Subgroups — #15889508

Order 3328.

664 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88123	C₂	1	1	yes	maximal, characteristic
2	#13115770	C₂	1	4	yes	maximal
3	#13115394	C₂	1	4	yes	maximal
4	#13115392	C₂	1	4	yes	maximal
5	#13114979	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1839	C₂	1	1	yes	maximal, characteristic
7	C₂×#88123	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₂₆	C₂²	1	2	yes	—
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88054	C₂²	1	4	yes	—
11	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
12	#11718596	C₄	1	1	yes	characteristic
13	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
14	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
15	#11718590	C₄	1	2	yes	—
16	#11718589	C₄	1	2	yes	—
17	#11718589	C₄	1	2	yes	—
18	C₁₃×#336	C₂²	1	2	yes	—
19	C₁₃×#336	C₂²	1	2	yes	—
20	C₁₃×#336	C₂²	1	2	yes	—
21	#11718142	C₂²	1	8	yes	—
22	#11718116	C₂²	1	2	yes	—
23	#11718116	C₂²	1	2	yes	—
24	#11718116	C₂²	1	2	yes	—
25	C₂²×Dic₂₆	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
39	C₂×#6101	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₁₃	C₂³	1	4	yes	—
41	C₂²×#1055	C₂³	1	4	yes	—
42	#88123	C₈	1	8	yes	complemented
43	C₄×Dic₂₆	—	4	2	no	—
44	C₄×Dic₂₆	—	4	2	no	—
45	#88071	C₂×C₄	1	4	yes	—
46	#88071	C₂×C₄	1	4	yes	—
47	#88054	—	4	4	no	—
48	C₈×Dic₁₃	—	4	4	no	—
49	#23363	—	13	1	no	maximal, Sylow
50	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
57	C₂²×Dic₁₃	C₂²×C₄	1	4	yes	—
58	C₂²×Dic₁₃	C₂²×C₄	1	4	yes	—
59	C₂×Dic₂₆	—	4	2	no	—
60	C₂×Dic₂₆	—	4	2	no	—
61	C₂×Dic₂₆	C₂×C₈	1	8	yes	—
62	C₂×C₈×C₁₃	—	2	4	no	—
63	C₂×C₈×C₁₃	—	2	4	no	—
64	C₂×C₈×C₁₃	—	2	4	no	—
65	C₄²×C₁₃	—	2	2	no	—
66	C₄²×C₁₃	—	2	2	no	—
67	C₄²×C₁₃	—	2	2	no	—
68	C₄²×C₁₃	—	4	1	no	—
69	C₄²×C₁₃	—	4	1	no	—
70	C₄²×C₁₃	—	4	1	no	—
71	C₄²×C₁₃	C₂×C₈	1	4	yes	—
72	C₄⋊Dic₁₃	—	4	2	no	—
73	C₄⋊Dic₁₃	—	4	2	no	—
74	C₄⋊Dic₁₃	—	4	2	no	—
75	C₄⋊Dic₁₃	C₂×C₈	1	8	yes	—
76	C₄⋊Dic₁₃	C₂×C₈	1	8	yes	—
77	#6101	—	4	4	no	—
78	C₄×Dic₁₃	—	2	4	no	—
79	C₄×Dic₁₃	—	2	4	no	—
80	C₂×#1055	—	2	4	no	—
81	C₂×#1055	—	2	4	no	—
82	C₂×#1055	—	2	4	no	—
83	C₄×C₄⋊Q₈	—	13	1	no	—
84	#1984	—	13	4	no	—
85	#1916	—	13	4	no	—
86	#1916	—	13	4	no	—
87	#1839	—	13	1	no	—
88	#1839	D₁₃	1	1	yes	characteristic
89	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	1	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	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
109	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
110	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
111	C₂×Dic₁₃	—	2	4	no	—
112	C₂×Dic₁₃	—	2	4	no	—
113	C₂×Dic₁₃	—	2	4	no	—
114	C₂×Dic₁₃	—	2	4	no	—
115	C₂×Dic₁₃	—	2	4	no	—
116	C₂×Dic₁₃	C₂²×C₈	1	8	yes	—
117	Dic₂₆	—	4	8	no	—
118	C₈×C₁₃	—	4	4	no	—
119	#1055	—	4	4	no	—
120	C₂×C₄⋊Q₈	—	13	1	no	—
121	C₂×C₄×Q₈	—	13	2	no	—
122	C₂×C₄⋊C₈	—	13	4	no	—
123	C₂×C₄×C₈	—	13	4	no	—
124	#391	—	13	2	no	—
125	#389	—	13	2	no	—
126	#389	—	13	2	no	—
127	#384	—	13	1	no	—
128	C₄×C₄⋊C₄	—	13	2	no	—
129	C₄×C₄⋊C₄	—	13	2	no	—
130	C₄³	C₂×D₁₃	1	1	yes	characteristic
131	#336	—	13	8	no	—
132	#336	—	13	2	no	—
133	#336	—	13	2	no	—
134	#336	—	13	2	no	—
135	#336	C₂×D₁₃	1	2	yes	—
136	#336	C₂×D₁₃	1	2	yes	—
137	#336	C₂×D₁₃	1	2	yes	—
138	C₂²×C₁₃	#518	1	1	yes	characteristic
139	C₂²×C₁₃	#519	1	1	yes	characteristic
140	C₂²×C₁₃	#519	1	1	yes	characteristic
141	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
142	C₂²×C₁₃	#568	1	1	yes	characteristic
143	C₂²×C₁₃	#568	1	1	yes	characteristic
144	C₂²×C₁₃	#568	1	1	yes	characteristic
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	—	2	2	no	—
148	C₄×C₁₃	—	4	1	no	—
149	C₄×C₁₃	—	4	1	no	—
150	C₄×C₁₃	—	4	1	no	—
151	C₄×C₁₃	#563	1	4	yes	—
152	Dic₁₃	—	2	8	no	—
153	Dic₁₃	—	4	4	no	—
154	C₂²×Q₈	—	13	2	no	—
155	C₂²×C₈	—	13	4	no	—
156	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
157	C₄⋊Q₈	—	13	8	no	—
158	C₄×Q₈	—	52	2	no	—
159	C₄×Q₈	—	52	2	no	—
160	C₂×C₄⋊C₄	—	13	2	no	—
161	C₂×C₄⋊C₄	—	13	2	no	—
162	C₂×C₄⋊C₄	—	13	4	no	—
163	C₂×C₄⋊C₄	—	13	2	no	—
164	C₂×C₄⋊C₄	—	13	2	no	—
165	C₂×C₄⋊C₄	—	13	2	no	—
166	C₂×C₄²	—	13	4	no	—
167	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
168	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
169	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
170	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
174	C₄⋊C₈	—	52	4	no	—
175	C₄×C₈	—	52	4	no	—
176	#95	—	13	4	no	—
177	#95	—	13	4	no	—
178	C₂×C₁₃	#2381	1	1	yes	characteristic
179	C₂×C₁₃	#2972	1	1	yes	characteristic
180	C₂×C₁₃	#2980	1	1	yes	characteristic
181	C₂×C₁₃	#2982	1	1	yes	characteristic
182	C₂×C₁₃	#2982	1	1	yes	characteristic
183	C₂×C₁₃	#2987	1	1	yes	characteristic
184	C₂×C₁₃	#2987	1	1	yes	characteristic
185	C₂×Q₈	—	13	8	no	—
186	C₂×Q₈	—	52	2	no	—
187	C₂×Q₈	—	52	2	no	—
188	C₂²×C₄	—	13	4	no	—
189	C₂²×C₄	—	13	4	no	—
190	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
196	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	26	4	no	—
201	C₂×C₈	—	26	4	no	—
202	C₂×C₈	—	26	4	no	—
203	C₄⋊C₄	—	13	8	no	—
204	C₄⋊C₄	—	13	8	no	—
205	C₄⋊C₄	—	52	4	no	—
206	C₄⋊C₄	—	52	2	no	—
207	C₄⋊C₄	—	52	2	no	—
208	C₄⋊C₄	—	52	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	—	4	1	no	—
215	C₄²	—	26	4	no	—
216	C₄²	—	26	4	no	—
217	C₄²	C₈×D₁₃	1	4	yes	—
218	C₁₃	#23363	1	1	yes	characteristic, complemented, Sylow
219	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
220	Q₈	—	52	8	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	13	8	no	—
237	C₂×C₄	—	26	4	no	—
238	C₂×C₄	—	26	4	no	—
239	C₂×C₄	—	26	4	no	—
240	C₂×C₄	—	26	4	no	—
241	C₂×C₄	—	26	4	no	—
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	#88256	1	2	yes	—
246	C₂×C₄	#88257	1	2	yes	—
247	C₂×C₄	#88257	1	2	yes	—
248	C₈	—	4	4	no	—
249	C₈	—	52	4	no	—
250	C₂²	#11719221	1	1	yes	characteristic
251	C₂²	#11719221	1	1	yes	characteristic
252	C₂²	#11719226	1	1	yes	characteristic
253	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
254	C₂²	#11719450	1	1	yes	characteristic
255	C₂²	#11719450	1	1	yes	characteristic
256	C₂²	#11719450	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	1	no	—
263	C₄	—	26	8	no	—
264	C₄	—	52	4	no	—
265	C₄	#11719417	1	4	yes	—
266	C₂	#13118016	1	1	yes	characteristic
267	C₂	#13123859	1	1	yes	characteristic
268	C₂	#13123906	1	1	yes	characteristic
269	C₂	#13123916	1	1	yes	characteristic
270	C₂	#13123916	1	1	yes	characteristic
271	C₂	#13123941	1	1	yes	characteristic
272	C₂	#13123941	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#1839	C₂
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1839	D₁₃
hypocenter	C₁₃	#23363
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1839

Complements

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

N	H ≅ G/N	type
42. #88123	C₈	semidirect
218. C₁₃	#23363	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88123	2
#13115770	2
#13115394	2
#13115392	2
#13114979	2
C₁₃×#1839	2
#23363	13