Subgroups — #16789617

Order 3328.

334 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718261	C₂	1	1	yes	maximal, characteristic
2	#13131047	C₂	1	2	yes	maximal
3	C₂×#11718257	C₂	1	1	yes	maximal, characteristic
4	#13129891	C₂	1	2	yes	maximal
5	#13114718	C₂	1	1	yes	maximal, characteristic
6	C₂×#88106	C₂²	1	2	yes	—
7	C₂²×#6092	C₂²	1	1	yes	characteristic
8	C₂²×#6092	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#108	C₄	1	1	yes	characteristic
10	#11718783	C₂²	1	2	yes	—
11	#11718767	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
13	#11718261	C₄	1	4	yes	complemented
14	#11718258	—	2	2	no	—
15	#11718257	—	2	2	no	—
16	#11718257	C₄	1	4	yes	complemented
17	#11718147	C₄	1	2	yes	—
18	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	#88180	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×#1055	C₂³	1	1	yes	characteristic
23	#88106	—	2	2	no	—
24	#88106	C₂×C₄	1	4	yes	—
25	C₂×#6092	—	2	2	no	—
26	C₂×#6092	—	2	1	no	—
27	C₂×#6092	—	2	1	no	—
28	C₂×#6092	C₂×C₄	1	2	yes	—
29	C₂×#6092	C₂×C₄	1	2	yes	—
30	C₂×#6092	D₄	1	1	yes	characteristic
31	C₂×#6092	D₄	1	1	yes	characteristic
32	C₂×#6092	Q₈	1	2	yes	—
33	C₁₃×#108	C₈	1	4	yes	—
34	#88058	C₈	1	2	yes	—
35	#88057	C₈	1	2	yes	—
36	#88055	—	2	2	no	—
37	C₈×Dic₁₃	—	2	2	no	—
38	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
39	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
40	#12664	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₃	—	2	1	no	—
43	C₁₃×M₁₆	—	2	2	no	—
44	C₁₃×M₁₆	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₁₃	—	2	2	no	—
46	C₂×C₈×C₁₃	C₄²	1	2	yes	—
47	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
50	C₂²⋊Dic₁₃	—	2	2	no	—
51	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
52	C₄×Dic₁₃	—	2	2	no	—
53	C₄×Dic₁₃	—	2	1	no	—
54	C₄×Dic₁₃	C₄²	1	2	yes	—
55	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
56	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
57	C₂×#1055	C₂×D₄	1	1	yes	characteristic
58	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
59	C₂×#1055	#41	1	1	yes	characteristic
60	C₂×#1055	#41	1	1	yes	characteristic
61	#6092	—	2	2	no	—
62	#6092	—	2	4	no	—
63	#6092	C₄⋊C₄	1	2	yes	—
64	#6092	C₄⋊C₄	1	2	yes	—
65	C₂×#364	—	13	1	no	—
66	#2108	—	13	2	no	—
67	C₂×C₄×C₁₆	—	13	1	no	—
68	#1772	—	13	1	no	—
69	#1379	—	13	2	no	—
70	C₂³×C₁₃	#131	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
77	C₂×Dic₁₃	—	2	4	no	—
78	C₂×Dic₁₃	—	2	2	no	—
79	C₂×Dic₁₃	—	2	4	no	—
80	C₈×C₁₃	—	2	4	no	—
81	C₈×C₁₃	C₄×C₈	1	2	yes	—
82	C₈×C₁₃	C₄×C₈	1	2	yes	—
83	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
84	#1055	#117	1	1	yes	characteristic
85	#1055	C₄×D₄	1	1	yes	characteristic
86	#1055	C₄×Q₈	1	1	yes	characteristic
87	C₂×M₃₂	—	13	2	no	—
88	C₂²×C₁₆	—	13	1	no	—
89	C₂²×C₁₆	—	13	1	no	—
90	C₂×#108	#261	1	1	yes	characteristic
91	#427	—	13	1	no	—
92	#405	—	13	2	no	—
93	C₂×C₄×C₈	—	13	1	no	—
94	#364	—	13	4	no	—
95	#346	—	26	2	no	—
96	C₄×C₁₆	—	13	4	no	—
97	C₄×C₁₆	—	26	2	no	—
98	#339	—	13	2	no	—
99	C₂²×C₁₃	—	2	2	no	—
100	C₂²×C₁₃	#405	1	1	yes	characteristic
101	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
102	C₂²×C₁₃	C₈×Q₈	1	1	yes	characteristic
103	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
105	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
106	C₄×C₁₃	C₈×Q₈	1	1	yes	characteristic
107	Dic₁₃	—	2	4	no	—
108	Dic₁₃	—	4	4	no	—
109	C₂×M₁₆	C₂×#261	1	2	yes	—
110	C₂²×C₈	—	13	1	no	—
111	C₂²×C₈	C₂×#261	1	1	yes	characteristic
112	#117	—	13	2	no	—
113	C₂×C₄²	—	13	1	no	—
114	M₃₂	—	13	4	no	—
115	M₃₂	—	26	2	no	—
116	C₂×C₁₆	—	13	2	no	—
117	C₂×C₁₆	—	13	1	no	—
118	C₂×C₁₆	—	13	1	no	—
119	C₂×C₁₆	—	13	2	no	—
120	C₂×C₁₆	—	13	2	no	—
121	C₂×C₁₆	—	26	2	no	—
122	C₂×C₁₆	—	26	1	no	—
123	C₂×C₁₆	—	26	1	no	—
124	#108	#1057	1	4	yes	—
125	#107	—	13	2	no	—
126	#106	—	13	2	no	—
127	#97	—	26	2	no	—
128	C₄×C₈	—	13	2	no	—
129	C₄×C₈	—	13	2	no	—
130	C₄×C₈	—	26	2	no	—
131	C₂×C₁₃	—	2	1	no	—
132	C₂×C₁₃	—	2	1	no	—
133	C₂×C₁₃	C₈×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₁₃	#2170	1	2	yes	—
135	C₂²×C₄	—	26	1	no	—
136	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
137	M₁₆	—	2	2	no	—
138	M₁₆	C₂×#1057	1	4	yes	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	13	1	no	—
141	C₂×C₈	—	13	1	no	—
142	C₂×C₈	—	13	1	no	—
143	C₂×C₈	—	13	1	no	—
144	C₂×C₈	—	13	1	no	—
145	C₂×C₈	—	13	1	no	—
146	C₂×C₈	D₁₃⋊C₈	1	1	yes	characteristic
147	C₂×C₈	D₁₃⋊C₈	1	1	yes	characteristic
148	C₂×C₈	C₄×#261	1	2	yes	—
149	C₂×C₈	C₂×#1057	1	2	yes	—
150	C₄⋊C₄	—	13	4	no	—
151	C₂²⋊C₄	—	26	2	no	—
152	C₄²	—	13	2	no	—
153	C₄²	—	26	1	no	—
154	C₄²	—	26	2	no	—
155	C₁₆	—	13	2	no	—
156	C₁₆	—	13	2	no	—
157	C₁₆	—	26	2	no	—
158	C₁₆	—	26	4	no	—
159	C₁₃	#12664	1	1	yes	characteristic, complemented, Sylow
160	C₂³	#88240	1	1	yes	characteristic
161	C₂×C₄	—	26	4	no	—
162	C₂×C₄	—	26	2	no	—
163	C₂×C₄	—	26	4	no	—
164	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
165	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
166	C₂×C₄	#88238	1	1	yes	characteristic
167	C₂×C₄	D₄×#261	1	1	yes	characteristic
168	C₂×C₄	Q₈×#261	1	1	yes	characteristic
169	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
170	C₈	—	2	4	no	—
171	C₈	—	13	1	no	—
172	C₈	—	13	1	no	—
173	C₈	—	13	1	no	—
174	C₈	—	13	1	no	—
175	C₈	C₄×#1057	1	2	yes	—
176	C₈	C₄×#1057	1	2	yes	—
177	C₂²	—	2	2	no	—
178	C₂²	#11719072	1	1	yes	characteristic
179	C₂²	#11719073	1	1	yes	characteristic
180	C₂²	Q₈×#1057	1	1	yes	characteristic
181	C₄	—	26	4	no	—
182	C₄	—	52	4	no	—
183	C₄	#11719074	1	1	yes	characteristic
184	C₄	C₄⋊C₄×#261	1	1	yes	characteristic
185	C₄	C₂×C₄×#1057	1	1	yes	characteristic
186	C₄	D₄×#1057	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	C₄⋊C₄×#1057	1	1	yes	characteristic
190	C₂	#13131225	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×#108	C₄
socle	C₂²×C₁₃	C₈×D₄
prime radical	C₂³×C₁₃	#131
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×#108	#261
hypocenter	C₁₃	#12664
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
13. #11718261	C₄	semidirect
16. #11718257	C₄	semidirect
159. C₁₃	#12664	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718261	2
#13131047	2
C₂×#11718257	2
#13129891	2
#13114718	2
#12664	13