Subgroups — #16790814

Order 3328.

284 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131177	C₂	1	2	yes	maximal
2	C₂×#11718261	C₂	1	1	yes	maximal, characteristic
3	#13131042	C₂	1	1	yes	maximal, characteristic
4	#13129887	C₂	1	2	yes	maximal
5	#13114718	C₂	1	1	yes	maximal, characteristic
6	C₂×#88106	C₂²	1	2	yes	—
7	C₂×#88106	C₂²	1	1	yes	characteristic
8	C₂×#88106	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#108	C₄	1	1	yes	characteristic
10	#11718783	C₂²	1	2	yes	—
11	#11718767	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
13	#11718261	—	2	1	no	—
14	#11718261	—	2	1	no	—
15	#11718261	—	2	4	no	—
16	#11718258	—	2	1	no	—
17	#11718257	—	2	1	no	—
18	#11718147	C₄	1	2	yes	—
19	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	#88180	C₂×C₄	1	2	yes	—
22	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×#1055	C₂³	1	1	yes	characteristic
24	#88106	—	2	1	no	—
25	#88106	—	2	1	no	—
26	#88106	—	2	2	no	—
27	#88106	—	2	2	no	—
28	#88106	D₄	1	2	yes	—
29	#88106	Q₈	1	2	yes	—
30	C₂×#6092	—	2	1	no	—
31	C₂×#6092	—	2	1	no	—
32	C₂×#6092	—	2	2	no	—
33	C₁₃×#108	C₈	1	2	yes	—
34	C₁₃×#108	C₈	1	2	yes	—
35	#88058	—	2	1	no	—
36	#88057	—	2	1	no	—
37	#88055	—	2	2	no	—
38	C₈×Dic₁₃	—	2	2	no	—
39	C₈×Dic₁₃	D₄	1	1	yes	characteristic
40	C₈×Dic₁₃	D₄	1	1	yes	characteristic
41	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
42	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
43	#13164	—	13	1	no	maximal, Sylow
44	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₁₃	—	2	1	no	—
46	C₁₃×M₁₆	—	2	2	no	—
47	C₁₃×M₁₆	C₂×C₈	1	4	yes	—
48	C₂×C₈×C₁₃	—	2	2	no	—
49	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
53	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
55	C₂²⋊Dic₁₃	—	2	2	no	—
56	C₄⋊Dic₁₃	—	2	2	no	—
57	C₄×Dic₁₃	—	2	1	no	—
58	C₄×Dic₁₃	—	2	2	no	—
59	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
60	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂×#1055	C₂×D₄	1	1	yes	characteristic
62	C₂×#1055	C₂×D₄	1	1	yes	characteristic
63	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
64	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
65	C₂×#1055	#41	1	1	yes	characteristic
66	C₂×#1055	#41	1	1	yes	characteristic
67	#6092	—	2	2	no	—
68	#6092	—	2	2	no	—
69	#6092	—	4	2	no	—
70	#2153	—	13	2	no	—
71	C₂×#364	—	13	1	no	—
72	C₄×M₃₂	—	13	1	no	—
73	#1772	—	13	1	no	—
74	#1378	—	13	2	no	—
75	C₂³×C₁₃	#131	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
82	C₂×Dic₁₃	—	2	1	no	—
83	C₂×Dic₁₃	—	2	1	no	—
84	C₂×Dic₁₃	—	4	2	no	—
85	C₂×Dic₁₃	—	4	2	no	—
86	C₈×C₁₃	—	2	4	no	—
87	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
88	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
89	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
90	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
91	#1055	#121	1	1	yes	characteristic
92	#1055	C₂²⋊Q₈	1	1	yes	characteristic
93	#1055	#125	1	1	yes	characteristic
94	#1055	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×M₃₂	—	13	1	no	—
96	C₂×M₃₂	—	13	1	no	—
97	C₂×M₃₂	—	13	2	no	—
98	C₂×#108	#261	1	1	yes	characteristic
99	#427	—	13	1	no	—
100	#405	—	13	2	no	—
101	C₂×C₄×C₈	—	13	1	no	—
102	#364	—	26	1	no	—
103	#364	—	26	1	no	—
104	#364	—	26	4	no	—
105	#346	—	26	1	no	—
106	C₄×C₁₆	—	26	1	no	—
107	#339	—	13	2	no	—
108	C₂²×C₁₃	—	4	1	no	—
109	C₂²×C₁₃	#424	1	1	yes	characteristic
110	C₂²×C₁₃	#436	1	1	yes	characteristic
111	C₂²×C₁₃	C₈×Q₈	1	1	yes	characteristic
112	C₄×C₁₃	#389	1	1	yes	characteristic
113	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
115	C₄×C₁₃	#446	1	1	yes	characteristic
116	Dic₁₃	—	2	2	no	—
117	Dic₁₃	—	4	1	no	—
118	Dic₁₃	—	8	2	no	—
119	C₂×M₁₆	C₂×#261	1	2	yes	—
120	C₂²×C₈	—	13	1	no	—
121	C₂²×C₈	C₂×#261	1	1	yes	characteristic
122	#117	—	13	2	no	—
123	C₂×C₄²	—	13	1	no	—
124	M₃₂	—	13	2	no	—
125	M₃₂	—	13	2	no	—
126	M₃₂	—	26	1	no	—
127	M₃₂	—	26	1	no	—
128	M₃₂	—	26	2	no	—
129	M₃₂	—	26	2	no	—
130	C₂×C₁₆	—	26	1	no	—
131	C₂×C₁₆	—	26	1	no	—
132	C₂×C₁₆	—	26	2	no	—
133	#108	#1057	1	2	yes	—
134	#108	#1057	1	2	yes	—
135	#107	—	26	1	no	—
136	#106	—	26	1	no	—
137	#97	—	26	2	no	—
138	C₄×C₈	—	13	1	no	—
139	C₄×C₈	—	13	1	no	—
140	C₄×C₈	—	13	1	no	—
141	C₄×C₈	—	13	1	no	—
142	C₄×C₈	—	26	2	no	—
143	C₂×C₁₃	—	2	1	no	—
144	C₂×C₁₃	—	2	1	no	—
145	C₂×C₁₃	—	2	1	no	—
146	C₂×C₁₃	#1939	1	1	yes	characteristic
147	C₂²×C₄	—	26	1	no	—
148	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
149	M₁₆	—	2	2	no	—
150	M₁₆	C₂×#1057	1	4	yes	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	13	1	no	—
153	C₂×C₈	—	13	1	no	—
154	C₂×C₈	—	13	1	no	—
155	C₂×C₈	—	13	1	no	—
156	C₂×C₈	—	13	1	no	—
157	C₂×C₈	—	13	1	no	—
158	C₂×C₈	#6118	1	1	yes	characteristic
159	C₂×C₈	#6118	1	1	yes	characteristic
160	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
161	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
162	C₂×C₈	C₂×#1057	1	1	yes	characteristic
163	C₂×C₈	C₂×#1057	1	1	yes	characteristic
164	C₄⋊C₄	—	26	2	no	—
165	C₂²⋊C₄	—	26	2	no	—
166	C₄²	—	13	1	no	—
167	C₄²	—	13	1	no	—
168	C₄²	—	26	1	no	—
169	C₄²	—	26	2	no	—
170	C₁₆	—	26	2	no	—
171	C₁₆	—	26	2	no	—
172	C₁₆	—	52	2	no	—
173	C₁₃	#13164	1	1	yes	characteristic, complemented, Sylow
174	C₂³	#88240	1	1	yes	characteristic
175	C₂×C₄	—	26	1	no	—
176	C₂×C₄	—	26	1	no	—
177	C₂×C₄	—	52	2	no	—
178	C₂×C₄	—	52	2	no	—
179	C₂×C₄	C₂×#6118	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
181	C₂×C₄	#88238	1	1	yes	characteristic
182	C₂×C₄	D₄×#261	1	1	yes	characteristic
183	C₂×C₄	Q₈×#261	1	1	yes	characteristic
184	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	13	1	no	—
187	C₈	—	13	1	no	—
188	C₈	—	13	1	no	—
189	C₈	—	13	1	no	—
190	C₈	C₄⋊#1057	1	1	yes	characteristic
191	C₈	C₄⋊#1057	1	1	yes	characteristic
192	C₈	C₄⋊#1057	1	1	yes	characteristic
193	C₈	C₄⋊#1057	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#11719076	1	1	yes	characteristic
196	C₂²	#11719079	1	1	yes	characteristic
197	C₂²	Q₈×#1057	1	1	yes	characteristic
198	C₄	—	26	2	no	—
199	C₄	—	52	1	no	—
200	C₄	—	104	2	no	—
201	C₄	#11719080	1	1	yes	characteristic
202	C₄	#11719083	1	1	yes	characteristic
203	C₄	C₂×C₄⋊#1057	1	1	yes	characteristic
204	C₄	D₄×#1057	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#13130933	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11719083
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×#108	C₄
socle	C₂×C₁₃	#1939
prime radical	C₂³×C₁₃	#131
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×#108	#261
hypocenter	C₁₃	#13164
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
173. C₁₃	#13164	semidirect

Maximal subgroups

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

subgroup	index
#13131177	2
C₂×#11718261	2
#13131042	2
#13129887	2
#13114718	2
#13164	13