Subgroups — #16019808

Order 3328.

548 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127230	C₂	1	1	yes	maximal, characteristic
2	#13126586	C₂	1	1	yes	maximal, characteristic
3	#13117063	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117037	C₂	1	2	yes	maximal, complemented
5	#13116989	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116924	C₂	1	2	yes	maximal, complemented
7	#13116768	C₂	1	2	yes	maximal, complemented
8	C₄×#88069	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116655	C₂	1	1	yes	maximal, characteristic
10	D₁₃×#358	C₂	1	2	yes	maximal, complemented
11	C₁₃×#2209	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₁₃	C₂²	1	2	yes	—
13	#11719432	—	2	1	no	—
14	#11719284	—	2	1	no	—
15	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
16	C₁₃×#494	C₂²	1	1	yes	characteristic
17	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
18	#11718849	C₂²	1	2	yes	—
19	#11718849	C₂²	1	2	yes	—
20	#11718847	—	2	2	no	—
21	C₂×#88069	C₂²	1	1	yes	characteristic
22	#11718751	—	2	1	no	—
23	#11718682	—	2	1	no	—
24	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
25	D₁₃×#107	C₂²	1	2	yes	—
26	#11718470	—	2	2	no	—
27	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
28	C₈×D₅₂	C₂²	1	1	yes	characteristic
29	#11718343	C₂²	1	1	yes	characteristic
30	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
31	C₁₃×#358	C₂²	1	2	yes	—
32	#11718156	C₂²	1	1	yes	characteristic
33	#11718156	C₂²	1	2	yes	—
34	#11718156	C₂²	1	2	yes	—
35	#11718099	C₂²	1	2	yes	—
36	#11718096	C₂²	1	2	yes	—
37	#11718095	C₂²	1	2	yes	—
38	#11718084	C₂²	1	1	yes	characteristic
39	#11718081	C₂²	1	1	yes	characteristic
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₁₃	—	2	2	no	—
42	C₂×C₁₃×Q₁₆	—	2	1	no	—
43	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
44	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
45	C₁₃×C₄⋊Q₈	—	2	1	no	—
46	C₄×Q₈×C₁₃	—	2	1	no	—
47	#88200	—	2	2	no	—
48	#88200	—	2	2	no	—
49	C₂×#6107	—	2	2	no	—
50	Q₁₆×D₁₃	—	2	4	no	—
51	Q₁₆×D₁₃	—	4	2	no	—
52	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
53	C₂×C₈×D₁₃	C₂³	1	2	yes	—
54	#88150	—	4	2	no	—
55	#88150	—	4	2	no	—
56	#88147	—	4	1	no	—
57	C₄⋊C₄×D₁₃	—	2	2	no	—
58	#88142	—	4	1	no	—
59	C₄×D₅₂	—	4	1	no	—
60	C₄×D₅₂	D₄	1	1	yes	characteristic
61	#88124	—	4	1	no	—
62	#88123	—	2	1	no	—
63	C₄×Dic₂₆	—	2	1	no	—
64	C₁₃×Q₃₂	—	4	2	no	—
65	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
66	C₁₃×#107	C₂³	1	1	yes	characteristic
67	C₁₃×#103	—	2	1	no	—
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	#88075	—	2	2	no	—
70	#88075	—	2	2	no	—
71	#88069	—	2	2	no	—
72	#88060	D₄	1	2	yes	—
73	#88058	—	2	1	no	—
74	#88058	C₂³	1	2	yes	—
75	#88058	C₂³	1	1	yes	characteristic
76	#88056	—	2	1	no	—
77	C₂×#6090	C₂³	1	1	yes	characteristic
78	C₂×#6090	C₂³	1	1	yes	characteristic
79	#88047	—	2	2	no	—
80	C₄⋊#1055	D₄	1	1	yes	characteristic
81	C₁₆×D₁₃	—	4	2	no	—
82	#33866	—	13	1	no	maximal, Sylow
83	C₂×Q₈×C₁₃	—	2	1	no	—
84	C₂×Q₈×C₁₃	—	2	2	no	—
85	Q₈×D₁₃	—	4	4	no	—
86	Q₈×D₁₃	—	8	2	no	—
87	C₂×D₅₂	D₈	1	1	yes	characteristic
88	C₂×C₄×D₁₃	—	4	2	no	—
89	C₂×C₄×D₁₃	—	4	2	no	—
90	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
91	C₂×Dic₂₆	—	2	1	no	—
92	C₂×Dic₂₆	—	4	2	no	—
93	C₂×Dic₂₆	—	4	1	no	—
94	C₁₃×Q₁₆	—	2	1	no	—
95	C₁₃×Q₁₆	—	2	2	no	—
96	C₁₃×Q₁₆	—	2	1	no	—
97	C₁₃×Q₁₆	—	2	2	no	—
98	C₁₃×Q₁₆	—	8	1	no	—
99	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
101	C₁₃×C₄⋊C₄	—	2	1	no	—
102	C₁₃×C₄⋊C₄	—	4	1	no	—
103	C₁₃×C₄⋊C₄	—	4	1	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₄²×C₁₃	—	4	1	no	—
106	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	#6107	—	4	4	no	—
108	#6103	—	4	2	no	—
109	#6103	—	4	2	no	—
110	#6103	D₈	1	2	yes	—
111	C₄⋊Dic₁₃	—	2	1	no	—
112	C₄⋊Dic₁₃	—	2	2	no	—
113	C₄⋊Dic₁₃	—	4	1	no	—
114	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
115	#6101	—	4	1	no	—
116	#6101	—	4	2	no	—
117	#6101	—	4	2	no	—
118	#6101	—	4	2	no	—
119	C₄×Dic₁₃	—	4	1	no	—
120	C₂×#1055	C₂×D₄	1	2	yes	—
121	Dic₅₂	—	2	1	no	—
122	Dic₅₂	#41	1	2	yes	—
123	C₈×D₁₃	—	2	2	no	—
124	C₈×D₁₃	—	2	2	no	—
125	C₁₃×C₁₆	—	2	2	no	—
126	#6090	—	2	1	no	—
127	#6090	—	2	1	no	—
128	#3338	—	13	1	no	—
129	D₄×Q₁₆	—	13	1	no	—
130	#2241	—	13	1	no	—
131	#2236	—	13	2	no	—
132	#2227	—	13	1	no	—
133	#2216	—	13	2	no	—
134	#2209	D₁₃	1	1	yes	characteristic, complemented
135	#2188	—	13	2	no	—
136	C₄×Q₃₂	—	13	1	no	—
137	D₄×C₁₆	—	13	1	no	—
138	C₂×#358	—	13	2	no	—
139	C₂²×D₁₃	C₂×D₈	1	2	yes	—
140	Q₈×C₁₃	—	4	2	no	—
141	Q₈×C₁₃	—	4	1	no	—
142	Q₈×C₁₃	—	4	2	no	—
143	Q₈×C₁₃	—	4	1	no	—
144	C₂×C₄×C₁₃	—	4	1	no	—
145	C₂×C₄×C₁₃	—	4	1	no	—
146	C₂×C₄×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₁₃	—	4	1	no	—
151	C₂×Dic₁₃	—	4	1	no	—
152	C₂×Dic₁₃	—	4	2	no	—
153	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
154	D₅₂	Q₃₂	1	4	yes	complemented
155	C₄×D₁₃	—	2	2	no	—
156	C₄×D₁₃	—	2	2	no	—
157	C₄×D₁₃	—	8	4	no	—
158	C₄×D₁₃	—	16	2	no	—
159	Dic₂₆	—	2	2	no	—
160	Dic₂₆	—	4	1	no	—
161	Dic₂₆	—	8	4	no	—
162	Dic₂₆	—	16	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	C₂²×Q₁₆	—	13	2	no	—
168	#554	—	26	1	no	—
169	D₄×Q₈	—	26	1	no	—
170	C₂×Q₃₂	—	13	1	no	—
171	C₂×Q₃₂	C₂×D₁₃	1	2	yes	—
172	C₂²×C₁₆	—	13	2	no	—
173	#500	—	13	1	no	—
174	#494	C₂×D₁₃	1	1	yes	characteristic
175	#484	—	26	2	no	—
176	#477	—	26	1	no	—
177	#467	—	13	2	no	—
178	#467	—	13	2	no	—
179	#462	—	26	1	no	—
180	#451	—	26	2	no	—
181	C₄×Q₁₆	—	13	1	no	—
182	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
183	C₈×D₄	—	13	1	no	—
184	C₂×#107	—	13	2	no	—
185	#366	—	13	2	no	—
186	#366	—	13	1	no	—
187	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
188	#358	—	13	2	no	—
189	#358	—	13	1	no	—
190	#358	—	13	2	no	—
191	#358	—	13	2	no	—
192	#358	—	13	1	no	—
193	#358	C₂×D₁₃	1	2	yes	—
194	C₂²⋊C₁₆	—	13	2	no	—
195	C₄×C₁₆	—	13	1	no	—
196	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
197	C₂×D₁₃	—	2	2	no	—
198	C₂×D₁₃	—	2	2	no	—
199	C₂×D₁₃	C₂×Q₃₂	1	4	yes	—
200	C₄×C₁₃	—	2	1	no	—
201	C₄×C₁₃	—	8	1	no	—
202	C₄×C₁₃	—	8	2	no	—
203	C₄×C₁₃	—	8	1	no	—
204	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
205	C₄×C₁₃	#546	1	1	yes	characteristic
206	C₄×C₁₃	#577	1	1	yes	characteristic
207	Dic₁₃	—	2	2	no	—
208	Dic₁₃	—	4	2	no	—
209	Dic₁₃	—	8	2	no	—
210	Dic₁₃	—	8	1	no	—
211	C₂²×Q₈	—	26	2	no	—
212	C₂×Q₁₆	—	2	1	no	—
213	C₂×Q₁₆	—	13	1	no	—
214	C₂×Q₁₆	—	26	2	no	—
215	C₂×Q₁₆	—	26	4	no	—
216	C₂×Q₁₆	—	52	2	no	—
217	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
218	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
219	C₂²×C₈	—	13	2	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	26	1	no	—
222	C₄⋊Q₈	—	52	1	no	—
223	#125	—	52	1	no	—
224	C₂²⋊Q₈	—	26	2	no	—
225	C₂²⋊Q₈	—	26	2	no	—
226	C₂²⋊Q₈	—	52	2	no	—
227	C₂²⋊Q₈	—	52	2	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	—	26	1	no	—
230	C₄×D₄	—	13	1	no	—
231	C₄×D₄	—	52	1	no	—
232	C₄×D₄	—	52	1	no	—
233	C₂×C₄⋊C₄	—	26	2	no	—
234	Q₃₂	—	4	2	no	—
235	Q₃₂	—	26	2	no	—
236	C₂×C₁₆	—	13	1	no	—
237	C₂×C₁₆	—	13	1	no	—
238	C₂×C₁₆	—	52	2	no	—
239	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
240	#107	—	13	2	no	—
241	#107	—	13	1	no	—
242	#107	—	26	2	no	—
243	#107	—	26	1	no	—
244	#107	C₂²×D₁₃	1	1	yes	characteristic
245	C₄⋊C₈	—	13	1	no	—
246	#103	—	2	1	no	—
247	#103	—	26	2	no	—
248	#103	—	26	2	no	—
249	#103	—	26	1	no	—
250	C₂²⋊C₈	—	13	2	no	—
251	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
252	C₂×C₁₃	#3294	1	1	yes	characteristic
253	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
254	C₂×C₁₃	#3416	1	1	yes	characteristic
255	D₁₃	—	2	4	no	—
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	26	1	no	—
259	C₂×Q₈	—	52	2	no	—
260	C₂×Q₈	—	52	4	no	—
261	C₂×Q₈	—	52	1	no	—
262	C₂×Q₈	—	104	2	no	—
263	C₂×D₄	—	13	1	no	—
264	C₂²×C₄	—	13	2	no	—
265	C₂²×C₄	—	52	2	no	—
266	C₂²×C₄	—	52	2	no	—
267	Q₁₆	—	2	1	no	—
268	Q₁₆	—	2	2	no	—
269	Q₁₆	—	2	1	no	—
270	Q₁₆	—	2	2	no	—
271	Q₁₆	—	8	1	no	—
272	Q₁₆	—	13	2	no	—
273	Q₁₆	—	26	1	no	—
274	Q₁₆	—	52	4	no	—
275	C₂×C₈	—	13	2	no	—
276	C₂×C₈	—	26	2	no	—
277	C₂×C₈	—	26	2	no	—
278	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
279	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	13	1	no	—
285	C₄⋊C₄	—	26	1	no	—
286	C₄⋊C₄	—	26	2	no	—
287	C₄⋊C₄	—	52	2	no	—
288	C₄⋊C₄	—	52	2	no	—
289	C₄⋊C₄	—	52	2	no	—
290	C₄⋊C₄	—	52	1	no	—
291	C₄⋊C₄	—	52	1	no	—
292	C₂²⋊C₄	—	13	2	no	—
293	C₂²⋊C₄	—	52	2	no	—
294	C₂²⋊C₄	—	52	2	no	—
295	C₄²	—	4	1	no	—
296	C₄²	—	52	1	no	—
297	C₄²	D₄×D₁₃	1	1	yes	characteristic
298	C₁₆	—	2	2	no	—
299	C₁₆	—	26	1	no	—
300	C₁₆	—	26	1	no	—
301	C₁₃	#33866	1	1	yes	characteristic, complemented, Sylow
302	C₂³	—	13	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	4	1	no	—
307	Q₈	—	26	2	no	—
308	Q₈	—	52	1	no	—
309	Q₈	—	104	4	no	—
310	Q₈	—	208	1	no	—
311	D₄	—	13	4	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	13	2	no	—
316	C₂×C₄	—	26	2	no	—
317	C₂×C₄	—	26	2	no	—
318	C₂×C₄	—	52	1	no	—
319	C₂×C₄	—	52	1	no	—
320	C₂×C₄	—	52	2	no	—
321	C₂×C₄	—	104	4	no	—
322	C₂×C₄	—	208	2	no	—
323	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
324	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
325	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	26	2	no	—
328	C₈	#88251	1	1	yes	characteristic
329	C₈	#41×D₁₃	1	1	yes	characteristic
330	C₂²	—	13	4	no	—
331	C₂²	—	26	2	no	—
332	C₂²	—	26	2	no	—
333	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	1	no	—
338	C₄	—	26	2	no	—
339	C₄	—	52	2	no	—
340	C₄	—	104	1	no	—
341	C₄	—	104	2	no	—
342	C₄	D₁₃×Q₃₂	1	2	yes	—
343	C₄	#11719317	1	1	yes	characteristic
344	C₄	#11719480	1	1	yes	characteristic
345	C₂	—	26	4	no	—
346	C₂	#13126652	1	1	yes	characteristic
347	C₂	C₂×D₁₃×Q₃₂	1	1	yes	characteristic
348	C₂	#13127981	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×#2209	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×D₅₂	D₈
hypercenter	#2209	D₁₃
hypocenter	C₁₃	#33866
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2209

Complements

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

N	H ≅ G/N	type
3. #13117063	C₂	semidirect
4. #13117037	C₂	semidirect
5. #13116989	C₂	semidirect
6. #13116924	C₂	semidirect
7. #13116768	C₂	semidirect
8. C₄×#88069	C₂	semidirect
10. D₁₃×#358	C₂	semidirect
11. C₁₃×#2209	C₂	semidirect
134. #2209	D₁₃	semidirect
154. D₅₂	Q₃₂	semidirect
301. C₁₃	#33866	semidirect

Maximal subgroups

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

subgroup	index
#13127230	2
#13126586	2
#13117063	2
#13117037	2
#13116989	2
#13116924	2
#13116768	2
C₄×#88069	2
#13116655	2
D₁₃×#358	2
C₁₃×#2209	2
#33866	13