Subgroups — #16016063

Order 3328.

528 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88155	C₂	1	1	yes	maximal, characteristic
2	#13116651	C₂	1	4	yes	maximal, complemented
3	#13116647	C₂	1	4	yes	maximal, complemented
4	C₂×#11718055	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116456	C₂	1	2	yes	maximal
6	#13116440	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88155	C₄	1	1	yes	characteristic
9	C₂×#88155	C₂²	1	1	yes	characteristic
10	C₄×#6127	C₄	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718762	C₄	1	1	yes	characteristic
14	#11718762	C₂²	1	1	yes	characteristic
15	C₂×#88052	C₂²	1	2	yes	—
16	C₄×#6099	C₄	1	1	yes	characteristic
17	#11718350	C₄	1	2	yes	—
18	#11718350	C₂²	1	2	yes	—
19	C₄×#6094	C₂²	1	2	yes	—
20	C₄×C₈×D₁₃	C₄	1	2	yes	—
21	C₁₃×#364	C₂²	1	2	yes	—
22	C₁₃×#364	C₂²	1	2	yes	—
23	#11718106	C₂²	1	8	yes	—
24	#11718104	C₂²	1	8	yes	—
25	#11718055	C₂²	1	2	yes	—
26	#11718055	C₂²	1	2	yes	—
27	C₂×#6127	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₂²⋊D₁₃	—	2	2	no	—
32	#88180	C₂×C₄	1	1	yes	characteristic
33	C₂×#6099	C₂×C₄	1	1	yes	characteristic
34	C₂×#6099	C₂×C₄	1	1	yes	characteristic
35	#88155	—	2	2	no	—
36	#88155	—	2	2	no	—
37	#88155	—	2	1	no	—
38	#88155	—	2	1	no	—
39	#88155	—	2	1	no	—
40	#88155	—	2	1	no	—
41	C₂×#6094	C₂×C₄	1	2	yes	—
42	C₂×#6094	C₂³	1	2	yes	—
43	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
45	C₄×D₅₂	—	2	1	no	—
46	#88126	C₂×C₄	1	2	yes	—
47	C₄²×D₁₃	C₂×C₄	1	2	yes	—
48	C₄×Dic₂₆	—	2	1	no	—
49	C₁₃×M₃₂	—	2	2	no	—
50	C₁₃×M₃₂	C₂³	1	4	yes	—
51	C₂×C₁₃×C₁₆	—	2	2	no	—
52	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	#88055	C₂×C₄	1	2	yes	—
57	#88055	C₂³	1	2	yes	—
58	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
59	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
60	#88052	—	2	2	no	—
61	#88052	C₂³	1	4	yes	—
62	C₂×#6090	—	2	2	no	—
63	#88043	C₂×C₄	1	2	yes	—
64	C₄×#1055	C₂×C₄	1	2	yes	—
65	#33428	—	13	1	no	maximal, Sylow
66	C₂²×C₄×C₁₃	—	2	1	no	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₁₃	M₁₆	1	2	yes	—
69	#6127	—	2	1	no	—
70	#6127	—	2	1	no	—
71	#6127	—	2	1	no	—
72	#6127	—	2	1	no	—
73	C₂×D₅₂	M₁₆	1	1	yes	characteristic
74	C₂×C₄×D₁₃	—	2	2	no	—
75	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
76	C₂×Dic₂₆	M₁₆	1	1	yes	characteristic
77	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
78	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₄²×C₁₃	—	2	1	no	—
88	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²⋊Dic₁₃	—	2	1	no	—
91	#6103	—	2	2	no	—
92	C₄⋊Dic₁₃	—	2	1	no	—
93	#6101	—	2	2	no	—
94	C₄×Dic₁₃	—	2	2	no	—
95	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
96	#6099	—	2	1	no	—
97	#6099	—	2	1	no	—
98	#6099	—	2	1	no	—
99	#6099	—	2	1	no	—
100	C₂×#1055	C₂²×C₄	1	2	yes	—
101	C₂×#1055	C₂²×C₄	1	2	yes	—
102	#6094	—	2	4	no	—
103	#6094	—	2	2	no	—
104	#6094	—	2	2	no	—
105	C₈×D₁₃	—	2	2	no	—
106	C₈×D₁₃	—	2	2	no	—
107	C₈×D₁₃	—	2	4	no	—
108	C₁₃×C₁₆	—	2	4	no	—
109	#6090	—	2	4	no	—
110	C₄×#131	—	13	1	no	—
111	#2166	—	13	4	no	—
112	#2165	—	13	4	no	—
113	C₂×#364	—	13	1	no	—
114	C₂×#364	D₁₃	1	1	yes	characteristic, complemented
115	#2115	—	13	2	no	—
116	#2110	—	13	2	no	—
117	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
118	C₂²×D₁₃	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
128	C₂²⋊D₁₃	—	2	2	no	—
129	C₂²⋊D₁₃	—	2	2	no	—
130	C₂×Dic₁₃	—	2	2	no	—
131	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
132	D₅₂	—	2	1	no	—
133	D₅₂	—	2	1	no	—
134	C₄×D₁₃	—	2	2	no	—
135	C₄×D₁₃	—	2	2	no	—
136	C₄×D₁₃	—	4	4	no	—
137	Dic₂₆	—	2	1	no	—
138	Dic₂₆	—	2	1	no	—
139	C₈×C₁₃	#117	1	1	yes	characteristic
140	C₈×C₁₃	#117	1	1	yes	characteristic
141	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
142	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
143	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
144	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
145	#1055	—	2	2	no	—
146	#1055	—	2	2	no	—
147	C₂×#131	—	13	1	no	—
148	C₂×#131	—	13	1	no	—
149	C₄×#41	—	13	1	no	—
150	C₂×M₃₂	—	13	2	no	—
151	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
152	#405	—	13	1	no	—
153	#405	—	13	1	no	—
154	#405	—	13	2	no	—
155	#405	—	13	2	no	—
156	C₄×M₁₆	—	13	1	no	—
157	C₄×M₁₆	—	13	2	no	—
158	C₂×C₄×C₈	—	13	2	no	—
159	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
160	#364	—	13	2	no	—
161	#364	—	13	2	no	—
162	#364	C₂×D₁₃	1	2	yes	—
163	#364	C₂×D₁₃	1	2	yes	—
164	#349	—	13	8	no	—
165	#347	—	13	8	no	—
166	C₂²×C₁₃	—	4	1	no	—
167	C₂²×C₁₃	#432	1	1	yes	characteristic
168	C₂²×C₁₃	#432	1	1	yes	characteristic
169	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
170	C₂×D₁₃	—	2	2	no	—
171	C₂×D₁₃	—	4	2	no	—
172	C₄×C₁₃	—	2	2	no	—
173	C₄×C₁₃	—	4	1	no	—
174	C₄×C₁₃	#432	1	1	yes	characteristic
175	C₄×C₁₃	#432	1	1	yes	characteristic
176	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
177	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
178	Dic₁₃	—	2	2	no	—
179	Dic₁₃	—	4	2	no	—
180	C₂×#41	—	13	1	no	—
181	#131	—	26	2	no	—
182	#131	—	26	2	no	—
183	#131	—	26	1	no	—
184	#131	—	26	1	no	—
185	#131	—	26	1	no	—
186	#131	—	26	1	no	—
187	C₂×M₁₆	—	13	1	no	—
188	C₂×M₁₆	—	13	1	no	—
189	C₂×M₁₆	—	13	2	no	—
190	C₂×M₁₆	—	13	2	no	—
191	C₂²×C₈	—	13	2	no	—
192	C₂²×C₈	—	13	2	no	—
193	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
195	C₄×Q₈	—	26	1	no	—
196	C₄×D₄	—	26	1	no	—
197	C₄×D₄	—	26	2	no	—
198	#117	—	13	1	no	—
199	#117	—	13	2	no	—
200	C₂×C₄²	—	13	2	no	—
201	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
202	M₃₂	—	2	2	no	—
203	M₃₂	—	13	4	no	—
204	M₃₂	—	26	2	no	—
205	M₃₂	C₂²×D₁₃	1	4	yes	—
206	C₂×C₁₆	—	2	2	no	—
207	C₂×C₁₆	—	26	2	no	—
208	#97	—	13	2	no	—
209	#97	—	13	2	no	—
210	#97	—	13	2	no	—
211	C₄×C₈	—	13	2	no	—
212	C₄×C₈	—	13	2	no	—
213	C₄×C₈	—	13	2	no	—
214	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
215	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
218	C₂×C₁₃	—	2	1	no	—
219	C₂×C₁₃	—	2	1	no	—
220	C₂×C₁₃	—	2	1	no	—
221	C₂×C₁₃	C₂×#432	1	1	yes	characteristic
222	D₁₃	—	4	2	no	—
223	#41	—	26	1	no	—
224	#41	—	26	1	no	—
225	#41	—	26	1	no	—
226	#41	—	26	1	no	—
227	C₂×Q₈	—	13	1	no	—
228	C₂×D₄	—	13	2	no	—
229	C₂×D₄	—	13	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	13	2	no	—
232	C₂²×C₄	—	26	2	no	—
233	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
234	M₁₆	—	26	1	no	—
235	M₁₆	—	26	1	no	—
236	M₁₆	—	26	1	no	—
237	M₁₆	—	26	1	no	—
238	M₁₆	—	26	4	no	—
239	M₁₆	—	26	2	no	—
240	M₁₆	—	26	2	no	—
241	C₂×C₈	—	13	2	no	—
242	C₂×C₈	—	13	2	no	—
243	C₂×C₈	—	26	2	no	—
244	C₂×C₈	—	26	2	no	—
245	C₂×C₈	—	26	4	no	—
246	C₂×C₈	#6094	1	2	yes	—
247	C₂×C₈	#6094	1	2	yes	—
248	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
249	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
251	C₂×C₈	#6129	1	1	yes	characteristic
252	C₂×C₈	#6129	1	1	yes	characteristic
253	C₂×C₈	#6131	1	1	yes	characteristic
254	C₂×C₈	#6131	1	1	yes	characteristic
255	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
256	C₄⋊C₄	—	26	2	no	—
257	C₄⋊C₄	—	26	1	no	—
258	C₂²⋊C₄	—	26	2	no	—
259	C₂²⋊C₄	—	26	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	13	2	no	—
262	C₄²	—	26	2	no	—
263	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
264	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
265	C₁₆	—	2	4	no	—
266	C₁₆	—	26	4	no	—
267	C₁₃	#33428	1	1	yes	characteristic, complemented, Sylow
268	C₂³	—	13	2	no	—
269	C₂³	M₁₆×D₁₃	1	1	yes	characteristic
270	Q₈	—	26	1	no	—
271	Q₈	—	26	1	no	—
272	D₄	—	26	2	no	—
273	D₄	—	26	2	no	—
274	D₄	—	26	1	no	—
275	D₄	—	26	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	13	2	no	—
280	C₂×C₄	—	26	2	no	—
281	C₂×C₄	—	26	2	no	—
282	C₂×C₄	—	26	2	no	—
283	C₂×C₄	—	52	4	no	—
284	C₂×C₄	#88146	1	1	yes	characteristic
285	C₂×C₄	#88146	1	1	yes	characteristic
286	C₂×C₄	C₂×#6094	1	1	yes	characteristic
287	C₂×C₄	C₂×#6094	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
290	C₈	—	26	2	no	—
291	C₈	—	26	2	no	—
292	C₈	#88146	1	1	yes	characteristic
293	C₈	#88146	1	1	yes	characteristic
294	C₈	C₂×#6094	1	2	yes	—
295	C₈	C₂×#6094	1	2	yes	—
296	C₈	C₂×#6129	1	1	yes	characteristic
297	C₈	C₂×#6131	1	1	yes	characteristic
298	C₂²	—	4	1	no	—
299	C₂²	—	26	2	no	—
300	C₂²	—	52	2	no	—
301	C₂²	#11718496	1	1	yes	characteristic
302	C₂²	#11718496	1	1	yes	characteristic
303	C₂²	C₂×M₁₆×D₁₃	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	26	2	no	—
307	C₄	—	52	2	no	—
308	C₄	#11718496	1	1	yes	characteristic
309	C₄	#11718496	1	1	yes	characteristic
310	C₄	C₂×#88146	1	1	yes	characteristic
311	C₄	C₂²×#6094	1	1	yes	characteristic
312	C₂	—	2	1	no	—
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	—	52	2	no	—
316	C₂	C₂×#11718496	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#88146
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#364	C₂
socle	C₂×C₁₃	C₂×#432
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#364	D₁₃
hypocenter	C₁₃	#33428
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. #13116651	C₂	semidirect
3. #13116647	C₂	semidirect
4. C₂×#11718055	C₂	semidirect
6. #13116440	C₂	semidirect
7. C₂×C₁₃×#364	C₂	semidirect
114. C₂×#364	D₁₃	semidirect
267. C₁₃	#33428	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88155	2
#13116651	2
#13116647	2
C₂×#11718055	2
#13116456	2
#13116440	2
C₂×C₁₃×#364	2
#33428	13