Subgroups — #16777128

Order 3328.

262 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#6092	C₂	1	2	yes	maximal
2	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
3	C₂²×#6092	—	2	2	no	—
4	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂×C₄×#1055	C₄	1	1	yes	characteristic
6	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
7	C₄⋊#6092	—	2	2	no	—
8	C₄⋊#6092	C₄	1	4	yes	—
9	C₈×#1055	—	2	2	no	—
10	C₂²×C₈×C₁₃	—	2	1	no	—
11	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×#1055	C₈	1	2	yes	—
13	C₂²×#1055	D₄	1	1	yes	characteristic
14	C₂²×#1055	Q₈	1	1	yes	characteristic
15	C₂×#6092	—	2	4	no	—
16	C₂×#6092	—	4	2	no	—
17	C₂×#6092	—	4	2	no	—
18	C₄×C₈×C₁₃	—	2	1	no	—
19	C₄×C₈×C₁₃	C₈	1	2	yes	—
20	C₄×#1055	—	2	1	no	—
21	C₄×#1055	—	2	1	no	—
22	C₄×#1055	C₂×C₄	1	2	yes	—
23	C₄×#1055	D₄	1	2	yes	—
24	#7151	—	13	1	no	maximal, Sylow
25	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
28	C₂×C₈×C₁₃	—	2	4	no	—
29	C₂×C₈×C₁₃	—	2	2	no	—
30	C₄²×C₁₃	C₄²	1	1	yes	characteristic
31	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
33	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
34	C₂×#1055	—	2	2	no	—
35	C₂×#1055	—	2	1	no	—
36	C₂×#1055	—	2	1	no	—
37	C₂×#1055	—	2	2	no	—
38	C₂×#1055	C₁₆	1	4	yes	complemented
39	C₂×#1055	C₂²⋊C₄	1	2	yes	—
40	C₂×#1055	C₄⋊C₄	1	2	yes	—
41	C₂×#1055	D₈	1	1	yes	characteristic
42	C₂×#1055	SD₁₆	1	2	yes	—
43	C₂×#1055	Q₁₆	1	1	yes	characteristic
44	#6092	—	4	4	no	—
45	C₂×C₄⋊C₁₆	—	13	2	no	—
46	C₂×C₈²	—	13	1	no	—
47	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₁₃	—	2	2	no	—
49	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
52	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
57	C₂×C₄×C₁₃	#108	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#108	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
60	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
61	C₈×C₁₃	—	2	4	no	—
62	#1055	—	2	2	no	—
63	#1055	—	2	4	no	—
64	#1055	#102	1	2	yes	—
65	#1055	#103	1	2	yes	—
66	C₂²×C₁₆	—	26	2	no	—
67	C₂×C₄×C₈	—	13	1	no	—
68	C₂×C₄×C₈	—	13	1	no	—
69	C₂×C₄×C₈	#261	1	1	yes	characteristic
70	C₄⋊C₁₆	—	13	4	no	—
71	C₄⋊C₁₆	—	26	2	no	—
72	C₈²	—	26	2	no	—
73	C₂²×C₁₃	#334	1	1	yes	characteristic
74	C₂²×C₁₃	#335	1	1	yes	characteristic
75	C₂²×C₁₃	#336	1	1	yes	characteristic
76	C₂²×C₁₃	C₄⋊C₁₆	1	2	yes	—
77	C₂²×C₁₃	#364	1	2	yes	—
78	C₄×C₁₃	—	2	1	no	—
79	C₄×C₁₃	—	2	1	no	—
80	C₄×C₁₃	#334	1	1	yes	characteristic
81	C₄×C₁₃	#335	1	1	yes	characteristic
82	C₄×C₁₃	#340	1	1	yes	characteristic
83	C₄×C₁₃	#341	1	1	yes	characteristic
84	C₄×C₁₃	C₄×C₁₆	1	1	yes	characteristic
85	C₄×C₁₃	#346	1	1	yes	characteristic
86	C₄×C₁₃	C₂²⋊C₁₆	1	2	yes	—
87	C₂²×C₈	—	2	1	no	—
88	C₂²×C₈	—	13	1	no	—
89	C₂²×C₈	—	13	2	no	—
90	C₂²×C₈	—	13	1	no	—
91	C₂×C₄²	C₂×#261	1	1	yes	characteristic
92	C₂×C₁₆	—	26	4	no	—
93	C₂×C₁₆	—	52	2	no	—
94	C₂×C₁₆	—	52	2	no	—
95	C₄×C₈	—	2	1	no	—
96	C₄×C₈	—	13	2	no	—
97	C₄×C₈	—	13	2	no	—
98	C₄×C₈	—	26	1	no	—
99	C₄×C₈	—	26	1	no	—
100	C₄×C₈	#1057	1	2	yes	—
101	C₂×C₁₃	#1277	1	1	yes	characteristic
102	C₂×C₁₃	#1367	1	2	yes	—
103	C₂×C₁₃	#1370	1	2	yes	—
104	C₂×C₁₃	#1374	1	1	yes	characteristic
105	C₂×C₁₃	#1376	1	1	yes	characteristic
106	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
107	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
108	C₂²×C₄	#6122	1	1	yes	characteristic
109	C₂×C₈	—	2	4	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	13	2	no	—
112	C₂×C₈	—	13	1	no	—
113	C₂×C₈	—	13	1	no	—
114	C₂×C₈	—	13	2	no	—
115	C₂×C₈	—	13	4	no	—
116	C₂×C₈	—	13	2	no	—
117	C₂×C₈	—	26	2	no	—
118	C₂×C₈	—	26	1	no	—
119	C₂×C₈	—	26	1	no	—
120	C₂×C₈	—	26	2	no	—
121	C₄²	#6118	1	1	yes	characteristic
122	C₄²	C₄×#261	1	1	yes	characteristic
123	C₄²	C₂×#1057	1	1	yes	characteristic
124	C₄²	C₂²⋊#261	1	1	yes	characteristic
125	C₁₆	—	52	4	no	—
126	C₁₃	#7151	1	1	yes	characteristic, complemented, Sylow
127	C₂³	#88112	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	D₁₃⋊C₁₆	1	2	yes	—
130	C₂×C₄	#88098	1	2	yes	—
131	C₂×C₄	#88101	1	1	yes	characteristic
132	C₂×C₄	#88102	1	1	yes	characteristic
133	C₂×C₄	#88103	1	1	yes	characteristic
134	C₂×C₄	#88104	1	1	yes	characteristic
135	C₂×C₄	C₄×#1057	1	1	yes	characteristic
136	C₂×C₄	C₄⋊#1057	1	1	yes	characteristic
137	C₂×C₄	#88110	1	1	yes	characteristic
138	C₂×C₄	#88111	1	1	yes	characteristic
139	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
140	C₂×C₄	C₂²⋊#1057	1	1	yes	characteristic
141	C₈	—	2	4	no	—
142	C₈	—	13	2	no	—
143	C₈	—	13	2	no	—
144	C₈	—	26	4	no	—
145	C₈	—	26	2	no	—
146	C₂²	#11718233	1	2	yes	—
147	C₂²	#11718255	1	1	yes	characteristic
148	C₂²	#11718256	1	1	yes	characteristic
149	C₂²	#11718260	1	2	yes	—
150	C₂²	#11718290	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	#11718253	1	1	yes	characteristic
154	C₄	#11718254	1	1	yes	characteristic
155	C₄	#11718257	1	1	yes	characteristic
156	C₄	#11718258	1	1	yes	characteristic
157	C₄	#11718259	1	2	yes	—
158	C₄	#11718265	1	1	yes	characteristic
159	C₄	#11718268	1	1	yes	characteristic
160	C₂	#13129666	1	1	yes	characteristic
161	C₂	#13129859	1	2	yes	—
162	C₂	#13129866	1	2	yes	—
163	C₂	#13129875	1	1	yes	characteristic
164	C₂	#13129880	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄⋊#261
commutator	C₄×C₁₃	C₄×C₁₆
Frattini	C₂×C₄²	C₂×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle, prime radical	C₂³×C₁₃	C₄⋊C₈
cosocle	C₂×C₄×#1055	C₂²
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#7151
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
38. C₂×#1055	C₁₆	semidirect
126. C₁₃	#7151	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#6092	2
C₂×C₈×#1055	2
#7151	13