Subgroups — #16016282

Order 3328.

718 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₀₈	C₂	1	1	yes	maximal, characteristic
2	#13123689	C₂	1	2	yes	maximal, complemented
3	#13116713	C₂	1	4	yes	maximal, complemented
4	C₄×D₂₀₈	C₂	1	4	yes	maximal, complemented
5	#13116620	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116553	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₀₄	C₂²	1	2	yes	—
9	#11719226	—	2	2	no	—
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₁₃×#405	C₂²	1	1	yes	characteristic
13	#11718792	—	2	2	no	—
14	#11718767	C₂²	1	2	yes	—
15	C₂×D₂₀₈	C₄	1	8	yes	complemented
16	C₂×D₂₀₈	C₂²	1	2	yes	—
17	C₂×D₂₀₈	C₂²	1	2	yes	—
18	#11718372	—	2	4	no	—
19	C₄×D₁₀₄	C₂²	1	8	yes	complemented
20	C₁₃×#346	C₂²	1	2	yes	complemented
21	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
22	#11718101	C₂²	1	8	yes	complemented
23	#11718097	C₂²	1	2	yes	complemented
24	#11718096	C₂²	1	2	yes	complemented
25	C₂²×D₅₂	—	2	2	no	—
26	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₁₃×#117	D₄	1	1	yes	characteristic
29	#88180	—	2	2	no	—
30	C₂×D₁₀₄	—	2	2	no	—
31	C₂×D₁₀₄	—	2	4	no	—
32	C₂×D₁₀₄	C₂×C₄	1	8	yes	complemented
33	C₂×D₁₀₄	C₂³	1	4	yes	—
34	#88147	—	4	4	no	—
35	C₂²⋊C₄×D₁₃	—	4	4	no	—
36	C₄×D₅₂	—	2	8	no	—
37	C₁₃×M₃₂	—	2	2	no	—
38	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
39	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
40	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
42	C₁₃×#97	D₄	1	2	yes	—
43	C₄×C₈×C₁₃	C₂³	1	2	yes	—
44	#88061	—	2	8	no	—
45	#88058	C₂³	1	4	yes	—
46	#88057	—	2	2	no	—
47	D₂₀₈	C₂×C₄	1	8	yes	complemented
48	D₂₀₈	C₂×C₄	1	8	yes	complemented
49	#33456	—	13	1	no	maximal, Sylow
50	C₂³×D₁₃	—	4	2	no	—
51	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂×D₅₂	—	2	8	no	—
53	C₂×D₅₂	—	2	4	no	—
54	C₂×D₅₂	—	4	2	no	—
55	C₂×D₅₂	—	4	4	no	—
56	C₂×C₄×D₁₃	—	4	8	no	—
57	C₁₃×M₁₆	—	2	2	no	—
58	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	#41	1	2	yes	—
62	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
64	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
65	C₄²×C₁₃	C₂×D₄	1	2	yes	—
66	C₂²⋊Dic₁₃	—	4	2	no	—
67	#6103	—	4	8	no	—
68	C₄⋊Dic₁₃	—	2	4	no	—
69	C₄×Dic₁₃	—	4	2	no	—
70	D₁₀₄	—	2	8	no	—
71	D₁₀₄	—	2	4	no	—
72	D₁₀₄	C₂²×C₄	1	8	yes	—
73	C₁₃×C₁₆	—	2	2	no	—
74	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
75	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
76	C₂²×D₁₆	—	13	1	no	—
77	#2948	—	13	2	no	—
78	#2177	—	13	4	no	—
79	C₄×D₁₆	—	13	4	no	—
80	#2158	—	13	1	no	—
81	#2140	—	13	2	no	—
82	#2108	D₁₃	1	1	yes	characteristic, complemented
83	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
84	C₂²×D₁₃	—	4	4	no	—
85	C₂²×D₁₃	—	4	8	no	—
86	C₂²×D₁₃	—	16	4	no	—
87	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
89	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₁₃	#135	1	2	yes	—
91	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×Dic₁₃	—	4	4	no	—
93	D₅₂	—	2	8	no	—
94	D₅₂	—	4	4	no	—
95	D₅₂	—	4	8	no	—
96	C₄×D₁₃	—	16	8	no	—
97	C₈×C₁₃	—	2	2	no	—
98	C₈×C₁₃	C₄×D₄	1	2	yes	—
99	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
100	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
101	C₂²×D₈	—	13	2	no	—
102	#518	—	26	2	no	—
103	C₂×D₁₆	—	13	8	no	—
104	C₂×D₁₆	—	13	2	no	—
105	C₂×D₁₆	—	13	2	no	—
106	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
107	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
108	#442	—	26	4	no	—
109	C₄×D₈	—	13	8	no	—
110	#427	—	13	2	no	—
111	#418	—	26	2	no	—
112	#405	C₂×D₁₃	1	1	yes	characteristic
113	#367	—	13	2	no	—
114	#366	—	13	2	no	—
115	#357	—	13	8	no	—
116	#346	C₂×D₁₃	1	2	yes	complemented
117	C₄×C₁₆	C₂×D₁₃	1	2	yes	complemented
118	C₂²×C₁₃	—	2	2	no	—
119	C₂²×C₁₃	C₄×D₈	1	2	yes	—
120	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
121	C₂×D₁₃	—	4	8	no	—
122	C₂×D₁₃	—	8	8	no	—
123	C₂×D₁₃	—	16	8	no	—
124	C₄×C₁₃	—	2	4	no	—
125	C₄×C₁₃	C₄×D₈	1	2	yes	—
126	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
128	Dic₁₃	—	8	4	no	—
129	C₂²×D₄	—	26	2	no	—
130	C₂×D₈	—	13	8	no	—
131	C₂×D₈	—	13	4	no	—
132	C₂×D₈	—	26	2	no	—
133	C₂×D₈	—	26	4	no	—
134	C₂×M₁₆	D₅₂	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
136	C₄×D₄	—	26	8	no	—
137	C₄×D₄	—	52	4	no	—
138	#117	—	26	2	no	—
139	#117	D₅₂	1	1	yes	characteristic
140	C₂×C₂²⋊C₄	—	52	4	no	—
141	D₁₆	—	13	8	no	—
142	D₁₆	—	13	8	no	—
143	M₃₂	—	2	2	no	—
144	C₂×C₁₆	C₄×D₁₃	1	4	yes	complemented
145	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
146	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
148	#107	—	13	4	no	—
149	#106	—	26	2	no	—
150	#102	—	26	8	no	—
151	#97	D₅₂	1	2	yes	—
152	C₄×C₈	C₂²×D₁₃	1	2	yes	—
153	C₂×C₁₃	—	2	2	no	—
154	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
155	C₂×C₁₃	#3415	1	2	yes	—
156	D₁₃	—	8	8	no	—
157	C₂⁴	—	52	2	no	—
158	C₂×D₄	—	26	8	no	—
159	C₂×D₄	—	26	4	no	—
160	C₂×D₄	—	52	2	no	—
161	C₂×D₄	—	52	4	no	—
162	C₂²×C₄	—	52	8	no	—
163	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
164	D₈	—	13	8	no	—
165	D₈	—	26	8	no	—
166	D₈	—	26	4	no	—
167	M₁₆	—	2	2	no	—
168	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
169	C₂×C₈	C₂×D₅₂	1	2	yes	—
170	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
171	C₂×C₈	#6127	1	2	yes	—
172	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
173	C₄⋊C₄	—	26	4	no	—
174	C₄⋊C₄	D₁₀₄	1	2	yes	—
175	C₂²⋊C₄	—	52	2	no	—
176	C₂²⋊C₄	—	52	8	no	—
177	C₂²⋊C₄	D₁₀₄	1	2	yes	—
178	C₄²	—	52	2	no	—
179	C₄²	C₂×D₅₂	1	2	yes	—
180	C₁₆	—	2	2	no	—
181	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
182	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
183	C₁₃	#33456	1	1	yes	characteristic, complemented, Sylow
184	C₂³	—	52	4	no	—
185	C₂³	—	52	8	no	—
186	C₂³	—	208	4	no	—
187	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
188	D₄	—	26	8	no	—
189	D₄	—	52	4	no	—
190	D₄	—	52	8	no	—
191	C₂×C₄	—	52	4	no	—
192	C₂×C₄	—	208	8	no	—
193	C₂×C₄	C₄×D₅₂	1	2	yes	—
194	C₂×C₄	C₂×D₁₀₄	1	4	yes	—
195	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
196	C₂×C₄	#88158	1	2	yes	—
197	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	C₄×D₅₂	1	2	yes	—
200	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
201	C₈	C₂×#6127	1	1	yes	characteristic
202	C₂²	—	2	2	no	—
203	C₂²	—	52	8	no	—
204	C₂²	—	104	8	no	—
205	C₂²	—	208	8	no	—
206	C₂²	C₄×D₁₀₄	1	2	yes	—
207	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
208	C₄	—	2	4	no	—
209	C₄	—	104	4	no	—
210	C₄	C₄×D₁₀₄	1	2	yes	—
211	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
212	C₄	C₂×#88158	1	1	yes	characteristic
213	C₂	—	2	2	no	—
214	C₂	—	104	8	no	—
215	C₂	C₂×C₄×D₁₀₄	1	1	yes	characteristic
216	C₂	#13127977	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2108	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₂₀₈	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2108	D₁₃
hypocenter	C₁₃	#33456
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #13123689	C₂	semidirect
3. #13116713	C₂	semidirect
4. C₄×D₂₀₈	C₂	semidirect
5. #13116620	C₂	semidirect
6. #13116553	C₂	semidirect
7. C₁₃×#2108	C₂	semidirect
15. C₂×D₂₀₈	C₄	semidirect
19. C₄×D₁₀₄	C₂²	semidirect
20. C₁₃×#346	C₂²	semidirect
21. C₄×C₁₃×C₁₆	C₂²	semidirect
22. #11718101	C₂²	semidirect
23. #11718097	C₂²	semidirect
24. #11718096	C₂²	semidirect
32. C₂×D₁₀₄	C₂×C₄	semidirect
38. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
47. D₂₀₈	C₂×C₄	semidirect
48. D₂₀₈	C₂×C₄	semidirect
82. #2108	D₁₃	semidirect
116. #346	C₂×D₁₃	semidirect
117. C₄×C₁₆	C₂×D₁₃	semidirect
144. C₂×C₁₆	C₄×D₁₃	semidirect
183. C₁₃	#33456	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₂₀₈	2
#13123689	2
#13116713	2
C₄×D₂₀₈	2
#13116620	2
#13116553	2
C₁₃×#2108	2
#33456	13