Subgroups — #15873412

Order 3328.

770 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×#11718587	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
3	#13115560	C₂	1	4	yes	maximal, complemented
4	#13115021	C₂	1	4	yes	maximal
5	#13114990	C₂	1	2	yes	maximal, complemented
6	#13114956	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2283	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	2	yes	—
10	C₁₃×#383	C₄	1	2	yes	—
11	C₁₃×#382	C₄	1	2	yes	—
12	C₁₃×#381	C₄	1	2	yes	—
13	C₁₃×#380	C₄	1	2	yes	—
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₄×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
16	C₁₃×#376	C₂²	1	2	yes	—
17	C₂×C₂²⋊#1055	C₂²	1	2	yes	—
18	#11718587	—	2	4	no	—
19	#11718585	—	2	4	no	—
20	C₂×C₄⋊#1055	C₂²	1	2	yes	—
21	C₂×C₄⋊#1055	C₂²	1	2	yes	—
22	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#1055	C₂²	1	8	yes	—
25	C₂×#88043	C₂²	1	2	yes	—
26	#11718116	C₂²	1	8	yes	—
27	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	D₄	1	4	yes	—
35	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×#6099	—	2	4	no	—
40	C₂×#6099	—	4	4	no	—
41	C₂²×#1055	C₂³	1	4	yes	—
42	C₂²×#1055	C₂³	1	2	yes	—
43	C₂²×#1055	C₂³	1	2	yes	—
44	C₁₃×#95	C₂×C₄	1	4	yes	—
45	C₁₃×#95	C₂×C₄	1	4	yes	—
46	C₁₃×#95	C₂×C₄	1	4	yes	—
47	C₁₃×#95	C₂×C₄	1	4	yes	—
48	C₂²⋊#1055	—	2	4	no	—
49	C₄⋊#1055	—	2	4	no	—
50	C₄⋊#1055	—	2	4	no	—
51	C₄⋊#1055	—	2	2	no	—
52	C₄⋊#1055	—	2	2	no	—
53	C₄⋊#1055	—	4	8	no	—
54	#88043	—	2	4	no	—
55	#21979	—	13	1	no	maximal, Sylow
56	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	—	4	2	no	—
60	C₂²×C₄×C₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
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×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₁₃×C₄⋊C₄	—	4	4	no	—
73	C₁₃×C₂²⋊C₄	—	4	4	no	—
74	C₄²×C₁₃	—	2	4	no	—
75	C₄²×C₁₃	—	2	4	no	—
76	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
77	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
78	C₄²×C₁₃	C₂×D₄	1	2	yes	—
79	C₄²×C₁₃	C₂×D₄	1	2	yes	—
80	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
81	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
82	#6099	—	4	8	no	—
83	C₂×#1055	—	2	2	no	—
84	C₂×#1055	—	2	2	no	—
85	C₂×#1055	—	2	4	no	—
86	C₂×#1055	—	2	8	no	—
87	C₂×#1055	—	2	4	no	—
88	C₂×#1055	—	2	4	no	—
89	C₂×#432	—	13	1	no	—
90	C₂×#423	—	13	1	no	—
91	#2283	D₁₃	1	1	yes	characteristic
92	#1940	—	13	4	no	—
93	#1851	—	13	4	no	—
94	#1843	—	13	2	no	—
95	#1832	—	13	2	no	—
96	C₂³×C₁₃	—	2	2	no	—
97	C₂³×C₁₃	—	2	2	no	—
98	C₂³×C₁₃	—	2	2	no	—
99	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	—	2	2	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	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	2	8	no	—
110	C₂×C₄×C₁₃	—	2	8	no	—
111	C₂×C₄×C₁₃	—	2	8	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	—	4	4	no	—
114	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
122	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
123	#1055	—	4	2	no	—
124	#1055	—	4	4	no	—
125	#1055	—	4	2	no	—
126	C₂²×M₁₆	—	13	2	no	—
127	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
128	#432	—	26	4	no	—
129	#423	—	26	4	no	—
130	C₂×C₄⋊C₈	—	13	1	no	—
131	C₂×C₄⋊C₈	—	13	2	no	—
132	C₂×C₄⋊C₈	—	13	1	no	—
133	C₂×C₄⋊C₈	—	13	2	no	—
134	C₂×C₄⋊C₈	—	13	8	no	—
135	C₂×C₂²⋊C₈	—	13	2	no	—
136	C₂×#97	—	13	2	no	—
137	#383	Dic₁₃	1	2	yes	—
138	#382	Dic₁₃	1	2	yes	—
139	#381	Dic₁₃	1	2	yes	—
140	#380	Dic₁₃	1	2	yes	—
141	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
142	C₄×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
143	#376	C₂×D₁₃	1	2	yes	—
144	#336	—	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	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₁₃	#520	1	2	yes	—
151	C₂²×C₁₃	#568	1	2	yes	—
152	C₂²×C₁₃	#568	1	2	yes	—
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	—	2	2	no	—
155	C₄×C₁₃	—	2	2	no	—
156	C₄×C₁₃	—	2	2	no	—
157	C₄×C₁₃	—	2	2	no	—
158	C₄×C₁₃	—	4	8	no	—
159	C₄×C₁₃	#535	1	1	yes	characteristic
160	C₄×C₁₃	#536	1	1	yes	characteristic
161	C₄×C₁₃	#544	1	2	yes	—
162	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
163	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
164	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
165	C₂×M₁₆	—	26	4	no	—
166	C₂×M₁₆	—	52	4	no	—
167	C₂²×C₈	—	13	4	no	—
168	C₂²×C₈	—	13	2	no	—
169	C₂²×C₈	—	13	2	no	—
170	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
171	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
172	C₂×C₄²	Dic₂₆	1	4	yes	—
173	C₂×C₄²	D₅₂	1	4	yes	—
174	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
175	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
176	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
177	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
178	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
179	C₄⋊C₈	—	26	4	no	—
180	C₄⋊C₈	—	26	4	no	—
181	C₄⋊C₈	—	26	2	no	—
182	C₄⋊C₈	—	26	2	no	—
183	C₄⋊C₈	—	52	8	no	—
184	C₂²⋊C₈	—	26	4	no	—
185	#97	—	26	4	no	—
186	#95	C₂×Dic₁₃	1	4	yes	—
187	#95	C₂×Dic₁₃	1	4	yes	—
188	#95	C₂×Dic₁₃	1	4	yes	—
189	#95	C₂×Dic₁₃	1	4	yes	—
190	C₂×C₁₃	—	4	2	no	—
191	C₂×C₁₃	#2317	1	1	yes	characteristic
192	C₂×C₁₃	#2905	1	1	yes	characteristic
193	C₂×C₁₃	#2905	1	1	yes	characteristic
194	C₂×C₁₃	#2995	1	4	yes	—
195	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
201	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
202	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
203	C₂²×C₄	C₂×D₅₂	1	2	yes	—
204	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
205	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
210	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
211	M₁₆	—	52	8	no	—
212	C₂×C₈	—	26	2	no	—
213	C₂×C₈	—	26	2	no	—
214	C₂×C₈	—	26	4	no	—
215	C₂×C₈	—	26	4	no	—
216	C₂×C₈	—	26	8	no	—
217	C₂×C₈	—	26	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₂²⋊C₄	—	4	4	no	—
220	C₄²	—	2	4	no	—
221	C₄²	—	2	4	no	—
222	C₄²	C₄⋊Dic₁₃	1	4	yes	—
223	C₄²	C₄⋊Dic₁₃	1	4	yes	—
224	C₄²	C₂×Dic₂₆	1	2	yes	—
225	C₄²	C₂×Dic₂₆	1	2	yes	—
226	C₄²	C₂×D₅₂	1	2	yes	—
227	C₄²	C₂×D₅₂	1	2	yes	—
228	C₁₃	#21979	1	1	yes	characteristic, complemented, Sylow
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
233	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
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	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
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	C₂×C₄⋊Dic₁₃	1	2	yes	—
252	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
253	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
254	C₂×C₄	#88256	1	2	yes	—
255	C₂×C₄	#88257	1	2	yes	—
256	C₈	—	52	2	no	—
257	C₈	—	52	4	no	—
258	C₈	—	52	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	4	4	no	—
263	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
264	C₂²	#11719524	1	2	yes	—
265	C₂²	#11719524	1	2	yes	—
266	C₂²	#11719532	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	8	no	—
273	C₄	#11719224	1	2	yes	—
274	C₄	#11719230	1	1	yes	characteristic
275	C₄	#11719231	1	1	yes	characteristic
276	C₂	—	4	2	no	—
277	C₂	#13117459	1	1	yes	characteristic
278	C₂	#13123400	1	1	yes	characteristic
279	C₂	#13123400	1	1	yes	characteristic
280	C₂	#13124006	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle	C₂³×C₁₃	C₂³×C₄
Fitting	C₁₃×#2283	C₂
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#2283	D₁₃
hypocenter	C₁₃	#21979
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2283

Complements

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

N	H ≅ G/N	type
3. #13115560	C₂	semidirect
5. #13114990	C₂	semidirect
6. #13114956	C₂	semidirect
228. C₁₃	#21979	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718587	2
C₂×#11718585	2
#13115560	2
#13115021	2
#13114990	2
#13114956	2
C₁₃×#2283	2
#21979	13