Subgroups — #15774519

Order 3328.

262 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	#13116635	C₂	1	2	yes	maximal
2	#13112906	C₂	1	2	yes	maximal, complemented
3	#13112900	C₂	1	2	yes	maximal, complemented
4	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₈²×C₁₃	C₄	1	1	yes	characteristic
9	C₈²×C₁₃	C₄	1	1	yes	characteristic
10	C₈²×C₁₃	C₂²	1	1	yes	characteristic
11	C₈²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂²⋊#6090	—	2	2	no	—
13	C₄⋊#6090	C₂²	1	4	yes	—
14	C₄×#6090	—	2	2	no	—
15	C₂²×C₈×C₁₃	—	2	1	no	—
16	C₂²×C₈×C₁₃	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₈	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₃	—	2	1	no	—
22	C₄×C₈×C₁₃	C₈	1	1	yes	characteristic
23	C₄×C₈×C₁₃	C₈	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×#6090	—	2	4	no	—
33	#12012	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₃	—	2	4	no	—
38	C₂×C₈×C₁₃	—	2	2	no	—
39	C₂×C₈×C₁₃	—	2	1	no	—
40	C₂×C₈×C₁₃	—	2	1	no	—
41	C₂×C₈×C₁₃	—	2	2	no	—
42	C₂×C₈×C₁₃	—	2	2	no	—
43	C₂×C₈×C₁₃	C₁₆	1	4	yes	complemented
44	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
55	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	#6090	—	4	4	no	—
57	#2162	—	13	2	no	—
58	C₂×C₈²	D₁₃	1	1	yes	characteristic
59	#1370	—	13	2	no	—
60	#1367	—	13	2	no	—
61	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	—	2	1	no	—
64	C₂×C₄×C₁₃	—	2	1	no	—
65	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
69	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
70	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
73	C₈×C₁₃	—	2	2	no	—
74	C₈×C₁₃	—	2	2	no	—
75	C₈×C₁₃	—	2	1	no	—
76	C₈×C₁₃	—	2	1	no	—
77	C₈×C₁₃	—	2	1	no	—
78	C₈×C₁₃	—	2	1	no	—
79	C₈×C₁₃	#108	1	2	yes	—
80	C₈×C₁₃	C₂×C₁₆	1	2	yes	—
81	C₈×C₁₃	C₂×C₁₆	1	2	yes	—
82	C₈×C₁₃	#135	1	2	yes	—
83	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
84	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
86	C₄⋊C₁₆	—	13	4	no	—
87	C₂²⋊C₁₆	—	26	2	no	—
88	C₄×C₁₆	—	26	2	no	—
89	C₈²	Dic₁₃	1	1	yes	characteristic
90	C₈²	Dic₁₃	1	1	yes	characteristic
91	C₈²	C₂×D₁₃	1	1	yes	characteristic
92	C₈²	C₂×D₁₃	1	1	yes	characteristic
93	C₂²×C₁₃	—	2	1	no	—
94	C₂²×C₁₃	—	2	1	no	—
95	C₂²×C₁₃	C₄⋊C₁₆	1	2	yes	—
96	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₄×C₁₃	—	2	1	no	—
98	C₄×C₁₃	—	2	1	no	—
99	C₄×C₁₃	C₄⋊C₁₆	1	2	yes	—
100	C₄×C₁₃	#364	1	2	yes	—
101	C₄×C₁₃	#427	1	1	yes	characteristic
102	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
103	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
104	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
105	C₂²×C₈	—	2	1	no	—
106	C₂²×C₈	#1055	1	1	yes	characteristic
107	C₂²×C₈	#1055	1	1	yes	characteristic
108	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
109	C₂²×C₈	D₅₂	1	1	yes	characteristic
110	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
111	C₂×C₁₆	—	26	4	no	—
112	C₄×C₈	—	2	1	no	—
113	C₄×C₈	#1055	1	1	yes	characteristic
114	C₄×C₈	#1055	1	1	yes	characteristic
115	C₄×C₈	Dic₂₆	1	1	yes	characteristic
116	C₄×C₈	D₅₂	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
118	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
119	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
121	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
122	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
123	C₂×C₁₃	—	2	2	no	—
124	C₂×C₁₃	#1564	1	1	yes	characteristic
125	C₂×C₁₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
126	C₂×C₁₃	C₂×#364	1	1	yes	characteristic
127	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
128	C₂²×C₄	#6099	1	1	yes	characteristic
129	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	#6090	1	4	yes	complemented
137	C₂×C₈	C₂×#1055	1	1	yes	characteristic
138	C₂×C₈	C₂×#1055	1	1	yes	characteristic
139	C₂×C₈	C₂×#1055	1	1	yes	characteristic
140	C₂×C₈	C₂×#1055	1	1	yes	characteristic
141	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
142	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
143	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
144	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
145	C₄²	C₂×#1055	1	1	yes	characteristic
146	C₄²	#6099	1	1	yes	characteristic
147	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
148	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
149	C₁₆	—	52	4	no	—
150	C₁₃	#12012	1	1	yes	characteristic, complemented, Sylow
151	C₂³	C₄⋊#1055	1	1	yes	characteristic
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
156	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
157	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
158	C₂×C₄	C₂×#6090	1	2	yes	—
159	C₂×C₄	#88052	1	2	yes	—
160	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
161	C₂×C₄	C₂×#6099	1	1	yes	characteristic
162	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
163	C₈	—	2	2	no	—
164	C₈	—	2	2	no	—
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	C₂×#6090	1	2	yes	—
170	C₈	C₂×#6090	1	2	yes	—
171	C₈	#88059	1	2	yes	—
172	C₈	#88158	1	2	yes	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	C₄⋊#6090	1	2	yes	—
176	C₂²	C₂×C₄⋊#1055	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	C₄⋊#6090	1	2	yes	—
180	C₄	#11718055	1	2	yes	—
181	C₄	C₂²×#6090	1	1	yes	characteristic
182	C₄	C₂×#88052	1	1	yes	characteristic
183	C₄	#11718767	1	1	yes	characteristic
184	C₄	C₂×#88059	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#13113584	1	1	yes	characteristic
187	C₂	C₂×C₄⋊#6090	1	1	yes	characteristic
188	C₂	C₂×#11718055	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	D₅₂
commutator	C₄×C₁₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₂×C₈²×C₁₃	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₄⋊C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₂×C₈²	D₁₃
hypocenter	C₁₃	#12012
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #13112906	C₂	semidirect
3. #13112900	C₂	semidirect
43. C₂×C₈×C₁₃	C₁₆	semidirect
136. C₂×C₈	#6090	semidirect
150. C₁₃	#12012	semidirect

Maximal subgroups

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

subgroup	index
#13116635	2
#13112906	2
#13112900	2
C₂×C₈²×C₁₃	2
#12012	13