Subgroups — #15763011

Order 3328.

484 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114689	C₂	1	1	yes	maximal, characteristic
2	C₄×#88043	C₂	1	1	yes	maximal, characteristic
3	#13112717	C₂	1	4	yes	maximal
4	C₁₃×#2862	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₁₃×#391	—	2	1	no	—
8	C₁₃×#391	—	2	1	no	—
9	C₁₃×#391	—	2	1	no	—
10	C₁₃×#391	—	2	1	no	—
11	C₁₃×#382	C₄	1	2	yes	—
12	C₁₃×#382	C₂²	1	2	yes	—
13	C₄³×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#1055	—	2	1	no	—
15	C₂×#88043	—	2	1	no	—
16	C₂×#88043	C₂²	1	2	yes	—
17	C₂×#88043	C₂²	1	2	yes	—
18	C₂×C₄×#1055	—	2	1	no	—
19	#11718116	—	2	1	no	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂²×Q₈×C₁₃	—	2	1	no	—
22	C₁₃×C₄⋊Q₈	—	4	2	no	—
23	C₁₃×C₄⋊Q₈	—	4	2	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
26	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
27	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×#1055	—	2	2	no	—
36	C₂²×#1055	—	2	2	no	—
37	C₁₃×#95	—	2	2	no	—
38	C₁₃×#95	—	2	2	no	—
39	C₁₃×#95	—	2	2	no	—
40	C₁₃×#95	—	2	2	no	—
41	C₄⋊#1055	—	2	4	no	—
42	#88043	—	2	4	no	—
43	#88043	—	2	2	no	—
44	#88043	—	2	4	no	—
45	#88043	Q₈	1	4	yes	complemented
46	C₄×#1055	—	2	4	no	—
47	#9890	—	13	1	no	maximal, Sylow
48	C₂×Q₈×C₁₃	—	4	2	no	—
49	C₂×Q₈×C₁₃	—	4	2	no	—
50	C₂×Q₈×C₁₃	—	4	2	no	—
51	C₂×Q₈×C₁₃	—	4	2	no	—
52	C₂×Q₈×C₁₃	—	4	2	no	—
53	C₂×Q₈×C₁₃	—	4	2	no	—
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	—	4	2	no	—
64	C₁₃×C₄⋊C₄	—	4	2	no	—
65	C₁₃×C₄⋊C₄	—	4	2	no	—
66	C₁₃×C₄⋊C₄	—	4	2	no	—
67	C₄²×C₁₃	—	2	2	no	—
68	C₄²×C₁₃	—	2	2	no	—
69	C₄²×C₁₃	—	2	2	no	—
70	C₄²×C₁₃	—	2	2	no	—
71	C₄²×C₁₃	—	2	2	no	—
72	C₄²×C₁₃	—	4	1	no	—
73	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
74	C₄²×C₁₃	#41	1	2	yes	—
75	C₂×#1055	—	2	4	no	—
76	C₂×#1055	—	2	4	no	—
77	C₂×#1055	—	2	4	no	—
78	C₂×#1055	—	2	4	no	—
79	C₂×#1055	—	4	2	no	—
80	C₂×#1055	—	4	2	no	—
81	#2862	D₁₃	1	1	yes	characteristic
82	#1767	—	13	1	no	—
83	C₄×#97	—	13	1	no	—
84	#1295	—	13	4	no	—
85	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	Q₈×C₁₃	—	8	4	no	—
87	Q₈×C₁₃	—	8	4	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
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	—	4	2	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	#100	1	2	yes	—
108	C₂×C₄×C₁₃	#101	1	2	yes	—
109	C₂×C₄×C₁₃	#104	1	2	yes	—
110	C₂×C₄×C₁₃	#104	1	2	yes	—
111	C₂×C₄×C₁₃	#104	1	2	yes	—
112	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
116	#1055	—	2	8	no	—
117	#1055	—	4	4	no	—
118	C₂×C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
119	C₂×C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
120	C₂×C₄⋊C₈	—	26	1	no	—
121	C₂×#97	—	13	2	no	—
122	C₂×#97	—	13	2	no	—
123	C₂×#97	—	26	1	no	—
124	C₂×C₄×C₈	—	26	1	no	—
125	#391	—	2	1	no	—
126	#391	—	2	1	no	—
127	#391	—	2	1	no	—
128	#391	—	2	1	no	—
129	#382	Dic₁₃	1	2	yes	—
130	#382	C₂×D₁₃	1	2	yes	—
131	C₄³	C₂×D₁₃	1	1	yes	characteristic
132	#336	—	26	1	no	—
133	C₂²×C₁₃	#391	1	1	yes	characteristic
134	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
135	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
136	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
137	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
138	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
139	C₂²×C₁₃	#421	1	1	yes	characteristic
140	C₄×C₁₃	—	2	1	no	—
141	C₄×C₁₃	—	2	1	no	—
142	C₄×C₁₃	—	2	2	no	—
143	C₄×C₁₃	—	2	2	no	—
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	—	4	1	no	—
147	C₄×C₁₃	—	8	2	no	—
148	C₄×C₁₃	—	8	2	no	—
149	C₄×C₁₃	#329	1	4	yes	—
150	C₂²×Q₈	—	2	1	no	—
151	C₂²×Q₈	—	2	1	no	—
152	C₂²×C₈	—	26	2	no	—
153	C₂²×C₈	—	26	2	no	—
154	C₄⋊Q₈	—	4	2	no	—
155	C₄⋊Q₈	—	4	2	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
159	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
160	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
161	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
162	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
163	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
167	C₄⋊C₈	—	26	4	no	—
168	#97	—	13	4	no	—
169	#97	—	26	4	no	—
170	#97	—	26	2	no	—
171	#97	—	26	4	no	—
172	C₄×C₈	—	26	4	no	—
173	#95	—	2	2	no	—
174	#95	—	2	2	no	—
175	#95	—	2	2	no	—
176	#95	—	2	2	no	—
177	C₂×C₁₃	C₂×#329	1	1	yes	characteristic
178	C₂×C₁₃	#1526	1	1	yes	characteristic
179	C₂×C₁₃	#1527	1	1	yes	characteristic
180	C₂×C₁₃	#1531	1	1	yes	characteristic
181	C₂×C₁₃	#1986	1	1	yes	characteristic
182	C₂×C₁₃	#1994	1	1	yes	characteristic
183	C₂×C₁₃	#1997	1	1	yes	characteristic
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
194	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
195	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
196	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
199	C₂×C₈	—	26	4	no	—
200	C₂×C₈	—	26	4	no	—
201	C₂×C₈	—	26	4	no	—
202	C₂×C₈	—	26	4	no	—
203	C₂×C₈	—	52	2	no	—
204	C₂×C₈	—	52	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	4	1	no	—
215	C₄²	Q₈×D₁₃	1	2	yes	—
216	C₄²	#6131	1	2	yes	—
217	C₁₃	#9890	1	1	yes	characteristic, complemented, Sylow
218	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
219	Q₈	—	8	4	no	—
220	Q₈	—	8	4	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	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	#88073	1	2	yes	—
241	C₂×C₄	#88076	1	2	yes	—
242	C₂×C₄	#88077	1	2	yes	—
243	C₂×C₄	#88077	1	2	yes	—
244	C₂×C₄	#88077	1	2	yes	—
245	C₂×C₄	#88198	1	1	yes	characteristic
246	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
247	C₂×C₄	#88200	1	2	yes	—
248	C₂×C₄	#88201	1	1	yes	characteristic
249	C₈	—	26	8	no	—
250	C₈	—	52	4	no	—
251	C₂²	C₂×#88073	1	1	yes	characteristic
252	C₂²	C₂×#88076	1	1	yes	characteristic
253	C₂²	#11718891	1	1	yes	characteristic
254	C₂²	C₂×#88077	1	1	yes	characteristic
255	C₂²	C₂×#88077	1	1	yes	characteristic
256	C₂²	C₂×#88077	1	1	yes	characteristic
257	C₂²	#11718897	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	8	2	no	—
266	C₄	—	8	2	no	—
267	C₄	#11718132	1	4	yes	—
268	C₂	C₂×#11718132	1	1	yes	characteristic
269	C₂	#13113431	1	1	yes	characteristic
270	C₂	#13113434	1	1	yes	characteristic
271	C₂	#13113451	1	1	yes	characteristic
272	C₂	#13115781	1	1	yes	characteristic
273	C₂	#13115814	1	1	yes	characteristic
274	C₂	#13115826	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2862	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2862	D₁₃
hypocenter	C₁₃	#9890
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

Complements

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

N	H ≅ G/N	type
45. #88043	Q₈	semidirect
217. C₁₃	#9890	semidirect

Maximal subgroups

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

subgroup	index
#13114689	2
C₄×#88043	2
#13112717	2
C₁₃×#2862	2
#9890	13