Subgroups — #15772950

Order 3328.

366 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116635	C₂	1	1	yes	maximal, characteristic
2	#13113595	C₂	1	1	yes	maximal, characteristic
3	#13112800	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
7	C₄×C₁₃×Q₁₆	—	2	1	no	—
8	C₄×C₁₃×Q₁₆	—	2	1	no	—
9	C₄×C₁₃×Q₁₆	C₄	1	2	yes	—
10	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
11	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
12	C₂×C₁₃×#103	—	2	1	no	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718585	—	2	1	no	—
15	C₂²⋊#6090	—	2	1	no	—
16	C₄⋊#6090	—	2	1	no	—
17	C₄×#6090	C₂²	1	2	yes	—
18	#11718050	—	2	1	no	—
19	#11718050	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂×C₁₃×Q₁₆	—	2	4	no	—
22	C₂×C₁₃×Q₁₆	—	2	1	no	—
23	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₄×Q₈×C₁₃	—	2	2	no	—
27	C₄×Q₈×C₁₃	—	2	2	no	—
28	C₄×Q₈×C₁₃	—	4	1	no	—
29	C₄×Q₈×C₁₃	—	4	1	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
32	C₂×C₄²×C₁₃	—	4	1	no	—
33	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
34	C₂×#6099	—	4	1	no	—
35	C₁₃×#107	—	2	1	no	—
36	C₁₃×#107	C₂×C₄	1	2	yes	—
37	C₁₃×#103	—	2	2	no	—
38	C₁₃×#103	—	2	2	no	—
39	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×#6090	—	2	2	no	—
44	C₄⋊#1055	—	2	2	no	—
45	C₄⋊#1055	—	4	1	no	—
46	#11658	—	13	1	no	maximal, Sylow
47	C₂×Q₈×C₁₃	—	2	4	no	—
48	C₂×Q₈×C₁₃	—	2	2	no	—
49	C₂×Q₈×C₁₃	—	4	2	no	—
50	C₂×Q₈×C₁₃	—	4	1	no	—
51	C₂²×C₄×C₁₃	—	4	1	no	—
52	C₂²×C₄×C₁₃	—	4	1	no	—
53	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
56	C₁₃×Q₁₆	—	2	4	no	—
57	C₁₃×Q₁₆	—	2	2	no	—
58	C₁₃×Q₁₆	M₁₆	1	4	yes	complemented
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	—	4	1	no	—
61	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₁₃×C₄⋊C₄	—	2	2	no	—
66	C₁₃×C₄⋊C₄	—	4	1	no	—
67	C₁₃×C₄⋊C₄	—	4	2	no	—
68	C₁₃×C₄⋊C₄	—	4	2	no	—
69	C₄²×C₁₃	—	4	2	no	—
70	C₄²×C₁₃	—	4	2	no	—
71	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₁₃	D₈	1	1	yes	characteristic
73	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
74	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
75	#6099	—	8	2	no	—
76	C₂×#1055	—	4	2	no	—
77	#6090	—	4	2	no	—
78	C₂×C₄×Q₁₆	D₁₃	1	1	yes	characteristic
79	#2162	—	13	1	no	—
80	#1567	—	13	1	no	—
81	#1332	—	13	4	no	—
82	C₂³×C₁₃	#102	1	1	yes	characteristic
83	Q₈×C₁₃	—	2	4	no	—
84	Q₈×C₁₃	—	4	4	no	—
85	Q₈×C₁₃	—	4	2	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	4	1	no	—
89	C₂×C₄×C₁₃	—	4	4	no	—
90	C₂×C₄×C₁₃	—	4	2	no	—
91	C₂×C₄×C₁₃	—	4	2	no	—
92	C₂×C₄×C₁₃	—	8	2	no	—
93	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
96	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
97	C₂×C₄×C₁₃	Q₃₂	1	2	yes	—
98	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₈×C₁₃	—	2	1	no	—
102	C₈×C₁₃	—	2	1	no	—
103	C₈×C₁₃	—	2	1	no	—
104	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
105	#1055	—	8	2	no	—
106	C₂²×Q₁₆	Dic₁₃	1	1	yes	characteristic
107	C₂×C₄×Q₈	—	2	1	no	—
108	C₄×Q₁₆	—	2	1	no	—
109	C₄×Q₁₆	—	2	1	no	—
110	C₄×Q₁₆	Dic₁₃	1	2	yes	—
111	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
112	C₂×#107	Dic₁₃	1	1	yes	characteristic
113	#423	—	26	1	no	—
114	C₂×#103	—	2	1	no	—
115	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
116	C₄⋊C₁₆	—	26	1	no	—
117	C₂²⋊C₁₆	—	26	1	no	—
118	C₄×C₁₆	—	13	2	no	—
119	#335	—	13	2	no	—
120	#335	—	26	1	no	—
121	C₂²×C₁₃	—	2	1	no	—
122	C₂²×C₁₃	—	2	1	no	—
123	C₂²×C₁₃	#358	1	2	yes	—
124	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
125	C₄×C₁₃	—	2	1	no	—
126	C₄×C₁₃	—	2	1	no	—
127	C₄×C₁₃	—	2	1	no	—
128	C₄×C₁₃	—	4	4	no	—
129	C₄×C₁₃	—	8	2	no	—
130	C₄×C₁₃	#358	1	2	yes	—
131	C₄×C₁₃	#407	1	1	yes	characteristic
132	C₄×C₁₃	#421	1	1	yes	characteristic
133	C₄×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
134	C₄×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
135	C₂²×Q₈	—	2	1	no	—
136	C₂×Q₁₆	—	2	4	no	—
137	C₂×Q₁₆	—	2	1	no	—
138	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
139	C₂×M₁₆	—	52	1	no	—
140	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
141	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
142	C₄×Q₈	—	2	2	no	—
143	C₄×Q₈	—	2	2	no	—
144	C₄×Q₈	—	4	1	no	—
145	C₄×Q₈	—	4	1	no	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄⋊C₄	—	4	1	no	—
148	C₂×C₄²	—	4	1	no	—
149	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
150	C₂×C₁₆	—	26	2	no	—
151	#107	—	2	1	no	—
152	#107	C₂×Dic₁₃	1	2	yes	—
153	C₄⋊C₈	—	26	2	no	—
154	C₄⋊C₈	—	52	1	no	—
155	#103	—	2	2	no	—
156	#103	—	2	2	no	—
157	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
161	C₂×C₁₃	—	2	2	no	—
162	C₂×C₁₃	#1486	1	1	yes	characteristic
163	C₂×C₁₃	C₂×#358	1	1	yes	characteristic
164	C₂×C₁₃	#2143	1	1	yes	characteristic
165	C₂×Q₈	—	2	4	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂×Q₈	—	4	1	no	—
169	C₂²×C₄	—	4	1	no	—
170	C₂²×C₄	—	4	1	no	—
171	C₂²×C₄	#6104	1	1	yes	characteristic
172	C₂²×C₄	#6105	1	1	yes	characteristic
173	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
174	Q₁₆	—	2	4	no	—
175	Q₁₆	—	2	2	no	—
176	Q₁₆	#6099	1	4	yes	complemented
177	M₁₆	—	104	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	52	2	no	—
181	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
182	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
183	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
184	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄²	—	4	2	no	—
190	C₄²	—	4	2	no	—
191	C₄²	#6104	1	1	yes	characteristic
192	C₄²	#6105	1	1	yes	characteristic
193	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
194	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
195	C₁₆	—	52	2	no	—
196	C₁₃	#11658	1	1	yes	characteristic, complemented, Sylow
197	C₂³	#88072	1	1	yes	characteristic
198	Q₈	—	2	4	no	—
199	Q₈	—	4	4	no	—
200	Q₈	—	4	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	8	2	no	—
208	C₂×C₄	#88068	1	2	yes	—
209	C₂×C₄	#88069	1	2	yes	—
210	C₂×C₄	#88072	1	1	yes	characteristic
211	C₂×C₄	#88072	1	1	yes	characteristic
212	C₂×C₄	#88072	1	1	yes	characteristic
213	C₂×C₄	C₂×#6104	1	1	yes	characteristic
214	C₂×C₄	C₂×#6105	1	1	yes	characteristic
215	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	104	2	no	—
220	C₈	C₂×#6099	1	2	yes	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	#11718156	1	2	yes	—
224	C₂²	C₂×#88072	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	4	4	no	—
229	C₄	—	8	2	no	—
230	C₄	#11718156	1	2	yes	—
231	C₄	C₂×#88068	1	1	yes	characteristic
232	C₄	C₂×#88069	1	1	yes	characteristic
233	C₄	#11718868	1	1	yes	characteristic
234	C₄	#11718897	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	#13113261	1	1	yes	characteristic
237	C₂	C₂×#11718156	1	1	yes	characteristic
238	C₂	#13116564	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂³×C₁₃	#102
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×Q₁₆	D₁₃
hypocenter	C₁₃	#11658
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #13112800	C₂	semidirect
58. C₁₃×Q₁₆	M₁₆	semidirect
176. Q₁₆	#6099	semidirect
196. C₁₃	#11658	semidirect

Maximal subgroups

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

subgroup	index
#13116635	2
#13113595	2
#13112800	2
C₂×C₄×C₁₃×Q₁₆	2
#11658	13