Finite Group Databasebeta

Subgroups — #16788718

Order 3328.

238 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 98 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	#13131197	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#6092	C₂	1	1	yes	maximal, characteristic
3	#13129866	C₂	1	2	yes	maximal, complemented
4	#13129859	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	#11718231	—	2	1	no	—
12	C₄⋊#6092	—	2	1	no	—
13	C₄⋊#6092	C₂²	1	1	yes	characteristic
14	C₄⋊#6092	C₂²	1	2	yes	—
15	C₄⋊#6092	C₂²	1	1	yes	characteristic
16	#11718047	C₄	1	2	yes	—
17	C₈×#1055	C₂²	1	2	yes	—
18	C₂×C₁₃×M₁₆	—	2	1	no	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×#1055	D₄	1	1	yes	characteristic
21	C₂²×#1055	Q₈	1	1	yes	characteristic
22	C₂×#6092	—	2	2	no	—
23	C₂×#6092	—	2	1	no	—
24	C₂×#6092	—	2	1	no	—
25	C₂×#6092	—	4	2	no	—
26	C₁₃×#97	—	2	1	no	—
27	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₂²⋊#1055	C₈	1	2	yes	—
29	C₄⋊#1055	C₈	1	2	yes	—
30	C₄×#1055	C₂×C₄	1	1	yes	characteristic
31	C₄×#1055	C₂×C₄	1	2	yes	—
32	C₄×#1055	D₄	1	1	yes	characteristic
33	C₄×#1055	Q₈	1	1	yes	characteristic
34	C₄×#1055	C₂³	1	1	yes	characteristic
35	#12016	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
39	C₁₃×M₁₆	—	2	4	no	—
40	C₂×C₈×C₁₃	—	2	2	no	—
41	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
44	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×#1055	—	2	1	no	—
46	C₂×#1055	—	2	1	no	—
47	C₂×#1055	—	4	1	no	—
48	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
49	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
50	C₂×#1055	C₂×C₈	1	4	yes	—
51	C₂×#1055	C₂×D₄	1	1	yes	characteristic
52	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
53	#6092	—	4	1	no	—
54	#6092	—	4	2	no	—
55	#6092	—	4	1	no	—
56	#2163	—	13	1	no	—
57	C₂×C₄⋊C₁₆	—	13	1	no	—
58	C₈×M₁₆	—	13	1	no	—
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	—	2	2	no	—
66	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
72	C₈×C₁₃	—	2	4	no	—
73	#1055	—	2	1	no	—
74	#1055	—	2	1	no	—
75	#1055	—	2	1	no	—
76	#1055	—	2	4	no	—
77	#1055	AGL(1,ℤ/8)	1	1	yes	characteristic
78	#1055	#137	1	1	yes	characteristic
79	C₂²×C₁₆	—	26	1	no	—
80	#431	—	13	1	no	—
81	C₄×M₁₆	#261	1	1	yes	characteristic
82	C₂×C₄×C₈	—	13	1	no	—
83	C₄⋊C₁₆	—	13	1	no	—
84	C₄⋊C₁₆	—	13	2	no	—
85	C₄⋊C₁₆	—	13	1	no	—
86	C₄⋊C₁₆	—	26	1	no	—
87	C₂²⋊C₁₆	—	26	1	no	—
88	#346	—	26	1	no	—
89	#322	—	13	2	no	—
90	C₈²	—	13	2	no	—
91	C₂²×C₁₃	—	2	1	no	—
92	C₂²×C₁₃	—	2	1	no	—
93	C₂²×C₁₃	#364	1	2	yes	—
94	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₄×C₁₃	—	2	1	no	—
96	C₄×C₁₃	—	2	1	no	—
97	C₄×C₁₃	—	2	1	no	—
98	C₄×C₁₃	—	2	1	no	—
99	C₄×C₁₃	#428	1	1	yes	characteristic
100	C₄×C₁₃	#430	1	1	yes	characteristic
101	C₄×C₁₃	#504	1	2	yes	—
102	C₂×M₁₆	—	2	1	no	—
103	C₂²×C₈	—	13	1	no	—
104	C₂²×C₈	—	13	1	no	—
105	C₂×C₄²	C₂×#261	1	1	yes	characteristic
106	C₂×C₁₆	—	26	1	no	—
107	C₂×C₁₆	—	26	1	no	—
108	C₂×C₁₆	—	26	2	no	—
109	C₂×C₁₆	—	52	2	no	—
110	C₄⋊C₈	—	13	2	no	—
111	C₂²⋊C₈	—	13	2	no	—
112	#97	—	2	1	no	—
113	C₄×C₈	—	13	1	no	—
114	C₄×C₈	—	13	1	no	—
115	C₄×C₈	—	13	1	no	—
116	C₄×C₈	—	13	2	no	—
117	C₄×C₈	—	13	1	no	—
118	C₄×C₈	C₂×#261	1	2	yes	—
119	C₂×C₁₃	—	2	2	no	—
120	C₂×C₁₃	#1565	1	1	yes	characteristic
121	C₂×C₁₃	#2151	1	1	yes	characteristic
122	C₂×C₁₃	C₂×#364	1	1	yes	characteristic
123	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
124	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
125	C₂²×C₄	#6122	1	1	yes	characteristic
126	M₁₆	—	2	4	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	13	1	no	—
129	C₂×C₈	—	13	4	no	—
130	C₂×C₈	—	13	1	no	—
131	C₂×C₈	—	13	1	no	—
132	C₂×C₈	—	13	1	no	—
133	C₂×C₈	—	26	1	no	—
134	C₂×C₈	—	26	1	no	—
135	C₂×C₈	—	52	1	no	—
136	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
137	C₄²	C₄⋊#261	1	1	yes	characteristic
138	C₄²	#6122	1	1	yes	characteristic
139	C₄²	C₂²×#261	1	1	yes	characteristic
140	C₁₆	—	52	2	no	—
141	C₁₆	—	52	1	no	—
142	C₁₆	—	52	1	no	—
143	C₁₃	#12016	1	1	yes	characteristic, complemented, Sylow
144	C₂³	#88112	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	#88112	1	1	yes	characteristic
150	C₂×C₄	#88112	1	1	yes	characteristic
151	C₂×C₄	#88112	1	1	yes	characteristic
152	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
154	C₂×C₄	C₂×#6122	1	1	yes	characteristic
155	C₈	—	2	4	no	—
156	C₈	—	13	1	no	—
157	C₈	—	13	1	no	—
158	C₈	—	26	4	no	—
159	C₈	—	26	1	no	—
160	C₈	—	26	1	no	—
161	C₈	—	26	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#11718233	1	2	yes	—
165	C₂²	C₂×#88112	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	#11719096	1	2	yes	—
171	C₄	#11719098	1	1	yes	characteristic
172	C₄	#11719100	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	#13130389	1	1	yes	characteristic
175	C₂	#13131173	1	1	yes	characteristic
176	C₂	C₂×#11718233	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
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	#12016
LCS	G ▷ C₂×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. #13129866	C₂	semidirect
4. #13129859	C₂	semidirect
143. C₁₃	#12016	semidirect

Maximal subgroups

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

subgroup	index
#13131197	2
C₂×C₄⋊#6092	2
#13129866	2
#13129859	2
M₁₆×#1055	2
#12016	13