Subgroups — #15889524

Order 3328.

636 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115777	C₂	1	2	yes	maximal
2	#13115773	C₂	1	2	yes	maximal
3	#13115402	C₂	1	4	yes	maximal
4	#13115395	C₂	1	4	yes	maximal
5	#13114979	C₂	1	2	yes	maximal
6	C₄×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#391	C₄	1	1	yes	characteristic
11	C₁₃×#391	C₄	1	1	yes	characteristic
12	C₁₃×#389	C₄	1	4	yes	—
13	C₁₃×#384	C₄	1	1	yes	characteristic
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
15	C₄³×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊#1055	C₂²	1	2	yes	—
17	C₂×C₄⋊#1055	C₂²	1	2	yes	—
18	C₂×#88043	C₂²	1	2	yes	—
19	C₂×C₄×#1055	C₂²	1	2	yes	—
20	#11718116	C₂²	1	2	yes	—
21	#11718116	C₂²	1	4	yes	—
22	#11718116	C₂²	1	4	yes	—
23	#11718116	C₂²	1	4	yes	—
24	#11718116	C₂²	1	2	yes	—
25	#11718116	C₂²	1	4	yes	—
26	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₁₃×C₄⋊Q₈	—	2	4	no	—
29	C₄×Q₈×C₁₃	—	4	1	no	—
30	C₄×Q₈×C₁₃	—	4	1	no	—
31	C₄×Q₈×C₁₃	—	4	1	no	—
32	C₄×Q₈×C₁₃	—	4	1	no	—
33	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×#1055	C₂³	1	4	yes	—
47	C₂²×#1055	C₂³	1	4	yes	—
48	C₁₃×#95	C₂×C₄	1	4	yes	—
49	C₁₃×#95	C₂×C₄	1	4	yes	—
50	C₄⋊#1055	—	4	2	no	—
51	C₄⋊#1055	—	4	2	no	—
52	#88043	—	4	2	no	—
53	C₄×#1055	—	4	2	no	—
54	#23365	—	13	1	no	maximal, Sylow
55	C₂×Q₈×C₁₃	—	2	2	no	—
56	C₂×Q₈×C₁₃	—	4	1	no	—
57	C₂×Q₈×C₁₃	—	4	1	no	—
58	C₂×Q₈×C₁₃	—	4	1	no	—
59	C₂×Q₈×C₁₃	—	4	1	no	—
60	C₂×Q₈×C₁₃	M₁₆	1	4	yes	—
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	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₁₃×C₄⋊C₄	—	2	8	no	—
71	C₁₃×C₄⋊C₄	—	2	8	no	—
72	C₁₃×C₄⋊C₄	—	4	1	no	—
73	C₁₃×C₄⋊C₄	—	4	1	no	—
74	C₁₃×C₄⋊C₄	—	4	2	no	—
75	C₁₃×C₄⋊C₄	—	4	2	no	—
76	C₄²×C₁₃	—	2	2	no	—
77	C₄²×C₁₃	—	2	4	no	—
78	C₄²×C₁₃	—	2	1	no	—
79	C₄²×C₁₃	—	2	1	no	—
80	C₄²×C₁₃	—	2	2	no	—
81	C₄²×C₁₃	—	2	2	no	—
82	C₄²×C₁₃	—	2	2	no	—
83	C₄²×C₁₃	—	2	2	no	—
84	C₄²×C₁₃	—	2	4	no	—
85	C₄²×C₁₃	—	4	1	no	—
86	C₂×#1055	—	2	4	no	—
87	C₂×#1055	—	2	4	no	—
88	C₂×#1055	—	2	8	no	—
89	C₂×#1055	—	2	8	no	—
90	C₄×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
91	#1985	—	13	2	no	—
92	#1984	—	13	2	no	—
93	#1917	—	13	4	no	—
94	#1916	—	13	4	no	—
95	#1839	—	13	2	no	—
96	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
97	Q₈×C₁₃	—	4	4	no	—
98	Q₈×C₁₃	—	4	4	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	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	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
119	C₂×C₄×C₁₃	#131	1	4	yes	—
120	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂×C₄×C₁₃	2₋¹⁺⁴	1	4	yes	—
123	#1055	—	4	4	no	—
124	#1055	—	4	4	no	—
125	C₂×C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
126	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
127	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
128	C₂×C₄⋊C₈	—	13	2	no	—
129	C₂×C₄⋊C₈	—	13	2	no	—
130	C₂×#97	—	13	2	no	—
131	C₂×C₄×C₈	—	13	2	no	—
132	#391	Dic₁₃	1	1	yes	characteristic
133	#391	Dic₁₃	1	1	yes	characteristic
134	#389	Dic₁₃	1	4	yes	—
135	#384	Dic₁₃	1	1	yes	characteristic
136	C₄×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
137	C₄³	C₂×D₁₃	1	1	yes	characteristic
138	#336	—	13	2	no	—
139	#336	—	13	4	no	—
140	#336	—	13	2	no	—
141	#336	—	13	4	no	—
142	#336	—	13	4	no	—
143	#336	—	13	4	no	—
144	C₂²×C₁₃	#518	1	1	yes	characteristic
145	C₂²×C₁₃	#519	1	2	yes	—
146	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
147	C₂²×C₁₃	C₂×#131	1	2	yes	—
148	C₂²×C₁₃	#568	1	1	yes	characteristic
149	C₄×C₁₃	—	2	4	no	—
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	—	2	4	no	—
152	C₄×C₁₃	—	4	2	no	—
153	C₄×C₁₃	—	4	1	no	—
154	C₄×C₁₃	—	4	2	no	—
155	C₄×C₁₃	—	4	2	no	—
156	C₄×C₁₃	—	4	2	no	—
157	C₄×C₁₃	#563	1	2	yes	—
158	C₄×C₁₃	#563	1	2	yes	—
159	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
160	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
161	C₂²×C₈	—	13	4	no	—
162	C₂²×C₈	—	13	4	no	—
163	C₄⋊Q₈	—	2	4	no	—
164	C₄×Q₈	—	4	1	no	—
165	C₄×Q₈	—	4	1	no	—
166	C₄×Q₈	—	4	1	no	—
167	C₄×Q₈	—	4	1	no	—
168	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
169	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
170	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
173	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
174	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
175	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
176	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
177	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
178	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
179	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
180	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
181	C₄⋊C₈	—	52	2	no	—
182	C₄⋊C₈	—	52	2	no	—
183	#97	—	52	2	no	—
184	C₄×C₈	—	52	2	no	—
185	#95	C₂×Dic₁₃	1	4	yes	—
186	#95	C₂×Dic₁₃	1	4	yes	—
187	C₂×C₁₃	#2381	1	1	yes	characteristic
188	C₂×C₁₃	#2973	1	1	yes	characteristic
189	C₂×C₁₃	#2979	1	1	yes	characteristic
190	C₂×C₁₃	#2984	1	2	yes	—
191	C₂×C₁₃	#2985	1	2	yes	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	4	1	no	—
194	C₂×Q₈	—	4	1	no	—
195	C₂×Q₈	—	4	1	no	—
196	C₂×Q₈	—	4	1	no	—
197	C₂×Q₈	#6099	1	4	yes	—
198	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
200	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
201	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
202	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
204	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
206	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
207	C₂×C₈	—	26	4	no	—
208	C₂×C₈	—	26	4	no	—
209	C₂×C₈	—	26	8	no	—
210	C₂×C₈	—	26	8	no	—
211	C₄⋊C₄	—	2	8	no	—
212	C₄⋊C₄	—	2	8	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	4	no	—
226	C₄²	—	4	1	no	—
227	C₁₃	#23365	1	1	yes	characteristic, complemented, Sylow
228	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
229	Q₈	—	4	4	no	—
230	Q₈	—	4	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	C₂×#6099	1	2	yes	—
250	C₂×C₄	C₂×#6099	1	4	yes	—
251	C₂×C₄	#88202	1	4	yes	—
252	C₂×C₄	#88251	1	1	yes	characteristic
253	C₂×C₄	#88251	1	1	yes	characteristic
254	C₂×C₄	#88254	1	4	yes	—
255	C₈	—	52	4	no	—
256	C₈	—	52	4	no	—
257	C₂²	C₂²×#6099	1	1	yes	characteristic
258	C₂²	#11719503	1	1	yes	characteristic
259	C₂²	#11719517	1	2	yes	—
260	C₂²	C₂×#88202	1	2	yes	—
261	C₂²	#11719524	1	1	yes	characteristic
262	C₄	—	2	4	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	4	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	2	no	—
270	C₄	#11719440	1	2	yes	—
271	C₄	#11719440	1	2	yes	—
272	C₂	#13118017	1	1	yes	characteristic
273	C₂	#13123865	1	1	yes	characteristic
274	C₂	#13123903	1	1	yes	characteristic
275	C₂	#13123925	1	2	yes	—
276	C₂	#13123931	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle, prime radical	C₂³×C₁₃	C₂³×C₄
Fitting	C₄×C₁₃×C₄⋊Q₈	C₂
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#23365
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
227. C₁₃	#23365	semidirect

Maximal subgroups

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

subgroup	index
#13115777	2
#13115773	2
#13115402	2
#13115395	2
#13114979	2
C₄×C₁₃×C₄⋊Q₈	2
#23365	13