Subgroups — #15774521

Order 3328.

250 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116639	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
3	#13112905	C₂	1	2	yes	maximal, complemented
4	#13112899	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2163	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×#1055	C₄	1	1	yes	characteristic
8	C₂×C₄×#1055	C₄	1	1	yes	characteristic
9	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
10	C₁₃×C₂²⋊C₁₆	—	2	1	no	—
11	C₁₃×#346	—	2	1	no	—
12	C₂²⋊#6090	—	2	1	no	—
13	C₄⋊#6090	C₂²	1	2	yes	—
14	#11718053	—	2	1	no	—
15	C₈×#1055	C₄	1	1	yes	characteristic
16	C₈×#1055	C₄	1	1	yes	characteristic
17	C₈×#1055	C₂²	1	1	yes	characteristic
18	C₈×#1055	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×#1055	—	2	1	no	—
23	C₂²×#1055	C₈	1	1	yes	characteristic
24	C₂²×#1055	C₈	1	1	yes	characteristic
25	C₂×C₁₃×C₁₆	—	2	2	no	—
26	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
29	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×#6090	—	2	2	no	—
31	C₄×#1055	—	2	1	no	—
32	C₄×#1055	C₈	1	1	yes	characteristic
33	C₄×#1055	C₈	1	1	yes	characteristic
34	C₄×#1055	C₂×C₄	1	1	yes	characteristic
35	C₄×#1055	C₂×C₄	1	1	yes	characteristic
36	C₄×#1055	C₂×C₄	1	1	yes	characteristic
37	C₄×#1055	C₂×C₄	1	1	yes	characteristic
38	#12013	—	13	1	no	maximal, Sylow
39	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
42	C₂×C₈×C₁₃	—	2	2	no	—
43	C₂×C₈×C₁₃	—	2	2	no	—
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×D₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
51	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×#1055	—	2	4	no	—
53	C₂×#1055	—	2	1	no	—
54	C₂×#1055	—	2	1	no	—
55	C₂×#1055	—	2	2	no	—
56	C₂×#1055	—	2	2	no	—
57	C₂×#1055	C₂×C₈	1	1	yes	characteristic
58	C₂×#1055	C₂×C₈	1	1	yes	characteristic
59	C₂×#1055	C₂×C₈	1	1	yes	characteristic
60	C₂×#1055	C₂×C₈	1	1	yes	characteristic
61	C₁₃×C₁₆	—	4	2	no	—
62	#6090	—	4	2	no	—
63	#2163	—	13	1	no	—
64	#2163	D₁₃	1	1	yes	characteristic
65	C₂×C₈²	—	13	1	no	—
66	#1370	—	13	2	no	—
67	#1367	—	13	2	no	—
68	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	1	no	—
71	C₂×C₄×C₁₃	—	2	1	no	—
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
80	C₈×C₁₃	—	2	1	no	—
81	C₈×C₁₃	—	2	1	no	—
82	C₈×C₁₃	#108	1	2	yes	—
83	C₈×C₁₃	#135	1	2	yes	—
84	#1055	—	2	2	no	—
85	#1055	—	2	2	no	—
86	#1055	—	2	1	no	—
87	#1055	—	2	1	no	—
88	#1055	M₃₂	1	2	yes	—
89	#1055	M₃₂	1	2	yes	—
90	C₂×C₄×C₈	—	13	1	no	—
91	C₂×C₄×C₈	—	13	1	no	—
92	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
93	C₄⋊C₁₆	—	13	2	no	—
94	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
95	C₂²⋊C₁₆	—	2	1	no	—
96	C₂²⋊C₁₆	—	26	1	no	—
97	#346	—	2	1	no	—
98	#346	—	26	1	no	—
99	C₈²	—	13	1	no	—
100	C₈²	—	13	1	no	—
101	C₈²	—	13	1	no	—
102	C₈²	—	13	1	no	—
103	C₂²×C₁₃	—	2	1	no	—
104	C₂²×C₁₃	—	2	1	no	—
105	C₂²×C₁₃	#364	1	2	yes	—
106	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
107	C₄×C₁₃	—	2	1	no	—
108	C₄×C₁₃	—	2	1	no	—
109	C₄×C₁₃	—	2	1	no	—
110	C₄×C₁₃	—	2	1	no	—
111	C₄×C₁₃	#427	1	1	yes	characteristic
112	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
113	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
114	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
115	C₂²×C₈	—	13	1	no	—
116	C₂²×C₈	—	13	1	no	—
117	C₂²×C₈	—	26	1	no	—
118	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
119	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
120	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	26	2	no	—
123	C₄×C₈	—	13	1	no	—
124	C₄×C₈	—	13	1	no	—
125	C₄×C₈	—	13	1	no	—
126	C₄×C₈	—	13	1	no	—
127	C₄×C₈	—	13	1	no	—
128	C₄×C₈	—	13	1	no	—
129	C₄×C₈	—	26	1	no	—
130	C₄×C₈	Dic₂₆	1	1	yes	characteristic
131	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
132	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
133	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
134	C₂×C₁₃	—	2	2	no	—
135	C₂×C₁₃	#1564	1	1	yes	characteristic
136	C₂×C₁₃	#2150	1	1	yes	characteristic
137	C₂×C₁₃	C₂×#364	1	1	yes	characteristic
138	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
139	C₂²×C₄	#6094	1	1	yes	characteristic
140	C₂²×C₄	#6101	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	13	1	no	—
144	C₂×C₈	—	13	1	no	—
145	C₂×C₈	—	13	1	no	—
146	C₂×C₈	—	13	1	no	—
147	C₂×C₈	—	26	2	no	—
148	C₂×C₈	—	26	2	no	—
149	C₂×C₈	—	26	4	no	—
150	C₂×C₈	—	26	1	no	—
151	C₂×C₈	—	26	1	no	—
152	C₂×C₈	#6101	1	1	yes	characteristic
153	C₂×C₈	#6101	1	1	yes	characteristic
154	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
155	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
156	C₄²	C₈×D₁₃	1	1	yes	characteristic
157	C₄²	#6094	1	1	yes	characteristic
158	C₄²	#6101	1	1	yes	characteristic
159	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
160	C₁₆	—	4	2	no	—
161	C₁₆	—	52	2	no	—
162	C₁₃	#12013	1	1	yes	characteristic, complemented, Sylow
163	C₂³	#88054	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	#88054	1	1	yes	characteristic
170	C₂×C₄	#88054	1	1	yes	characteristic
171	C₂×C₄	#88054	1	1	yes	characteristic
172	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
173	C₂×C₄	C₂×#6094	1	1	yes	characteristic
174	C₂×C₄	C₂×#6101	1	1	yes	characteristic
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	13	2	no	—
178	C₈	—	13	2	no	—
179	C₈	—	26	2	no	—
180	C₈	—	26	2	no	—
181	C₈	—	26	1	no	—
182	C₈	—	26	1	no	—
183	C₈	#88062	1	2	yes	—
184	C₈	#88204	1	2	yes	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	#11718103	1	2	yes	—
188	C₂²	C₂×#88054	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	D₁₃×M₃₂	1	1	yes	characteristic
194	C₄	D₁₃×M₃₂	1	1	yes	characteristic
195	C₄	#11718659	1	1	yes	characteristic
196	C₄	C₂×#88062	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	#13113582	1	1	yes	characteristic
199	C₂	#13116591	1	1	yes	characteristic
200	C₂	C₂×#11718103	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#2163	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₄⋊C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#2163	D₁₃
hypocenter	C₁₃	#12013
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
3. #13112905	C₂	semidirect
4. #13112899	C₂	semidirect
162. C₁₃	#12013	semidirect

Maximal subgroups

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

subgroup	index
#13116639	2
C₂×C₈×#1055	2
#13112905	2
#13112899	2
C₁₃×#2163	2
#12013	13