Subgroups — #16018407

Order 3328.

490 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	#13125350	C₂	1	1	yes	maximal, characteristic
2	#13117116	C₂	1	2	yes	maximal, complemented
3	#13117112	C₂	1	4	yes	maximal, complemented
4	#13117109	C₂	1	2	yes	maximal, complemented
5	#13116618	C₂	1	4	yes	maximal
6	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#533	—	2	1	no	—
9	#11719175	—	2	1	no	—
10	C₁₃×#500	C₂²	1	2	yes	—
11	C₁₃×#499	—	2	1	no	—
12	C₁₃×#499	C₂²	1	2	yes	—
13	C₁₃×#498	—	2	1	no	—
14	C₁₃×#427	—	2	1	no	—
15	C₁₃×#427	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	#11718767	—	2	1	no	—
18	#11718767	C₂²	1	2	yes	—
19	C₂²×#6090	C₂²	1	2	yes	—
20	#11718344	—	2	1	no	—
21	#11718344	C₂²	1	2	yes	—
22	#11718343	C₂²	1	2	yes	—
23	#11718341	—	2	1	no	—
24	#11718082	C₂²	1	8	yes	—
25	#11718081	C₂²	1	8	yes	—
26	C₄×#6090	C₂²	1	1	yes	characteristic
27	C₄×#6090	C₂²	1	2	yes	—
28	C₄×#6090	C₂²	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₁₃×C₄⋊Q₈	—	2	2	no	—
32	C₁₃×#125	—	2	2	no	—
33	C₁₃×C₂²⋊Q₈	—	4	2	no	—
34	C₄×Q₈×C₁₃	—	4	2	no	—
35	C₁₃×#117	—	2	2	no	—
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	#88180	—	2	2	no	—
38	#88178	—	4	2	no	—
39	#88124	—	2	2	no	—
40	#88123	—	2	2	no	—
41	C₄×Dic₂₆	—	4	2	no	—
42	C₁₃×#107	—	2	2	no	—
43	C₁₃×#107	C₂³	1	4	yes	—
44	C₁₃×#106	—	2	2	no	—
45	C₁₃×#106	—	2	2	no	—
46	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	#88058	—	2	2	no	—
51	#88058	C₂³	1	4	yes	—
52	#88057	—	2	2	no	—
53	#88057	—	2	2	no	—
54	C₂×#6090	Q₈	1	8	yes	complemented
55	C₂×#6090	C₂³	1	2	yes	—
56	C₂×#6090	C₂³	1	2	yes	—
57	#33726	—	13	1	no	maximal, Sylow
58	C₂×Q₈×C₁₃	—	4	2	no	—
59	C₂²×C₄×C₁₃	D₈	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₂₆	—	4	2	no	—
62	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
63	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
67	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	—	2	4	no	—
69	C₁₃×C₄⋊C₄	—	4	4	no	—
70	C₁₃×C₄⋊C₄	—	4	2	no	—
71	C₁₃×C₂²⋊C₄	—	4	2	no	—
72	C₄²×C₁₃	—	4	2	no	—
73	C₄²×C₁₃	D₈	1	2	yes	—
74	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²⋊Dic₁₃	—	4	2	no	—
77	C₄⋊Dic₁₃	—	2	4	no	—
78	C₄⋊Dic₁₃	—	4	2	no	—
79	#6101	—	4	4	no	—
80	C₄×Dic₁₃	—	4	2	no	—
81	#6090	C₂×Q₈	1	4	yes	—
82	#6090	C₂×Q₈	1	4	yes	—
83	#3160	—	13	1	no	—
84	#3160	D₁₃	1	1	yes	characteristic
85	#2254	—	13	2	no	—
86	#2253	—	13	4	no	—
87	#2252	—	13	2	no	—
88	#2158	—	13	4	no	—
89	C₂×C₄×C₁₆	—	13	1	no	—
90	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
91	Q₈×C₁₃	—	8	4	no	—
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	—	4	4	no	—
94	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
99	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
100	C₂×Dic₁₃	—	4	4	no	—
101	Dic₂₆	—	8	4	no	—
102	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
103	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
104	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
105	C₈×C₁₃	C₂²×Q₈	1	2	yes	—
106	#533	—	2	1	no	—
107	#533	—	26	1	no	—
108	C₂²×C₁₆	—	13	2	no	—
109	#500	—	13	2	no	—
110	#500	C₂×D₁₃	1	2	yes	—
111	#499	—	2	1	no	—
112	#499	—	13	2	no	—
113	#499	—	26	1	no	—
114	#499	C₂×D₁₃	1	2	yes	—
115	#498	—	2	1	no	—
116	#498	—	26	1	no	—
117	#427	—	2	1	no	—
118	#427	—	13	2	no	—
119	#427	—	26	1	no	—
120	#427	C₂×D₁₃	1	2	yes	—
121	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
122	#367	—	13	8	no	—
123	#366	—	13	8	no	—
124	C₄×C₁₆	—	13	1	no	—
125	C₄×C₁₆	—	13	2	no	—
126	C₄×C₁₆	—	13	1	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	#500	1	2	yes	—
129	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
130	C₄×C₁₃	—	2	2	no	—
131	C₄×C₁₃	—	8	4	no	—
132	C₄×C₁₃	#500	1	2	yes	—
133	C₄×C₁₃	#508	1	4	yes	—
134	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
135	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
136	Dic₁₃	—	8	4	no	—
137	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
138	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
139	C₄⋊Q₈	—	2	2	no	—
140	C₄⋊Q₈	—	26	2	no	—
141	#125	—	2	2	no	—
142	#125	—	26	2	no	—
143	C₂²⋊Q₈	—	4	2	no	—
144	C₂²⋊Q₈	—	52	2	no	—
145	C₄×Q₈	—	4	2	no	—
146	C₄×Q₈	—	52	2	no	—
147	#117	—	2	2	no	—
148	#117	—	26	2	no	—
149	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
150	C₂×C₁₆	—	13	8	no	—
151	C₂×C₁₆	—	13	2	no	—
152	C₂×C₁₆	—	13	2	no	—
153	#107	—	2	2	no	—
154	#107	—	13	4	no	—
155	#107	—	26	2	no	—
156	#107	C₂²×D₁₃	1	4	yes	—
157	#106	—	2	2	no	—
158	#106	—	2	2	no	—
159	#106	—	26	2	no	—
160	#106	—	26	2	no	—
161	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
163	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
164	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
165	C₂×C₁₃	—	2	2	no	—
166	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
167	C₂×C₁₃	C₂×#508	1	2	yes	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	—	52	2	no	—
170	C₂²×C₄	#6104	1	2	yes	—
171	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
172	C₂×C₈	#6107	1	4	yes	—
173	C₂×C₈	Q₈×D₁₃	1	4	yes	—
174	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
175	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
177	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	4	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	26	4	no	—
182	C₄⋊C₄	—	52	4	no	—
183	C₄⋊C₄	—	52	2	no	—
184	C₂²⋊C₄	—	4	2	no	—
185	C₂²⋊C₄	—	52	2	no	—
186	C₄²	—	4	2	no	—
187	C₄²	—	52	2	no	—
188	C₄²	#6104	1	2	yes	—
189	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
190	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
191	C₁₆	—	13	4	no	—
192	C₁₆	—	13	4	no	—
193	C₁₃	#33726	1	1	yes	characteristic, complemented, Sylow
194	C₂³	C₂×#6104	1	1	yes	characteristic
195	Q₈	—	8	4	no	—
196	Q₈	—	104	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	52	4	no	—
200	C₂×C₄	C₂×#6104	1	2	yes	—
201	C₂×C₄	C₂×#6104	1	2	yes	—
202	C₂×C₄	C₂×#6104	1	1	yes	characteristic
203	C₂×C₄	C₂×#6107	1	2	yes	—
204	C₂×C₄	#88198	1	2	yes	—
205	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
206	C₈	C₂×#6107	1	2	yes	—
207	C₈	C₂×#6107	1	2	yes	—
208	C₈	#88198	1	2	yes	—
209	C₈	C₂×Q₈×D₁₃	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	#11718739	1	2	yes	—
212	C₂²	C₂²×#6104	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	8	4	no	—
215	C₄	—	104	4	no	—
216	C₄	#11718739	1	2	yes	—
217	C₄	#11718855	1	4	yes	—
218	C₄	C₂²×#6107	1	1	yes	characteristic
219	C₄	C₂×#88198	1	1	yes	characteristic
220	C₂	—	2	2	no	—
221	C₂	C₂×#11718739	1	1	yes	characteristic
222	C₂	C₂×#11718855	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3160	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	#3160	D₁₃
hypocenter	C₁₃	#33726
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #13117116	C₂	semidirect
3. #13117112	C₂	semidirect
4. #13117109	C₂	semidirect
54. C₂×#6090	Q₈	semidirect
193. C₁₃	#33726	semidirect

Maximal subgroups

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

subgroup	index
#13125350	2
#13117116	2
#13117112	2
#13117109	2
#13116618	2
C₂×C₄×#6090	2
C₁₃×#3160	2
#33726	13