Subgroups — #16018450

Order 3328.

482 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718344	C₂	1	1	yes	maximal, characteristic
2	#13117119	C₂	1	4	yes	maximal, complemented
3	#13117112	C₂	1	4	yes	maximal, complemented
4	#13116625	C₂	1	4	yes	maximal
5	C₄×#88052	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#533	—	2	1	no	—
8	C₂×#88124	—	2	1	no	—
9	C₁₃×#500	C₂²	1	2	yes	—
10	C₁₃×#499	—	2	1	no	—
11	C₁₃×#499	C₂²	1	2	yes	—
12	C₁₃×#498	—	2	1	no	—
13	C₁₃×#427	—	2	1	no	—
14	C₁₃×#427	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88058	C₂²	1	2	yes	—
17	C₂×#88057	—	2	1	no	—
18	C₂×#88052	C₂²	1	2	yes	—
19	#11718344	—	2	1	no	—
20	#11718344	—	2	1	no	—
21	#11718344	C₂²	1	2	yes	—
22	#11718344	C₂²	1	2	yes	—
23	#11718082	C₂²	1	8	yes	—
24	#11718081	C₂²	1	8	yes	—
25	#11718053	C₂²	1	2	yes	—
26	C₄×#6090	C₂²	1	2	yes	—
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₁₃×C₄⋊Q₈	—	2	2	no	—
30	C₁₃×#125	—	2	2	no	—
31	C₁₃×C₂²⋊Q₈	—	4	2	no	—
32	C₄×Q₈×C₁₃	—	4	2	no	—
33	C₁₃×#117	—	2	2	no	—
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₃	—	2	2	no	—
36	C₂×#6101	—	4	2	no	—
37	#88124	—	2	2	no	—
38	#88124	—	2	2	no	—
39	#88124	—	4	2	no	—
40	C₁₃×#107	—	2	2	no	—
41	C₁₃×#107	C₂³	1	4	yes	—
42	C₁₃×#106	—	2	2	no	—
43	C₁₃×#106	—	2	2	no	—
44	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	#88058	—	2	2	no	—
49	#88058	C₂³	1	4	yes	—
50	#88057	—	2	2	no	—
51	#88057	—	2	2	no	—
52	#88052	Q₈	1	8	yes	complemented
53	C₂×#6090	C₂³	1	4	yes	—
54	#33732	—	13	1	no	maximal, Sylow
55	C₂×Q₈×C₁₃	—	4	2	no	—
56	C₂²×C₄×C₁₃	D₈	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	—	4	2	no	—
59	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
60	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
64	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₁₃×C₄⋊C₄	—	2	4	no	—
66	C₁₃×C₄⋊C₄	—	4	4	no	—
67	C₁₃×C₄⋊C₄	—	4	2	no	—
68	C₁₃×C₂²⋊C₄	—	4	2	no	—
69	C₄²×C₁₃	—	4	2	no	—
70	C₄²×C₁₃	D₈	1	2	yes	—
71	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₄⋊Dic₁₃	—	2	4	no	—
74	C₄⋊Dic₁₃	—	4	2	no	—
75	#6101	—	4	4	no	—
76	#6101	—	4	4	no	—
77	#6090	C₂×Q₈	1	8	yes	—
78	#3160	D₁₃	1	1	yes	characteristic
79	C₂×#499	—	13	1	no	—
80	#2255	—	13	4	no	—
81	#2253	—	13	4	no	—
82	#2159	—	13	4	no	—
83	C₄×M₃₂	—	13	1	no	—
84	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
85	Q₈×C₁₃	—	8	4	no	—
86	C₂×C₄×C₁₃	—	4	2	no	—
87	C₂×C₄×C₁₃	—	4	4	no	—
88	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
89	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
91	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×Dic₁₃	—	4	4	no	—
95	C₂×Dic₁₃	—	8	4	no	—
96	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
97	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
98	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
99	C₈×C₁₃	C₂²×Q₈	1	2	yes	—
100	#533	—	2	1	no	—
101	C₂×#125	—	26	1	no	—
102	C₂×M₃₂	—	13	2	no	—
103	#500	C₂×D₁₃	1	2	yes	—
104	#499	—	2	1	no	—
105	#499	—	13	2	no	—
106	#499	—	13	2	no	—
107	#499	—	26	1	no	—
108	#499	—	26	1	no	—
109	#499	C₂×D₁₃	1	2	yes	—
110	#498	—	2	1	no	—
111	#427	—	2	1	no	—
112	#427	C₂×D₁₃	1	2	yes	—
113	C₂×#107	—	13	2	no	—
114	C₂×#106	—	26	1	no	—
115	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
116	#367	—	13	8	no	—
117	#366	—	13	8	no	—
118	#346	—	13	2	no	—
119	C₄×C₁₆	—	13	2	no	—
120	C₂²×C₁₃	—	2	2	no	—
121	C₂²×C₁₃	#500	1	2	yes	—
122	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	2	2	no	—
125	C₄×C₁₃	—	8	4	no	—
126	C₄×C₁₃	#500	1	2	yes	—
127	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
128	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
129	Dic₁₃	—	8	4	no	—
130	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
132	C₄⋊Q₈	—	2	2	no	—
133	#125	—	2	2	no	—
134	#125	—	26	2	no	—
135	#125	—	26	2	no	—
136	#125	—	52	2	no	—
137	C₂²⋊Q₈	—	4	2	no	—
138	C₄×Q₈	—	4	2	no	—
139	#117	—	2	2	no	—
140	C₂×C₄⋊C₄	—	26	2	no	—
141	C₂×C₄⋊C₄	—	52	2	no	—
142	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
143	M₃₂	—	13	8	no	—
144	C₂×C₁₆	—	13	4	no	—
145	#107	—	2	2	no	—
146	#107	—	13	4	no	—
147	#107	—	26	2	no	—
148	#107	C₂²×D₁₃	1	4	yes	—
149	#106	—	2	2	no	—
150	#106	—	2	2	no	—
151	#106	—	26	2	no	—
152	#106	—	26	2	no	—
153	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
155	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
156	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
157	C₂×C₁₃	—	2	2	no	—
158	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
159	C₂×C₁₃	#3414	1	2	yes	—
160	C₂×Q₈	—	4	2	no	—
161	C₂²×C₄	—	52	2	no	—
162	C₂²×C₄	#6104	1	2	yes	—
163	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
164	C₂×C₈	#6107	1	4	yes	—
165	C₂×C₈	Q₈×D₁₃	1	4	yes	—
166	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
167	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
168	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
169	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	4	4	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	26	4	no	—
174	C₄⋊C₄	—	52	2	no	—
175	C₄⋊C₄	—	52	4	no	—
176	C₄⋊C₄	—	52	4	no	—
177	C₂²⋊C₄	—	4	2	no	—
178	C₄²	—	4	2	no	—
179	C₄²	#6104	1	2	yes	—
180	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
181	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
182	C₁₆	—	13	8	no	—
183	C₁₃	#33732	1	1	yes	characteristic, complemented, Sylow
184	C₂³	C₂×#6104	1	1	yes	characteristic
185	Q₈	—	8	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	52	4	no	—
189	C₂×C₄	—	104	4	no	—
190	C₂×C₄	C₂×#6104	1	2	yes	—
191	C₂×C₄	C₂×#6104	1	2	yes	—
192	C₂×C₄	C₂×#6104	1	1	yes	characteristic
193	C₂×C₄	C₂×#6107	1	2	yes	—
194	C₂×C₄	#88198	1	2	yes	—
195	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
196	C₈	C₂×#6107	1	2	yes	—
197	C₈	C₂×#6107	1	2	yes	—
198	C₈	#88198	1	2	yes	—
199	C₈	C₂×Q₈×D₁₃	1	2	yes	—
200	C₂²	—	2	2	no	—
201	C₂²	#11718739	1	2	yes	—
202	C₂²	C₂²×#6104	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	8	4	no	—
206	C₄	—	104	4	no	—
207	C₄	#11718739	1	2	yes	—
208	C₄	C₂²×#6107	1	1	yes	characteristic
209	C₄	C₂×#88198	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	C₂×#11718739	1	1	yes	characteristic
212	C₂	#13127970	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3160	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	#3160	D₁₃
hypocenter	C₁₃	#33732
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #13117119	C₂	semidirect
3. #13117112	C₂	semidirect
52. #88052	Q₈	semidirect
183. C₁₃	#33732	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718344	2
#13117119	2
#13117112	2
#13116625	2
C₄×#88052	2
C₁₃×#3160	2
#33732	13