Subgroups — #15755808

Order 3328.

372 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	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718049	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113223	C₂	1	2	yes	maximal
4	#13112687	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2922	C₂	1	1	yes	maximal, characteristic
6	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
7	C₁₃×#433	—	2	1	no	—
8	C₁₃×#433	—	2	1	no	—
9	C₁₃×#431	—	2	2	no	—
10	C₁₃×#424	C₄	1	2	yes	—
11	C₁₃×#408	C₄	1	2	yes	—
12	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718586	C₂²	1	2	yes	—
15	C₂×C₄⋊#1055	C₂²	1	2	yes	—
16	Q₈⋊#1055	—	2	2	no	—
17	D₄⋊#1055	—	2	2	no	—
18	#11718050	—	2	1	no	—
19	#11718049	—	2	2	no	—
20	#11718049	—	2	1	no	—
21	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
25	C₄×Q₈×C₁₃	Q₈	1	1	yes	characteristic
26	C₄×D₄×C₁₃	—	2	1	no	—
27	C₄×D₄×C₁₃	—	2	1	no	—
28	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
29	C₄×D₄×C₁₃	Q₈	1	1	yes	characteristic
30	C₁₃×#117	—	2	1	no	—
31	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	—	2	1	no	—
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×#1055	—	2	2	no	—
35	C₁₃×C₄⋊C₈	—	2	2	no	—
36	C₁₃×C₄⋊C₈	—	2	2	no	—
37	C₁₃×C₂²⋊C₈	—	2	2	no	—
38	C₁₃×C₂²⋊C₈	—	2	2	no	—
39	C₁₃×#97	—	2	1	no	—
40	C₄×C₈×C₁₃	—	2	1	no	—
41	C₄×C₈×C₁₃	D₄	1	2	yes	—
42	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₂²⋊#1055	—	2	2	no	—
45	C₄⋊#1055	—	2	2	no	—
46	C₄⋊#1055	—	2	2	no	—
47	C₄⋊#1055	—	2	2	no	—
48	C₄×#1055	—	2	2	no	—
49	#8607	—	13	1	no	maximal, Sylow
50	C₁₃×#41	—	2	4	no	—
51	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×D₄×C₁₃	—	2	1	no	—
53	C₂×D₄×C₁₃	C₄⋊C₄	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	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
60	C₁₃×M₁₆	—	2	4	no	—
61	C₂×C₈×C₁₃	—	2	4	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
65	C₁₃×C₄⋊C₄	—	2	1	no	—
66	C₁₃×C₄⋊C₄	—	2	1	no	—
67	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
69	C₁₃×C₂²⋊C₄	—	2	1	no	—
70	C₁₃×C₂²⋊C₄	—	2	1	no	—
71	C₁₃×C₂²⋊C₄	—	2	1	no	—
72	C₄²×C₁₃	—	2	1	no	—
73	C₄²×C₁₃	—	2	1	no	—
74	C₄²×C₁₃	—	2	1	no	—
75	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
79	C₂×#1055	—	4	2	no	—
80	C₂×#1055	—	4	2	no	—
81	C₂×#1055	—	8	2	no	—
82	#2922	D₁₃	1	1	yes	characteristic
83	#1564	—	13	1	no	—
84	C₂×#334	—	13	1	no	—
85	#1477	—	13	2	no	—
86	#1283	—	13	2	no	—
87	C₂³×C₁₃	—	2	1	no	—
88	C₂³×C₁₃	#117	1	1	yes	characteristic
89	Q₈×C₁₃	—	2	2	no	—
90	D₄×C₁₃	—	2	2	no	—
91	D₄×C₁₃	—	4	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	4	1	no	—
97	C₂×C₄×C₁₃	—	4	2	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
103	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#131	1	4	yes	—
107	C₈×C₁₃	—	2	2	no	—
108	C₈×C₁₃	—	2	2	no	—
109	C₈×C₁₃	—	4	2	no	—
110	#1055	—	8	2	no	—
111	#1055	—	8	2	no	—
112	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
113	#433	—	2	1	no	—
114	#433	—	2	1	no	—
115	#431	—	2	2	no	—
116	#431	—	13	2	no	—
117	#424	Dic₁₃	1	2	yes	—
118	C₂×C₄⋊C₈	—	13	2	no	—
119	#408	Dic₁₃	1	2	yes	—
120	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
122	#335	—	26	1	no	—
123	#334	—	26	2	no	—
124	#334	—	26	1	no	—
125	Q₈⋊C₈	—	26	2	no	—
126	D₄⋊C₈	—	26	2	no	—
127	C₂²×C₁₃	—	2	1	no	—
128	C₂²×C₁₃	—	2	1	no	—
129	C₂²×C₁₃	—	4	1	no	—
130	C₂²×C₁₃	—	4	1	no	—
131	C₂²×C₁₃	—	4	1	no	—
132	C₂²×C₁₃	#380	1	1	yes	characteristic
133	C₂²×C₁₃	#433	1	2	yes	—
134	C₄×C₁₃	—	2	1	no	—
135	C₄×C₁₃	—	2	1	no	—
136	C₄×C₁₃	—	2	1	no	—
137	C₄×C₁₃	—	4	1	no	—
138	C₄×C₁₃	—	4	1	no	—
139	C₄×C₁₃	—	4	1	no	—
140	C₄×C₁₃	#408	1	2	yes	—
141	C₄×C₁₃	#424	1	2	yes	—
142	C₄×C₁₃	#471	1	2	yes	—
143	C₂×#41	C₂×Dic₁₃	1	1	yes	characteristic
144	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
145	C₂²×C₈	—	26	2	no	—
146	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
147	C₄×Q₈	Dic₂₆	1	1	yes	characteristic
148	C₄×Q₈	D₅₂	1	1	yes	characteristic
149	C₄×D₄	—	2	1	no	—
150	C₄×D₄	—	2	1	no	—
151	C₄×D₄	Dic₂₆	1	1	yes	characteristic
152	C₄×D₄	D₅₂	1	1	yes	characteristic
153	#117	—	2	1	no	—
154	#117	C₂×Dic₁₃	1	1	yes	characteristic
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	26	2	no	—
160	C₄⋊C₈	—	26	2	no	—
161	C₄⋊C₈	—	26	2	no	—
162	C₂²⋊C₈	—	2	2	no	—
163	C₂²⋊C₈	—	2	2	no	—
164	C₂²⋊C₈	—	26	2	no	—
165	#97	—	2	1	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	—	26	2	no	—
168	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
169	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
170	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
171	C₂×C₁₃	—	2	2	no	—
172	C₂×C₁₃	—	8	1	no	—
173	C₂×C₁₃	#1798	1	1	yes	characteristic
174	C₂×C₁₃	#1814	1	1	yes	characteristic
175	C₂×C₁₃	#1815	1	1	yes	characteristic
176	#41	—	2	4	no	—
177	C₂×Q₈	C₄⋊Dic₁₃	1	1	yes	characteristic
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	C₄⋊Dic₁₃	1	1	yes	characteristic
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	#6127	1	1	yes	characteristic
184	C₂²×C₄	#6127	1	1	yes	characteristic
185	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
186	M₁₆	—	2	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	52	2	no	—
190	C₂×C₈	—	52	2	no	—
191	C₂×C₈	—	104	2	no	—
192	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
193	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
197	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
198	C₂²⋊C₄	—	2	1	no	—
199	C₂²⋊C₄	—	2	1	no	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
205	C₄²	C₂×D₅₂	1	1	yes	characteristic
206	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
207	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
208	C₁₃	#8607	1	1	yes	characteristic, complemented, Sylow
209	C₂³	—	2	1	no	—
210	C₂³	#88180	1	1	yes	characteristic
211	Q₈	—	2	2	no	—
212	D₄	—	2	2	no	—
213	D₄	—	4	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	#88178	1	1	yes	characteristic
223	C₂×C₄	#88178	1	1	yes	characteristic
224	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
225	C₂×C₄	#88184	1	1	yes	characteristic
226	C₂×C₄	#88184	1	1	yes	characteristic
227	C₂×C₄	#88202	1	4	yes	—
228	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	—	2	2	no	—
231	C₈	—	4	2	no	—
232	C₈	—	104	2	no	—
233	C₈	—	104	2	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	1	no	—
239	C₂²	#11718661	1	2	yes	—
240	C₂²	#11718873	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	1	no	—
246	C₄	—	4	1	no	—
247	C₄	#11718660	1	2	yes	—
248	C₄	#11718802	1	2	yes	—
249	C₄	#11718895	1	2	yes	—
250	C₂	—	2	2	no	—
251	C₂	—	8	1	no	—
252	C₂	#13114820	1	1	yes	characteristic
253	C₂	#13114890	1	1	yes	characteristic
254	C₂	#13114895	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88184
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2922	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2922	D₁₃
hypocenter	C₁₃	#8607
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2922

Complements

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

N	H ≅ G/N	type
1. #13113584	C₂	semidirect
2. C₂×#11718049	C₂	semidirect
4. #13112687	C₂	semidirect
208. C₁₃	#8607	semidirect

Maximal subgroups

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

subgroup	index
#13113584	2
C₂×#11718049	2
#13113223	2
#13112687	2
C₁₃×#2922	2
#8607	13