Subgroups — #16019035

Order 3328.

504 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125415	C₂	1	1	yes	maximal, characteristic
2	#13125350	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117118	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117111	C₂	1	1	yes	maximal, characteristic, complemented
5	#13117060	C₂	1	1	yes	maximal, characteristic, complemented
6	#13117060	C₂	1	1	yes	maximal, characteristic, complemented
7	#13117028	C₂	1	4	yes	maximal, complemented
8	#13116994	C₂	1	4	yes	maximal, complemented
9	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719182	—	2	1	no	—
11	#11719175	—	2	1	no	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718776	—	2	1	no	—
14	#11718776	C₂²	1	2	yes	—
15	#11718767	—	2	1	no	—
16	#11718767	C₂²	1	2	yes	—
17	#11718356	C₂²	1	1	yes	characteristic
18	#11718355	—	2	1	no	—
19	C₄×D₁₀₄	—	2	1	no	—
20	C₄×D₁₀₄	C₂²	1	2	yes	—
21	#11718344	—	2	1	no	—
22	#11718344	C₂²	1	2	yes	—
23	#11718343	C₂²	1	1	yes	characteristic
24	#11718343	C₂²	1	1	yes	characteristic
25	#11718343	C₂²	1	1	yes	characteristic
26	#11718341	—	2	1	no	—
27	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
28	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
29	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
30	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
31	#11718101	C₂²	1	4	yes	—
32	#11718101	C₂²	1	4	yes	—
33	#11718097	C₂²	1	4	yes	—
34	#11718096	C₂²	1	4	yes	—
35	C₂×#6127	—	4	1	no	—
36	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
39	#88184	—	2	2	no	—
40	#88184	—	4	1	no	—
41	#88180	—	2	2	no	—
42	#88178	—	4	1	no	—
43	#88178	—	4	1	no	—
44	C₂×D₁₀₄	—	2	1	no	—
45	C₂×D₁₀₄	C₂³	1	2	yes	—
46	#88129	—	4	1	no	—
47	#88128	—	2	1	no	—
48	C₄×D₅₂	—	2	2	no	—
49	C₄×D₅₂	—	4	1	no	—
50	#88124	—	2	2	no	—
51	#88123	—	2	1	no	—
52	#88123	—	2	1	no	—
53	#88123	—	2	1	no	—
54	C₄×Dic₂₆	—	4	1	no	—
55	C₄×Dic₂₆	—	4	1	no	—
56	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
57	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
58	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	#88061	—	2	2	no	—
63	#88061	—	2	2	no	—
64	#88058	—	2	1	no	—
65	#88058	—	2	1	no	—
66	#88058	—	2	1	no	—
67	#88058	C₂³	1	2	yes	—
68	#88058	C₂³	1	2	yes	—
69	#88058	C₂³	1	2	yes	—
70	#88057	—	2	2	no	—
71	#88057	—	2	2	no	—
72	#33794	—	13	1	no	maximal, Sylow
73	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₂²⋊D₁₃	—	4	2	no	—
77	#6127	—	8	4	no	—
78	C₂×D₅₂	—	2	2	no	—
79	C₂×D₅₂	—	4	1	no	—
80	C₂×D₅₂	—	4	1	no	—
81	C₂×C₄×D₁₃	—	4	2	no	—
82	C₂×Dic₂₆	—	4	1	no	—
83	C₂×Dic₂₆	—	4	1	no	—
84	C₂×Dic₂₆	—	4	1	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₄²×C₁₃	D₈	1	1	yes	characteristic
92	C₄²×C₁₃	D₈	1	1	yes	characteristic
93	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂²⋊Dic₁₃	—	4	2	no	—
96	#6103	—	4	2	no	—
97	#6103	—	4	2	no	—
98	C₄⋊Dic₁₃	—	2	2	no	—
99	C₄⋊Dic₁₃	—	2	2	no	—
100	C₄⋊Dic₁₃	—	2	2	no	—
101	C₄⋊Dic₁₃	—	4	1	no	—
102	C₄⋊Dic₁₃	—	4	1	no	—
103	C₄⋊Dic₁₃	—	4	1	no	—
104	#6101	—	4	2	no	—
105	#6101	—	4	2	no	—
106	C₄×Dic₁₃	—	4	2	no	—
107	D₁₀₄	—	2	4	no	—
108	D₁₀₄	—	4	2	no	—
109	C₁₃×C₁₆	—	2	2	no	—
110	C₁₃×C₁₆	—	2	2	no	—
111	#3169	—	13	1	no	—
112	#3160	—	13	1	no	—
113	#2254	—	13	1	no	—
114	#2252	—	13	1	no	—
115	#2240	—	13	1	no	—
116	#2240	—	13	1	no	—
117	#2234	—	13	4	no	—
118	#2228	—	13	4	no	—
119	#2162	D₁₃	1	1	yes	characteristic, complemented
120	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
121	C₂²×D₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₂²⋊D₁₃	—	8	4	no	—
133	C₂×Dic₁₃	—	4	2	no	—
134	C₂×Dic₁₃	—	4	2	no	—
135	C₂×Dic₁₃	—	4	2	no	—
136	D₅₂	—	4	2	no	—
137	D₅₂	—	4	2	no	—
138	D₅₂	—	8	2	no	—
139	D₅₂	—	8	2	no	—
140	C₄×D₁₃	—	8	4	no	—
141	Dic₂₆	—	8	2	no	—
142	Dic₂₆	—	8	2	no	—
143	Dic₂₆	—	8	2	no	—
144	C₈×C₁₃	—	2	1	no	—
145	C₈×C₁₃	—	2	1	no	—
146	C₈×C₁₃	#137	1	2	yes	—
147	C₈×C₁₃	2₋¹⁺⁴	1	2	yes	—
148	#533	—	26	1	no	—
149	#532	—	26	1	no	—
150	#500	—	13	1	no	—
151	#500	—	13	1	no	—
152	#500	—	13	1	no	—
153	#499	—	13	2	no	—
154	#499	—	26	1	no	—
155	#498	—	26	1	no	—
156	#493	—	13	1	no	—
157	#486	—	26	1	no	—
158	#466	—	13	2	no	—
159	#466	—	26	1	no	—
160	C₄×D₈	—	13	2	no	—
161	C₄×D₈	—	26	1	no	—
162	#427	—	13	2	no	—
163	#427	—	26	1	no	—
164	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
165	#367	—	13	4	no	—
166	#366	—	13	4	no	—
167	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
168	#357	—	13	4	no	—
169	#357	—	13	4	no	—
170	C₂²⋊C₁₆	C₂×D₁₃	1	2	yes	—
171	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
172	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	—	2	1	no	—
175	C₂²×C₁₃	—	2	1	no	—
176	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
177	C₂×D₁₃	—	8	2	no	—
178	C₂×D₁₃	—	8	2	no	—
179	C₂×D₁₃	—	8	2	no	—
180	C₄×C₁₃	—	2	1	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	#508	1	2	yes	—
184	C₄×C₁₃	#509	1	2	yes	—
185	C₄×C₁₃	#536	1	1	yes	characteristic
186	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
187	Dic₁₃	—	8	2	no	—
188	Dic₁₃	—	8	2	no	—
189	Dic₁₃	—	8	2	no	—
190	C₂×#41	—	52	1	no	—
191	C₂×D₈	—	13	2	no	—
192	C₂×D₈	—	26	1	no	—
193	C₂²×C₈	D₅₂	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
195	C₄⋊Q₈	—	26	1	no	—
196	C₄⋊Q₈	—	26	1	no	—
197	C₄⋊Q₈	—	26	1	no	—
198	#127	—	26	1	no	—
199	#125	—	26	2	no	—
200	#124	—	52	1	no	—
201	C₂²⋊Q₈	—	52	1	no	—
202	C₂²⋊Q₈	—	52	1	no	—
203	#121	—	26	2	no	—
204	#121	—	52	1	no	—
205	C₄×Q₈	—	52	1	no	—
206	C₄×Q₈	—	52	1	no	—
207	C₄×D₄	—	26	2	no	—
208	C₄×D₄	—	52	1	no	—
209	#117	—	26	2	no	—
210	C₂×C₄²	D₅₂	1	1	yes	characteristic
211	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
212	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
213	#107	—	13	2	no	—
214	#107	—	13	2	no	—
215	#107	—	13	2	no	—
216	#107	—	26	1	no	—
217	#107	—	26	1	no	—
218	#107	—	26	1	no	—
219	#106	—	26	2	no	—
220	#106	—	26	2	no	—
221	#102	—	26	2	no	—
222	#102	—	26	2	no	—
223	C₄×C₈	D₅₂	1	1	yes	characteristic
224	C₄×C₈	D₅₂	1	1	yes	characteristic
225	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
226	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
227	C₂×C₁₃	—	4	1	no	—
228	C₂×C₁₃	#3226	1	1	yes	characteristic
229	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
230	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
231	D₁₃	—	16	2	no	—
232	#41	—	104	4	no	—
233	C₂×Q₈	—	52	1	no	—
234	C₂×Q₈	—	52	1	no	—
235	C₂×Q₈	—	52	1	no	—
236	C₂×D₄	—	26	2	no	—
237	C₂×D₄	—	52	2	no	—
238	C₂×D₄	—	52	1	no	—
239	C₂×D₄	—	52	1	no	—
240	C₂²×C₄	—	52	2	no	—
241	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
242	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
244	D₈	—	26	4	no	—
245	D₈	—	52	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
249	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
250	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
252	C₄⋊C₄	—	26	2	no	—
253	C₄⋊C₄	—	26	2	no	—
254	C₄⋊C₄	—	26	2	no	—
255	C₄⋊C₄	—	52	1	no	—
256	C₄⋊C₄	—	52	2	no	—
257	C₄⋊C₄	—	52	2	no	—
258	C₄⋊C₄	—	52	1	no	—
259	C₄⋊C₄	—	52	1	no	—
260	C₂²⋊C₄	—	52	2	no	—
261	C₂²⋊C₄	—	52	2	no	—
262	C₂²⋊C₄	—	52	2	no	—
263	C₄²	—	52	2	no	—
264	C₄²	D₁₀₄	1	1	yes	characteristic
265	C₄²	D₁₀₄	1	1	yes	characteristic
266	C₄²	C₂×D₅₂	1	1	yes	characteristic
267	C₄²	C₂×D₅₂	1	1	yes	characteristic
268	C₁₆	—	2	2	no	—
269	C₁₆	—	2	2	no	—
270	C₁₃	#33794	1	1	yes	characteristic, complemented, Sylow
271	C₂³	—	52	2	no	—
272	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
273	Q₈	—	104	2	no	—
274	Q₈	—	104	2	no	—
275	Q₈	—	104	2	no	—
276	D₄	—	52	2	no	—
277	D₄	—	52	2	no	—
278	D₄	—	104	4	no	—
279	D₄	—	104	2	no	—
280	D₄	—	104	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	52	2	no	—
286	C₂×C₄	—	52	2	no	—
287	C₂×C₄	—	52	2	no	—
288	C₂×C₄	—	104	4	no	—
289	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
290	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
291	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
292	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
293	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	#88163	1	2	yes	—
298	C₈	#88257	1	2	yes	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	104	2	no	—
303	C₂²	—	104	2	no	—
304	C₂²	—	104	2	no	—
305	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	104	2	no	—
310	C₄	—	104	2	no	—
311	C₄	—	104	2	no	—
312	C₄	#11718565	1	2	yes	—
313	C₄	#11718569	1	2	yes	—
314	C₄	#11719231	1	1	yes	characteristic
315	C₄	C₂×#88163	1	1	yes	characteristic
316	C₂	—	4	1	no	—
317	C₂	—	208	2	no	—
318	C₂	#13125959	1	1	yes	characteristic
319	C₂	C₂×#11718565	1	1	yes	characteristic
320	C₂	C₂×#11718569	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2162	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13125415	C₂
hypercenter	#2162	D₁₃
hypocenter	C₁₃	#33794
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. #13125350	C₂	semidirect
3. #13117118	C₂	semidirect
4. #13117111	C₂	semidirect
5. #13117060	C₂	semidirect
6. #13117060	C₂	semidirect
7. #13117028	C₂	semidirect
8. #13116994	C₂	semidirect
9. C₁₃×#2162	C₂	semidirect
119. #2162	D₁₃	semidirect
270. C₁₃	#33794	semidirect

Maximal subgroups

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

subgroup	index
#13125415	2
#13125350	2
#13117118	2
#13117111	2
#13117060	2
#13117028	2
#13116994	2
C₁₃×#2162	2
#33794	13