₄×Dic₁₃	1	1	yes	characteristic
269	C₄	M₁₆×Dic₁₃	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	4	2	no	—
273	C₂	C₂²⋊C₈×Dic₁₃	1	1	yes	characteristic
274	C₂	#13116455	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₁₃
commutator	C₂²×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#349	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#349	D₁₃
hypocenter	C₁₃	#12602
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
3. M₃₂×Dic₁₃	C₂	semidirect
4. #13112919	C₂	semidirect
17. C₁₃×#349	C₄	semidirect
18. #11718145	C₄	semidirect
135. #349	Dic₁₃	semidirect
226. C₁₃	#12602	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718783	2
C₂×#11718145	2
M₃₂×Dic₁₃	2
#13112919	2
C₂×C₁₃×#349	2
#12602	13