Finite Group Databasebeta

Subgroups — #16791424

Order 3328.

194 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 78 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	#13131187	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718221	C₂	1	1	yes	maximal, characteristic
3	#13129998	C₂	1	2	yes	maximal, complemented
4	#13129997	C₂	1	2	yes	maximal, complemented
5	D₁₃×M₆₄	C₂	1	1	yes	maximal, characteristic
6	#11719092	—	2	1	no	—
7	C₂×#88102	—	2	1	no	—
8	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
9	#11718222	—	2	1	no	—
10	#11718221	—	2	1	no	—
11	#11718221	C₂²	1	1	yes	characteristic
12	#11718221	C₂²	1	2	yes	—
13	#11718221	C₂²	1	1	yes	characteristic
14	C₁₃×M₆₄	C₄	1	1	yes	characteristic, complemented
15	#11718091	C₄	1	1	yes	characteristic, complemented
16	#11718039	C₄	1	2	yes	complemented
17	D₁₃×C₃₂	C₂²	1	2	yes	—
18	#88237	—	4	1	no	—
19	C₂×C₄⋊#261	—	4	1	no	—
20	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
21	#88102	—	2	2	no	—
22	#88102	—	2	1	no	—
23	#88102	—	2	1	no	—
24	#88102	—	4	1	no	—
25	#88101	—	4	1	no	—
26	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×#6090	Q₈	1	1	yes	characteristic
28	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic
29	C₁₆×D₁₃	D₄	1	1	yes	characteristic
30	C₁₆×D₁₃	Q₈	1	1	yes	characteristic
31	C₁₆×D₁₃	C₂³	1	1	yes	characteristic
32	C₁₃×C₃₂	C₂×C₄	1	2	yes	complemented
33	#88034	C₂×C₄	1	2	yes	complemented
34	#13634	—	13	1	no	maximal, Sylow
35	C₂²×#261	—	8	1	no	—
36	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
37	C₂²⋊#261	—	8	1	no	—
38	C₄⋊#261	—	4	1	no	—
39	C₄⋊#261	—	4	1	no	—
40	C₄⋊#261	—	4	2	no	—
41	C₄⋊#261	—	8	1	no	—
42	C₄×#261	—	8	1	no	—
43	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×#1055	Q₁₆	1	1	yes	characteristic
45	C₈×D₁₃	C₄⋊C₄	1	1	yes	characteristic
46	C₈×D₁₃	D₈	1	1	yes	characteristic
47	C₈×D₁₃	Q₁₆	1	1	yes	characteristic
48	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
49	C₁₃×C₁₆	C₄⋊C₄	1	1	yes	characteristic
50	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
51	#6090	C₄⋊C₄	1	1	yes	characteristic
52	#6090	C₂×Q₈	1	1	yes	characteristic
53	C₂×M₆₄	—	13	1	no	—
54	#2158	—	13	1	no	—
55	C₂×#366	—	13	1	no	—
56	#1424	—	13	2	no	—
57	#1423	—	13	2	no	—
58	C₂²×D₁₃	D₁₆	1	1	yes	characteristic
59	C₂×#261	—	8	1	no	—
60	C₂×#261	—	8	1	no	—
61	C₂×#261	—	8	2	no	—
62	C₂×#261	—	16	2	no	—
63	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
64	C₂×Dic₁₃	Q₃₂	1	1	yes	characteristic
65	C₄×D₁₃	#107	1	1	yes	characteristic
66	C₄×D₁₃	D₁₆	1	1	yes	characteristic
67	C₄×D₁₃	Q₃₂	1	1	yes	characteristic
68	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
69	C₈×C₁₃	#107	1	1	yes	characteristic
70	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	#1055	#107	1	1	yes	characteristic
72	#1055	C₂×Q₁₆	1	1	yes	characteristic
73	C₂²×C₁₆	—	13	1	no	—
74	#427	—	26	1	no	—
75	C₂×#107	—	26	1	no	—
76	M₆₄	—	13	1	no	—
77	M₆₄	—	13	2	no	—
78	M₆₄	#261	1	1	yes	characteristic, complemented
79	C₂×C₃₂	—	13	2	no	—
80	#367	—	26	1	no	—
81	#366	—	13	2	no	—
82	#366	—	13	1	no	—
83	#366	—	13	1	no	—
84	#366	—	26	1	no	—
85	C₂²×C₁₃	#366	1	1	yes	characteristic
86	C₂×D₁₃	—	2	1	no	—
87	C₂×D₁₃	—	2	1	no	—
88	C₂×D₁₃	#366	1	1	yes	characteristic
89	C₂×D₁₃	C₂×D₁₆	1	1	yes	characteristic
90	#261	—	16	1	no	—
91	#261	—	16	2	no	—
92	#261	—	16	1	no	—
93	C₄×C₁₃	#366	1	1	yes	characteristic
94	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
95	Dic₁₃	#366	1	1	yes	characteristic
96	Dic₁₃	C₂×Q₃₂	1	1	yes	characteristic
97	C₂²×C₈	—	13	1	no	—
98	#117	—	52	1	no	—
99	C₂×C₄⋊C₄	—	52	1	no	—
100	C₂×C₁₆	—	13	1	no	—
101	C₂×C₁₆	—	13	1	no	—
102	C₂×C₁₆	—	13	1	no	—
103	C₂×C₁₆	—	13	1	no	—
104	C₂×C₁₆	—	13	1	no	—
105	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
106	#107	—	26	2	no	—
107	#107	—	26	1	no	—
108	#107	—	26	1	no	—
109	#107	—	52	1	no	—
110	#106	—	52	1	no	—
111	C₃₂	—	13	2	no	—
112	C₃₂	C₂×#261	1	2	yes	complemented
113	C₂×C₁₃	—	2	1	no	—
114	C₂×C₁₃	C₂×#366	1	1	yes	characteristic
115	D₁₃	—	2	1	no	—
116	D₁₃	#2263	1	1	yes	characteristic
117	D₁₃	#2264	1	1	yes	characteristic
118	C₂²×C₄	—	13	1	no	—
119	C₂²×C₄	—	104	1	no	—
120	C₂×C₈	—	13	1	no	—
121	C₂×C₈	—	13	1	no	—
122	C₂×C₈	—	13	1	no	—
123	C₂×C₈	—	13	1	no	—
124	C₂×C₈	—	13	1	no	—
125	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
126	C₄⋊C₄	—	52	1	no	—
127	C₄⋊C₄	—	52	1	no	—
128	C₄⋊C₄	—	52	2	no	—
129	C₄⋊C₄	—	104	1	no	—
130	C₂²⋊C₄	—	104	1	no	—
131	C₄²	—	104	1	no	—
132	C₁₆	—	13	1	no	—
133	C₁₆	—	13	1	no	—
134	C₁₆	C₄⋊#261	1	1	yes	characteristic
135	C₁₆	C₂²×#261	1	1	yes	characteristic
136	C₁₃	#13634	1	1	yes	characteristic, complemented, Sylow
137	C₂³	—	13	1	no	—
138	C₂×C₄	—	13	1	no	—
139	C₂×C₄	—	13	1	no	—
140	C₂×C₄	—	13	1	no	—
141	C₂×C₄	—	13	1	no	—
142	C₂×C₄	—	13	1	no	—
143	C₂×C₄	—	104	1	no	—
144	C₂×C₄	—	104	1	no	—
145	C₂×C₄	—	104	2	no	—
146	C₂×C₄	—	208	2	no	—
147	C₂×C₄	#88102	1	1	yes	characteristic
148	C₈	—	13	1	no	—
149	C₈	—	13	1	no	—
150	C₈	#88102	1	1	yes	characteristic
151	C₈	C₂×C₄⋊#261	1	1	yes	characteristic
152	C₂²	—	13	1	no	—
153	C₂²	—	13	1	no	—
154	C₂²	—	26	1	no	—
155	C₂²	—	26	1	no	—
156	C₂²	#11718221	1	1	yes	characteristic
157	C₄	—	13	1	no	—
158	C₄	—	13	1	no	—
159	C₄	—	208	1	no	—
160	C₄	—	208	1	no	—
161	C₄	—	208	2	no	—
162	C₄	#11718221	1	1	yes	characteristic
163	C₄	C₂×#88102	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	13	1	no	—
166	C₂	—	13	1	no	—
167	C₂	—	26	1	no	—
168	C₂	C₂×#11718221	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#11718221
commutator	C₁₃×C₁₆	C₂²×C₄
Frattini	C₁₆	C₂²×#261
Fitting	C₁₃×M₆₄	C₄
socle	C₂×C₁₃	C₂×#366
prime radical	C₂²×D₁₃	D₁₆
cosocle	C₁₆×D₁₃	C₂³
hypercenter	M₆₄	#261
hypocenter	C₁₃	#13634
LCS	G ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
3. #13129998	C₂	semidirect
4. #13129997	C₂	semidirect
14. C₁₃×M₆₄	C₄	semidirect
15. #11718091	C₄	semidirect
16. #11718039	C₄	semidirect
32. C₁₃×C₃₂	C₂×C₄	semidirect
33. #88034	C₂×C₄	semidirect
78. M₆₄	#261	semidirect
112. C₃₂	C₂×#261	semidirect
136. C₁₃	#13634	semidirect

Maximal subgroups

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

subgroup	index
#13131187	2
C₂×#11718221	2
#13129998	2
#13129997	2
D₁₃×M₆₄	2
#13634	13