Subgroups — #16789098

Order 3328.

606 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719092	C₂	1	2	yes	maximal
2	#13129894	C₂	1	4	yes	maximal, complemented
3	C₂²×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#88237	—	2	2	no	—
5	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
6	#11719092	—	2	4	no	—
7	#11719092	C₄	1	8	yes	complemented
8	C₂×#88102	C₂²	1	4	yes	—
9	C₂×#88101	—	2	2	no	—
10	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic, complemented
11	C₂²×#6090	C₄	1	1	yes	characteristic, complemented
12	C₂×C₁₆×D₁₃	—	2	4	no	—
13	C₂×C₁₆×D₁₃	C₄	1	2	yes	complemented
14	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
16	C₂×C₂²⋊#261	—	4	2	no	—
17	#88237	—	2	8	no	—
18	#88237	—	4	4	no	—
19	C₂×C₄⋊#261	—	2	4	no	—
20	C₂×C₄×#261	—	4	2	no	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×#1055	Q₈	1	1	yes	characteristic
23	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
25	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
26	C₂×C₈×D₁₃	D₄	1	4	yes	—
27	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
28	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
29	#88102	—	2	4	no	—
30	#88102	C₂×C₄	1	8	yes	—
31	#88101	—	2	8	no	—
32	C₂×C₁₃×C₁₆	—	2	2	no	—
33	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
34	C₂×#6090	—	2	2	no	—
35	C₂×#6090	C₂×C₄	1	2	yes	—
36	C₁₆×D₁₃	—	2	4	no	—
37	C₁₆×D₁₃	—	2	4	no	—
38	#12556	—	13	1	no	maximal, Sylow
39	C₂³×D₁₃	D₈	1	1	yes	characteristic
40	C₂²×#261	—	4	4	no	—
41	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×Dic₁₃	Q₁₆	1	1	yes	characteristic
43	C₂×C₄×D₁₃	C₂²⋊C₄	1	4	yes	—
44	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
46	C₂×C₄×D₁₃	SD₁₆	1	4	yes	—
47	C₂×C₄×D₁₃	Q₁₆	1	1	yes	characteristic
48	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂²⋊#261	—	4	8	no	—
50	C₄⋊#261	—	2	8	no	—
51	C₄⋊#261	—	4	4	no	—
52	C₄×#261	—	4	8	no	—
53	C₂×C₈×C₁₃	C₄²	1	2	yes	—
54	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
58	C₂×#1055	C₄⋊C₄	1	4	yes	—
59	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
60	C₈×D₁₃	C₄²	1	2	yes	—
61	C₈×D₁₃	C₂²⋊C₄	1	4	yes	—
62	C₈×D₁₃	C₂²⋊C₄	1	2	yes	—
63	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
64	C₈×D₁₃	C₂²×C₄	1	2	yes	—
65	C₈×D₁₃	C₂×D₄	1	2	yes	—
66	C₁₃×C₁₆	—	2	2	no	—
67	#6090	—	2	2	no	—
68	C₂³×C₁₆	—	13	1	no	—
69	C₂×#427	—	13	2	no	—
70	#1380	—	13	4	no	—
71	C₂³×C₁₃	#107	1	1	yes	characteristic
72	C₂²×D₁₃	—	2	4	no	—
73	C₂²×D₁₃	#102	1	4	yes	—
74	C₂²×D₁₃	#107	1	1	yes	characteristic
75	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
76	C₂×#261	—	4	8	no	—
77	C₂×#261	—	8	8	no	—
78	C₂×C₄×C₁₃	#95	1	2	yes	—
79	C₂×C₄×C₁₃	#106	1	2	yes	—
80	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×Dic₁₃	#103	1	4	yes	—
83	C₂×Dic₁₃	#107	1	1	yes	characteristic
84	C₂×Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
85	C₄×D₁₃	#95	1	2	yes	—
86	C₄×D₁₃	#102	1	4	yes	—
87	C₄×D₁₃	#103	1	4	yes	—
88	C₄×D₁₃	#106	1	2	yes	—
89	C₄×D₁₃	C₂×C₂²⋊C₄	1	2	yes	—
90	C₄×D₁₃	C₂×SD₁₆	1	2	yes	—
91	C₈×C₁₃	#95	1	2	yes	—
92	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
93	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	#1055	#95	1	2	yes	—
95	#1055	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₈	—	13	1	no	—
97	C₂×#117	—	26	2	no	—
98	C₂²×C₁₆	—	13	1	no	—
99	C₂²×C₁₆	—	13	2	no	—
100	C₂²×C₁₆	—	13	2	no	—
101	C₂²×C₁₆	—	26	4	no	—
102	C₂²×C₁₆	#261	1	1	yes	characteristic, complemented
103	#427	—	13	8	no	—
104	#427	—	26	4	no	—
105	C₂×#107	—	13	4	no	—
106	C₂×#106	—	26	2	no	—
107	C₂²×C₁₃	—	2	2	no	—
108	C₂²×C₁₃	#340	1	2	yes	—
109	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
110	C₂×D₁₃	—	2	2	no	—
111	C₂×D₁₃	—	2	8	no	—
112	C₂×D₁₃	#340	1	2	yes	—
113	C₂×D₁₃	C₂×#102	1	2	yes	—
114	C₂×D₁₃	#508	1	4	yes	—
115	#261	—	8	8	no	—
116	C₄×C₁₃	#340	1	2	yes	—
117	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
118	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
119	Dic₁₃	#340	1	2	yes	—
120	Dic₁₃	C₂×#103	1	2	yes	—
121	C₂³×C₄	—	13	1	no	—
122	C₂²×C₈	—	13	1	no	—
123	C₂²×C₈	—	13	4	no	—
124	C₂²×C₈	—	13	1	no	—
125	C₂²×C₈	—	13	1	no	—
126	C₂²×C₈	—	13	4	no	—
127	C₂²×C₈	—	13	1	no	—
128	C₂²×C₈	—	13	1	no	—
129	C₂²×C₈	C₂×#261	1	1	yes	characteristic
130	#117	—	26	8	no	—
131	#117	—	52	4	no	—
132	C₂×C₄⋊C₄	—	26	4	no	—
133	C₂×C₂²⋊C₄	—	52	2	no	—
134	C₂×C₄²	—	52	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	13	2	no	—
137	C₂×C₁₆	—	26	4	no	—
138	C₂×C₁₆	—	26	4	no	—
139	C₂×C₁₆	—	26	2	no	—
140	C₂×C₁₆	C₂×#261	1	2	yes	—
141	#107	—	13	8	no	—
142	#107	—	26	4	no	—
143	#106	—	26	8	no	—
144	C₂×C₁₃	—	2	2	no	—
145	C₂×C₁₃	C₂×#340	1	1	yes	characteristic
146	C₂×C₁₃	#2158	1	2	yes	—
147	D₁₃	—	2	2	no	—
148	D₁₃	#2138	1	4	yes	—
149	C₂⁴	—	13	1	no	—
150	C₂²×C₄	—	13	1	no	—
151	C₂²×C₄	—	13	4	no	—
152	C₂²×C₄	—	13	1	no	—
153	C₂²×C₄	—	13	1	no	—
154	C₂²×C₄	—	13	4	no	—
155	C₂²×C₄	—	13	1	no	—
156	C₂²×C₄	—	13	1	no	—
157	C₂²×C₄	—	52	4	no	—
158	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
159	C₂×C₈	—	13	4	no	—
160	C₂×C₈	—	13	1	no	—
161	C₂×C₈	—	13	1	no	—
162	C₂×C₈	—	13	4	no	—
163	C₂×C₈	—	13	4	no	—
164	C₂×C₈	—	13	2	no	—
165	C₂×C₈	—	13	2	no	—
166	C₂×C₈	—	13	2	no	—
167	C₂×C₈	—	13	2	no	—
168	C₂×C₈	C₄×#261	1	2	yes	—
169	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
170	C₂×C₈	C₂²⋊#261	1	2	yes	—
171	C₂×C₈	C₂²×#261	1	1	yes	characteristic
172	C₄⋊C₄	—	26	8	no	—
173	C₄⋊C₄	—	52	4	no	—
174	C₂²⋊C₄	—	52	8	no	—
175	C₄²	—	52	8	no	—
176	C₁₆	—	2	2	no	—
177	C₁₆	—	26	2	no	—
178	C₁₃	#12556	1	1	yes	characteristic, complemented, Sylow
179	C₂³	—	13	1	no	—
180	C₂³	—	13	4	no	—
181	C₂³	—	13	1	no	—
182	C₂³	—	26	4	no	—
183	C₂³	#88102	1	1	yes	characteristic
184	C₂×C₄	—	13	4	no	—
185	C₂×C₄	—	13	1	no	—
186	C₂×C₄	—	13	1	no	—
187	C₂×C₄	—	13	4	no	—
188	C₂×C₄	—	13	4	no	—
189	C₂×C₄	—	13	2	no	—
190	C₂×C₄	—	13	2	no	—
191	C₂×C₄	—	13	2	no	—
192	C₂×C₄	—	13	2	no	—
193	C₂×C₄	—	52	8	no	—
194	C₂×C₄	—	104	8	no	—
195	C₂×C₄	#88101	1	2	yes	—
196	C₂×C₄	#88102	1	1	yes	characteristic
197	C₂×C₄	(C₂×#261)⋊C₄	1	2	yes	—
198	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
199	C₈	—	13	2	no	—
200	C₈	—	13	2	no	—
201	C₈	(C₂×#261)⋊C₄	1	2	yes	—
202	C₈	C₂×C₄×#261	1	1	yes	characteristic
203	C₈	C₂×C₂²⋊#261	1	1	yes	characteristic
204	C₂²	—	2	2	no	—
205	C₂²	—	13	2	no	—
206	C₂²	—	13	4	no	—
207	C₂²	—	13	2	no	—
208	C₂²	—	26	8	no	—
209	C₂²	—	26	2	no	—
210	C₂²	#11718265	1	2	yes	—
211	C₂²	C₂×#88102	1	1	yes	characteristic
212	C₄	—	13	2	no	—
213	C₄	—	13	2	no	—
214	C₄	—	104	8	no	—
215	C₄	#11718265	1	2	yes	—
216	C₄	C₂×#88101	1	1	yes	characteristic
217	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
218	C₂	—	2	2	no	—
219	C₂	—	13	4	no	—
220	C₂	—	26	2	no	—
221	C₂	C₂×#11718265	1	1	yes	characteristic
222	C₂	#13131187	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#88102
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×#261
Fitting	C₂²×C₁₃×C₁₆	C₄
socle	C₂²×C₁₃	C₂×#107
prime radical	C₂³×D₁₃	D₈
cosocle	C₂×C₈×D₁₃	C₂³
hypercenter	C₂²×C₁₆	#261
hypocenter	C₁₃	#12556
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #13129894	C₂	semidirect
7. #11719092	C₄	semidirect
10. C₂²×C₁₃×C₁₆	C₄	semidirect
11. C₂²×#6090	C₄	semidirect
13. C₂×C₁₆×D₁₃	C₄	semidirect
102. C₂²×C₁₆	#261	semidirect
178. C₁₃	#12556	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719092	2
#13129894	2
C₂²×C₁₆×D₁₃	2
#12556	13