Subgroups — #15751138

Order 3328.

462 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	#13114979	C₂	1	1	yes	maximal, characteristic
2	#13114689	C₂	1	1	yes	maximal, characteristic
3	#13112659	C₂	1	4	yes	maximal
4	C₄×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
7	C₁₃×#391	—	2	1	no	—
8	C₁₃×#389	—	2	1	no	—
9	C₁₃×#389	C₂²	1	2	yes	—
10	C₁₃×#384	C₄	1	1	yes	characteristic
11	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
12	C₄×C₁₃×C₄⋊C₄	C₄	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	#11718116	—	2	1	no	—
17	#11718116	—	2	1	no	—
18	#11718116	C₂²	1	2	yes	—
19	#11718116	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₁₃×C₄⋊Q₈	—	2	4	no	—
22	C₄×Q₈×C₁₃	—	4	2	no	—
23	C₄×Q₈×C₁₃	—	4	2	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
28	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
29	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃	—	2	2	no	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×#1055	—	2	2	no	—
39	C₂²×#1055	—	2	2	no	—
40	C₁₃×#95	—	2	2	no	—
41	C₁₃×#95	—	2	2	no	—
42	C₄⋊#1055	—	2	4	no	—
43	#88043	—	4	2	no	—
44	#7625	—	13	1	no	maximal, Sylow
45	C₂×Q₈×C₁₃	—	2	4	no	—
46	C₂×Q₈×C₁₃	—	4	2	no	—
47	C₂×Q₈×C₁₃	—	4	2	no	—
48	C₂²×C₄×C₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₃	—	2	2	no	—
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	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₁₃×C₄⋊C₄	—	2	4	no	—
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×C₄⋊C₄	—	4	2	no	—
60	C₁₃×C₄⋊C₄	—	4	2	no	—
61	C₁₃×C₄⋊C₄	—	4	2	no	—
62	C₁₃×C₄⋊C₄	—	4	2	no	—
63	C₁₃×C₄⋊C₄	—	4	2	no	—
64	C₁₃×C₄⋊C₄	M₁₆	1	4	yes	—
65	C₄²×C₁₃	—	2	2	no	—
66	C₄²×C₁₃	—	2	2	no	—
67	C₄²×C₁₃	—	2	2	no	—
68	C₄²×C₁₃	—	2	2	no	—
69	C₄²×C₁₃	—	4	1	no	—
70	C₄²×C₁₃	—	4	1	no	—
71	C₄²×C₁₃	—	4	4	no	—
72	C₄²×C₁₃	D₈	1	2	yes	—
73	C₄²×C₁₃	SD₁₆	1	2	yes	—
74	C₂×#1055	—	2	4	no	—
75	C₂×#1055	—	4	2	no	—
76	C₂×#1055	—	4	2	no	—
77	C₂×#1055	—	4	2	no	—
78	C₂×#1055	—	4	2	no	—
79	C₂×#1055	—	4	2	no	—
80	C₄×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
81	#1839	—	13	1	no	—
82	#1767	—	13	1	no	—
83	#1271	—	13	4	no	—
84	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	Q₈×C₁₃	—	4	8	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	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	4	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	#101	1	2	yes	—
107	C₂×C₄×C₁₃	#102	1	2	yes	—
108	C₂×C₄×C₁₃	#102	1	2	yes	—
109	C₂×C₄×C₁₃	#104	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#137	1	2	yes	—
114	#1055	—	4	4	no	—
115	#1055	—	8	2	no	—
116	C₂×C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
117	C₂×C₄×Q₈	—	2	1	no	—
118	C₂×C₄⋊C₈	—	26	1	no	—
119	C₂×#97	—	26	1	no	—
120	#391	—	2	1	no	—
121	#389	—	2	1	no	—
122	#389	C₂×D₁₃	1	2	yes	—
123	#384	Dic₁₃	1	1	yes	characteristic
124	C₄×C₄⋊C₄	—	2	1	no	—
125	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
126	C₄³	C₂×D₁₃	1	1	yes	characteristic
127	#336	—	13	2	no	—
128	#336	—	13	2	no	—
129	#336	—	26	1	no	—
130	#336	—	26	1	no	—
131	C₂²×C₁₃	#407	1	1	yes	characteristic
132	C₂²×C₁₃	#410	1	1	yes	characteristic
133	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
134	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
135	C₂²×C₁₃	#419	1	1	yes	characteristic
136	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
137	C₂²×C₁₃	#421	1	1	yes	characteristic
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	2	1	no	—
143	C₄×C₁₃	—	2	1	no	—
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	—	4	4	no	—
147	C₄×C₁₃	—	8	2	no	—
148	C₂²×Q₈	—	2	1	no	—
149	C₂²×C₈	—	26	2	no	—
150	C₂²×C₈	—	26	2	no	—
151	C₄⋊Q₈	—	2	4	no	—
152	C₄×Q₈	—	4	2	no	—
153	C₄×Q₈	—	4	2	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
159	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
160	C₂×C₄²	—	2	2	no	—
161	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
162	C₂×C₄²	C₂×Dic₁₃	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	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
168	C₄⋊C₈	—	26	4	no	—
169	#97	—	52	2	no	—
170	#95	—	2	2	no	—
171	#95	—	2	2	no	—
172	C₂×C₁₃	#1467	1	1	yes	characteristic
173	C₂×C₁₃	#1486	1	1	yes	characteristic
174	C₂×C₁₃	#1491	1	1	yes	characteristic
175	C₂×C₁₃	#1502	1	1	yes	characteristic
176	C₂×C₁₃	#1515	1	1	yes	characteristic
177	C₂×C₁₃	#1531	1	1	yes	characteristic
178	C₂×C₁₃	#1546	1	1	yes	characteristic
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
186	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
187	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
188	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
191	C₂×C₈	—	26	4	no	—
192	C₂×C₈	—	52	2	no	—
193	C₂×C₈	—	52	2	no	—
194	C₂×C₈	—	52	2	no	—
195	C₂×C₈	—	52	2	no	—
196	C₂×C₈	—	52	2	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	#6099	1	4	yes	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	4	1	no	—
210	C₄²	—	4	1	no	—
211	C₄²	—	4	4	no	—
212	C₄²	#6104	1	2	yes	—
213	C₄²	#6105	1	2	yes	—
214	C₁₃	#7625	1	1	yes	characteristic, complemented, Sylow
215	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
216	Q₈	—	4	8	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	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	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	#88072	1	2	yes	—
238	C₂×C₄	#88072	1	2	yes	—
239	C₂×C₄	#88076	1	2	yes	—
240	C₂×C₄	#88077	1	2	yes	—
241	C₂×C₄	C₂×#6099	1	2	yes	—
242	C₂×C₄	C₂×#6104	1	1	yes	characteristic
243	C₂×C₄	C₂×#6105	1	1	yes	characteristic
244	C₂×C₄	#88196	1	2	yes	—
245	C₈	—	52	4	no	—
246	C₈	—	104	2	no	—
247	C₂²	#11718868	1	1	yes	characteristic
248	C₂²	C₂×#88072	1	1	yes	characteristic
249	C₂²	#11718886	1	1	yes	characteristic
250	C₂²	C₂×#88076	1	1	yes	characteristic
251	C₂²	C₂×#88077	1	1	yes	characteristic
252	C₂²	#11718897	1	1	yes	characteristic
253	C₂²	#11718901	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	4	no	—
263	C₄	—	8	2	no	—
264	C₂	#13113180	1	1	yes	characteristic
265	C₂	#13113261	1	1	yes	characteristic
266	C₂	#13113283	1	1	yes	characteristic
267	C₂	#13113331	1	1	yes	characteristic
268	C₂	#13113387	1	1	yes	characteristic
269	C₂	#13113451	1	1	yes	characteristic
270	C₂	#13113508	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#7625
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
214. C₁₃	#7625	semidirect

Maximal subgroups

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

subgroup	index
#13114979	2
#13114689	2
#13112659	2
C₄×C₁₃×C₄⋊Q₈	2
#7625	13