Finite Group Databasebeta

Subgroups — #16790236

Order 3328.

234 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130755	C₂	1	2	yes	maximal
2	#13129906	C₂	1	1	yes	maximal, characteristic
3	#13129904	C₂	1	1	yes	maximal, characteristic
4	#13129898	C₂	1	2	yes	maximal
5	#13116434	C₂	1	1	yes	maximal, characteristic
6	D₁₃⋊#108	C₂²	1	2	yes	—
7	#11719092	C₂²	1	2	yes	—
8	C₂×#88038	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
10	#11718350	C₂²	1	1	yes	characteristic
11	#11718266	—	2	2	no	—
12	#11718235	—	2	2	no	—
13	C₁₃×#346	C₄	1	1	yes	characteristic, complemented
14	#11718094	C₄	1	1	yes	characteristic, complemented
15	C₁₆×Dic₁₃	C₄	1	1	yes	characteristic, complemented
16	C₄×#6090	C₄	1	1	yes	characteristic, complemented
17	#88237	—	2	2	no	—
18	D₁₃⋊M₁₆	—	2	2	no	—
19	C₂×#6094	D₄	1	1	yes	characteristic
20	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
21	#88126	D₄	1	1	yes	characteristic
22	#88104	—	2	2	no	—
23	#88103	—	2	2	no	—
24	#88102	—	2	2	no	—
25	#88101	—	2	2	no	—
26	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	#88055	Q₈	1	1	yes	characteristic
30	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×#6090	C₂×C₄	1	1	yes	characteristic
32	C₂×#6090	C₂×C₄	1	1	yes	characteristic
33	#88043	Q₈	1	1	yes	characteristic
34	#88038	—	2	2	no	—
35	C₁₆×D₁₃	—	2	2	no	—
36	#12893	—	13	1	no	maximal, Sylow
37	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂²⋊#261	—	4	2	no	—
39	#6122	—	4	2	no	—
40	C₄⋊#261	—	4	2	no	—
41	C₄×#261	—	4	2	no	—
42	#6118	—	4	2	no	—
43	D₁₃⋊C₈	—	4	2	no	—
44	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	#6103	—	2	1	no	—
48	#6101	—	2	1	no	—
49	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
51	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
52	#6094	—	2	2	no	—
53	C₈×D₁₃	#41	1	2	yes	—
54	C₈×D₁₃	#41	1	2	yes	—
55	C₁₃×C₁₆	—	2	1	no	—
56	C₁₃×C₁₆	—	2	1	no	—
57	#6090	—	2	1	no	—
58	#6090	—	2	1	no	—
59	#2108	—	13	1	no	—
60	#1833	—	13	2	no	—
61	#1385	—	13	1	no	—
62	#1384	—	13	1	no	—
63	#1382	—	13	2	no	—
64	C₂²×D₁₃	#135	1	1	yes	characteristic
65	C₂×#261	—	8	2	no	—
66	C₂×C₄×C₁₃	—	2	1	no	—
67	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×Dic₁₃	—	2	1	no	—
69	C₂×Dic₁₃	#135	1	1	yes	characteristic
70	C₄×D₁₃	#124	1	2	yes	—
71	C₄×D₁₃	#135	1	2	yes	—
72	#1057	—	8	2	no	—
73	C₈×C₁₃	—	2	1	no	—
74	C₈×C₁₃	#117	1	1	yes	characteristic
75	C₈×C₁₃	#117	1	1	yes	characteristic
76	#1055	—	2	1	no	—
77	#1055	#125	1	2	yes	—
78	C₂×M₃₂	—	13	1	no	—
79	C₂²×C₁₆	—	13	1	no	—
80	C₂×#108	—	13	2	no	—
81	#427	—	13	2	no	—
82	#405	—	13	1	no	—
83	#346	—	13	1	no	—
84	#346	#261	1	1	yes	characteristic, complemented
85	C₄×C₁₆	—	13	1	no	—
86	C₄×C₁₆	—	13	1	no	—
87	#344	—	26	2	no	—
88	H(3,ℤ/4)	—	26	2	no	—
89	C₂²×C₁₃	#427	1	1	yes	characteristic
90	C₂×D₁₃	—	4	1	no	—
91	C₂×D₁₃	#488	1	2	yes	—
92	#261	—	16	2	no	—
93	C₄×C₁₃	—	4	1	no	—
94	C₄×C₁₃	#382	1	1	yes	characteristic
95	C₄×C₁₃	#427	1	1	yes	characteristic
96	Dic₁₃	—	4	1	no	—
97	Dic₁₃	#488	1	2	yes	—
98	C₂×M₁₆	—	13	1	no	—
99	C₂×M₁₆	—	26	2	no	—
100	C₂²×C₈	—	13	1	no	—
101	#117	—	13	1	no	—
102	#117	—	26	2	no	—
103	M₃₂	—	26	2	no	—
104	C₂×C₁₆	—	13	1	no	—
105	C₂×C₁₆	—	13	1	no	—
106	C₂×C₁₆	—	26	2	no	—
107	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
108	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
109	#108	—	26	2	no	—
110	#108	—	26	2	no	—
111	#107	—	26	2	no	—
112	#106	—	26	2	no	—
113	#97	—	13	1	no	—
114	#97	—	13	1	no	—
115	C₄×C₈	—	13	1	no	—
116	C₄×C₈	C₂×#261	1	1	yes	characteristic
117	C₂×C₁₃	—	2	1	no	—
118	C₂×C₁₃	#1835	1	1	yes	characteristic
119	D₁₃	—	2	2	no	—
120	C₂²×C₄	—	13	1	no	—
121	M₁₆	—	26	2	no	—
122	M₁₆	—	52	2	no	—
123	M₁₆	—	52	2	no	—
124	C₂×C₈	—	13	1	no	—
125	C₂×C₈	—	13	2	no	—
126	C₂×C₈	—	13	2	no	—
127	C₂×C₈	—	13	1	no	—
128	C₂×C₈	—	52	2	no	—
129	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
130	C₂×C₈	C₂²×#261	1	1	yes	characteristic
131	C₄⋊C₄	—	26	1	no	—
132	C₄⋊C₄	—	52	2	no	—
133	C₂²⋊C₄	—	26	1	no	—
134	C₂²⋊C₄	—	52	2	no	—
135	C₄²	—	13	1	no	—
136	C₄²	—	52	2	no	—
137	C₄²	C₄⋊#261	1	1	yes	characteristic
138	C₁₆	—	2	1	no	—
139	C₁₆	—	2	1	no	—
140	C₁₆	—	26	1	no	—
141	C₁₆	—	26	1	no	—
142	C₁₃	#12893	1	1	yes	characteristic, complemented, Sylow
143	C₂³	—	13	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	13	1	no	—
146	C₂×C₄	—	13	2	no	—
147	C₂×C₄	—	13	2	no	—
148	C₂×C₄	—	26	1	no	—
149	C₂×C₄	—	104	2	no	—
150	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	13	2	no	—
153	C₈	—	26	1	no	—
154	C₈	—	104	2	no	—
155	C₈	#88237	1	1	yes	characteristic
156	C₈	#88237	1	1	yes	characteristic
157	C₂²	—	13	2	no	—
158	C₂²	—	52	1	no	—
159	C₂²	#11719092	1	1	yes	characteristic
160	C₄	—	4	1	no	—
161	C₄	—	13	2	no	—
162	C₄	—	52	1	no	—
163	C₄	—	208	2	no	—
164	C₄	#11719060	1	1	yes	characteristic
165	C₄	#11719092	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	26	2	no	—
168	C₂	#13130758	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11719092
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#261
Fitting	C₁₃×#346	C₄
socle	C₂×C₁₃	#1835
prime radical	C₂²×D₁₃	#135
cosocle	C₂×C₈×D₁₃	C₂³
hypercenter	#346	#261
hypocenter	C₁₃	#12893
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
13. C₁₃×#346	C₄	semidirect
14. #11718094	C₄	semidirect
15. C₁₆×Dic₁₃	C₄	semidirect
16. C₄×#6090	C₄	semidirect
84. #346	#261	semidirect
142. C₁₃	#12893	semidirect

Maximal subgroups

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

subgroup	index
#13130755	2
#13129906	2
#13129904	2
#13129898	2
#13116434	2
#12893	13