Subgroups — #16020595

Order 3328.

528 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127809	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117059	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117058	C₂	1	2	yes	maximal, complemented
5	#13116984	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116981	C₂	1	2	yes	maximal, complemented
7	#13116874	C₂	1	2	yes	maximal, complemented
8	#13116738	C₂	1	1	yes	maximal, characteristic, complemented
9	SD₃₂×Dic₁₃	C₂	1	2	yes	maximal, complemented
10	C₄×#88068	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2207	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719423	—	2	1	no	—
13	Q₈×Dic₂₆	—	2	1	no	—
14	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
15	C₁₃×#493	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
17	Q₁₆×Dic₁₃	C₂²	1	2	yes	—
18	C₂×#88068	C₂²	1	1	yes	characteristic
19	#11718812	C₂²	1	2	yes	—
20	#11718811	—	2	2	no	—
21	D₈×Dic₁₃	C₂²	1	2	yes	—
22	#11718733	—	2	1	no	—
23	#11718682	—	2	1	no	—
24	#11718535	C₂²	1	2	yes	—
25	#11718455	—	2	2	no	—
26	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
27	#11718343	C₂²	1	1	yes	characteristic
28	C₈×Dic₂₆	C₂²	1	1	yes	characteristic
29	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₁₃×#357	C₂²	1	2	yes	—
31	#11718156	C₂²	1	1	yes	characteristic
32	#11718154	C₂²	1	2	yes	—
33	#11718154	C₂²	1	2	yes	—
34	#11718101	C₂²	1	2	yes	—
35	#11718097	C₂²	1	2	yes	—
36	#11718093	C₂²	1	2	yes	—
37	C₁₆×Dic₁₃	C₂²	1	2	yes	—
38	#11718083	C₂²	1	1	yes	characteristic
39	#11718082	C₂²	1	1	yes	characteristic
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₂×#6129	—	4	1	no	—
42	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
43	C₂×C₁₃×D₈	—	2	1	no	—
44	C₂×C₁₃×D₈	C₂³	1	2	yes	—
45	C₁₃×#127	—	2	1	no	—
46	C₄×Q₈×C₁₃	—	2	1	no	—
47	Q₈×Dic₁₃	—	2	2	no	—
48	#88194	—	2	2	no	—
49	#88193	—	4	2	no	—
50	#88192	—	4	1	no	—
51	D₄×Dic₁₃	—	2	2	no	—
52	C₂×#6104	—	2	2	no	—
53	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
54	#88142	—	2	2	no	—
55	#88142	—	4	2	no	—
56	C₄×D₅₂	—	2	1	no	—
57	#88123	—	2	1	no	—
58	#88123	—	4	1	no	—
59	C₄×Dic₂₆	—	4	1	no	—
60	C₄×Dic₂₆	D₄	1	1	yes	characteristic
61	C₁₃×SD₃₂	—	2	4	no	—
62	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
63	C₁₃×#107	C₂³	1	1	yes	characteristic
64	C₁₃×#103	—	2	1	no	—
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	#88075	—	2	2	no	—
67	#88072	—	2	2	no	—
68	#88068	—	2	2	no	—
69	#88061	—	2	1	no	—
70	#88058	—	2	1	no	—
71	#88058	C₂³	1	2	yes	—
72	#88058	C₂³	1	1	yes	characteristic
73	#88054	D₄	1	2	yes	—
74	C₈×Dic₁₃	C₂³	1	2	yes	—
75	C₂×#6090	C₂³	1	1	yes	characteristic
76	C₂×#6090	C₂³	1	1	yes	characteristic
77	#88047	—	2	2	no	—
78	C₄⋊#1055	D₄	1	1	yes	characteristic
79	#33929	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	—	2	1	no	—
81	C₂×D₄×C₁₃	—	2	2	no	—
82	C₂×D₄×C₁₃	—	4	1	no	—
83	C₂×D₄×C₁₃	—	4	1	no	—
84	C₂×C₂²⋊D₁₃	—	4	2	no	—
85	C₂×C₂²⋊D₁₃	—	4	2	no	—
86	C₂²×Dic₁₃	—	4	2	no	—
87	#6129	—	8	4	no	—
88	C₂×D₅₂	—	2	1	no	—
89	C₂×C₄×D₁₃	—	4	1	no	—
90	C₂×Dic₂₆	—	4	1	no	—
91	C₂×Dic₂₆	—	4	2	no	—
92	C₂×Dic₂₆	D₈	1	1	yes	characteristic
93	C₁₃×Q₁₆	—	2	1	no	—
94	C₁₃×Q₁₆	#41	1	2	yes	complemented
95	C₁₃×D₈	—	2	2	no	—
96	C₁₃×D₈	—	2	2	no	—
97	C₁₃×D₈	—	8	1	no	—
98	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₁₃×C₄⋊C₄	—	2	1	no	—
101	C₁₃×C₄⋊C₄	—	4	1	no	—
102	C₄²×C₁₃	—	4	1	no	—
103	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂²⋊Dic₁₃	—	4	2	no	—
105	#6104	—	8	2	no	—
106	#6103	—	4	1	no	—
107	C₄⋊Dic₁₃	—	2	1	no	—
108	C₄⋊Dic₁₃	—	2	2	no	—
109	C₄⋊Dic₁₃	—	4	1	no	—
110	C₄⋊Dic₁₃	—	4	2	no	—
111	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
112	#6101	—	4	2	no	—
113	#6101	D₈	1	2	yes	—
114	C₄×Dic₁₃	—	4	2	no	—
115	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
116	C₂×#1055	C₂×D₄	1	2	yes	—
117	D₁₀₄	—	2	1	no	—
118	D₁₀₄	—	2	1	no	—
119	C₁₃×C₁₆	—	2	2	no	—
120	#6090	—	2	1	no	—
121	#6090	—	2	1	no	—
122	#3391	—	13	1	no	—
123	Q₈×Q₁₆	—	13	1	no	—
124	#2240	—	13	1	no	—
125	#2240	—	13	2	no	—
126	#2226	—	13	1	no	—
127	#2226	—	13	2	no	—
128	#2207	—	13	2	no	—
129	#2207	D₁₃	1	1	yes	characteristic, complemented
130	Q₈×C₁₆	—	13	1	no	—
131	C₄×SD₃₂	—	13	2	no	—
132	C₄×SD₃₂	—	13	1	no	—
133	C₂³×C₁₃	—	4	2	no	—
134	C₂²×D₁₃	—	4	1	no	—
135	Q₈×C₁₃	—	2	2	no	—
136	Q₈×C₁₃	—	4	1	no	—
137	D₄×C₁₃	—	4	2	no	—
138	D₄×C₁₃	—	4	2	no	—
139	D₄×C₁₃	—	8	2	no	—
140	D₄×C₁₃	—	16	1	no	—
141	C₂×C₄×C₁₃	—	4	1	no	—
142	C₂×C₄×C₁₃	—	4	1	no	—
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	C₂²⋊D₁₃	—	8	4	no	—
147	C₂²⋊D₁₃	—	16	2	no	—
148	C₂×Dic₁₃	—	4	2	no	—
149	C₂×Dic₁₃	—	4	1	no	—
150	C₂×Dic₁₃	—	8	4	no	—
151	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
152	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
153	D₅₂	—	4	1	no	—
154	D₅₂	—	4	1	no	—
155	C₄×D₁₃	—	8	2	no	—
156	Dic₂₆	—	16	2	no	—
157	Dic₂₆	—	16	1	no	—
158	Dic₂₆	SD₃₂	1	4	yes	complemented
159	C₈×C₁₃	—	2	1	no	—
160	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
161	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
162	#1055	—	2	2	no	—
163	Q₈²	—	26	1	no	—
164	#550	—	26	1	no	—
165	C₂×SD₃₂	—	13	1	no	—
166	C₂×SD₃₂	C₂×D₁₃	1	2	yes	—
167	#500	—	13	2	no	—
168	#500	—	13	1	no	—
169	#493	—	13	2	no	—
170	#493	C₂×D₁₃	1	1	yes	characteristic
171	#477	—	26	2	no	—
172	#477	—	26	1	no	—
173	#459	—	26	1	no	—
174	#459	—	26	2	no	—
175	C₈×Q₈	—	13	1	no	—
176	C₄×Q₁₆	—	13	2	no	—
177	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
178	C₄×D₈	—	13	2	no	—
179	C₄×D₈	—	13	1	no	—
180	#367	—	13	2	no	—
181	#367	—	13	1	no	—
182	C₄⋊C₁₆	—	13	2	no	—
183	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
184	#358	—	13	1	no	—
185	#357	—	13	2	no	—
186	#357	—	13	2	no	—
187	#357	—	13	2	no	—
188	#357	—	13	1	no	—
189	#357	C₂×D₁₃	1	2	yes	—
190	C₄×C₁₆	—	13	2	no	—
191	C₄×C₁₆	—	13	1	no	—
192	C₂²×C₁₃	—	8	2	no	—
193	C₂²×C₁₃	—	8	2	no	—
194	C₂²×C₁₃	—	8	2	no	—
195	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
196	C₂×D₁₃	—	8	1	no	—
197	C₂×D₁₃	—	8	1	no	—
198	C₂×D₁₃	—	8	1	no	—
199	C₄×C₁₃	—	2	1	no	—
200	C₄×C₁₃	—	4	2	no	—
201	C₄×C₁₃	—	8	1	no	—
202	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
203	C₄×C₁₃	#548	1	1	yes	characteristic
204	C₄×C₁₃	#577	1	1	yes	characteristic
205	Dic₁₃	—	2	2	no	—
206	Dic₁₃	—	8	1	no	—
207	Dic₁₃	—	8	2	no	—
208	Dic₁₃	C₂×SD₃₂	1	4	yes	—
209	C₂×#41	—	52	1	no	—
210	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	—	13	1	no	—
213	C₂×D₈	—	26	2	no	—
214	C₂×D₈	C₂²×D₁₃	1	2	yes	—
215	C₄⋊Q₈	—	26	2	no	—
216	C₄⋊Q₈	—	26	1	no	—
217	C₄⋊Q₈	—	52	1	no	—
218	C₄⋊Q₈	—	52	2	no	—
219	#127	—	2	1	no	—
220	#127	—	26	2	no	—
221	#121	—	52	1	no	—
222	#121	—	52	2	no	—
223	C₄×Q₈	—	2	1	no	—
224	C₄×Q₈	—	13	1	no	—
225	C₄×Q₈	—	26	2	no	—
226	C₄×Q₈	—	52	1	no	—
227	C₄×D₄	—	26	2	no	—
228	C₄×D₄	—	26	1	no	—
229	SD₃₂	—	2	4	no	—
230	SD₃₂	—	26	2	no	—
231	C₂×C₁₆	—	13	1	no	—
232	C₂×C₁₆	—	13	1	no	—
233	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
234	#107	—	13	2	no	—
235	#107	—	13	1	no	—
236	#107	—	26	2	no	—
237	#107	—	26	1	no	—
238	#107	C₂²×D₁₃	1	1	yes	characteristic
239	C₄⋊C₈	—	13	2	no	—
240	C₄⋊C₈	—	13	1	no	—
241	#103	—	2	1	no	—
242	#103	—	26	2	no	—
243	#102	—	26	2	no	—
244	#102	—	26	1	no	—
245	C₄×C₈	—	13	2	no	—
246	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
247	C₂×C₁₃	—	16	2	no	—
248	C₂×C₁₃	#3358	1	1	yes	characteristic
249	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
250	C₂×C₁₃	#3415	1	1	yes	characteristic
251	D₁₃	—	16	1	no	—
252	#41	—	104	4	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	13	1	no	—
255	C₂×Q₈	—	52	2	no	—
256	C₂×Q₈	—	52	1	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	—	26	1	no	—
261	C₂×D₄	—	52	2	no	—
262	C₂×D₄	—	52	2	no	—
263	C₂²×C₄	—	52	2	no	—
264	C₂²×C₄	—	52	1	no	—
265	Q₁₆	—	2	1	no	—
266	Q₁₆	#6129	1	2	yes	complemented
267	D₈	—	2	2	no	—
268	D₈	—	2	2	no	—
269	D₈	—	8	1	no	—
270	D₈	—	26	1	no	—
271	D₈	—	26	1	no	—
272	D₈	—	104	2	no	—
273	C₂×C₈	—	13	2	no	—
274	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
275	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	13	2	no	—
279	C₄⋊C₄	—	13	1	no	—
280	C₄⋊C₄	—	26	1	no	—
281	C₄⋊C₄	—	26	2	no	—
282	C₄⋊C₄	—	52	2	no	—
283	C₄⋊C₄	—	52	2	no	—
284	C₄⋊C₄	—	52	1	no	—
285	C₂²⋊C₄	—	52	2	no	—
286	C₂²⋊C₄	—	52	1	no	—
287	C₄²	—	4	1	no	—
288	C₄²	—	13	2	no	—
289	C₄²	—	52	2	no	—
290	C₄²	D₄×D₁₃	1	1	yes	characteristic
291	C₁₆	—	2	2	no	—
292	C₁₆	—	26	1	no	—
293	C₁₆	—	26	1	no	—
294	C₁₃	#33929	1	1	yes	characteristic, complemented, Sylow
295	C₂³	—	4	2	no	—
296	C₂³	—	52	1	no	—
297	Q₈	—	2	2	no	—
298	Q₈	—	4	1	no	—
299	Q₈	—	13	4	no	—
300	Q₈	—	208	1	no	—
301	Q₈	—	208	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	8	2	no	—
305	D₄	—	16	1	no	—
306	D₄	—	52	1	no	—
307	D₄	—	52	1	no	—
308	D₄	—	104	4	no	—
309	D₄	—	208	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	13	2	no	—
313	C₂×C₄	—	13	2	no	—
314	C₂×C₄	—	52	2	no	—
315	C₂×C₄	—	52	1	no	—
316	C₂×C₄	—	104	4	no	—
317	C₂×C₄	—	104	2	no	—
318	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
319	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
320	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	26	2	no	—
323	C₈	C₂×#6129	1	1	yes	characteristic
324	C₈	#88257	1	1	yes	characteristic
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	104	1	no	—
329	C₂²	—	104	1	no	—
330	C₂²	—	104	1	no	—
331	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	2	no	—
334	C₄	—	8	1	no	—
335	C₄	—	13	4	no	—
336	C₄	—	26	2	no	—
337	C₄	—	104	1	no	—
338	C₄	—	104	2	no	—
339	C₄	D₁₃×SD₃₂	1	2	yes	—
340	C₄	#11719303	1	1	yes	characteristic
341	C₄	#11719480	1	1	yes	characteristic
342	C₂	—	16	2	no	—
343	C₂	—	208	1	no	—
344	C₂	#13127466	1	1	yes	characteristic
345	C₂	C₂×D₁₃×SD₃₂	1	1	yes	characteristic
346	C₂	#13127976	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2207	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13127809	C₂
hypercenter	#2207	D₁₃
hypocenter	C₁₃	#33929
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2207

Complements

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

N	H ≅ G/N	type
2. Q₁₆×Dic₂₆	C₂	semidirect
3. #13117059	C₂	semidirect
4. #13117058	C₂	semidirect
5. #13116984	C₂	semidirect
6. #13116981	C₂	semidirect
7. #13116874	C₂	semidirect
8. #13116738	C₂	semidirect
9. SD₃₂×Dic₁₃	C₂	semidirect
10. C₄×#88068	C₂	semidirect
11. C₁₃×#2207	C₂	semidirect
94. C₁₃×Q₁₆	#41	semidirect
129. #2207	D₁₃	semidirect
158. Dic₂₆	SD₃₂	semidirect
266. Q₁₆	#6129	semidirect
294. C₁₃	#33929	semidirect

Maximal subgroups

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

subgroup	index
#13127809	2
Q₁₆×Dic₂₆	2
#13117059	2
#13117058	2
#13116984	2
#13116981	2
#13116874	2
#13116738	2
SD₃₂×Dic₁₃	2
C₄×#88068	2
C₁₃×#2207	2
#33929	13