Subgroups — #16016064

Order 3328.

528 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116652	C₂	1	4	yes	maximal, complemented
2	#13116648	C₂	1	4	yes	maximal, complemented
3	C₂×#11718055	C₂	1	2	yes	maximal, complemented
4	#13116457	C₂	1	2	yes	maximal
5	#13116441	C₂	1	2	yes	maximal, complemented
6	C₄×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
8	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
10	C₁₃×#405	C₄	1	1	yes	characteristic
11	C₁₃×#405	C₄	1	2	yes	—
12	C₁₃×#405	C₂²	1	1	yes	characteristic
13	C₁₃×#405	C₂²	1	2	yes	—
14	C₄×C₁₃×M₁₆	C₄	1	1	yes	characteristic
15	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×#88052	C₂²	1	4	yes	—
19	#11718145	C₂²	1	8	yes	—
20	#11718058	C₂²	1	8	yes	—
21	#11718055	C₂²	1	4	yes	—
22	#11718055	C₂²	1	4	yes	—
23	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
24	C₁₃×#131	—	2	4	no	—
25	C₁₃×#131	—	2	2	no	—
26	C₁₃×#131	—	2	2	no	—
27	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₁₃	—	2	1	no	—
36	C₄×D₄×C₁₃	—	2	2	no	—
37	C₄×D₄×C₁₃	—	2	1	no	—
38	C₁₃×#117	C₂×C₄	1	2	yes	—
39	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₁₃×#97	C₂×C₄	1	2	yes	—
43	C₁₃×#97	C₂×C₄	1	2	yes	—
44	C₁₃×#97	C₂³	1	2	yes	—
45	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	#88052	—	2	4	no	—
53	#88052	C₂³	1	8	yes	—
54	C₂×#6090	—	2	4	no	—
55	#33428	—	13	1	no	maximal, Sylow
56	C₁₃×#41	—	2	2	no	—
57	C₁₃×#41	—	2	2	no	—
58	C₂×Q₈×C₁₃	M₁₆	1	1	yes	characteristic
59	C₂×D₄×C₁₃	M₁₆	1	2	yes	—
60	C₂×D₄×C₁₃	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	—	2	1	no	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₁₃×M₁₆	—	2	2	no	—
66	C₁₃×M₁₆	—	2	2	no	—
67	C₁₃×M₁₆	—	2	4	no	—
68	C₁₃×M₁₆	—	2	4	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	4	no	—
72	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	#41	1	2	yes	—
79	C₂×C₈×C₁₃	#41	1	2	yes	—
80	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	—	2	1	no	—
83	C₁₃×C₂²⋊C₄	—	2	2	no	—
84	C₁₃×C₂²⋊C₄	—	2	1	no	—
85	C₄²×C₁₃	—	2	2	no	—
86	C₄²×C₁₃	—	2	1	no	—
87	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	#6090	—	2	8	no	—
91	C₄×#131	D₁₃	1	1	yes	characteristic
92	#2166	—	13	4	no	—
93	#2165	—	13	4	no	—
94	C₂×#364	—	13	2	no	—
95	#2115	—	13	2	no	—
96	#2110	—	13	2	no	—
97	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
98	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
99	Q₈×C₁₃	—	2	2	no	—
100	D₄×C₁₃	—	2	4	no	—
101	D₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	4	4	no	—
109	C₂×C₄×C₁₃	#117	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
114	C₈×C₁₃	—	2	2	no	—
115	C₈×C₁₃	—	2	2	no	—
116	C₈×C₁₃	#117	1	1	yes	characteristic
117	C₈×C₁₃	#117	1	1	yes	characteristic
118	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
119	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
120	C₈×C₁₃	C₂×#41	1	2	yes	—
121	C₂×#131	Dic₁₃	1	1	yes	characteristic
122	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
123	C₄×#41	Dic₁₃	1	1	yes	characteristic
124	C₂×M₃₂	—	13	4	no	—
125	#405	Dic₁₃	1	1	yes	characteristic
126	#405	Dic₁₃	1	2	yes	—
127	#405	C₂×D₁₃	1	2	yes	—
128	#405	C₂×D₁₃	1	1	yes	characteristic
129	C₄×M₁₆	Dic₁₃	1	1	yes	characteristic
130	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
131	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
132	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
133	#364	—	13	4	no	—
134	#364	—	13	4	no	—
135	#349	—	13	8	no	—
136	#347	—	13	8	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	—	4	1	no	—
139	C₂²×C₁₃	—	4	2	no	—
140	C₂²×C₁₃	#432	1	2	yes	—
141	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
142	C₄×C₁₃	—	2	2	no	—
143	C₄×C₁₃	—	2	2	no	—
144	C₄×C₁₃	—	4	2	no	—
145	C₄×C₁₃	—	4	1	no	—
146	C₄×C₁₃	#432	1	2	yes	—
147	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
148	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
149	C₂×#41	C₂×Dic₁₃	1	1	yes	characteristic
150	#131	—	2	4	no	—
151	#131	—	2	2	no	—
152	#131	—	2	2	no	—
153	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
154	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
155	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
156	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
157	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
158	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
159	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
160	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
161	C₄×Q₈	—	2	1	no	—
162	C₄×D₄	—	2	2	no	—
163	C₄×D₄	—	2	1	no	—
164	#117	C₂×Dic₁₃	1	2	yes	—
165	#117	C₂×Dic₁₃	1	1	yes	characteristic
166	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
167	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
168	M₃₂	—	13	8	no	—
169	M₃₂	—	26	4	no	—
170	C₂×C₁₆	—	26	4	no	—
171	#97	C₂×Dic₁₃	1	2	yes	—
172	#97	C₂×Dic₁₃	1	2	yes	—
173	#97	C₂²×D₁₃	1	2	yes	—
174	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
175	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
176	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
177	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
178	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
179	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
180	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
181	C₂×C₁₃	—	2	1	no	—
182	C₂×C₁₃	—	2	2	no	—
183	C₂×C₁₃	—	4	2	no	—
184	C₂×C₁₃	C₂×#432	1	1	yes	characteristic
185	#41	—	2	2	no	—
186	#41	—	2	2	no	—
187	C₂×Q₈	#6099	1	1	yes	characteristic
188	C₂×D₄	#6099	1	2	yes	—
189	C₂×D₄	#6099	1	1	yes	characteristic
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
193	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
194	M₁₆	—	2	2	no	—
195	M₁₆	—	2	2	no	—
196	M₁₆	—	2	4	no	—
197	M₁₆	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	#6099	1	4	yes	—
202	C₂×C₈	#6127	1	2	yes	—
203	C₂×C₈	#6127	1	2	yes	—
204	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
205	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
206	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
207	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
208	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
209	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	1	no	—
216	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
217	C₄²	C₂²×Dic₁₃	1	2	yes	—
218	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
219	C₁₆	—	26	8	no	—
220	C₁₃	#33428	1	1	yes	characteristic, complemented, Sylow
221	C₂³	C₂×#6099	1	2	yes	—
222	C₂³	C₂×#6099	1	1	yes	characteristic
223	Q₈	—	2	2	no	—
224	D₄	—	2	4	no	—
225	D₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	C₂×#6099	1	2	yes	—
234	C₂×C₄	C₂×#6099	1	2	yes	—
235	C₂×C₄	C₂×#6099	1	1	yes	characteristic
236	C₂×C₄	#88180	1	2	yes	—
237	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₈	C₂×#6099	1	2	yes	—
241	C₈	C₂×#6099	1	2	yes	—
242	C₈	#88180	1	1	yes	characteristic
243	C₈	#88180	1	1	yes	characteristic
244	C₈	C₂×#6127	1	2	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	#11718587	1	2	yes	—
249	C₂²	C₂²×#6099	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	1	no	—
254	C₄	#11718587	1	2	yes	—
255	C₄	C₂²×#6099	1	1	yes	characteristic
256	C₄	C₂×#88180	1	1	yes	characteristic
257	C₂	—	2	1	no	—
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	C₂	C₂×#11718587	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#88180
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×#131	C₂
socle	C₂×C₁₃	C₂×#432
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₄×#131	D₁₃
hypocenter	C₁₃	#33428
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ #405 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #13116652	C₂	semidirect
2. #13116648	C₂	semidirect
3. C₂×#11718055	C₂	semidirect
5. #13116441	C₂	semidirect
220. C₁₃	#33428	semidirect

Maximal subgroups

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

subgroup	index
#13116652	2
#13116648	2
C₂×#11718055	2
#13116457	2
#13116441	2
C₄×C₁₃×#131	2
#33428	13