Subgroups — #16788713

Order 3328.

246 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131193	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#6092	C₂	1	1	yes	maximal, characteristic
3	#13129865	C₂	1	2	yes	maximal, complemented
4	#13129858	C₂	1	2	yes	maximal, complemented
5	M₁₆×#1055	C₂	1	1	yes	maximal, characteristic
6	C₂²×#6092	—	2	1	no	—
7	C₄×C₁₃×M₁₆	C₄	1	1	yes	characteristic
8	#11718586	C₄	1	1	yes	characteristic
9	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
10	C₂²⋊#6092	—	2	1	no	—
11	C₄⋊#6092	—	2	1	no	—
12	C₄⋊#6092	C₂²	1	1	yes	characteristic
13	C₄⋊#6092	C₂²	1	2	yes	—
14	C₄⋊#6092	C₂²	1	1	yes	characteristic
15	C₄×#6092	—	2	1	no	—
16	#11718047	C₄	1	2	yes	—
17	C₈×#1055	C₂²	1	2	yes	—
18	C₂×C₁₃×M₁₆	C₈	1	1	yes	characteristic
19	C₂×C₁₃×M₁₆	C₈	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×#1055	D₄	1	1	yes	characteristic
22	C₂²×#1055	Q₈	1	1	yes	characteristic
23	C₂×#6092	—	2	1	no	—
24	C₂×#6092	—	2	1	no	—
25	C₂×#6092	—	2	2	no	—
26	C₂×#6092	—	4	2	no	—
27	C₁₃×#97	C₈	1	2	yes	—
28	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂²⋊#1055	—	2	1	no	—
30	C₄⋊#1055	—	2	1	no	—
31	C₄×#1055	C₂×C₄	1	2	yes	—
32	C₄×#1055	C₂×C₄	1	1	yes	characteristic
33	C₄×#1055	D₄	1	1	yes	characteristic
34	C₄×#1055	Q₈	1	1	yes	characteristic
35	C₄×#1055	C₂³	1	1	yes	characteristic
36	#12015	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
40	C₁₃×M₁₆	—	2	2	no	—
41	C₁₃×M₁₆	C₁₆	1	4	yes	complemented
42	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
44	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
47	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×#1055	—	2	1	no	—
49	C₂×#1055	—	2	1	no	—
50	C₂×#1055	—	2	2	no	—
51	C₂×#1055	—	4	1	no	—
52	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
53	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
54	C₂×#1055	C₂×D₄	1	1	yes	characteristic
55	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
56	#6092	—	4	1	no	—
57	#6092	—	4	2	no	—
58	#6092	—	4	1	no	—
59	#2162	—	13	1	no	—
60	C₂×C₄⋊C₁₆	—	13	1	no	—
61	C₈×M₁₆	—	13	1	no	—
62	#1370	—	13	2	no	—
63	#1367	—	13	2	no	—
64	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	—	2	1	no	—
66	C₂×C₄×C₁₃	—	2	1	no	—
67	C₂×C₄×C₁₃	—	4	1	no	—
68	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
72	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
73	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
76	C₈×C₁₃	—	2	2	no	—
77	C₈×C₁₃	C₂×C₁₆	1	4	yes	complemented
78	#1055	—	2	1	no	—
79	#1055	—	2	1	no	—
80	#1055	—	2	1	no	—
81	#1055	—	4	2	no	—
82	#1055	AGL(1,ℤ/8)	1	1	yes	characteristic
83	#1055	#137	1	1	yes	characteristic
84	C₂²×C₁₆	—	26	1	no	—
85	#431	—	13	1	no	—
86	C₄×M₁₆	#261	1	1	yes	characteristic
87	C₂×C₄×C₈	—	13	1	no	—
88	C₄⋊C₁₆	—	13	1	no	—
89	C₄⋊C₁₆	—	13	2	no	—
90	C₄⋊C₁₆	—	13	1	no	—
91	C₄⋊C₁₆	—	26	1	no	—
92	C₂²⋊C₁₆	—	26	1	no	—
93	C₄×C₁₆	—	26	1	no	—
94	#322	—	13	2	no	—
95	C₈²	—	13	2	no	—
96	C₂²×C₁₃	—	2	1	no	—
97	C₂²×C₁₃	—	2	1	no	—
98	C₂²×C₁₃	C₄⋊C₁₆	1	2	yes	—
99	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₄×C₁₃	—	2	1	no	—
101	C₄×C₁₃	—	2	1	no	—
102	C₄×C₁₃	—	2	1	no	—
103	C₄×C₁₃	C₄⋊C₁₆	1	2	yes	—
104	C₄×C₁₃	#428	1	1	yes	characteristic
105	C₄×C₁₃	#430	1	1	yes	characteristic
106	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
107	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
108	C₂×M₁₆	#1057	1	1	yes	characteristic
109	C₂×M₁₆	#1057	1	1	yes	characteristic
110	C₂²×C₈	—	13	1	no	—
111	C₂²×C₈	—	13	1	no	—
112	C₂×C₄²	C₂×#261	1	1	yes	characteristic
113	C₂×C₁₆	—	26	1	no	—
114	C₂×C₁₆	—	26	1	no	—
115	C₂×C₁₆	—	26	2	no	—
116	C₂×C₁₆	—	52	2	no	—
117	C₄⋊C₈	—	26	1	no	—
118	C₂²⋊C₈	—	26	1	no	—
119	#97	#1057	1	2	yes	—
120	C₄×C₈	—	13	1	no	—
121	C₄×C₈	—	13	1	no	—
122	C₄×C₈	—	13	1	no	—
123	C₄×C₈	—	13	1	no	—
124	C₄×C₈	—	13	2	no	—
125	C₄×C₈	C₂×#261	1	2	yes	—
126	C₂×C₁₃	—	2	2	no	—
127	C₂×C₁₃	#1565	1	1	yes	characteristic
128	C₂×C₁₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
129	C₂×C₁₃	#2153	1	1	yes	characteristic
130	C₂²×C₄	#6118	1	1	yes	characteristic
131	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
132	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
133	M₁₆	—	2	2	no	—
134	M₁₆	#6092	1	4	yes	complemented
135	C₂×C₈	—	13	1	no	—
136	C₂×C₈	—	13	1	no	—
137	C₂×C₈	—	13	1	no	—
138	C₂×C₈	—	13	1	no	—
139	C₂×C₈	—	26	1	no	—
140	C₂×C₈	—	26	1	no	—
141	C₂×C₈	—	26	2	no	—
142	C₂×C₈	—	52	1	no	—
143	C₂×C₈	C₂×#1057	1	2	yes	—
144	C₂×C₈	C₂×#1057	1	2	yes	—
145	C₄²	#6118	1	1	yes	characteristic
146	C₄²	C₄⋊#261	1	1	yes	characteristic
147	C₄²	C₂×#1057	1	1	yes	characteristic
148	C₄²	C₂²×#261	1	1	yes	characteristic
149	C₁₆	—	52	2	no	—
150	C₁₆	—	52	1	no	—
151	C₁₆	—	52	1	no	—
152	C₁₃	#12015	1	1	yes	characteristic, complemented, Sylow
153	C₂³	C₄⋊#1057	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	C₂×#6092	1	2	yes	—
158	C₂×C₄	#88106	1	2	yes	—
159	C₂×C₄	C₄⋊#1057	1	1	yes	characteristic
160	C₂×C₄	C₄⋊#1057	1	1	yes	characteristic
161	C₂×C₄	C₄⋊#1057	1	1	yes	characteristic
162	C₂×C₄	C₂×#6118	1	1	yes	characteristic
163	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
164	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	13	1	no	—
167	C₈	—	13	1	no	—
168	C₈	—	26	1	no	—
169	C₈	—	26	1	no	—
170	C₈	—	26	1	no	—
171	C₈	—	52	2	no	—
172	C₈	C₂×#6092	1	4	yes	complemented
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	C₄⋊#6092	1	2	yes	—
176	C₂²	C₂×C₄⋊#1057	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₄⋊#6092	1	2	yes	—
181	C₄	#11719098	1	1	yes	characteristic
182	C₄	#11719100	1	1	yes	characteristic
183	C₄	C₂²×#6092	1	1	yes	characteristic
184	C₄	C₂×#88106	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#13130387	1	1	yes	characteristic
187	C₂	C₂×C₄⋊#6092	1	1	yes	characteristic
188	C₂	#13131177	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#261
commutator	C₄×C₁₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#261
Fitting	C₄×C₁₃×M₁₆	C₄
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₄⋊C₈
cosocle	C₄×#1055	C₂³
hypercenter	C₄×M₁₆	#261
hypocenter	C₁₃	#12015
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13129865	C₂	semidirect
4. #13129858	C₂	semidirect
41. C₁₃×M₁₆	C₁₆	semidirect
77. C₈×C₁₃	C₂×C₁₆	semidirect
134. M₁₆	#6092	semidirect
152. C₁₃	#12015	semidirect
172. C₈	C₂×#6092	semidirect

Maximal subgroups

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

subgroup	index
#13131193	2
C₂×C₄⋊#6092	2
#13129865	2
#13129858	2
M₁₆×#1055	2
#12015	13