Subgroups — #15861991

Order 3328.

898 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718685	C₂	1	2	yes	maximal
2	#13114711	C₂	1	4	yes	maximal
3	C₄⋊C₄×#1055	C₂	1	2	yes	maximal
4	#13114689	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊#1055	C₂	1	2	yes	maximal
6	#13114585	C₂	1	2	yes	maximal
7	#13114582	C₂	1	1	yes	maximal, characteristic
8	C₄²×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₃	C₄	1	2	yes	—
10	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₄×C₁₃×C₄⋊C₄	C₄	1	4	yes	—
12	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
13	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₄³×C₁₃	C₂²	1	2	yes	—
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	#11718685	C₄	1	16	yes	complemented
18	C₂×C₄⋊#1055	C₂²	1	4	yes	—
19	C₂×C₄⋊#1055	C₂²	1	2	yes	—
20	C₂×#88043	C₂²	1	2	yes	—
21	C₂×#88043	C₂²	1	4	yes	—
22	C₂×C₄×#1055	C₂²	1	4	yes	—
23	C₂×C₄×#1055	C₂²	1	2	yes	—
24	#11718116	C₂²	1	1	yes	characteristic
25	#11718116	C₂²	1	2	yes	—
26	#11718116	C₂²	1	4	yes	—
27	#11718116	C₂²	1	2	yes	—
28	#11718116	C₂²	1	1	yes	characteristic
29	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Q₈×C₁₃	—	2	8	no	—
31	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
33	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
45	C₂²×#1055	C₂³	1	2	yes	—
46	C₂²×#1055	C₂³	1	4	yes	—
47	C₂²×#1055	C₂³	1	2	yes	—
48	C₁₃×#95	C₂×C₄	1	4	yes	—
49	C₁₃×#95	C₂×C₄	1	2	yes	—
50	C₄⋊#1055	C₂×C₄	1	16	yes	—
51	C₄⋊#1055	C₂×C₄	1	8	yes	—
52	#88043	—	4	2	no	—
53	#88043	C₂×C₄	1	16	yes	—
54	C₄×#1055	—	4	4	no	—
55	C₄×#1055	C₂×C₄	1	8	yes	—
56	#20573	—	13	1	no	maximal, Sylow
57	C₂×Q₈×C₁₃	—	2	4	no	—
58	C₂×Q₈×C₁₃	C₄²	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
67	C₁₃×C₄⋊C₄	—	2	8	no	—
68	C₁₃×C₄⋊C₄	C₄²	1	8	yes	—
69	C₁₃×C₄⋊C₄	C₄²	1	4	yes	—
70	C₁₃×C₄⋊C₄	M₁₆	1	8	yes	—
71	C₄²×C₁₃	—	2	8	no	—
72	C₄²×C₁₃	—	2	4	no	—
73	C₄²×C₁₃	—	2	8	no	—
74	C₄²×C₁₃	—	2	4	no	—
75	C₄²×C₁₃	—	2	2	no	—
76	C₄²×C₁₃	—	2	1	no	—
77	C₄²×C₁₃	—	2	2	no	—
78	C₄²×C₁₃	—	2	2	no	—
79	C₄²×C₁₃	—	2	1	no	—
80	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
81	C₄²×C₁₃	C₂²×C₄	1	8	yes	—
82	C₂×#1055	—	2	2	no	—
83	C₂×#1055	—	2	4	no	—
84	C₂×#1055	—	2	2	no	—
85	C₂×#1055	—	2	2	no	—
86	C₂×#1055	—	2	2	no	—
87	C₂×#1055	—	2	4	no	—
88	C₂×#1055	C₂²×C₄	1	4	yes	—
89	C₂×#1055	C₂²×C₄	1	4	yes	—
90	C₂×#1055	C₂²×C₄	1	8	yes	—
91	C₂×#446	—	13	2	no	—
92	C₄²×Q₈	D₁₃	1	1	yes	characteristic
93	#1771	—	13	4	no	—
94	C₈×C₄⋊C₄	—	13	2	no	—
95	#1767	—	13	1	no	—
96	C₄×C₄⋊C₈	—	13	2	no	—
97	#1745	—	13	2	no	—
98	#1744	—	13	1	no	—
99	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
100	Q₈×C₁₃	—	2	8	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	4	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
113	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
114	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×M₁₆	1	8	yes	—
116	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
117	C₂×C₄×C₁₃	#131	1	4	yes	—
118	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
119	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
121	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
122	#1055	—	2	4	no	—
123	#1055	—	2	4	no	—
124	#1055	—	2	8	no	—
125	C₂×C₄×Q₈	Dic₁₃	1	2	yes	—
126	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
127	#446	—	13	16	no	—
128	C₂×C₄⋊C₈	—	13	4	no	—
129	C₂×C₄⋊C₈	—	13	2	no	—
130	C₂×#97	—	13	4	no	—
131	C₂×#97	—	13	2	no	—
132	C₂×C₄×C₈	—	13	2	no	—
133	C₂×C₄×C₈	—	13	4	no	—
134	C₄×C₄⋊C₄	Dic₁₃	1	4	yes	—
135	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
136	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
137	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
138	C₄³	C₂×D₁₃	1	1	yes	characteristic
139	C₄³	C₂×D₁₃	1	2	yes	—
140	#336	—	13	1	no	—
141	#336	—	13	1	no	—
142	#336	—	13	2	no	—
143	#336	—	13	2	no	—
144	#336	—	13	4	no	—
145	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
146	C₂²×C₁₃	#519	1	1	yes	characteristic
147	C₂²×C₁₃	#520	1	1	yes	characteristic
148	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
149	C₂²×C₁₃	C₂×#131	1	2	yes	—
150	C₄×C₁₃	—	2	4	no	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	—	2	4	no	—
153	C₄×C₁₃	—	2	4	no	—
154	C₄×C₁₃	—	2	8	no	—
155	C₄×C₁₃	—	4	2	no	—
156	C₄×C₁₃	C₄×M₁₆	1	4	yes	—
157	C₄×C₁₃	#519	1	2	yes	—
158	C₄×C₁₃	#520	1	2	yes	—
159	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
160	C₂²×C₈	—	13	2	no	—
161	C₂²×C₈	—	13	2	no	—
162	C₂²×C₈	—	13	4	no	—
163	C₄×Q₈	—	2	8	no	—
164	C₄×Q₈	C₄×D₁₃	1	8	yes	—
165	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
166	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
167	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
168	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
169	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
170	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
171	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
172	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
173	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
175	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
176	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
177	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
178	C₄⋊C₈	—	13	8	no	—
179	C₄⋊C₈	—	13	16	no	—
180	#97	—	13	16	no	—
181	#97	—	52	2	no	—
182	C₄×C₈	—	13	8	no	—
183	C₄×C₈	—	52	4	no	—
184	#95	C₂×Dic₁₃	1	4	yes	—
185	#95	C₂×Dic₁₃	1	2	yes	—
186	C₂×C₁₃	#2276	1	1	yes	characteristic
187	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
188	C₂×C₁₃	C₄×#131	1	1	yes	characteristic
189	C₂×C₁₃	#2984	1	2	yes	—
190	C₂×C₁₃	#2990	1	2	yes	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	C₄×Dic₁₃	1	4	yes	—
193	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
194	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
195	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
196	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
197	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
198	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
201	C₂×C₈	—	13	4	no	—
202	C₂×C₈	—	13	4	no	—
203	C₂×C₈	—	13	8	no	—
204	C₂×C₈	—	26	2	no	—
205	C₂×C₈	—	26	2	no	—
206	C₂×C₈	—	26	4	no	—
207	C₂×C₈	—	26	2	no	—
208	C₂×C₈	—	26	4	no	—
209	C₂×C₈	—	26	2	no	—
210	C₄⋊C₄	—	2	8	no	—
211	C₄⋊C₄	#6099	1	8	yes	—
212	C₄⋊C₄	C₄×Dic₁₃	1	8	yes	—
213	C₄⋊C₄	C₄×Dic₁₃	1	4	yes	—
214	C₄²	—	2	8	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	2	8	no	—
217	C₄²	—	2	4	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	1	no	—
223	C₄²	C₂×C₄×D₁₃	1	4	yes	—
224	C₄²	C₂×C₄×D₁₃	1	8	yes	—
225	C₁₃	#20573	1	1	yes	characteristic, complemented, Sylow
226	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
227	Q₈	—	2	8	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	C₂×#6099	1	8	yes	—
239	C₂×C₄	C₂×#6099	1	4	yes	—
240	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
241	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
242	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
243	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
244	C₂×C₄	#88202	1	4	yes	—
245	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
246	C₂×C₄	#88254	1	2	yes	—
247	C₂×C₄	#88256	1	1	yes	characteristic
248	C₂×C₄	#88257	1	1	yes	characteristic
249	C₈	—	26	4	no	—
250	C₈	—	26	4	no	—
251	C₈	—	26	8	no	—
252	C₂²	C₂²×#6099	1	2	yes	—
253	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
254	C₂²	#11719517	1	1	yes	characteristic
255	C₂²	C₂×#88202	1	2	yes	—
256	C₂²	#11719532	1	1	yes	characteristic
257	C₄	—	2	4	no	—
258	C₄	—	2	8	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	2	no	—
263	C₄	C₄×#6099	1	4	yes	—
264	C₄	#11719277	1	2	yes	—
265	C₄	#11719279	1	2	yes	—
266	C₂	#13117201	1	1	yes	characteristic
267	C₂	C₂×C₄×#6099	1	1	yes	characteristic
268	C₂	C₄×#88202	1	1	yes	characteristic
269	C₂	#13123925	1	2	yes	—
270	C₂	#13123961	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂²×C₄²
Fitting	C₄²×Q₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄²×Q₈	D₁₃
hypocenter	C₁₃	#20573
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
17. #11718685	C₄	semidirect
225. C₁₃	#20573	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718685	2
#13114711	2
C₄⋊C₄×#1055	2
#13114689	2
C₄×C₄⋊#1055	2
#13114585	2
#13114582	2
C₄²×Q₈×C₁₃	2
#20573	13