Subgroups — #16020669

Order 3328.

474 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127825	C₂	1	1	yes	maximal, characteristic
2	#13117075	C₂	1	2	yes	maximal
3	#13117065	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116980	C₂	1	1	yes	maximal, characteristic
5	#13116975	C₂	1	2	yes	maximal, complemented
6	#13116898	C₂	1	2	yes	maximal
7	#13116886	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116745	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116705	C₂	1	2	yes	maximal, complemented
10	C₄×#88068	C₂	1	1	yes	maximal, characteristic
11	Q₈×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈²×C₁₃	—	2	1	no	—
13	#11719289	—	2	1	no	—
14	C₁₃×#500	C₂²	1	2	yes	—
15	C₁₃×#500	C₂²	1	1	yes	characteristic
16	C₁₃×#477	—	2	1	no	—
17	C₁₃×#477	—	2	2	no	—
18	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
20	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
21	C₂×#88069	C₂²	1	2	yes	—
22	C₂×#88068	C₂²	1	1	yes	characteristic
23	#11718506	—	2	2	no	—
24	#11718502	—	2	1	no	—
25	#11718360	C₂²	1	1	yes	characteristic
26	C₄×Dic₅₂	C₂²	1	2	yes	—
27	#11718357	C₂²	1	2	yes	—
28	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
29	#11718156	C₂²	1	1	yes	characteristic
30	#11718084	C₂²	1	2	yes	—
31	#11718084	C₂²	1	2	yes	—
32	#11718084	C₂²	1	2	yes	—
33	#11718083	C₂²	1	1	yes	characteristic
34	#11718083	C₂²	1	2	yes	—
35	#11718082	C₂²	1	1	yes	characteristic
36	#11718082	C₂²	1	2	yes	—
37	C₄⋊#6090	C₂²	1	2	yes	—
38	C₄⋊#6090	C₂²	1	1	yes	characteristic
39	#11718053	C₂²	1	2	yes	—
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
42	C₁₃×C₄⋊Q₈	—	2	2	no	—
43	C₁₃×C₄⋊Q₈	—	2	1	no	—
44	C₁₃×C₄⋊Q₈	—	4	1	no	—
45	C₁₃×C₄⋊Q₈	—	4	2	no	—
46	C₄×Q₈×C₁₃	—	2	2	no	—
47	C₄×Q₈×C₁₃	—	2	1	no	—
48	C₄×Q₈×C₁₃	—	4	1	no	—
49	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
50	#88200	—	4	1	no	—
51	C₂×Dic₅₂	—	2	1	no	—
52	C₂×Dic₅₂	C₂³	1	2	yes	—
53	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
54	C₂×#6095	—	2	2	no	—
55	#88152	—	4	2	no	—
56	#88146	—	4	1	no	—
57	#88129	—	2	2	no	—
58	C₄×D₅₂	—	2	1	no	—
59	#88123	—	2	1	no	—
60	C₄×Dic₂₆	—	2	2	no	—
61	C₁₃×#107	—	2	2	no	—
62	C₁₃×#107	—	2	1	no	—
63	C₁₃×#107	C₂³	1	1	yes	characteristic
64	C₁₃×#107	C₂³	1	2	yes	—
65	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
66	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
67	C₁₃×#103	—	2	2	no	—
68	C₁₃×#103	—	2	1	no	—
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	2	yes	—
71	#88069	—	2	4	no	—
72	#88068	—	2	2	no	—
73	#88061	—	2	1	no	—
74	#88058	C₂³	1	1	yes	characteristic
75	#88056	—	2	2	no	—
76	C₂×#6090	C₂³	1	1	yes	characteristic
77	C₂×#6090	C₂³	1	1	yes	characteristic
78	C₂×#6090	C₂³	1	2	yes	—
79	#33936	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	—	2	1	no	—
81	C₂×Q₈×C₁₃	—	4	2	no	—
82	C₂×Q₈×C₁₃	—	4	1	no	—
83	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
84	C₂×D₅₂	—	2	1	no	—
85	C₂×C₄×D₁₃	—	4	1	no	—
86	C₂×Dic₂₆	—	2	2	no	—
87	C₁₃×Q₁₆	—	2	1	no	—
88	C₁₃×Q₁₆	#41	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	2	1	no	—
94	C₁₃×C₄⋊C₄	—	4	2	no	—
95	C₁₃×C₄⋊C₄	—	4	1	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₄⋊C₄	—	4	1	no	—
98	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
99	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
100	C₄²×C₁₃	—	4	2	no	—
101	C₄²×C₁₃	—	4	1	no	—
102	C₄²×C₁₃	C₂×D₄	1	2	yes	—
103	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
104	#6103	—	4	1	no	—
105	#6103	—	4	2	no	—
106	#6103	—	4	2	no	—
107	C₄⋊Dic₁₃	—	2	1	no	—
108	C₄⋊Dic₁₃	—	4	1	no	—
109	C₄⋊Dic₁₃	—	4	1	no	—
110	#6101	—	4	2	no	—
111	C₄×Dic₁₃	—	4	2	no	—
112	Dic₅₂	—	2	2	no	—
113	Dic₅₂	—	2	2	no	—
114	Dic₅₂	—	8	1	no	—
115	D₁₀₄	—	2	1	no	—
116	D₁₀₄	—	2	1	no	—
117	#6095	—	8	2	no	—
118	#6090	—	2	1	no	—
119	#6090	—	2	2	no	—
120	#6090	—	2	1	no	—
121	#3393	—	13	1	no	—
122	Q₈×Q₁₆	D₁₃	1	1	yes	characteristic, complemented
123	#2243	—	13	2	no	—
124	#2241	—	13	1	no	—
125	#2226	—	13	1	no	—
126	#2225	—	13	2	no	—
127	#2211	—	13	2	no	—
128	#2209	—	13	1	no	—
129	#2183	—	13	1	no	—
130	#2176	—	13	2	no	—
131	C₄×SD₃₂	—	13	1	no	—
132	C₂²×D₁₃	—	4	1	no	—
133	Q₈×C₁₃	—	2	2	no	—
134	Q₈×C₁₃	—	2	2	no	—
135	Q₈×C₁₃	—	4	1	no	—
136	Q₈×C₁₃	—	16	2	no	—
137	Q₈×C₁₃	—	16	1	no	—
138	C₂×C₄×C₁₃	—	4	1	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	1	no	—
141	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×Dic₁₃	—	4	1	no	—
147	C₂×Dic₁₃	—	4	2	no	—
148	D₅₂	—	4	1	no	—
149	D₅₂	—	4	1	no	—
150	C₄×D₁₃	—	8	2	no	—
151	Dic₂₆	—	4	2	no	—
152	Dic₂₆	—	4	2	no	—
153	C₈×C₁₃	—	2	2	no	—
154	C₈×C₁₃	—	2	1	no	—
155	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
156	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
157	Q₈²	—	2	1	no	—
158	#556	—	26	1	no	—
159	C₂×Q₃₂	—	13	2	no	—
160	C₂×SD₃₂	—	13	1	no	—
161	#500	C₂×D₁₃	1	2	yes	—
162	#500	C₂×D₁₃	1	1	yes	characteristic
163	#495	—	13	2	no	—
164	#494	—	13	1	no	—
165	#477	—	2	1	no	—
166	#477	—	2	2	no	—
167	#464	—	26	2	no	—
168	#461	—	26	1	no	—
169	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
170	C₄×Q₁₆	—	13	2	no	—
171	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
172	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
173	C₄×D₈	—	13	1	no	—
174	#367	—	13	1	no	—
175	#367	—	13	2	no	—
176	C₄⋊C₁₆	—	13	1	no	—
177	C₄⋊C₁₆	—	13	2	no	—
178	#358	—	13	1	no	—
179	#358	—	13	2	no	—
180	#358	—	13	2	no	—
181	#358	—	13	2	no	—
182	#357	—	13	1	no	—
183	#357	—	13	2	no	—
184	#346	—	13	2	no	—
185	C₄×C₁₆	—	13	1	no	—
186	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
187	C₂×D₁₃	—	8	1	no	—
188	C₂×D₁₃	—	8	1	no	—
189	C₂×D₁₃	—	8	1	no	—
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	2	1	no	—
193	C₄×C₁₃	—	4	2	no	—
194	C₄×C₁₃	—	8	2	no	—
195	C₄×C₁₃	—	8	1	no	—
196	C₄×C₁₃	#510	1	2	yes	—
197	C₄×C₁₃	#548	1	1	yes	characteristic
198	C₄×C₁₃	#577	1	1	yes	characteristic
199	Dic₁₃	—	8	2	no	—
200	Dic₁₃	—	8	1	no	—
201	C₂×Q₁₆	—	13	2	no	—
202	C₂×Q₁₆	—	26	1	no	—
203	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
204	C₂×SD₁₆	—	26	2	no	—
205	C₂×D₈	—	13	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	2	no	—
208	C₄⋊Q₈	—	4	1	no	—
209	C₄⋊Q₈	—	4	2	no	—
210	C₄⋊Q₈	—	26	1	no	—
211	#126	—	52	2	no	—
212	#124	—	26	2	no	—
213	C₂²⋊Q₈	—	52	1	no	—
214	C₄×Q₈	—	2	1	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₄×Q₈	—	4	1	no	—
217	C₄×Q₈	—	26	2	no	—
218	C₄×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
219	C₄×D₄	—	26	1	no	—
220	#117	—	52	1	no	—
221	Q₃₂	—	26	4	no	—
222	SD₃₂	—	26	2	no	—
223	C₂×C₁₆	—	13	1	no	—
224	C₂×C₁₆	—	13	1	no	—
225	C₂×C₁₆	—	13	2	no	—
226	#107	—	2	1	no	—
227	#107	—	2	2	no	—
228	#107	—	13	1	no	—
229	#107	C₂²×D₁₃	1	1	yes	characteristic
230	#107	C₂²×D₁₃	1	2	yes	—
231	C₄⋊C₈	C₂²⋊D₁₃	1	1	yes	characteristic
232	C₄⋊C₈	C₂²⋊D₁₃	1	2	yes	—
233	#103	—	2	1	no	—
234	#103	—	2	2	no	—
235	#103	—	26	2	no	—
236	#102	—	26	1	no	—
237	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₁₃	1	2	yes	—
239	C₂×C₁₃	#3358	1	1	yes	characteristic
240	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
241	C₂×C₁₃	#3416	1	1	yes	characteristic
242	D₁₃	—	16	1	no	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×Q₈	—	26	2	no	—
247	C₂×Q₈	#6104	1	1	yes	characteristic
248	C₂×D₄	—	26	1	no	—
249	C₂²×C₄	—	52	1	no	—
250	Q₁₆	—	2	1	no	—
251	Q₁₆	—	26	2	no	—
252	Q₁₆	—	26	2	no	—
253	Q₁₆	—	104	1	no	—
254	Q₁₆	#6131	1	2	yes	—
255	SD₁₆	—	104	2	no	—
256	D₈	—	26	1	no	—
257	D₈	—	26	1	no	—
258	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
260	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	26	1	no	—
268	C₄⋊C₄	—	52	1	no	—
269	C₄⋊C₄	—	52	2	no	—
270	C₄⋊C₄	—	52	1	no	—
271	C₄⋊C₄	#6104	1	1	yes	characteristic
272	C₄⋊C₄	#6104	1	2	yes	—
273	C₂²⋊C₄	—	52	2	no	—
274	C₂²⋊C₄	—	52	1	no	—
275	C₂²⋊C₄	—	52	2	no	—
276	C₄²	—	4	2	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	52	2	no	—
279	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
280	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
281	C₁₆	—	26	1	no	—
282	C₁₆	—	26	2	no	—
283	C₁₆	—	26	1	no	—
284	C₁₃	#33936	1	1	yes	characteristic, complemented, Sylow
285	C₂³	—	52	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	16	2	no	—
290	Q₈	—	16	1	no	—
291	Q₈	—	52	2	no	—
292	Q₈	—	52	2	no	—
293	D₄	—	52	1	no	—
294	D₄	—	52	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	52	1	no	—
299	C₂×C₄	—	52	2	no	—
300	C₂×C₄	—	104	2	no	—
301	C₂×C₄	C₂×#6104	1	2	yes	—
302	C₂×C₄	C₂×#6104	1	2	yes	—
303	C₂×C₄	C₂×#6104	1	1	yes	characteristic
304	C₂×C₄	C₂×#6104	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	C₂×#6131	1	1	yes	characteristic
309	C₈	#88254	1	1	yes	characteristic
310	C₂²	—	104	1	no	—
311	C₂²	—	104	1	no	—
312	C₂²	—	104	1	no	—
313	C₂²	C₂²×#6104	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	1	no	—
320	C₄	—	104	1	no	—
321	C₄	—	104	2	no	—
322	C₄	#11718840	1	2	yes	—
323	C₄	#11719522	1	1	yes	characteristic
324	C₄	#11719542	1	1	yes	characteristic
325	C₂	—	208	1	no	—
326	C₂	#13127471	1	1	yes	characteristic
327	C₂	C₂×#11718840	1	1	yes	characteristic
328	C₂	#13127983	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	Q₈×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×D₁₀₄	C₂³
hypercenter	Q₈×Q₁₆	D₁₃
hypocenter	C₁₃	#33936
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #13117065	C₂	semidirect
5. #13116975	C₂	semidirect
7. #13116886	C₂	semidirect
8. #13116745	C₂	semidirect
9. #13116705	C₂	semidirect
11. Q₈×C₁₃×Q₁₆	C₂	semidirect
122. Q₈×Q₁₆	D₁₃	semidirect
284. C₁₃	#33936	semidirect

Maximal subgroups

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

subgroup	index
#13127825	2
#13117075	2
#13117065	2
#13116980	2
#13116975	2
#13116898	2
#13116886	2
#13116745	2
#13116705	2
C₄×#88068	2
Q₈×C₁₃×Q₁₆	2
#33936	13