Subgroups — #15828635

Order 3328.

806 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123469	C₂	1	2	yes	maximal
2	#13113139	C₂	1	4	yes	maximal, complemented
3	#13113133	C₂	1	4	yes	maximal, complemented
4	#13113128	C₂	1	2	yes	maximal, complemented
5	#13113119	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₄	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×M₁₆	C₄	1	2	yes	—
10	C₄×C₁₃×M₁₆	C₄	1	2	yes	—
11	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
12	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
13	C₂×C₁₃×#97	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718868	C₄	1	4	yes	—
17	#11718587	C₄	1	4	yes	—
18	#11718587	C₄	1	4	yes	—
19	#11718586	C₂²	1	4	yes	—
20	#11718586	C₂²	1	4	yes	—
21	#11718585	C₄	1	4	yes	—
22	C₄×#6099	C₂²	1	4	yes	—
23	#11718047	C₂²	1	4	yes	complemented
24	#11718046	C₂²	1	8	yes	complemented
25	C₈×#1055	C₂²	1	4	yes	complemented
26	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₃×M₁₆	—	2	8	no	—
29	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×#6099	C₂×C₄	1	8	yes	—
43	C₁₃×#97	C₂×C₄	1	4	yes	—
44	C₁₃×#97	C₂×C₄	1	4	yes	—
45	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₁₃	C₂³	1	2	yes	—
48	C₄×C₈×C₁₃	C₂³	1	2	yes	—
49	C₂²⋊#1055	C₂×C₄	1	8	yes	—
50	C₂²⋊#1055	C₂×C₄	1	8	yes	—
51	C₄⋊#1055	C₂×C₄	1	8	yes	—
52	C₄⋊#1055	C₂×C₄	1	8	yes	—
53	#88043	C₂×C₄	1	8	yes	—
54	C₄×#1055	C₂³	1	8	yes	—
55	#17334	—	13	1	no	maximal, Sylow
56	C₂⁴×C₁₃	C₄²	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	—	2	4	no	—
58	C₂²×C₄×C₁₃	—	4	4	no	—
59	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
63	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
64	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₁₃×M₁₆	—	2	8	no	—
72	C₁₃×M₁₆	—	2	8	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	M₁₆	1	8	yes	complemented
75	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
77	C₄²×C₁₃	C₄²	1	2	yes	—
78	C₄²×C₁₃	C₄²	1	2	yes	—
79	C₄²×C₁₃	C₄²	1	2	yes	—
80	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
81	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
82	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
83	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
87	#6099	—	4	8	no	—
88	C₂×#1055	C₂²×C₄	1	16	yes	—
89	#2920	—	13	2	no	—
90	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
91	#1455	—	13	4	no	—
92	#1453	—	13	4	no	—
93	#1452	—	13	2	no	—
94	#1450	—	13	2	no	—
95	C₂³×C₁₃	—	2	2	no	—
96	C₂³×C₁₃	—	4	2	no	—
97	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
98	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
99	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	2	4	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
118	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
120	C₈×C₁₃	—	2	4	no	—
121	C₈×C₁₃	C₂×M₁₆	1	8	yes	—
122	#1055	—	2	16	no	—
123	C₂²×M₁₆	Dic₁₃	1	2	yes	—
124	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
125	#432	—	13	4	no	—
126	#432	—	13	4	no	—
127	#431	—	13	4	no	—
128	#431	—	13	4	no	—
129	#423	—	13	4	no	—
130	#407	—	13	4	no	—
131	C₄×M₁₆	—	13	4	no	—
132	C₄×M₁₆	Dic₁₃	1	2	yes	—
133	C₄×M₁₆	Dic₁₃	1	2	yes	—
134	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
135	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
136	C₂×#97	Dic₁₃	1	2	yes	—
137	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
139	#322	—	13	8	no	—
140	#322	—	13	4	no	—
141	C₈²	—	13	4	no	—
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	—	2	1	no	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	—	2	2	no	—
147	C₂²×C₁₃	—	4	4	no	—
148	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
149	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	—	2	1	no	—
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	—	2	2	no	—
155	C₄×C₁₃	C₄×M₁₆	1	2	yes	—
156	C₄×C₁₃	C₂²×M₁₆	1	2	yes	—
157	C₄×C₁₃	#568	1	4	yes	—
158	C₂³×C₄	C₄×D₁₃	1	2	yes	—
159	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
160	C₂×M₁₆	—	2	8	no	—
161	C₂×M₁₆	—	13	8	no	—
162	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
163	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
164	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
165	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
166	C₂×C₄²	C₄×D₁₃	1	2	yes	—
167	C₂×C₄²	C₄×D₁₃	1	2	yes	—
168	C₂×C₄²	C₄×D₁₃	1	2	yes	—
169	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
170	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
171	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
175	C₄⋊C₈	—	13	8	no	—
176	C₄⋊C₈	—	13	8	no	—
177	C₂²⋊C₈	—	13	8	no	—
178	C₂²⋊C₈	—	13	8	no	—
179	#97	—	13	8	no	—
180	#97	C₂×Dic₁₃	1	4	yes	—
181	#97	C₂×Dic₁₃	1	4	yes	—
182	C₄×C₈	—	13	8	no	—
183	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
184	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
185	C₄×C₈	C₂²×D₁₃	1	2	yes	—
186	C₄×C₈	C₂²×D₁₃	1	2	yes	—
187	C₂×C₁₃	—	2	2	no	—
188	C₂×C₁₃	—	4	1	no	—
189	C₂×C₁₃	—	4	1	no	—
190	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
191	C₂×C₁₃	#2873	1	2	yes	—
192	C₂⁴	C₄×Dic₁₃	1	1	yes	characteristic
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	4	no	—
195	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
196	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
197	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
198	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
199	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
200	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
201	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
202	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
203	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
204	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
207	M₁₆	—	2	8	no	—
208	M₁₆	—	2	8	no	—
209	M₁₆	—	52	8	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	13	16	no	—
212	C₂×C₈	#6099	1	8	yes	complemented
213	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
214	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
215	C₄²	C₄×Dic₁₃	1	2	yes	—
216	C₄²	C₄×Dic₁₃	1	2	yes	—
217	C₄²	C₄×Dic₁₃	1	2	yes	—
218	C₄²	C₂×C₄×D₁₃	1	2	yes	—
219	C₄²	C₂×C₄×D₁₃	1	2	yes	—
220	C₄²	C₂×C₄×D₁₃	1	2	yes	—
221	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
222	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
223	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
224	C₄²	C₂³×D₁₃	1	1	yes	characteristic
225	C₁₃	#17334	1	1	yes	characteristic, complemented, Sylow
226	C₂³	—	2	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
229	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
230	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	C₂×#6099	1	4	yes	—
243	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
244	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
246	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
247	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
248	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
249	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
250	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
251	C₈	—	2	4	no	—
252	C₈	—	26	16	no	—
253	C₈	C₂×#6099	1	8	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	4	4	no	—
260	C₂²	C₄×#6099	1	2	yes	—
261	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	C₄×#6099	1	2	yes	—
268	C₄	#11719450	1	4	yes	—
269	C₄	C₂²×#6099	1	2	yes	—
270	C₂	—	2	2	no	—
271	C₂	—	4	1	no	—
272	C₂	—	4	1	no	—
273	C₂	C₂×C₄×#6099	1	1	yes	characteristic
274	C₂	#13123254	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂²×C₄²
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
prime radical	C₂⁴×C₁₃	C₄²
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	#17334
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13113139	C₂	semidirect
3. #13113133	C₂	semidirect
4. #13113128	C₂	semidirect
5. #13113119	C₂	semidirect
23. #11718047	C₂²	semidirect
24. #11718046	C₂²	semidirect
25. C₈×#1055	C₂²	semidirect
74. C₂×C₈×C₁₃	M₁₆	semidirect
212. C₂×C₈	#6099	semidirect
225. C₁₃	#17334	semidirect

Maximal subgroups

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

subgroup	index
#13123469	2
#13113139	2
#13113133	2
#13113128	2
#13113119	2
C₂×C₄×C₁₃×M₁₆	2
#17334	13