Subgroups — #15778280

Order 3328.

334 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718055	C₂	1	1	yes	maximal, characteristic
2	#13116433	C₂	1	2	yes	maximal
3	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic
4	#13112925	C₂	1	2	yes	maximal
5	C₁₃×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#108	C₄	1	1	yes	characteristic
7	C₁₃×#427	C₄	1	1	yes	characteristic
8	C₁₃×#405	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88052	C₂²	1	2	yes	—
11	C₂²×#6090	C₂²	1	1	yes	characteristic
12	C₂²×#6090	C₂²	1	1	yes	characteristic
13	C₁₃×#339	C₄	1	2	yes	—
14	#11718055	C₄	1	4	yes	complemented
15	#11718053	—	2	2	no	—
16	C₄×#6090	—	2	2	no	—
17	C₄×#6090	C₄	1	4	yes	complemented
18	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₁₃×#117	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₁₃×#108	C₈	1	4	yes	—
24	C₁₃×#107	C₈	1	2	yes	—
25	C₁₃×#106	C₈	1	2	yes	—
26	C₁₃×#97	—	2	2	no	—
27	C₄×C₈×C₁₃	—	2	2	no	—
28	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	#88052	—	2	2	no	—
31	#88052	C₂×C₄	1	4	yes	—
32	C₂×#6090	—	2	2	no	—
33	C₂×#6090	—	2	1	no	—
34	C₂×#6090	—	2	1	no	—
35	C₂×#6090	C₂×C₄	1	2	yes	—
36	C₂×#6090	C₂×C₄	1	2	yes	—
37	C₂×#6090	D₄	1	1	yes	characteristic
38	C₂×#6090	D₄	1	1	yes	characteristic
39	C₂×#6090	Q₈	1	2	yes	—
40	#12664	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	—	2	1	no	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₁₃×M₁₆	—	2	2	no	—
44	C₁₃×M₁₆	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₁₃	—	2	2	no	—
46	C₂×C₈×C₁₃	C₄²	1	2	yes	—
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	2	yes	—
50	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
55	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
56	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
57	C₁₃×C₂²⋊C₄	—	2	2	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₄²×C₁₃	—	2	1	no	—
60	C₄²×C₁₃	C₄²	1	2	yes	—
61	#6090	—	2	2	no	—
62	#6090	—	2	4	no	—
63	#6090	C₄⋊C₄	1	2	yes	—
64	#6090	C₄⋊C₄	1	2	yes	—
65	C₂×#364	—	13	1	no	—
66	#2108	—	13	2	no	—
67	C₂×C₄×C₁₆	—	13	1	no	—
68	#1772	D₁₃	1	1	yes	characteristic
69	#1379	—	13	2	no	—
70	C₂³×C₁₃	#131	1	1	yes	characteristic
71	C₂×C₄×C₁₃	—	2	4	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	4	no	—
74	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
80	C₈×C₁₃	—	2	4	no	—
81	C₈×C₁₃	C₄×C₈	1	2	yes	—
82	C₈×C₁₃	C₄×C₈	1	2	yes	—
83	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₈×C₁₃	#117	1	1	yes	characteristic
85	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
86	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
87	C₂×M₃₂	—	13	2	no	—
88	C₂²×C₁₆	—	13	1	no	—
89	C₂²×C₁₆	—	13	1	no	—
90	C₂×#108	Dic₁₃	1	1	yes	characteristic
91	#427	Dic₁₃	1	1	yes	characteristic
92	#405	C₂×D₁₃	1	2	yes	—
93	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
94	#364	—	13	4	no	—
95	#346	—	26	2	no	—
96	C₄×C₁₆	—	13	4	no	—
97	C₄×C₁₆	—	26	2	no	—
98	#339	Dic₁₃	1	2	yes	—
99	C₂²×C₁₃	—	2	2	no	—
100	C₂²×C₁₃	#405	1	1	yes	characteristic
101	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
102	C₂²×C₁₃	C₈×Q₈	1	1	yes	characteristic
103	C₄×C₁₃	—	2	4	no	—
104	C₄×C₁₃	—	4	4	no	—
105	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
106	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
107	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
108	C₄×C₁₃	C₈×Q₈	1	1	yes	characteristic
109	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
110	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
111	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
112	#117	C₂×Dic₁₃	1	2	yes	—
113	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
114	M₃₂	—	13	4	no	—
115	M₃₂	—	26	2	no	—
116	C₂×C₁₆	—	13	2	no	—
117	C₂×C₁₆	—	13	1	no	—
118	C₂×C₁₆	—	13	1	no	—
119	C₂×C₁₆	—	13	2	no	—
120	C₂×C₁₆	—	13	2	no	—
121	C₂×C₁₆	—	26	2	no	—
122	C₂×C₁₆	—	26	1	no	—
123	C₂×C₁₆	—	26	1	no	—
124	#108	#1055	1	4	yes	—
125	#107	#1055	1	2	yes	—
126	#106	#1055	1	2	yes	—
127	#97	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	C₄×D₁₃	1	2	yes	—
130	C₄×C₈	C₄×D₁₃	1	2	yes	—
131	C₂×C₁₃	—	2	1	no	—
132	C₂×C₁₃	—	2	1	no	—
133	C₂×C₁₃	C₈×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₁₃	#2170	1	2	yes	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
137	M₁₆	—	2	2	no	—
138	M₁₆	C₂×#1055	1	4	yes	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	C₂×#1055	1	1	yes	characteristic
141	C₂×C₈	C₂×#1055	1	1	yes	characteristic
142	C₂×C₈	C₂×#1055	1	2	yes	—
143	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
144	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
145	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
146	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
147	C₂×C₈	#6129	1	1	yes	characteristic
148	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
149	C₂×C₈	#6131	1	1	yes	characteristic
150	C₄⋊C₄	C₂×#1055	1	4	yes	—
151	C₂²⋊C₄	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	1	no	—
154	C₄²	C₄×Dic₁₃	1	2	yes	—
155	C₁₆	—	13	2	no	—
156	C₁₆	—	13	2	no	—
157	C₁₆	—	26	2	no	—
158	C₁₆	—	26	4	no	—
159	C₁₃	#12664	1	1	yes	characteristic, complemented, Sylow
160	C₂³	#88202	1	1	yes	characteristic
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
165	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
167	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
168	C₂×C₄	Q₈×Dic₁₃	1	1	yes	characteristic
169	C₂×C₄	#88202	1	1	yes	characteristic
170	C₈	—	2	4	no	—
171	C₈	C₄×#1055	1	2	yes	—
172	C₈	C₄×#1055	1	2	yes	—
173	C₈	#88141	1	1	yes	characteristic
174	C₈	C₄⋊C₄×D₁₃	1	1	yes	characteristic
175	C₈	#88146	1	1	yes	characteristic
176	C₈	#88147	1	1	yes	characteristic
177	C₂²	—	2	2	no	—
178	C₂²	#11718658	1	1	yes	characteristic
179	C₂²	D₄×#1055	1	1	yes	characteristic
180	C₂²	Q₈×#1055	1	1	yes	characteristic
181	C₄	—	2	4	no	—
182	C₄	—	4	4	no	—
183	C₄	C₂×C₄×#1055	1	1	yes	characteristic
184	C₄	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
185	C₄	D₄×#1055	1	1	yes	characteristic
186	C₄	Q₈×#1055	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	C₄⋊C₄×#1055	1	1	yes	characteristic
190	C₂	#13116673	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1772	C₂
socle	C₂²×C₁₃	C₈×D₄
prime radical	C₂³×C₁₃	#131
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1772	D₁₃
hypocenter	C₁₃	#12664
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
14. #11718055	C₄	semidirect
17. C₄×#6090	C₄	semidirect
159. C₁₃	#12664	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718055	2
#13116433	2
C₂×C₄×#6090	2
#13112925	2
C₁₃×#1772	2
#12664	13