Subgroups — #16019845

Order 3328.

546 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127266	C₂	1	1	yes	maximal, characteristic, complemented
2	#13126556	C₂	1	1	yes	maximal, characteristic
3	#13117042	C₂	1	2	yes	maximal, complemented
4	#13116970	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116926	C₂	1	2	yes	maximal, complemented
6	#13116883	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116779	C₂	1	2	yes	maximal, complemented
8	C₄×#88040	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₆×D₅₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116542	C₂	1	2	yes	maximal, complemented
11	C₁₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88173	C₂²	1	2	yes	—
13	#11719437	—	2	1	no	—
14	#11719431	—	2	1	no	—
15	C₁₃×#500	C₂²	1	1	yes	characteristic
16	C₁₃×#494	C₂²	1	1	yes	characteristic
17	#11718849	C₂²	1	2	yes	—
18	#11718848	—	2	2	no	—
19	C₂×#88069	C₂²	1	2	yes	—
20	#11718755	—	2	1	no	—
21	#11718744	—	2	1	no	—
22	C₂×#88040	C₂²	1	1	yes	characteristic
23	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
24	#11718549	—	2	2	no	—
25	#11718548	C₂²	1	2	yes	—
26	#11718541	C₂²	1	2	yes	—
27	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
28	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
29	C₈×D₅₂	C₂²	1	1	yes	characteristic
30	C₁₃×#358	C₂²	1	2	yes	—
31	C₁₃×#358	C₂²	1	2	yes	—
32	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
33	#11718156	C₂²	1	2	yes	—
34	#11718101	C₂²	1	1	yes	characteristic
35	#11718099	C₂²	1	2	yes	—
36	#11718097	C₂²	1	1	yes	characteristic
37	#11718095	C₂²	1	1	yes	characteristic
38	#11718083	C₂²	1	2	yes	—
39	#11718081	C₂²	1	2	yes	—
40	C₄⋊#6090	C₂²	1	1	yes	characteristic
41	C₂×#6131	—	2	2	no	—
42	C₂×C₁₃×Q₁₆	—	2	1	no	—
43	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
44	C₁₃×C₄⋊Q₈	—	2	1	no	—
45	C₁₃×C₄⋊Q₈	—	2	1	no	—
46	#88201	—	4	1	no	—
47	#88200	—	2	2	no	—
48	Q₈×Dic₁₃	—	4	1	no	—
49	C₂×#6106	—	2	2	no	—
50	#88173	—	2	4	no	—
51	#88173	—	4	2	no	—
52	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
53	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
54	C₂×C₈×D₁₃	C₂³	1	2	yes	—
55	#88152	—	4	2	no	—
56	#88151	—	2	2	no	—
57	#88149	—	4	1	no	—
58	#88148	—	4	2	no	—
59	#88146	—	2	2	no	—
60	C₄⋊C₄×D₁₃	—	4	1	no	—
61	C₄×D₅₂	—	2	1	no	—
62	C₄×D₅₂	D₄	1	1	yes	characteristic
63	C₄×Dic₂₆	—	2	1	no	—
64	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
65	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
66	C₁₃×#107	—	2	1	no	—
67	C₁₃×#107	C₂³	1	2	yes	—
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	#88075	—	2	2	no	—
70	#88069	—	4	2	no	—
71	#88061	—	2	1	no	—
72	#88060	D₄	1	2	yes	—
73	#88058	C₂³	1	1	yes	characteristic
74	#88058	C₂³	1	1	yes	characteristic
75	#88056	—	2	1	no	—
76	C₂×#6090	C₂³	1	2	yes	—
77	#88050	—	2	2	no	—
78	#88048	—	2	2	no	—
79	C₄⋊#1055	D₄	1	1	yes	characteristic
80	#88040	—	2	2	no	—
81	C₁₆×D₁₃	—	4	2	no	—
82	#33869	—	13	1	no	maximal, Sylow
83	C₂×Q₈×C₁₃	—	2	2	no	—
84	C₂×Q₈×C₁₃	—	4	1	no	—
85	#6131	—	4	4	no	—
86	#6131	—	8	2	no	—
87	C₂×D₅₂	—	2	1	no	—
88	C₂×D₅₂	—	4	2	no	—
89	C₂×D₅₂	D₈	1	1	yes	characteristic
90	C₂×C₄×D₁₃	—	4	1	no	—
91	C₂×C₄×D₁₃	—	4	2	no	—
92	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
93	C₂×Dic₂₆	—	2	1	no	—
94	C₁₃×Q₁₆	—	2	2	no	—
95	C₁₃×Q₁₆	—	2	2	no	—
96	C₁₃×Q₁₆	—	8	1	no	—
97	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	4	1	no	—
101	C₁₃×C₄⋊C₄	—	4	1	no	—
102	C₁₃×C₄⋊C₄	—	4	1	no	—
103	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
104	#6106	—	4	4	no	—
105	#6103	—	4	2	no	—
106	#6103	—	4	2	no	—
107	#6103	—	4	2	no	—
108	#6103	—	4	1	no	—
109	#6103	D₈	1	2	yes	—
110	C₄⋊Dic₁₃	—	2	1	no	—
111	C₄⋊Dic₁₃	—	2	1	no	—
112	C₄⋊Dic₁₃	—	4	2	no	—
113	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
114	#6101	—	4	1	no	—
115	#6101	—	4	2	no	—
116	C₄×Dic₁₃	—	4	1	no	—
117	C₄×Dic₁₃	—	4	2	no	—
118	C₂×#1055	C₂×D₄	1	2	yes	—
119	Dic₅₂	—	2	1	no	—
120	Dic₅₂	—	2	1	no	—
121	D₁₀₄	—	2	1	no	—
122	D₁₀₄	#41	1	2	yes	—
123	C₈×D₁₃	—	2	2	no	—
124	C₈×D₁₃	—	2	2	no	—
125	C₁₃×C₁₆	—	2	1	no	—
126	C₁₃×C₁₆	—	2	1	no	—
127	#6090	—	2	2	no	—
128	#3342	—	13	1	no	—
129	#3283	—	13	1	no	—
130	#2241	D₁₃	1	1	yes	characteristic, complemented
131	#2237	—	13	2	no	—
132	#2224	—	13	1	no	—
133	#2216	—	13	2	no	—
134	#2209	—	13	1	no	—
135	#2189	—	13	2	no	—
136	C₄×SD₃₂	—	13	1	no	—
137	D₄×C₁₆	—	13	1	no	—
138	#2138	—	13	2	no	—
139	C₂²×D₁₃	—	4	1	no	—
140	C₂²×D₁₃	C₂×D₈	1	2	yes	—
141	Q₈×C₁₃	—	4	2	no	—
142	Q₈×C₁₃	—	4	2	no	—
143	Q₈×C₁₃	—	8	2	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×Dic₁₃	—	4	1	no	—
150	C₂×Dic₁₃	—	4	1	no	—
151	C₂×Dic₁₃	—	4	1	no	—
152	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
153	D₅₂	—	2	2	no	—
154	D₅₂	—	2	2	no	—
155	D₅₂	—	4	1	no	—
156	D₅₂	—	8	4	no	—
157	C₄×D₁₃	—	2	2	no	—
158	C₄×D₁₃	—	2	2	no	—
159	C₄×D₁₃	—	8	4	no	—
160	C₄×D₁₃	—	8	2	no	—
161	Dic₂₆	—	4	1	no	—
162	Dic₂₆	—	4	1	no	—
163	C₈×C₁₃	—	2	1	no	—
164	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
165	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
166	#1055	—	2	2	no	—
167	C₂×#135	—	13	2	no	—
168	#556	—	26	1	no	—
169	#547	—	26	1	no	—
170	C₂×Q₃₂	—	13	2	no	—
171	C₂×SD₃₂	—	13	1	no	—
172	C₂²×C₁₆	—	13	2	no	—
173	#500	C₂×D₁₃	1	1	yes	characteristic
174	#494	C₂×D₁₃	1	1	yes	characteristic
175	#485	—	26	2	no	—
176	#475	—	26	1	no	—
177	#467	—	13	2	no	—
178	#467	—	13	2	no	—
179	#461	—	26	1	no	—
180	#452	—	26	2	no	—
181	C₄×Q₁₆	—	13	1	no	—
182	C₄×D₈	—	13	1	no	—
183	C₈×D₄	—	13	1	no	—
184	#427	—	13	2	no	—
185	#367	—	13	1	no	—
186	#366	—	13	2	no	—
187	C₄⋊C₁₆	—	13	1	no	—
188	#358	—	13	2	no	—
189	#358	—	13	1	no	—
190	#358	C₂×D₁₃	1	2	yes	—
191	#358	C₂×D₁₃	1	2	yes	—
192	#357	—	13	1	no	—
193	#357	—	13	2	no	—
194	C₂²⋊C₁₆	—	13	2	no	—
195	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
196	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
197	C₂×D₁₃	—	2	2	no	—
198	C₂×D₁₃	—	2	2	no	—
199	C₂×D₁₃	—	2	2	no	—
200	C₂×D₁₃	—	4	2	no	—
201	C₂×D₁₃	—	8	1	no	—
202	C₂×D₁₃	—	8	1	no	—
203	C₄×C₁₃	—	2	1	no	—
204	C₄×C₁₃	—	8	2	no	—
205	C₄×C₁₃	—	8	2	no	—
206	C₄×C₁₃	#508	1	2	yes	—
207	C₄×C₁₃	#546	1	1	yes	characteristic
208	C₄×C₁₃	#577	1	1	yes	characteristic
209	Dic₁₃	—	2	2	no	—
210	Dic₁₃	—	8	1	no	—
211	Dic₁₃	—	8	1	no	—
212	Dic₁₃	—	8	1	no	—
213	C₂×#41	—	26	2	no	—
214	#135	—	26	4	no	—
215	#135	—	52	2	no	—
216	C₂×Q₁₆	—	2	1	no	—
217	C₂×Q₁₆	—	13	1	no	—
218	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
219	C₂×SD₁₆	—	26	2	no	—
220	C₂×D₈	—	13	1	no	—
221	C₂²×C₈	—	13	2	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	—	2	1	no	—
224	#126	—	52	2	no	—
225	#124	—	52	1	no	—
226	#123	—	52	2	no	—
227	C₂²⋊Q₈	—	26	2	no	—
228	C₂²⋊Q₈	—	26	2	no	—
229	#121	—	52	1	no	—
230	C₄×Q₈	—	26	1	no	—
231	C₄×Q₈	—	52	1	no	—
232	C₄×D₄	—	13	1	no	—
233	C₄×D₄	—	26	1	no	—
234	#117	—	26	2	no	—
235	C₂×C₄⋊C₄	—	52	1	no	—
236	Q₃₂	—	52	2	no	—
237	SD₃₂	—	26	2	no	—
238	C₂×C₁₆	—	13	2	no	—
239	C₂×C₁₆	—	52	2	no	—
240	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
241	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
242	#107	—	2	1	no	—
243	#107	—	13	1	no	—
244	#107	—	13	1	no	—
245	#107	C₂²×D₁₃	1	2	yes	—
246	#106	—	26	2	no	—
247	C₄⋊C₈	—	13	1	no	—
248	#103	—	26	2	no	—
249	#103	—	26	2	no	—
250	#103	—	26	1	no	—
251	#102	—	26	1	no	—
252	C₂²⋊C₈	—	13	2	no	—
253	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
254	C₂×C₁₃	#3294	1	1	yes	characteristic
255	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
256	C₂×C₁₃	#3417	1	1	yes	characteristic
257	D₁₃	—	4	2	no	—
258	D₁₃	—	8	2	no	—
259	#41	—	52	4	no	—
260	#41	—	104	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	4	1	no	—
263	C₂×Q₈	—	26	1	no	—
264	C₂×D₄	—	13	1	no	—
265	C₂×D₄	—	26	1	no	—
266	C₂×D₄	—	52	2	no	—
267	C₂²×C₄	—	13	2	no	—
268	C₂²×C₄	—	52	2	no	—
269	C₂²×C₄	—	52	1	no	—
270	Q₁₆	—	2	2	no	—
271	Q₁₆	—	2	2	no	—
272	Q₁₆	—	8	1	no	—
273	Q₁₆	—	26	1	no	—
274	Q₁₆	—	26	1	no	—
275	SD₁₆	—	52	4	no	—
276	D₈	—	13	2	no	—
277	D₈	—	26	1	no	—
278	C₂×C₈	—	13	2	no	—
279	C₂×C₈	—	26	2	no	—
280	C₂×C₈	—	26	2	no	—
281	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
282	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	13	1	no	—
288	C₄⋊C₄	—	26	1	no	—
289	C₄⋊C₄	—	26	1	no	—
290	C₄⋊C₄	—	52	2	no	—
291	C₄⋊C₄	—	52	2	no	—
292	C₄⋊C₄	—	52	1	no	—
293	C₂²⋊C₄	—	13	2	no	—
294	C₂²⋊C₄	—	52	2	no	—
295	C₂²⋊C₄	—	52	2	no	—
296	C₂²⋊C₄	—	52	2	no	—
297	C₂²⋊C₄	—	52	1	no	—
298	C₄²	—	52	2	no	—
299	C₄²	—	52	1	no	—
300	C₄²	D₄×D₁₃	1	1	yes	characteristic
301	C₁₆	—	2	1	no	—
302	C₁₆	—	2	1	no	—
303	C₁₆	—	26	2	no	—
304	C₁₃	#33869	1	1	yes	characteristic, complemented, Sylow
305	C₂³	—	13	2	no	—
306	C₂³	—	52	1	no	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	8	2	no	—
310	Q₈	—	52	1	no	—
311	Q₈	—	52	1	no	—
312	D₄	—	26	2	no	—
313	D₄	—	26	2	no	—
314	D₄	—	52	1	no	—
315	D₄	—	104	4	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	13	2	no	—
319	C₂×C₄	—	26	2	no	—
320	C₂×C₄	—	26	2	no	—
321	C₂×C₄	—	52	1	no	—
322	C₂×C₄	—	52	1	no	—
323	C₂×C₄	—	52	1	no	—
324	C₂×C₄	—	104	4	no	—
325	C₂×C₄	—	104	2	no	—
326	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
327	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
329	C₈	—	2	1	no	—
330	C₈	—	26	2	no	—
331	C₈	#41×D₁₃	1	1	yes	characteristic
332	C₈	#88256	1	1	yes	characteristic
333	C₂²	—	26	2	no	—
334	C₂²	—	26	2	no	—
335	C₂²	—	26	2	no	—
336	C₂²	—	52	2	no	—
337	C₂²	—	104	1	no	—
338	C₂²	—	104	1	no	—
339	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	—	26	2	no	—
344	C₄	—	104	1	no	—
345	C₄	—	104	1	no	—
346	C₄	—	104	1	no	—
347	C₄	#11718577	1	2	yes	—
348	C₄	#11719381	1	1	yes	characteristic
349	C₄	#11719467	1	1	yes	characteristic
350	C₂	—	52	2	no	—
351	C₂	—	104	2	no	—
352	C₂	#13126648	1	1	yes	characteristic
353	C₂	C₂×#11718577	1	1	yes	characteristic
354	C₂	#13127985	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×#2241	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13126556	C₂
hypercenter	#2241	D₁₃
hypocenter	C₁₃	#33869
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
1. #13127266	C₂	semidirect
3. #13117042	C₂	semidirect
4. #13116970	C₂	semidirect
5. #13116926	C₂	semidirect
6. #13116883	C₂	semidirect
7. #13116779	C₂	semidirect
8. C₄×#88040	C₂	semidirect
9. C₁₆×D₅₂	C₂	semidirect
10. #13116542	C₂	semidirect
11. C₁₃×#2241	C₂	semidirect
130. #2241	D₁₃	semidirect
304. C₁₃	#33869	semidirect

Maximal subgroups

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

subgroup	index
#13127266	2
#13126556	2
#13117042	2
#13116970	2
#13116926	2
#13116883	2
#13116779	2
C₄×#88040	2
C₁₆×D₅₂	2
#13116542	2
C₁₃×#2241	2
#33869	13