Finite Group Databasebeta

Subgroups — #16788655

Order 3328.

242 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 116 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	#13131168	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#6092	C₂	1	1	yes	maximal, characteristic
3	#13129859	C₂	1	4	yes	maximal, complemented
4	#13113120	C₂	1	1	yes	maximal, characteristic
5	C₂×#88106	—	2	1	no	—
6	C₂²×#6092	—	2	1	no	—
7	C₁₃×#431	C₄	1	1	yes	characteristic
8	#11718586	C₄	1	1	yes	characteristic
9	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
10	C₄⋊#6092	—	2	1	no	—
11	C₄⋊#6092	—	2	1	no	—
12	C₄⋊#6092	C₂²	1	2	yes	—
13	C₄⋊#6092	C₂²	1	2	yes	—
14	#11718046	C₄	1	2	yes	—
15	C₈×#1055	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×#1055	D₄	1	1	yes	characteristic
18	C₂²×#1055	Q₈	1	1	yes	characteristic
19	#88106	—	4	2	no	—
20	C₂×#6092	—	2	2	no	—
21	C₂×#6092	—	2	2	no	—
22	C₂×#6092	—	4	2	no	—
23	C₁₃×C₄⋊C₈	—	2	1	no	—
24	C₁₃×C₂²⋊C₈	—	2	1	no	—
25	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²⋊#1055	C₈	1	2	yes	—
27	C₄⋊#1055	C₈	1	2	yes	—
28	C₄×#1055	C₂×C₄	1	1	yes	characteristic
29	C₄×#1055	C₂×C₄	1	2	yes	—
30	C₄×#1055	D₄	1	1	yes	characteristic
31	C₄×#1055	Q₈	1	1	yes	characteristic
32	C₄×#1055	C₂³	1	1	yes	characteristic
33	#11997	—	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	2	no	—
38	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
41	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×#1055	—	2	2	no	—
43	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
44	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
45	C₂×#1055	C₂×C₈	1	4	yes	—
46	C₂×#1055	SD₁₆	1	2	yes	—
47	C₂×#1055	SD₁₆	1	2	yes	—
48	C₂×#1055	C₂×D₄	1	1	yes	characteristic
49	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
50	#6092	—	4	2	no	—
51	#6092	—	4	2	no	—
52	#2150	—	13	1	no	—
53	C₂×C₄⋊C₁₆	—	13	1	no	—
54	#1450	—	13	1	no	—
55	#1367	—	13	4	no	—
56	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	—	4	1	no	—
59	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₃	#106	1	2	yes	—
63	C₂×C₄×C₁₃	#108	1	2	yes	—
64	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
67	C₈×C₁₃	—	4	2	no	—
68	#1055	—	2	1	no	—
69	#1055	—	2	1	no	—
70	#1055	—	2	4	no	—
71	#1055	#106	1	2	yes	—
72	#1055	C₂×SD₁₆	1	1	yes	characteristic
73	#1055	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×M₃₂	—	26	1	no	—
75	C₂²×C₁₆	—	26	1	no	—
76	#431	—	13	1	no	—
77	#431	#261	1	1	yes	characteristic
78	C₂×C₄×C₈	—	13	1	no	—
79	C₄⋊C₁₆	—	13	2	no	—
80	C₄⋊C₁₆	—	13	2	no	—
81	C₄⋊C₁₆	—	26	1	no	—
82	C₄⋊C₁₆	—	26	1	no	—
83	#322	—	13	2	no	—
84	C₈²	—	13	2	no	—
85	C₂²×C₁₃	—	2	1	no	—
86	C₂²×C₁₃	—	2	1	no	—
87	C₂²×C₁₃	#334	1	2	yes	—
88	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₄×C₁₃	—	2	1	no	—
90	C₄×C₁₃	—	2	1	no	—
91	C₄×C₁₃	—	2	1	no	—
92	C₄×C₁₃	#334	1	2	yes	—
93	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
94	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
95	C₄×C₁₃	#504	1	2	yes	—
96	C₂²×C₈	—	13	1	no	—
97	C₂²×C₈	—	13	1	no	—
98	C₂×C₄²	C₂×#261	1	1	yes	characteristic
99	M₃₂	—	52	2	no	—
100	C₂×C₁₆	—	26	2	no	—
101	C₂×C₁₆	—	26	2	no	—
102	C₂×C₁₆	—	52	2	no	—
103	C₄⋊C₈	—	2	1	no	—
104	C₄⋊C₈	—	13	2	no	—
105	C₂²⋊C₈	—	2	1	no	—
106	C₂²⋊C₈	—	13	2	no	—
107	C₄×C₈	—	13	1	no	—
108	C₄×C₈	—	13	1	no	—
109	C₄×C₈	—	13	1	no	—
110	C₄×C₈	—	13	2	no	—
111	C₄×C₈	—	13	1	no	—
112	C₄×C₈	C₂×#261	1	2	yes	—
113	C₂×C₁₃	—	2	2	no	—
114	C₂×C₁₃	C₂×#334	1	1	yes	characteristic
115	C₂×C₁₃	#2151	1	1	yes	characteristic
116	C₂×C₁₃	#2153	1	1	yes	characteristic
117	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
118	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
119	C₂²×C₄	#6122	1	1	yes	characteristic
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	13	2	no	—
122	C₂×C₈	—	13	2	no	—
123	C₂×C₈	—	13	1	no	—
124	C₂×C₈	—	13	1	no	—
125	C₂×C₈	—	13	4	no	—
126	C₂×C₈	—	13	1	no	—
127	C₂×C₈	—	13	1	no	—
128	C₂×C₈	—	26	2	no	—
129	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
130	C₄²	C₄⋊#261	1	1	yes	characteristic
131	C₄²	#6122	1	1	yes	characteristic
132	C₄²	C₂²×#261	1	1	yes	characteristic
133	C₁₆	—	52	2	no	—
134	C₁₆	—	52	2	no	—
135	C₁₃	#11997	1	1	yes	characteristic, complemented, Sylow
136	C₂³	#88112	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	#88101	1	2	yes	—
140	C₂×C₄	#88104	1	2	yes	—
141	C₂×C₄	#88112	1	1	yes	characteristic
142	C₂×C₄	#88112	1	1	yes	characteristic
143	C₂×C₄	#88112	1	1	yes	characteristic
144	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
146	C₂×C₄	C₂×#6122	1	1	yes	characteristic
147	C₈	—	4	2	no	—
148	C₈	—	13	1	no	—
149	C₈	—	13	1	no	—
150	C₈	—	13	2	no	—
151	C₈	—	26	1	no	—
152	C₈	—	26	1	no	—
153	C₈	—	26	4	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#11718255	1	2	yes	—
157	C₂²	C₂×#88112	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#11718255	1	2	yes	—
162	C₄	C₂×#88101	1	1	yes	characteristic
163	C₄	C₂×#88104	1	1	yes	characteristic
164	C₄	#11719096	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	C₂×#11718255	1	1	yes	characteristic
167	C₂	#13131173	1	1	yes	characteristic
168	C₂	#13131179	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₄²	C₂²×#261
Fitting	C₁₃×#431	C₄
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₄⋊C₈
cosocle	C₄×#1055	C₂³
hypercenter	#431	#261
hypocenter	C₁₃	#11997
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
3. #13129859	C₂	semidirect
135. C₁₃	#11997	semidirect

Maximal subgroups

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

subgroup	index
#13131168	2
C₂×C₄⋊#6092	2
#13129859	2
#13113120	2
#11997	13