Subgroups — #16016411

Order 3328.

506 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123768	C₂	1	1	yes	maximal, characteristic
2	#13116742	C₂	1	4	yes	maximal, complemented
3	C₁₆×Dic₂₆	C₂	1	4	yes	maximal, complemented
4	#13116596	C₂	1	1	yes	maximal, characteristic
5	#13116595	C₂	1	1	yes	maximal, characteristic
6	#13116591	C₂	1	1	yes	maximal, characteristic
7	#13116432	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic
10	#11719222	C₄	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₁₃×#405	C₂²	1	1	yes	characteristic
14	#11718780	C₄	1	1	yes	characteristic
15	#11718779	C₄	1	1	yes	characteristic
16	#11718762	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×#88052	C₂²	1	1	yes	characteristic
19	C₂²×#6090	C₂²	1	1	yes	characteristic
20	#11718660	C₄	1	1	yes	characteristic
21	#11718362	C₄	1	4	yes	—
22	C₈×Dic₂₆	C₂²	1	4	yes	—
23	C₁₃×#346	C₂²	1	2	yes	—
24	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
25	#11718094	C₂²	1	2	yes	—
26	#11718093	C₂²	1	4	yes	—
27	#11718093	C₂²	1	4	yes	—
28	C₁₆×Dic₁₃	C₂²	1	2	yes	—
29	C₁₆×Dic₁₃	C₂²	1	2	yes	—
30	C₁₆×Dic₁₃	C₂²	1	2	yes	—
31	C₄⋊#6090	C₂²	1	4	yes	—
32	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
35	#88180	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×#6099	C₂×C₄	1	1	yes	characteristic
38	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
39	#88144	C₈	1	2	yes	—
40	#88142	C₈	1	2	yes	—
41	#88132	C₈	1	4	yes	—
42	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
43	C₁₃×M₃₂	—	2	2	no	—
44	C₂×C₁₃×C₁₆	Q₈	1	4	yes	complemented
45	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
46	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
47	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
48	C₁₃×#97	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₁₃	C₂³	1	2	yes	—
50	#88055	C₂×C₄	1	2	yes	—
51	#88054	C₂×C₄	1	4	yes	—
52	#88054	C₂×C₄	1	4	yes	—
53	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
54	C₈×Dic₁₃	C₂³	1	2	yes	—
55	C₈×Dic₁₃	C₂³	1	2	yes	—
56	#88052	—	2	2	no	—
57	C₂×#6090	—	2	2	no	—
58	C₂×#6090	C₂³	1	2	yes	—
59	C₂×#6090	C₂³	1	2	yes	—
60	C₄⋊#1055	C₂×C₄	1	4	yes	—
61	#33474	—	13	1	no	maximal, Sylow
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
64	C₂×Dic₂₆	C₂×C₈	1	4	yes	—
65	C₁₃×M₁₆	—	2	2	no	—
66	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₁₃	#41	1	2	yes	—
70	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
72	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
73	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
75	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
76	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
77	#6101	C₂×C₈	1	4	yes	—
78	#6101	C₂×C₈	1	4	yes	—
79	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
80	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
81	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
82	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
83	#6099	—	2	2	no	—
84	C₂×#1055	—	2	2	no	—
85	C₂×#1055	C₂²×C₄	1	2	yes	—
86	C₂×#1055	C₂²×C₄	1	2	yes	—
87	C₁₃×C₁₆	—	2	2	no	—
88	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
89	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
90	#6090	—	2	2	no	—
91	#6090	—	2	2	no	—
92	#2960	—	13	1	no	—
93	#2183	—	13	4	no	—
94	Q₈×C₁₆	—	13	4	no	—
95	#2151	—	13	1	no	—
96	#2151	—	13	1	no	—
97	#2150	—	13	1	no	—
98	#2108	—	13	1	no	—
99	#2108	D₁₃	1	1	yes	characteristic
100	C₂×C₄×C₁₆	—	13	1	no	—
101	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
103	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
104	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#131	1	2	yes	—
106	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×Dic₁₃	—	2	4	no	—
108	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
109	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
110	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
111	Dic₂₆	—	2	8	no	—
112	C₈×C₁₃	—	2	2	no	—
113	C₈×C₁₃	C₄×Q₈	1	2	yes	—
114	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
115	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
116	#1055	—	2	2	no	—
117	#1055	—	2	2	no	—
118	#533	—	13	1	no	—
119	C₂×M₃₂	—	13	1	no	—
120	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
121	C₂²×C₁₆	—	13	1	no	—
122	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
123	#446	—	13	4	no	—
124	C₈×Q₈	—	13	4	no	—
125	#424	—	13	1	no	—
126	#424	—	13	1	no	—
127	#423	—	13	1	no	—
128	#405	—	13	1	no	—
129	#405	C₂×D₁₃	1	1	yes	characteristic
130	C₂×C₄×C₈	—	13	1	no	—
131	C₄⋊C₁₆	—	13	4	no	—
132	C₄⋊C₁₆	—	13	4	no	—
133	C₄⋊C₁₆	—	13	4	no	—
134	#346	—	13	2	no	—
135	#346	C₂×D₁₃	1	2	yes	—
136	C₄×C₁₆	—	13	2	no	—
137	C₄×C₁₆	—	13	2	no	—
138	C₄×C₁₆	—	13	2	no	—
139	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
140	C₂²×C₁₃	—	2	2	no	—
141	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
142	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
143	C₄×C₁₃	—	2	4	no	—
144	C₄×C₁₃	C₈×Q₈	1	2	yes	—
145	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
146	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
147	Dic₁₃	—	2	4	no	—
148	Dic₁₃	—	2	2	no	—
149	Dic₁₃	#504	1	4	yes	—
150	C₂×M₁₆	—	13	1	no	—
151	C₂×M₁₆	C₄×D₁₃	1	1	yes	characteristic
152	C₂²×C₈	—	13	1	no	—
153	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
154	C₄⋊Q₈	—	13	2	no	—
155	#125	—	13	2	no	—
156	C₂²⋊Q₈	—	13	4	no	—
157	C₄×Q₈	—	13	4	no	—
158	#117	—	13	1	no	—
159	#117	C₄×D₁₃	1	1	yes	characteristic
160	C₂×C₄²	—	13	1	no	—
161	M₃₂	—	2	2	no	—
162	M₃₂	—	26	2	no	—
163	C₂×C₁₆	—	13	2	no	—
164	C₂×C₁₆	—	13	2	no	—
165	C₂×C₁₆	—	26	2	no	—
166	C₂×C₁₆	Dic₂₆	1	4	yes	complemented
167	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
168	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
169	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
170	C₄⋊C₈	—	13	4	no	—
171	C₄⋊C₈	—	13	4	no	—
172	C₄⋊C₈	—	13	4	no	—
173	#97	—	13	2	no	—
174	#97	C₄×D₁₃	1	2	yes	—
175	C₄×C₈	—	13	2	no	—
176	C₄×C₈	—	13	2	no	—
177	C₄×C₈	—	13	2	no	—
178	C₄×C₈	C₂²×D₁₃	1	2	yes	—
179	C₂×C₁₃	—	2	2	no	—
180	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
181	C₂×C₁₃	C₂×#504	1	2	yes	—
182	C₂×Q₈	—	13	4	no	—
183	C₂²×C₄	—	13	2	no	—
184	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
185	M₁₆	—	2	2	no	—
186	M₁₆	—	26	2	no	—
187	C₂×C₈	—	13	2	no	—
188	C₂×C₈	—	13	2	no	—
189	C₂×C₈	—	26	2	no	—
190	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
191	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
192	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
193	C₂×C₈	#6127	1	2	yes	—
194	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
195	C₄⋊C₄	—	13	2	no	—
196	C₄⋊C₄	—	13	4	no	—
197	C₄⋊C₄	—	13	4	no	—
198	C₄⋊C₄	—	13	4	no	—
199	C₄⋊C₄	C₈×D₁₃	1	2	yes	—
200	C₂²⋊C₄	—	13	2	no	—
201	C₂²⋊C₄	C₈×D₁₃	1	2	yes	—
202	C₄²	—	13	1	no	—
203	C₄²	—	13	1	no	—
204	C₄²	—	13	2	no	—
205	C₄²	—	13	2	no	—
206	C₄²	C₂×C₄×D₁₃	1	2	yes	—
207	C₁₆	—	2	2	no	—
208	C₁₆	—	26	2	no	—
209	C₁₆	—	26	2	no	—
210	C₁₆	C₂×Dic₂₆	1	2	yes	—
211	C₁₆	C₂×Dic₂₆	1	2	yes	—
212	C₁₃	#33474	1	1	yes	characteristic, complemented, Sylow
213	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
214	Q₈	—	26	8	no	—
215	C₂×C₄	—	13	2	no	—
216	C₂×C₄	—	13	2	no	—
217	C₂×C₄	—	13	4	no	—
218	C₂×C₄	—	26	4	no	—
219	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
220	C₂×C₄	C₂×C₈×D₁₃	1	4	yes	—
221	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
222	C₂×C₄	#88155	1	2	yes	—
223	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	26	2	no	—
226	C₈	—	26	2	no	—
227	C₈	C₄×Dic₂₆	1	2	yes	—
228	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
229	C₈	C₂×#6127	1	1	yes	characteristic
230	C₂²	—	2	2	no	—
231	C₂²	C₈×Dic₂₆	1	2	yes	—
232	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
233	C₄	—	2	4	no	—
234	C₄	—	13	4	no	—
235	C₄	—	26	2	no	—
236	C₄	—	26	4	no	—
237	C₄	C₈×Dic₂₆	1	2	yes	—
238	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
239	C₄	C₂×#88155	1	1	yes	characteristic
240	C₂	—	2	2	no	—
241	C₂	C₂×C₈×Dic₂₆	1	1	yes	characteristic
242	C₂	D₁₃×#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2108	C₂
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2108	D₁₃
hypocenter	C₁₃	#33474
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #13116742	C₂	semidirect
3. C₁₆×Dic₂₆	C₂	semidirect
44. C₂×C₁₃×C₁₆	Q₈	semidirect
166. C₂×C₁₆	Dic₂₆	semidirect
212. C₁₃	#33474	semidirect

Maximal subgroups

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

subgroup	index
#13123768	2
#13116742	2
C₁₆×Dic₂₆	2
#13116596	2
#13116595	2
#13116591	2
#13116432	2
C₂×C₁₆×Dic₁₃	2
C₁₃×#2108	2
#33474	13