Subgroups — #16017582

Order 3328.

534 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	#13125350	C₂	1	1	yes	maximal, characteristic
2	#13117000	C₂	1	4	yes	maximal, complemented
3	#13116988	C₂	1	2	yes	maximal, complemented
4	#13116978	C₂	1	2	yes	maximal, complemented
5	#13116622	C₂	1	2	yes	maximal
6	#13116592	C₂	1	1	yes	maximal, characteristic
7	#13116556	C₂	1	2	yes	maximal
8	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
11	#11719175	—	2	1	no	—
12	C₄×C₁₃×Q₁₆	—	2	1	no	—
13	C₄×C₁₃×Q₁₆	—	2	1	no	—
14	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
15	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
16	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#103	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	#11718767	—	2	1	no	—
20	#11718767	C₂²	1	2	yes	—
21	C₂×#88052	C₂²	1	2	yes	—
22	#11718344	—	2	1	no	—
23	#11718344	C₂²	1	2	yes	—
24	#11718343	C₂²	1	1	yes	characteristic
25	#11718343	C₂²	1	1	yes	characteristic
26	#11718341	—	2	1	no	—
27	#11718156	C₂²	1	4	yes	—
28	#11718156	C₂²	1	4	yes	—
29	#11718082	C₂²	1	4	yes	—
30	#11718081	C₂²	1	4	yes	—
31	C₄⋊#6090	C₂²	1	2	yes	—
32	C₄⋊#6090	C₂²	1	2	yes	—
33	C₂²×Q₈×C₁₃	—	2	1	no	—
34	C₂×C₁₃×Q₁₆	—	2	1	no	—
35	C₂×C₁₃×Q₁₆	—	2	2	no	—
36	C₂×C₁₃×Q₁₆	Q₈	1	4	yes	complemented
37	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₄×Q₈×C₁₃	—	2	2	no	—
41	C₄×Q₈×C₁₃	—	2	2	no	—
42	C₄×Q₈×C₁₃	—	4	1	no	—
43	C₄×Q₈×C₁₃	—	4	1	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
45	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
46	C₂×C₄²×C₁₃	—	4	1	no	—
47	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
48	#88180	—	2	2	no	—
49	#88178	—	4	1	no	—
50	#88178	—	4	1	no	—
51	#88124	—	2	2	no	—
52	#88123	—	2	1	no	—
53	#88123	—	2	1	no	—
54	C₄×Dic₂₆	—	4	1	no	—
55	C₄×Dic₂₆	—	4	1	no	—
56	C₁₃×#107	—	2	1	no	—
57	C₁₃×#107	C₂³	1	2	yes	—
58	C₁₃×#103	—	2	2	no	—
59	C₁₃×#103	—	2	2	no	—
60	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	#88058	—	2	1	no	—
65	#88058	—	2	1	no	—
66	#88058	C₂³	1	2	yes	—
67	#88058	C₂³	1	2	yes	—
68	#88057	—	2	2	no	—
69	#88057	—	2	2	no	—
70	#88052	—	2	4	no	—
71	C₂×#6090	C₂³	1	4	yes	—
72	#33628	—	13	1	no	maximal, Sylow
73	C₂×Q₈×C₁₃	—	2	4	no	—
74	C₂×Q₈×C₁₃	—	2	2	no	—
75	C₂×Q₈×C₁₃	—	4	1	no	—
76	C₂×Q₈×C₁₃	—	4	2	no	—
77	C₂²×C₄×C₁₃	—	4	1	no	—
78	C₂²×C₄×C₁₃	—	4	1	no	—
79	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₂₆	—	4	1	no	—
83	C₂×Dic₂₆	—	4	1	no	—
84	C₁₃×Q₁₆	—	2	4	no	—
85	C₁₃×Q₁₆	—	2	2	no	—
86	C₁₃×Q₁₆	C₂×Q₈	1	4	yes	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₁₃	#41	1	2	yes	—
93	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	4	1	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₄⋊C₄	—	4	2	no	—
98	C₄²×C₁₃	—	4	2	no	—
99	C₄²×C₁₃	—	4	2	no	—
100	C₄²×C₁₃	D₈	1	1	yes	characteristic
101	C₄²×C₁₃	D₈	1	1	yes	characteristic
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂²⋊Dic₁₃	—	4	2	no	—
105	C₄⋊Dic₁₃	—	2	2	no	—
106	C₄⋊Dic₁₃	—	2	2	no	—
107	C₄⋊Dic₁₃	—	4	1	no	—
108	C₄⋊Dic₁₃	—	4	1	no	—
109	#6101	—	4	2	no	—
110	#6101	—	4	2	no	—
111	C₄×Dic₁₃	—	4	2	no	—
112	#6090	—	2	4	no	—
113	#3160	—	13	1	no	—
114	C₂×C₄×Q₁₆	D₁₃	1	1	yes	characteristic
115	#2229	—	13	4	no	—
116	#2227	—	13	2	no	—
117	#2225	—	13	2	no	—
118	#2159	—	13	2	no	—
119	#2150	—	13	1	no	—
120	#2141	—	13	2	no	—
121	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
122	Q₈×C₁₃	—	2	4	no	—
123	Q₈×C₁₃	—	4	2	no	—
124	Q₈×C₁₃	—	4	4	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	4	1	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	4	4	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	8	2	no	—
131	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#137	1	2	yes	—
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₁₃	—	4	2	no	—
140	C₂×Dic₁₃	—	4	2	no	—
141	Dic₂₆	—	8	2	no	—
142	Dic₂₆	—	8	2	no	—
143	C₈×C₁₃	—	2	1	no	—
144	C₈×C₁₃	—	2	1	no	—
145	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
146	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
147	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
148	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
149	#533	—	26	1	no	—
150	C₂×C₄×Q₈	—	2	1	no	—
151	C₂×M₃₂	—	13	2	no	—
152	#500	—	13	1	no	—
153	#500	—	13	1	no	—
154	#499	—	13	2	no	—
155	#499	—	26	1	no	—
156	#498	—	26	1	no	—
157	C₄×Q₁₆	—	2	1	no	—
158	C₄×Q₁₆	—	2	1	no	—
159	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
160	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
161	#427	—	13	2	no	—
162	#427	—	26	1	no	—
163	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
164	C₂×#103	—	2	1	no	—
165	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
166	#367	—	13	4	no	—
167	#366	—	13	4	no	—
168	C₄⋊C₁₆	—	13	2	no	—
169	C₄⋊C₁₆	—	13	2	no	—
170	#358	—	13	4	no	—
171	#358	—	13	4	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	#474	1	2	yes	—
175	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	2	1	no	—
178	C₄×C₁₃	—	2	1	no	—
179	C₄×C₁₃	—	4	4	no	—
180	C₄×C₁₃	—	8	2	no	—
181	C₄×C₁₃	#474	1	2	yes	—
182	C₄×C₁₃	#510	1	2	yes	—
183	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
184	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
185	Dic₁₃	—	8	2	no	—
186	Dic₁₃	—	8	2	no	—
187	C₂²×Q₈	—	2	1	no	—
188	C₂×Q₁₆	—	2	2	no	—
189	C₂×Q₁₆	—	2	1	no	—
190	C₂×Q₁₆	Dic₂₆	1	4	yes	complemented
191	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
192	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
194	C₄⋊Q₈	—	26	1	no	—
195	C₄⋊Q₈	—	26	1	no	—
196	#125	—	26	2	no	—
197	C₂²⋊Q₈	—	52	1	no	—
198	C₂²⋊Q₈	—	52	1	no	—
199	C₄×Q₈	—	2	2	no	—
200	C₄×Q₈	—	2	2	no	—
201	C₄×Q₈	—	4	1	no	—
202	C₄×Q₈	—	4	1	no	—
203	C₄×Q₈	—	52	1	no	—
204	C₄×Q₈	—	52	1	no	—
205	#117	—	26	2	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	4	1	no	—
208	C₂×C₄²	—	4	1	no	—
209	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
210	M₃₂	—	26	4	no	—
211	C₂×C₁₆	—	13	4	no	—
212	#107	—	2	1	no	—
213	#107	—	13	2	no	—
214	#107	—	13	2	no	—
215	#107	—	26	1	no	—
216	#107	—	26	1	no	—
217	#107	C₂²×D₁₃	1	2	yes	—
218	#106	—	26	2	no	—
219	#106	—	26	2	no	—
220	#103	—	2	2	no	—
221	#103	—	2	2	no	—
222	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
223	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
224	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
225	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
226	C₂×C₁₃	—	2	2	no	—
227	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
228	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
229	C₂×C₁₃	#3414	1	1	yes	characteristic
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×Q₈	—	4	2	no	—
234	C₂×Q₈	—	52	1	no	—
235	C₂×Q₈	—	52	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	#6104	1	1	yes	characteristic
239	C₂²×C₄	#6104	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
241	Q₁₆	—	2	4	no	—
242	Q₁₆	—	2	2	no	—
243	Q₁₆	C₂×Dic₂₆	1	4	yes	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
246	C₂×C₈	#6127	1	2	yes	—
247	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
248	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	26	2	no	—
256	C₄⋊C₄	—	26	2	no	—
257	C₄⋊C₄	—	52	2	no	—
258	C₄⋊C₄	—	52	2	no	—
259	C₄⋊C₄	—	52	1	no	—
260	C₄⋊C₄	—	52	1	no	—
261	C₂²⋊C₄	—	52	2	no	—
262	C₄²	—	4	2	no	—
263	C₄²	—	4	2	no	—
264	C₄²	—	52	2	no	—
265	C₄²	#6104	1	1	yes	characteristic
266	C₄²	#6104	1	1	yes	characteristic
267	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
268	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
269	C₁₆	—	26	4	no	—
270	C₁₃	#33628	1	1	yes	characteristic, complemented, Sylow
271	C₂³	C₂×#6104	1	1	yes	characteristic
272	Q₈	—	2	4	no	—
273	Q₈	—	4	4	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	104	2	no	—
276	Q₈	—	104	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	8	2	no	—
283	C₂×C₄	—	52	2	no	—
284	C₂×C₄	—	52	2	no	—
285	C₂×C₄	#88178	1	2	yes	—
286	C₂×C₄	C₂×#6104	1	1	yes	characteristic
287	C₂×C₄	C₂×#6104	1	1	yes	characteristic
288	C₂×C₄	C₂×#6104	1	1	yes	characteristic
289	C₂×C₄	C₂×#6104	1	1	yes	characteristic
290	C₂×C₄	C₂×#6104	1	1	yes	characteristic
291	C₂×C₄	#88205	1	2	yes	—
292	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	#88178	1	2	yes	—
296	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
297	C₈	C₂×#6127	1	1	yes	characteristic
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	#11718668	1	2	yes	—
301	C₂²	C₂²×#6104	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	4	no	—
306	C₄	—	8	2	no	—
307	C₄	—	104	2	no	—
308	C₄	—	104	2	no	—
309	C₄	#11718668	1	2	yes	—
310	C₄	#11718840	1	2	yes	—
311	C₄	C₂×#88178	1	1	yes	characteristic
312	C₄	C₂×#88205	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	C₂×#11718668	1	1	yes	characteristic
315	C₂	C₂×#11718840	1	1	yes	characteristic
316	C₂	#13127970	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	C₂×C₄×Q₁₆	D₁₃
hypocenter	C₁₃	#33628
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #13117000	C₂	semidirect
3. #13116988	C₂	semidirect
4. #13116978	C₂	semidirect
36. C₂×C₁₃×Q₁₆	Q₈	semidirect
190. C₂×Q₁₆	Dic₂₆	semidirect
270. C₁₃	#33628	semidirect

Maximal subgroups

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

subgroup	index
#13125350	2
#13117000	2
#13116988	2
#13116978	2
#13116622	2
#13116592	2
#13116556	2
C₂×C₄×C₁₃×Q₁₆	2
#33628	13