Subgroups — #15777950

Order 3328.

590 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#6090	C₂	1	2	yes	maximal
2	C₂×#11718053	C₂	1	2	yes	maximal, complemented
3	#13112914	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	—	2	4	no	—
9	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×#6090	C₂²	1	4	yes	—
14	C₂²⋊#6090	C₄	1	8	yes	complemented
15	#11718053	—	2	4	no	—
16	C₂³×C₄×C₁₃	C₈	1	2	yes	—
17	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₃	—	2	2	no	—
20	C₂²×C₈×C₁₃	—	2	2	no	—
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂²×C₈×C₁₃	C₈	1	4	yes	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	—	2	2	no	—
29	C₂×C₄²×C₁₃	—	2	2	no	—
30	C₂×C₄²×C₁₃	C₈	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₃	—	2	2	no	—
34	C₄×C₈×C₁₃	—	2	2	no	—
35	C₄×C₈×C₁₃	—	2	2	no	—
36	C₄×C₈×C₁₃	D₄	1	4	yes	—
37	C₂×#6090	—	2	4	no	—
38	C₂×#6090	—	2	4	no	—
39	C₂×#6090	C₂×C₄	1	8	yes	complemented
40	#12599	—	13	1	no	maximal, Sylow
41	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	—	2	4	no	—
43	C₂²×C₄×C₁₃	—	2	4	no	—
44	C₂²×C₄×C₁₃	—	2	4	no	—
45	C₂²×C₄×C₁₃	—	2	1	no	—
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₃	—	2	1	no	—
56	C₂×C₈×C₁₃	—	2	1	no	—
57	C₂×C₈×C₁₃	—	2	1	no	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	—	2	8	no	—
60	C₂×C₈×C₁₃	—	2	8	no	—
61	C₂×C₈×C₁₃	C₄²	1	2	yes	—
62	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
66	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₁₃	#41	1	2	yes	—
69	C₄²×C₁₃	—	2	4	no	—
70	C₄²×C₁₃	—	2	1	no	—
71	C₄²×C₁₃	—	2	1	no	—
72	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₁₃	M₁₆	1	2	yes	—
74	#6090	—	2	8	no	—
75	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
76	C₂×C₂²⋊C₁₆	—	13	2	no	—
77	C₂×#346	—	13	2	no	—
78	#1374	—	13	2	no	—
79	C₂³×C₁₃	—	2	1	no	—
80	C₂³×C₁₃	—	2	1	no	—
81	C₂³×C₁₃	—	2	2	no	—
82	C₂³×C₁₃	C₄×C₈	1	2	yes	—
83	C₂³×C₁₃	M₃₂	1	4	yes	—
84	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	—	2	16	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
97	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
98	C₂×C₄×C₁₃	M₃₂	1	4	yes	—
99	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
105	C₈×C₁₃	—	2	2	no	—
106	C₈×C₁₃	—	2	2	no	—
107	C₈×C₁₃	C₂²⋊C₈	1	4	yes	—
108	C₈×C₁₃	C₄×D₄	1	4	yes	—
109	C₂³×C₈	Dic₁₃	1	1	yes	characteristic
110	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
111	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
112	C₂²×C₁₆	—	13	4	no	—
113	C₂×C₄×C₈	—	2	4	no	—
114	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
115	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
118	C₂²⋊C₁₆	—	13	8	no	—
119	#346	—	26	4	no	—
120	C₂²×C₁₃	—	2	2	no	—
121	C₂²×C₁₃	—	2	4	no	—
122	C₂²×C₁₃	—	2	4	no	—
123	C₂²×C₁₃	—	2	2	no	—
124	C₂²×C₁₃	#346	1	4	yes	complemented
125	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
126	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
127	C₂²×C₁₃	#431	1	1	yes	characteristic
128	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
129	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
130	C₄×C₁₃	—	2	4	no	—
131	C₄×C₁₃	—	2	4	no	—
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
135	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
136	C₄×C₁₃	#431	1	1	yes	characteristic
137	C₂³×C₄	#1055	1	2	yes	—
138	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	2	2	no	—
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	#1055	1	4	yes	—
144	C₂²×C₈	C₄×D₁₃	1	4	yes	—
145	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
146	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
147	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
149	C₂×C₄²	—	2	2	no	—
150	C₂×C₄²	—	2	2	no	—
151	C₂×C₄²	#1055	1	2	yes	—
152	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
153	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
154	C₂×C₁₆	—	13	8	no	—
155	C₂×C₁₆	—	26	4	no	—
156	C₂×C₁₆	—	26	4	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	C₂²⋊D₁₃	1	4	yes	—
161	C₂×C₁₃	—	2	4	no	—
162	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
163	C₂×C₁₃	C₂×#346	1	1	yes	characteristic
164	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
165	C₂×C₁₃	#2112	1	2	yes	—
166	C₂×C₁₃	#2163	1	2	yes	—
167	C₂⁴	C₂×#1055	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	C₂×#1055	1	2	yes	—
176	C₂²×C₄	C₂×#1055	1	2	yes	—
177	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
178	C₂²×C₄	C₂×#1055	1	2	yes	—
179	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
180	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	8	no	—
186	C₂×C₈	—	2	8	no	—
187	C₂×C₈	C₂×#1055	1	2	yes	—
188	C₂×C₈	C₂×#1055	1	2	yes	—
189	C₂×C₈	#6099	1	2	yes	—
190	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
191	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
192	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
193	C₂×C₈	#6127	1	2	yes	—
194	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
195	C₄²	—	2	4	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	#6099	1	2	yes	—
199	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
200	C₁₆	—	26	8	no	—
201	C₁₃	#12599	1	1	yes	characteristic, complemented, Sylow
202	C₂³	—	2	2	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	C₄×#1055	1	2	yes	—
206	C₂³	#88052	1	4	yes	—
207	C₂³	C₂²×#1055	1	1	yes	characteristic
208	C₂×C₄	—	2	16	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	C₄×#1055	1	2	yes	—
220	C₂×C₄	#88052	1	4	yes	—
221	C₂×C₄	C₂²⋊#1055	1	4	yes	—
222	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
223	C₂×C₄	C₂×#6099	1	1	yes	characteristic
224	C₂×C₄	C₂×#6099	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
226	C₂×C₄	#88180	1	1	yes	characteristic
227	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	C₂²⋊#1055	1	4	yes	—
231	C₈	C₄×C₂²⋊D₁₃	1	4	yes	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	4	no	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	2	no	—
236	C₂²	#11718053	1	4	yes	complemented
237	C₂²	C₂×C₄×#1055	1	1	yes	characteristic
238	C₂²	#11718586	1	1	yes	characteristic
239	C₂²	C₂×#88052	1	2	yes	—
240	C₂²	C₂×#88052	1	2	yes	—
241	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
242	C₄	—	2	4	no	—
243	C₄	—	2	4	no	—
244	C₄	—	2	2	no	—
245	C₄	C₄×#6099	1	1	yes	characteristic
246	C₄	#11718586	1	1	yes	characteristic
247	C₄	C₂×C₂²⋊#1055	1	1	yes	characteristic
248	C₄	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
249	C₂	—	2	4	no	—
250	C₂	C₄×C₂²⋊#1055	1	1	yes	characteristic
251	C₂	C₂×#11718053	1	1	yes	characteristic
252	C₂	C₄×#88052	1	1	yes	characteristic
253	C₂	#13116447	1	2	yes	—
254	C₂	#13116639	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	#12599
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11718053	C₂	semidirect
3. #13112914	C₂	semidirect
14. C₂²⋊#6090	C₄	semidirect
39. C₂×#6090	C₂×C₄	semidirect
124. C₂²×C₁₃	#346	semidirect
201. C₁₃	#12599	semidirect
236. C₂²	#11718053	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#6090	2
C₂×#11718053	2
#13112914	2
C₂²×C₄×C₈×C₁₃	2
#12599	13