Subgroups — #16020591

Order 3328.

498 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127849	C₂	1	1	yes	maximal, characteristic, complemented
2	#13127704	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117068	C₂	1	2	yes	maximal, complemented
4	#13117060	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116994	C₂	1	2	yes	maximal, complemented
6	#13116989	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116892	C₂	1	2	yes	maximal, complemented
8	#13116872	C₂	1	1	yes	maximal, characteristic
9	C₄×#88040	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₂₀₈	C₂	1	2	yes	maximal, complemented
11	Q₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719290	—	2	1	no	—
13	#11719284	—	2	1	no	—
14	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88040	C₂²	1	1	yes	characteristic
16	C₂×D₂₀₈	C₂²	1	2	yes	—
17	#11718509	—	2	1	no	—
18	#11718508	—	2	2	no	—
19	#11718505	—	2	2	no	—
20	#11718503	—	2	1	no	—
21	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
22	#11718357	C₂²	1	2	yes	—
23	#11718356	C₂²	1	1	yes	characteristic
24	C₄×D₁₀₄	C₂²	1	2	yes	—
25	C₄×D₁₀₄	C₂²	1	2	yes	—
26	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
27	#11718344	C₂²	1	2	yes	—
28	#11718343	C₂²	1	1	yes	characteristic
29	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
32	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
33	#11718101	C₂²	1	1	yes	characteristic
34	#11718101	C₂²	1	2	yes	—
35	#11718101	C₂²	1	2	yes	—
36	#11718101	C₂²	1	2	yes	—
37	#11718097	C₂²	1	1	yes	characteristic
38	#11718096	C₂²	1	2	yes	—
39	#11718095	C₂²	1	1	yes	characteristic
40	#11718095	C₂²	1	2	yes	—
41	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
42	#88201	—	4	1	no	—
43	#88200	—	4	1	no	—
44	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
45	C₂×D₁₀₄	—	2	1	no	—
46	C₂×D₁₀₄	C₂³	1	2	yes	—
47	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
48	C₂×#6095	—	2	2	no	—
49	#88150	—	4	2	no	—
50	#88148	—	4	2	no	—
51	#88147	—	4	1	no	—
52	C₄⋊C₄×D₁₃	—	4	1	no	—
53	#88129	—	2	2	no	—
54	#88128	—	2	1	no	—
55	C₄×D₅₂	—	2	1	no	—
56	C₄×D₅₂	—	2	2	no	—
57	C₄×D₅₂	—	2	2	no	—
58	#88124	—	2	2	no	—
59	#88123	—	2	1	no	—
60	C₄×Dic₂₆	—	2	1	no	—
61	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
63	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
64	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
65	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂³	1	2	yes	—
68	#88061	—	2	2	no	—
69	#88061	—	2	2	no	—
70	#88061	—	2	1	no	—
71	#88058	—	2	1	no	—
72	#88058	C₂³	1	2	yes	—
73	#88058	C₂³	1	1	yes	characteristic
74	#88058	C₂³	1	1	yes	characteristic
75	#88057	—	2	2	no	—
76	#88056	—	2	1	no	—
77	#88040	—	2	2	no	—
78	D₂₀₈	—	2	4	no	—
79	#33928	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
81	C₂×D₅₂	—	2	1	no	—
82	C₂×D₅₂	—	2	2	no	—
83	C₂×D₅₂	—	4	1	no	—
84	C₂×D₅₂	—	4	1	no	—
85	C₂×C₄×D₁₃	—	4	1	no	—
86	C₂×C₄×D₁₃	—	4	2	no	—
87	C₂×C₄×D₁₃	—	4	2	no	—
88	C₂×Dic₂₆	—	2	1	no	—
89	C₂×Dic₂₆	—	4	1	no	—
90	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
95	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₂×D₄	1	2	yes	—
97	#6103	—	4	2	no	—
98	#6103	—	4	1	no	—
99	#6103	—	4	2	no	—
100	#6103	—	4	2	no	—
101	#6103	—	4	2	no	—
102	C₄⋊Dic₁₃	—	2	1	no	—
103	C₄⋊Dic₁₃	—	2	2	no	—
104	C₄⋊Dic₁₃	—	2	1	no	—
105	C₄⋊Dic₁₃	—	4	1	no	—
106	#6101	—	4	1	no	—
107	#6101	—	4	2	no	—
108	#6101	—	4	2	no	—
109	C₄×Dic₁₃	—	4	1	no	—
110	Dic₅₂	—	2	1	no	—
111	Dic₅₂	—	2	1	no	—
112	D₁₀₄	—	2	2	no	—
113	D₁₀₄	—	2	1	no	—
114	D₁₀₄	—	2	2	no	—
115	D₁₀₄	—	8	1	no	—
116	D₁₀₄	#41	1	2	yes	—
117	#6095	—	8	2	no	—
118	C₁₃×C₁₆	—	2	2	no	—
119	C₁₃×C₁₆	—	2	1	no	—
120	C₁₃×C₁₆	—	2	1	no	—
121	#3396	—	13	1	no	—
122	#3380	—	13	1	no	—
123	#2242	—	13	2	no	—
124	#2240	—	13	1	no	—
125	#2228	—	13	2	no	—
126	#2227	—	13	1	no	—
127	#2210	—	13	2	no	—
128	#2206	—	13	1	no	—
129	Q₈×C₁₆	D₁₃	1	1	yes	characteristic, complemented
130	C₄×SD₃₂	—	13	1	no	—
131	C₄×D₁₆	—	13	2	no	—
132	C₂²×D₁₃	—	4	1	no	—
133	C₂²×D₁₃	—	4	2	no	—
134	Q₈×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×Dic₁₃	—	4	1	no	—
141	C₂×Dic₁₃	—	4	1	no	—
142	C₂×Dic₁₃	—	4	1	no	—
143	C₂×Dic₁₃	—	4	2	no	—
144	D₅₂	—	2	2	no	—
145	D₅₂	—	4	2	no	—
146	D₅₂	—	4	1	no	—
147	D₅₂	—	4	2	no	—
148	D₅₂	—	8	2	no	—
149	D₅₂	—	16	1	no	—
150	C₄×D₁₃	—	8	2	no	—
151	C₄×D₁₃	—	16	2	no	—
152	C₄×D₁₃	—	16	2	no	—
153	Dic₂₆	—	4	1	no	—
154	Dic₂₆	—	4	1	no	—
155	Dic₂₆	—	16	1	no	—
156	C₈×C₁₃	—	2	2	no	—
157	C₈×C₁₃	—	2	1	no	—
158	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
159	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
160	#559	—	26	1	no	—
161	#554	—	26	1	no	—
162	C₂×SD₃₂	—	13	1	no	—
163	C₂×D₁₆	—	13	2	no	—
164	#500	—	13	1	no	—
165	#499	—	13	2	no	—
166	#495	—	13	2	no	—
167	#493	—	13	1	no	—
168	#478	—	26	2	no	—
169	#476	—	26	1	no	—
170	#463	—	26	2	no	—
171	#460	—	26	1	no	—
172	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
173	C₄×Q₁₆	—	13	1	no	—
174	C₄×D₈	—	13	2	no	—
175	C₄×D₈	—	13	1	no	—
176	C₄×D₈	—	13	2	no	—
177	#367	—	13	1	no	—
178	#366	—	13	2	no	—
179	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
180	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
181	#358	—	13	1	no	—
182	#358	—	13	2	no	—
183	#357	—	13	1	no	—
184	#357	—	13	2	no	—
185	#357	—	13	2	no	—
186	#357	—	13	2	no	—
187	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
188	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
189	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
190	C₂×D₁₃	—	4	2	no	—
191	C₂×D₁₃	—	8	1	no	—
192	C₂×D₁₃	—	8	1	no	—
193	C₂×D₁₃	—	8	2	no	—
194	C₂×D₁₃	—	8	2	no	—
195	C₂×D₁₃	—	8	2	no	—
196	C₄×C₁₃	—	2	2	no	—
197	C₄×C₁₃	—	2	1	no	—
198	C₄×C₁₃	—	2	2	no	—
199	C₄×C₁₃	#508	1	2	yes	—
200	C₄×C₁₃	#548	1	1	yes	characteristic
201	C₄×C₁₃	#577	1	1	yes	characteristic
202	Dic₁₃	—	8	1	no	—
203	Dic₁₃	—	8	1	no	—
204	Dic₁₃	—	8	2	no	—
205	Dic₁₃	—	8	1	no	—
206	C₂×Q₁₆	—	13	1	no	—
207	C₂×SD₁₆	—	26	2	no	—
208	C₂×D₈	—	13	1	no	—
209	C₂×D₈	—	13	2	no	—
210	C₂×D₈	—	26	1	no	—
211	C₄⋊Q₈	—	26	1	no	—
212	#127	—	26	1	no	—
213	#125	—	26	2	no	—
214	#124	—	26	2	no	—
215	#124	—	52	1	no	—
216	#123	—	52	2	no	—
217	C₂²⋊Q₈	—	52	1	no	—
218	C₂²⋊Q₈	—	52	2	no	—
219	C₄×Q₈	—	26	1	no	—
220	C₄×Q₈	D₅₂	1	1	yes	characteristic
221	C₄×D₄	—	26	1	no	—
222	C₄×D₄	—	26	2	no	—
223	C₄×D₄	—	26	2	no	—
224	C₄×D₄	—	52	1	no	—
225	C₂×C₄⋊C₄	—	52	1	no	—
226	SD₃₂	—	26	2	no	—
227	D₁₆	—	26	4	no	—
228	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
229	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
230	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
231	#107	—	13	2	no	—
232	#107	—	13	1	no	—
233	#107	—	13	1	no	—
234	#107	—	26	1	no	—
235	#106	—	26	2	no	—
236	C₄⋊C₈	D₅₂	1	1	yes	characteristic
237	C₄⋊C₈	D₅₂	1	2	yes	—
238	#103	—	26	1	no	—
239	#102	—	26	2	no	—
240	#102	—	26	2	no	—
241	#102	—	26	1	no	—
242	C₄×C₈	C₂²×D₁₃	1	2	yes	—
243	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
244	C₂×C₁₃	#3358	1	1	yes	characteristic
245	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
246	C₂×C₁₃	#3415	1	1	yes	characteristic
247	D₁₃	—	8	2	no	—
248	D₁₃	—	16	2	no	—
249	C₂×Q₈	—	26	1	no	—
250	C₂×Q₈	—	52	1	no	—
251	C₂×Q₈	D₁₀₄	1	1	yes	characteristic
252	C₂×D₄	—	26	1	no	—
253	C₂×D₄	—	26	2	no	—
254	C₂×D₄	—	52	1	no	—
255	C₂×D₄	—	52	1	no	—
256	C₂²×C₄	—	52	1	no	—
257	C₂²×C₄	—	52	2	no	—
258	C₂²×C₄	—	52	2	no	—
259	Q₁₆	—	26	1	no	—
260	Q₁₆	—	26	1	no	—
261	SD₁₆	—	104	2	no	—
262	D₈	—	13	2	no	—
263	D₈	—	26	1	no	—
264	D₈	—	26	2	no	—
265	D₈	—	26	2	no	—
266	D₈	—	104	1	no	—
267	C₂×C₈	C₂×D₅₂	1	2	yes	—
268	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
269	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
270	C₄⋊C₄	—	26	1	no	—
271	C₄⋊C₄	—	26	1	no	—
272	C₄⋊C₄	—	26	2	no	—
273	C₄⋊C₄	—	52	2	no	—
274	C₄⋊C₄	—	52	1	no	—
275	C₄⋊C₄	—	52	1	no	—
276	C₄⋊C₄	—	52	2	no	—
277	C₄⋊C₄	D₁₀₄	1	1	yes	characteristic
278	C₄⋊C₄	D₁₀₄	1	2	yes	—
279	C₂²⋊C₄	—	52	1	no	—
280	C₂²⋊C₄	—	52	2	no	—
281	C₂²⋊C₄	—	52	2	no	—
282	C₂²⋊C₄	—	52	2	no	—
283	C₂²⋊C₄	—	52	2	no	—
284	C₄²	—	52	1	no	—
285	C₄²	C₂×D₅₂	1	2	yes	—
286	C₄²	C₂×D₅₂	1	1	yes	characteristic
287	C₁₆	—	2	1	no	—
288	C₁₆	—	2	2	no	—
289	C₁₆	—	2	1	no	—
290	C₁₃	#33928	1	1	yes	characteristic, complemented, Sylow
291	C₂³	—	52	1	no	—
292	C₂³	—	52	2	no	—
293	Q₈	—	2	2	no	—
294	Q₈	—	52	1	no	—
295	Q₈	—	52	1	no	—
296	Q₈	—	208	1	no	—
297	D₄	—	26	2	no	—
298	D₄	—	52	1	no	—
299	D₄	—	52	2	no	—
300	D₄	—	52	2	no	—
301	D₄	—	104	2	no	—
302	D₄	—	208	1	no	—
303	C₂×C₄	—	52	1	no	—
304	C₂×C₄	—	52	1	no	—
305	C₂×C₄	—	52	1	no	—
306	C₂×C₄	—	52	2	no	—
307	C₂×C₄	—	104	2	no	—
308	C₂×C₄	—	208	2	no	—
309	C₂×C₄	—	208	2	no	—
310	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
311	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
312	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
313	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	2	no	—
317	C₈	#88254	1	1	yes	characteristic
318	C₈	#41×D₁₃	1	1	yes	characteristic
319	C₂²	—	52	2	no	—
320	C₂²	—	104	1	no	—
321	C₂²	—	104	1	no	—
322	C₂²	—	104	2	no	—
323	C₂²	—	104	2	no	—
324	C₂²	—	104	2	no	—
325	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	104	1	no	—
330	C₄	—	104	1	no	—
331	C₄	—	104	2	no	—
332	C₄	—	104	1	no	—
333	C₄	#11718565	1	2	yes	—
334	C₄	#11719281	1	1	yes	characteristic
335	C₄	#11719456	1	1	yes	characteristic
336	C₂	—	104	2	no	—
337	C₂	—	208	2	no	—
338	C₂	#13127472	1	1	yes	characteristic
339	C₂	C₂×#11718565	1	1	yes	characteristic
340	C₂	#13127977	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	Q₈×C₁₃×C₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13116872	C₂
hypercenter	Q₈×C₁₆	D₁₃
hypocenter	C₁₃	#33928
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
1. #13127849	C₂	semidirect
2. #13127704	C₂	semidirect
3. #13117068	C₂	semidirect
4. #13117060	C₂	semidirect
5. #13116994	C₂	semidirect
6. #13116989	C₂	semidirect
7. #13116892	C₂	semidirect
9. C₄×#88040	C₂	semidirect
10. C₄×D₂₀₈	C₂	semidirect
11. Q₈×C₁₃×C₁₆	C₂	semidirect
129. Q₈×C₁₆	D₁₃	semidirect
290. C₁₃	#33928	semidirect

Maximal subgroups

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

subgroup	index
#13127849	2
#13127704	2
#13117068	2
#13117060	2
#13116994	2
#13116989	2
#13116892	2
#13116872	2
C₄×#88040	2
C₄×D₂₀₈	2
Q₈×C₁₃×C₁₆	2
#33928	13