Subgroups — #15756078

Order 3328.

358 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114974	C₂	1	1	yes	maximal, characteristic
2	#13113584	C₂	1	1	yes	maximal, characteristic
3	#13113265	C₂	1	2	yes	maximal
4	#13112733	C₂	1	2	yes	maximal
5	C₂×C₁₃×#446	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₁₃×#446	—	2	1	no	—
8	C₁₃×#446	—	2	1	no	—
9	C₁₃×#446	C₄	1	2	yes	—
10	C₁₃×#446	C₄	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#97	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	#11718586	C₂²	1	2	yes	—
16	#11718585	C₂²	1	2	yes	—
17	#11718144	—	2	1	no	—
18	#11718144	—	2	1	no	—
19	Q₈⋊#1055	—	2	4	no	—
20	#11718050	—	2	1	no	—
21	#11718049	—	2	1	no	—
22	C₂²×Q₈×C₁₃	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	—	2	1	no	—
24	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₄×Q₈×C₁₃	—	2	1	no	—
27	C₄×Q₈×C₁₃	—	2	1	no	—
28	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₄×Q₈×C₁₃	D₄	1	2	yes	—
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₃×C₄⋊C₄	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	—	2	1	no	—
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×#6099	—	2	2	no	—
35	C₁₃×C₄⋊C₈	—	2	2	no	—
36	C₁₃×C₄⋊C₈	—	2	2	no	—
37	C₁₃×C₄⋊C₈	—	2	1	no	—
38	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₁₃×#97	—	2	2	no	—
40	C₁₃×#97	—	2	2	no	—
41	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₂²⋊#1055	—	2	2	no	—
46	C₄⋊#1055	—	2	2	no	—
47	C₄⋊#1055	—	2	2	no	—
48	C₄⋊#1055	—	2	2	no	—
49	C₄×#1055	—	2	2	no	—
50	#8665	—	13	1	no	maximal, Sylow
51	C₂×Q₈×C₁₃	—	2	1	no	—
52	C₂×Q₈×C₁₃	—	4	2	no	—
53	C₂×Q₈×C₁₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	—	4	2	no	—
63	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₁₃×C₄⋊C₄	—	2	2	no	—
68	C₁₃×C₄⋊C₄	—	2	2	no	—
69	C₁₃×C₄⋊C₄	—	2	1	no	—
70	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
71	C₄²×C₁₃	—	2	2	no	—
72	C₄²×C₁₃	—	2	2	no	—
73	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₁₃	#41	1	1	yes	characteristic
77	#6099	—	4	2	no	—
78	#6099	—	4	2	no	—
79	C₂×#1055	—	4	2	no	—
80	C₂×#1055	—	4	2	no	—
81	C₂×#446	D₁₃	1	1	yes	characteristic
82	#1837	—	13	1	no	—
83	#1564	—	13	1	no	—
84	#1487	—	13	2	no	—
85	#1301	—	13	2	no	—
86	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
87	Q₈×C₁₃	—	2	4	no	—
88	Q₈×C₁₃	—	4	2	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	4	1	no	—
93	C₂×C₄×C₁₃	—	4	2	no	—
94	C₂×C₄×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
99	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#135	1	2	yes	—
102	C₂×C₄×C₁₃	#137	1	2	yes	—
103	C₈×C₁₃	—	2	1	no	—
104	C₈×C₁₃	—	2	1	no	—
105	C₈×C₁₃	—	2	1	no	—
106	C₈×C₁₃	—	2	1	no	—
107	C₈×C₁₃	—	4	2	no	—
108	#1055	—	8	2	no	—
109	#1055	—	8	2	no	—
110	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
111	#446	—	2	1	no	—
112	#446	—	2	1	no	—
113	#446	Dic₁₃	1	2	yes	—
114	#446	Dic₁₃	1	2	yes	—
115	#431	—	13	2	no	—
116	#423	—	13	2	no	—
117	C₂×C₄⋊C₈	—	2	1	no	—
118	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
119	C₂×#97	—	2	1	no	—
120	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
121	#341	—	26	1	no	—
122	#341	—	26	1	no	—
123	#335	—	26	1	no	—
124	#334	—	26	1	no	—
125	Q₈⋊C₈	—	26	4	no	—
126	C₂²×C₁₃	—	2	1	no	—
127	C₂²×C₁₃	—	2	1	no	—
128	C₂²×C₁₃	#380	1	1	yes	characteristic
129	C₂²×C₁₃	#479	1	2	yes	—
130	C₄×C₁₃	—	2	1	no	—
131	C₄×C₁₃	—	2	1	no	—
132	C₄×C₁₃	—	2	1	no	—
133	C₄×C₁₃	—	4	2	no	—
134	C₄×C₁₃	—	4	2	no	—
135	C₄×C₁₃	#416	1	1	yes	characteristic
136	C₄×C₁₃	#419	1	1	yes	characteristic
137	C₄×C₁₃	#430	1	2	yes	—
138	C₄×C₁₃	#464	1	2	yes	—
139	C₂²×Q₈	Dic₂₆	1	1	yes	characteristic
140	C₂×M₁₆	—	26	2	no	—
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
143	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
144	C₄×Q₈	—	2	1	no	—
145	C₄×Q₈	—	2	1	no	—
146	C₄×Q₈	D₅₂	1	2	yes	—
147	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	Dic₂₆	1	1	yes	characteristic
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	1	no	—
155	C₄⋊C₈	—	26	2	no	—
156	C₄⋊C₈	—	26	2	no	—
157	C₄⋊C₈	—	26	2	no	—
158	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
159	C₂²⋊C₈	—	26	2	no	—
160	#97	—	2	2	no	—
161	#97	—	2	2	no	—
162	C₄×C₈	—	26	2	no	—
163	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
164	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
165	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
166	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
167	C₂×C₁₃	—	2	2	no	—
168	C₂×C₁₃	#1805	1	1	yes	characteristic
169	C₂×C₁₃	#1811	1	1	yes	characteristic
170	C₂×C₁₃	#1814	1	1	yes	characteristic
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	C₄⋊Dic₁₃	1	2	yes	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
177	C₂²×C₄	#6127	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
179	M₁₆	—	52	2	no	—
180	M₁₆	—	52	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	52	2	no	—
186	C₂×C₈	—	52	2	no	—
187	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
188	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	C₂×D₅₂	1	1	yes	characteristic
198	C₄²	#6127	1	1	yes	characteristic
199	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
200	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
201	C₁₃	#8665	1	1	yes	characteristic, complemented, Sylow
202	C₂³	#88178	1	1	yes	characteristic
203	Q₈	—	2	4	no	—
204	Q₈	—	4	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	#88178	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
213	C₂×C₄	#88180	1	1	yes	characteristic
214	C₂×C₄	#88184	1	1	yes	characteristic
215	C₂×C₄	#88184	1	1	yes	characteristic
216	C₂×C₄	#88196	1	2	yes	—
217	C₂×C₄	#88204	1	2	yes	—
218	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	4	2	no	—
224	C₈	—	104	2	no	—
225	C₈	—	104	2	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	#11718683	1	2	yes	—
229	C₂²	#11718873	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	#11718689	1	2	yes	—
236	C₄	#11718786	1	2	yes	—
237	C₄	#11718886	1	1	yes	characteristic
238	C₄	#11718894	1	1	yes	characteristic
239	C₂	—	2	2	no	—
240	C₂	#13114851	1	1	yes	characteristic
241	C₂	#13114878	1	1	yes	characteristic
242	C₂	#13114890	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88184
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#446	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂³×C₁₃	C₂²⋊Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#446	D₁₃
hypocenter	C₁₃	#8665
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
201. C₁₃	#8665	semidirect

Maximal subgroups

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

subgroup	index
#13114974	2
#13113584	2
#13113265	2
#13112733	2
C₂×C₁₃×#446	2
#8665	13