Subgroups — #16018849

Order 3328.

592 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	#13125264	C₂	1	1	yes	maximal, characteristic
2	#13117093	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117088	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117065	C₂	1	1	yes	maximal, characteristic, complemented
5	#13117061	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116898	C₂	1	4	yes	maximal, complemented
7	#13116774	C₂	1	4	yes	maximal, complemented
8	#13116635	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#533	—	2	1	no	—
11	C₂×#88158	—	2	1	no	—
12	C₂×#88158	C₂²	1	2	yes	—
13	#11719182	—	2	1	no	—
14	C₁₃×#500	C₂²	1	1	yes	characteristic
15	C₁₃×#494	C₂²	1	1	yes	characteristic
16	C₁₃×#487	—	2	1	no	—
17	C₁₃×#467	—	2	1	no	—
18	C₁₃×#467	C₂²	1	2	yes	—
19	C₄×C₁₃×Q₁₆	—	2	1	no	—
20	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₂×#88069	C₂²	1	4	yes	—
23	C₂×#88068	C₂²	1	4	yes	—
24	#11718360	C₂²	1	1	yes	characteristic
25	#11718357	—	2	1	no	—
26	#11718357	C₂²	1	2	yes	—
27	#11718356	C₂²	1	1	yes	characteristic
28	#11718354	—	2	1	no	—
29	C₂²⋊#6090	C₂²	1	2	yes	—
30	#11718084	C₂²	1	4	yes	—
31	#11718083	C₂²	1	4	yes	—
32	C₄⋊#6090	C₂²	1	2	yes	—
33	C₄×#6090	C₂²	1	1	yes	characteristic
34	C₄×#6090	C₂²	1	1	yes	characteristic
35	C₂×#6127	—	2	2	no	—
36	C₂×C₁₃×Q₁₆	—	2	1	no	—
37	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×C₄⋊Q₈	—	2	1	no	—
41	C₁₃×C₄⋊Q₈	—	2	1	no	—
42	C₁₃×#125	—	4	1	no	—
43	C₁₃×C₂²⋊Q₈	—	2	2	no	—
44	C₁₃×C₂²⋊Q₈	—	4	1	no	—
45	C₄×Q₈×C₁₃	—	2	2	no	—
46	C₄×Q₈×C₁₃	—	4	1	no	—
47	C₁₃×#117	—	4	1	no	—
48	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
49	#88184	—	4	1	no	—
50	#88184	—	4	1	no	—
51	C₂×Dic₅₂	—	2	1	no	—
52	C₂×Dic₅₂	C₂³	1	2	yes	—
53	#88158	—	4	4	no	—
54	#88158	—	4	4	no	—
55	C₂×D₁₀₄	—	2	1	no	—
56	C₂×D₁₀₄	C₂³	1	2	yes	—
57	C₂×#6095	—	2	2	no	—
58	C₂×#6095	—	2	2	no	—
59	#88129	—	2	2	no	—
60	#88128	—	2	1	no	—
61	C₄×D₅₂	—	4	1	no	—
62	C₄×D₅₂	—	4	1	no	—
63	#88123	—	2	1	no	—
64	C₁₃×#107	—	2	1	no	—
65	C₁₃×#107	C₂³	1	2	yes	—
66	C₁₃×#103	—	2	2	no	—
67	C₁₃×#103	—	2	2	no	—
68	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	#88069	—	4	4	no	—
73	#88068	—	4	4	no	—
74	C₂×#6090	C₂³	1	2	yes	—
75	C₂×#6090	C₂³	1	2	yes	—
76	#33776	—	13	1	no	maximal, Sylow
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	4	1	no	—
79	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₂²⋊D₁₃	—	4	2	no	—
83	#6127	—	4	4	no	—
84	#6127	—	4	4	no	—
85	C₂×D₅₂	—	2	2	no	—
86	C₂×D₅₂	—	4	1	no	—
87	C₂×D₅₂	—	4	1	no	—
88	C₂×C₄×D₁₃	—	4	2	no	—
89	C₂×Dic₂₆	—	2	2	no	—
90	C₁₃×Q₁₆	—	2	4	no	—
91	C₁₃×Q₁₆	—	4	2	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₄⋊C₄	—	4	1	no	—
101	C₁₃×C₄⋊C₄	—	4	2	no	—
102	C₁₃×C₄⋊C₄	—	4	2	no	—
103	C₁₃×C₄⋊C₄	—	4	1	no	—
104	C₁₃×C₂²⋊C₄	—	4	2	no	—
105	C₄²×C₁₃	—	4	2	no	—
106	C₄²×C₁₃	D₈	1	1	yes	characteristic
107	C₄²×C₁₃	D₈	1	1	yes	characteristic
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
110	#6103	—	4	2	no	—
111	#6103	—	4	2	no	—
112	C₄⋊Dic₁₃	—	4	1	no	—
113	C₄⋊Dic₁₃	—	4	1	no	—
114	Dic₅₂	—	2	4	no	—
115	Dic₅₂	—	4	2	no	—
116	D₁₀₄	—	2	4	no	—
117	D₁₀₄	—	4	2	no	—
118	#6095	—	4	4	no	—
119	#6095	—	4	4	no	—
120	#6090	—	2	2	no	—
121	#6090	—	2	2	no	—
122	#3170	D₁₃	1	1	yes	characteristic, complemented
123	#3147	—	13	1	no	—
124	#2248	—	13	1	no	—
125	#2247	—	13	1	no	—
126	#2241	—	13	1	no	—
127	#2240	—	13	1	no	—
128	#2211	—	13	4	no	—
129	#2189	—	13	4	no	—
130	#2162	—	13	1	no	—
131	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
132	C₂²×D₁₃	—	4	2	no	—
133	Q₈×C₁₃	—	4	2	no	—
134	Q₈×C₁₃	—	4	2	no	—
135	Q₈×C₁₃	—	8	2	no	—
136	C₂×C₄×C₁₃	—	2	1	no	—
137	C₂×C₄×C₁₃	—	2	1	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂²⋊D₁₃	—	8	4	no	—
149	C₂×Dic₁₃	—	4	2	no	—
150	D₅₂	—	4	2	no	—
151	D₅₂	—	4	2	no	—
152	D₅₂	—	8	2	no	—
153	D₅₂	—	8	2	no	—
154	C₄×D₁₃	—	8	4	no	—
155	Dic₂₆	—	4	2	no	—
156	Dic₂₆	—	4	2	no	—
157	C₈×C₁₃	—	2	1	no	—
158	C₈×C₁₃	—	2	1	no	—
159	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
160	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
161	C₂×#135	—	13	2	no	—
162	C₂×#135	—	26	1	no	—
163	#533	—	2	1	no	—
164	#532	—	26	1	no	—
165	C₂×Q₃₂	—	13	4	no	—
166	C₂×SD₃₂	—	13	4	no	—
167	#500	C₂×D₁₃	1	1	yes	characteristic
168	#495	—	13	2	no	—
169	#495	—	26	1	no	—
170	#494	—	13	1	no	—
171	#494	C₂×D₁₃	1	1	yes	characteristic
172	#493	—	13	1	no	—
173	#492	—	26	1	no	—
174	#487	—	2	1	no	—
175	#467	—	2	1	no	—
176	#467	C₂×D₁₃	1	2	yes	—
177	C₄×Q₁₆	—	2	1	no	—
178	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
179	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
180	C₄⋊C₁₆	—	13	2	no	—
181	#358	—	13	4	no	—
182	#357	—	13	4	no	—
183	C₂²⋊C₁₆	—	13	2	no	—
184	C₄×C₁₆	—	13	1	no	—
185	C₄×C₁₆	—	13	1	no	—
186	C₂²×C₁₃	—	2	1	no	—
187	C₂²×C₁₃	—	2	1	no	—
188	C₂²×C₁₃	—	2	1	no	—
189	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
190	C₂×D₁₃	—	8	2	no	—
191	C₂×D₁₃	—	8	2	no	—
192	C₂×D₁₃	—	8	2	no	—
193	C₄×C₁₃	—	2	1	no	—
194	C₄×C₁₃	—	2	1	no	—
195	C₄×C₁₃	—	2	1	no	—
196	C₄×C₁₃	—	8	2	no	—
197	C₄×C₁₃	—	8	2	no	—
198	C₄×C₁₃	#508	1	2	yes	—
199	C₄×C₁₃	#510	1	2	yes	—
200	C₄×C₁₃	#535	1	1	yes	characteristic
201	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
202	Dic₁₃	—	8	2	no	—
203	C₂×#41	—	26	2	no	—
204	#135	—	52	4	no	—
205	#135	—	52	4	no	—
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	—	13	2	no	—
208	C₂×Q₁₆	—	26	1	no	—
209	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
210	C₂×SD₁₆	—	26	2	no	—
211	C₂×SD₁₆	—	26	2	no	—
212	C₂×D₈	—	13	2	no	—
213	C₂×D₈	—	26	1	no	—
214	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	26	1	no	—
219	#127	—	26	1	no	—
220	#125	—	4	1	no	—
221	#124	—	26	2	no	—
222	C₂²⋊Q₈	—	2	2	no	—
223	C₂²⋊Q₈	—	4	1	no	—
224	#121	—	52	1	no	—
225	#121	—	52	1	no	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×Q₈	—	4	1	no	—
228	C₄×D₄	—	52	1	no	—
229	C₄×D₄	—	52	1	no	—
230	#117	—	4	1	no	—
231	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
232	Q₃₂	—	52	4	no	—
233	SD₃₂	—	52	4	no	—
234	C₂×C₁₆	—	13	2	no	—
235	C₂×C₁₆	—	13	2	no	—
236	#107	—	2	1	no	—
237	#107	C₂²×D₁₃	1	2	yes	—
238	#103	—	2	2	no	—
239	#103	—	2	2	no	—
240	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
241	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
244	C₂×C₁₃	—	4	1	no	—
245	C₂×C₁₃	#3205	1	1	yes	characteristic
246	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
247	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
248	D₁₃	—	16	2	no	—
249	#41	—	52	4	no	—
250	#41	—	52	4	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×Q₈	—	26	2	no	—
254	C₂×D₄	—	26	2	no	—
255	C₂×D₄	—	52	2	no	—
256	C₂×D₄	—	52	1	no	—
257	C₂×D₄	—	52	1	no	—
258	C₂²×C₄	—	52	2	no	—
259	C₂²×C₄	#6104	1	1	yes	characteristic
260	C₂²×C₄	#6104	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
262	Q₁₆	—	2	4	no	—
263	Q₁₆	—	4	2	no	—
264	Q₁₆	—	26	4	no	—
265	Q₁₆	—	52	2	no	—
266	SD₁₆	—	52	4	no	—
267	SD₁₆	—	52	4	no	—
268	D₈	—	26	4	no	—
269	D₈	—	52	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
273	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
274	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
275	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	52	1	no	—
283	C₄⋊C₄	—	52	1	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	52	2	no	—
286	C₂²⋊C₄	—	52	2	no	—
287	C₄²	—	4	2	no	—
288	C₄²	#6104	1	1	yes	characteristic
289	C₄²	#6104	1	1	yes	characteristic
290	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
291	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
292	C₁₆	—	26	2	no	—
293	C₁₆	—	26	2	no	—
294	C₁₃	#33776	1	1	yes	characteristic, complemented, Sylow
295	C₂³	—	52	2	no	—
296	C₂³	C₂×#6104	1	1	yes	characteristic
297	Q₈	—	4	2	no	—
298	Q₈	—	4	2	no	—
299	Q₈	—	8	2	no	—
300	Q₈	—	52	2	no	—
301	Q₈	—	52	2	no	—
302	D₄	—	52	2	no	—
303	D₄	—	52	2	no	—
304	D₄	—	104	4	no	—
305	D₄	—	104	2	no	—
306	D₄	—	104	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	52	2	no	—
314	C₂×C₄	—	104	4	no	—
315	C₂×C₄	C₂×#6104	1	1	yes	characteristic
316	C₂×C₄	C₂×#6104	1	1	yes	characteristic
317	C₂×C₄	C₂×#6104	1	1	yes	characteristic
318	C₂×C₄	C₂×#6104	1	1	yes	characteristic
319	C₂×C₄	C₂×#6104	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	#88203	1	2	yes	—
324	C₈	#88256	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	104	2	no	—
329	C₂²	—	104	2	no	—
330	C₂²	—	104	2	no	—
331	C₂²	C₂²×#6104	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	104	2	no	—
338	C₄	#11718840	1	2	yes	—
339	C₄	#11718855	1	2	yes	—
340	C₄	C₂×#88203	1	1	yes	characteristic
341	C₄	#11719535	1	1	yes	characteristic
342	C₂	—	4	1	no	—
343	C₂	—	208	2	no	—
344	C₂	#13125751	1	1	yes	characteristic
345	C₂	C₂×#11718855	1	1	yes	characteristic
346	C₂	C₂×#11718840	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3170	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13125264	C₂
hypercenter	#3170	D₁₃
hypocenter	C₁₃	#33776
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #13117093	C₂	semidirect
3. #13117088	C₂	semidirect
4. #13117065	C₂	semidirect
5. #13117061	C₂	semidirect
6. #13116898	C₂	semidirect
7. #13116774	C₂	semidirect
8. #13116635	C₂	semidirect
9. C₁₃×#3170	C₂	semidirect
122. #3170	D₁₃	semidirect
294. C₁₃	#33776	semidirect

Maximal subgroups

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

subgroup	index
#13125264	2
#13117093	2
#13117088	2
#13117065	2
#13117061	2
#13116898	2
#13116774	2
#13116635	2
C₁₃×#3170	2
#33776	13