Subgroups — #15768800

Order 3328.

830 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718597	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718146	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
8	C₂²×#6099	C₂²	1	2	yes	—
9	C₄×C₁₃×M₁₆	—	2	2	no	—
10	C₄×C₁₃×M₁₆	C₄	1	4	yes	complemented
11	C₂×C₁₃×#97	—	2	1	no	—
12	C₂×C₄×C₈×C₁₃	—	2	1	no	—
13	C₂×#88071	C₄	1	4	yes	—
14	#11718597	—	2	2	no	—
15	#11718597	—	2	1	no	—
16	#11718597	—	2	1	no	—
17	#11718585	—	2	2	no	—
18	#11718585	C₄	1	4	yes	complemented
19	C₂×C₄⋊#1055	—	2	2	no	—
20	#11718146	—	2	8	no	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂³×Dic₁₃	C₂×C₄	1	4	yes	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
28	C₂×C₁₃×M₁₆	D₄	1	4	yes	—
29	C₂×C₁₃×M₁₆	Q₈	1	4	yes	—
30	C₂²×C₈×C₁₃	—	2	2	no	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₂×C₄²×C₁₃	D₄	1	2	yes	—
35	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
38	C₂×C₄⋊Dic₁₃	—	2	4	no	—
39	C₂×C₄⋊Dic₁₃	—	2	2	no	—
40	C₂×C₄⋊Dic₁₃	—	2	2	no	—
41	C₂×C₄⋊Dic₁₃	—	2	4	no	—
42	C₂×#6099	—	2	2	no	—
43	C₂×#6099	—	2	4	no	—
44	C₂×#6099	—	2	4	no	—
45	C₂×#6099	C₂×C₄	1	4	yes	—
46	C₂²×#1055	—	2	2	no	—
47	C₁₃×#97	—	2	4	no	—
48	C₄×C₈×C₁₃	—	2	4	no	—
49	#88071	—	2	4	no	—
50	#88071	—	2	4	no	—
51	C₄⋊#1055	—	2	8	no	—
52	#10858	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	4	4	no	—
55	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
56	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₁₃	#41	1	2	yes	—
68	C₂²×C₄×C₁₃	#41	1	2	yes	—
69	C₂²×Dic₁₃	—	2	4	no	—
70	C₂²×Dic₁₃	—	2	8	no	—
71	C₂²×Dic₁₃	—	4	8	no	—
72	C₁₃×M₁₆	—	2	4	no	—
73	C₁₃×M₁₆	—	2	4	no	—
74	C₁₃×M₁₆	—	2	4	no	—
75	C₁₃×M₁₆	C₄⋊C₄	1	8	yes	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	2	4	no	—
79	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
80	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
85	C₄⋊Dic₁₃	—	4	4	no	—
86	C₄⋊Dic₁₃	—	4	2	no	—
87	C₄⋊Dic₁₃	—	4	2	no	—
88	#6099	—	2	8	no	—
89	#6099	—	4	4	no	—
90	C₂×#1055	—	2	4	no	—
91	C₂×#1055	—	4	2	no	—
92	C₂×#1055	—	4	2	no	—
93	C₂×#423	—	13	1	no	—
94	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
95	C₂×#382	—	13	1	no	—
96	C₂×#343	—	13	4	no	—
97	C₂³×C₁₃	—	2	4	no	—
98	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
99	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₁₃	C₄×D₄	1	2	yes	—
102	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	2	4	no	—
105	C₂×C₄×C₁₃	—	4	4	no	—
106	C₂×C₄×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	#95	1	4	yes	—
108	C₂×C₄×C₁₃	#95	1	4	yes	—
109	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#117	1	2	yes	—
115	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₁₃	#121	1	2	yes	—
118	C₂×C₄×C₁₃	#121	1	2	yes	—
119	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#125	1	2	yes	—
124	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
127	C₂×Dic₁₃	—	4	4	no	—
128	C₂×Dic₁₃	—	4	8	no	—
129	C₂×Dic₁₃	—	8	8	no	—
130	C₈×C₁₃	—	2	8	no	—
131	#1055	—	4	4	no	—
132	C₂²×M₁₆	—	13	2	no	—
133	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
134	C₂²×C₄⋊C₄	—	13	2	no	—
135	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
136	#423	—	13	4	no	—
137	#423	—	26	2	no	—
138	C₂×C₄⋊C₈	—	26	2	no	—
139	C₄×M₁₆	—	2	2	no	—
140	C₄×M₁₆	Dic₁₃	1	4	yes	complemented
141	C₂×#97	—	2	1	no	—
142	C₂×C₄×C₈	—	2	1	no	—
143	#382	—	26	2	no	—
144	#382	—	26	1	no	—
145	#382	—	26	1	no	—
146	C₂×#95	—	13	4	no	—
147	#343	—	26	8	no	—
148	C₂²×C₁₃	—	2	4	no	—
149	C₂²×C₁₃	—	2	2	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	—	2	4	no	—
152	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
153	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
154	C₂²×C₁₃	#389	1	2	yes	—
155	C₂²×C₁₃	#390	1	1	yes	characteristic
156	C₂²×C₁₃	#391	1	1	yes	characteristic
157	C₂²×C₁₃	#413	1	4	yes	—
158	C₄×C₁₃	—	2	4	no	—
159	C₄×C₁₃	—	2	4	no	—
160	C₄×C₁₃	#380	1	2	yes	—
161	C₄×C₁₃	#382	1	1	yes	characteristic
162	C₄×C₁₃	#384	1	1	yes	characteristic
163	C₄×C₁₃	#389	1	2	yes	—
164	C₄×C₁₃	#390	1	1	yes	characteristic
165	C₄×C₁₃	#391	1	1	yes	characteristic
166	Dic₁₃	—	8	4	no	—
167	C₂³×C₄	—	13	4	no	—
168	C₂³×C₄	C₄×D₁₃	1	2	yes	—
169	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	13	4	no	—
174	C₂×M₁₆	—	26	2	no	—
175	C₂×M₁₆	—	26	4	no	—
176	C₂×M₁₆	—	26	4	no	—
177	C₂×M₁₆	Dic₂₆	1	4	yes	—
178	C₂×M₁₆	D₅₂	1	4	yes	—
179	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	26	2	no	—
182	C₂×C₄⋊C₄	—	26	4	no	—
183	C₂×C₄⋊C₄	—	26	2	no	—
184	C₂×C₄⋊C₄	—	26	4	no	—
185	C₂×C₄⋊C₄	—	26	2	no	—
186	C₂×C₄²	Dic₂₆	1	2	yes	—
187	C₂×C₄²	C₄×D₁₃	1	2	yes	—
188	C₂×C₄²	D₅₂	1	2	yes	—
189	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
190	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
192	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
193	C₄⋊C₈	—	26	8	no	—
194	#97	—	2	4	no	—
195	C₄×C₈	—	2	4	no	—
196	#95	—	26	4	no	—
197	#95	—	26	4	no	—
198	C₂×C₁₃	—	2	2	no	—
199	C₂×C₁₃	—	2	2	no	—
200	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
201	C₂×C₁₃	#1742	1	2	yes	—
202	C₂×C₁₃	#1967	1	2	yes	—
203	C₂×C₁₃	#1998	1	2	yes	—
204	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	—	26	4	no	—
207	C₂²×C₄	—	26	8	no	—
208	C₂²×C₄	—	52	8	no	—
209	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
210	C₂²×C₄	#6101	1	4	yes	—
211	C₂²×C₄	#6103	1	2	yes	—
212	C₂²×C₄	#6103	1	2	yes	—
213	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
214	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₅₂	1	2	yes	—
217	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
218	C₂²×C₄	#6127	1	2	yes	—
219	C₂²×C₄	#6127	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	26	8	no	—
227	M₁₆	—	52	4	no	—
228	M₁₆	C₄⋊Dic₁₃	1	8	yes	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	26	4	no	—
233	C₂×C₈	—	52	2	no	—
234	C₂×C₈	—	52	2	no	—
235	C₄⋊C₄	—	52	2	no	—
236	C₄⋊C₄	—	52	2	no	—
237	C₄⋊C₄	—	52	4	no	—
238	C₄²	#6101	1	2	yes	—
239	C₄²	#6101	1	2	yes	—
240	C₄²	C₄⋊Dic₁₃	1	4	yes	—
241	C₄²	#6103	1	4	yes	—
242	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
243	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
244	C₁₃	#10858	1	1	yes	characteristic, complemented, Sylow
245	C₂³	—	2	4	no	—
246	C₂³	C₄×Dic₂₆	1	2	yes	—
247	C₂³	C₄×D₅₂	1	2	yes	—
248	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
249	C₂³	C₂×#6101	1	1	yes	characteristic
250	C₂³	C₂×#6103	1	1	yes	characteristic
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	52	8	no	—
256	C₂×C₄	—	52	4	no	—
257	C₂×C₄	—	104	8	no	—
258	C₂×C₄	#88071	1	4	yes	—
259	C₂×C₄	#88071	1	4	yes	—
260	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
261	C₂×C₄	#88123	1	2	yes	—
262	C₂×C₄	#88123	1	1	yes	characteristic
263	C₂×C₄	#88124	1	2	yes	—
264	C₂×C₄	C₄×D₅₂	1	2	yes	—
265	C₂×C₄	#88128	1	1	yes	characteristic
266	C₂×C₄	#88129	1	1	yes	characteristic
267	C₂×C₄	#88129	1	1	yes	characteristic
268	C₂×C₄	C₂×#6101	1	1	yes	characteristic
269	C₂×C₄	#88178	1	2	yes	—
270	C₂×C₄	#88178	1	2	yes	—
271	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
273	C₂×C₄	#88180	1	2	yes	—
274	C₂×C₄	C₂×#6103	1	1	yes	characteristic
275	C₂×C₄	#88184	1	2	yes	—
276	C₂×C₄	#88184	1	2	yes	—
277	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
278	C₈	—	2	8	no	—
279	C₈	—	52	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	#11718589	1	2	yes	—
285	C₂²	#11718590	1	1	yes	characteristic
286	C₂²	C₄×C₄⋊Dic₁₃	1	2	yes	—
287	C₂²	#11718600	1	1	yes	characteristic
288	C₂²	#11718793	1	4	yes	—
289	C₂²	C₂×#88071	1	1	yes	characteristic
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	104	4	no	—
293	C₄	#11718589	1	2	yes	—
294	C₄	#11718590	1	1	yes	characteristic
295	C₄	#11718596	1	1	yes	characteristic
296	C₄	#11718597	1	1	yes	characteristic
297	C₄	#11718600	1	1	yes	characteristic
298	C₄	#11718873	1	2	yes	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	#13113066	1	1	yes	characteristic
302	C₂	#13114574	1	2	yes	—
303	C₂	#13115685	1	2	yes	—
304	C₂	#13115829	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#6103
commutator	C₂³×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	#10858
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
10. C₄×C₁₃×M₁₆	C₄	semidirect
18. #11718585	C₄	semidirect
140. C₄×M₁₆	Dic₁₃	semidirect
244. C₁₃	#10858	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718585	2
C₂×#11718597	2
C₂×#11718146	2
C₂×C₄×C₁₃×M₁₆	2
#10858	13