Subgroups — #16795142

Order 3328.

616 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131766	C₂	1	2	yes	maximal
2	#13130414	C₂	1	2	yes	maximal, complemented
3	#13130410	C₂	1	4	yes	maximal, complemented
4	#13130399	C₂	1	2	yes	maximal, complemented
5	#13130395	C₂	1	1	yes	maximal, characteristic, complemented
6	#13130384	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₃⋊#335	C₂	1	2	yes	maximal, complemented
8	C₈×#6131	C₂	1	1	yes	maximal, characteristic
9	C₄×#6131	C₂²	1	1	yes	characteristic
10	#11719158	—	2	2	no	—
11	#11719075	—	2	4	no	—
12	#11719073	—	2	2	no	—
13	#11719054	C₂²	1	4	yes	—
14	#11719052	C₂²	1	2	yes	—
15	#11719051	C₂²	1	2	yes	—
16	D₁₃⋊(C₄⋊C₈)	C₂²	1	4	yes	—
17	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
18	Q₈×#1055	C₄	1	1	yes	characteristic
19	#11718495	C₄	1	2	yes	—
20	#11718495	C₄	1	1	yes	characteristic
21	#11718494	C₂²	1	2	yes	—
22	#11718494	C₂²	1	1	yes	characteristic
23	C₈×D₅₂	C₄	1	2	yes	—
24	C₈×D₅₂	C₄	1	1	yes	characteristic
25	C₄×C₈×D₁₃	C₂²	1	2	yes	—
26	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
27	#11718256	C₂²	1	1	yes	characteristic
28	#11718256	C₂²	1	2	yes	—
29	#11718256	C₂²	1	1	yes	characteristic
30	#11718256	C₂²	1	2	yes	—
31	#11718255	C₂²	1	4	yes	—
32	#11718254	C₂²	1	4	yes	—
33	#11718253	C₂²	1	2	yes	—
34	C₂×#6131	Q₈	1	1	yes	characteristic
35	C₂×#6118	—	2	2	no	—
36	C₂×D₁₃⋊C₈	—	2	4	no	—
37	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	Q₈×Dic₁₃	D₄	1	1	yes	characteristic
39	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
43	#88147	D₄	1	1	yes	characteristic
44	#88147	D₄	1	2	yes	—
45	#88146	Q₈	1	1	yes	characteristic
46	#88146	Q₈	1	2	yes	—
47	C₄×D₅₂	C₂×C₄	1	2	yes	—
48	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
49	C₄²×D₁₃	C₂³	1	2	yes	—
50	C₄²×D₁₃	C₂³	1	1	yes	characteristic
51	#88112	—	2	4	no	—
52	#88111	—	2	2	no	—
53	#88111	—	2	4	no	—
54	#88110	—	2	4	no	—
55	C₄⋊#1057	—	2	2	no	—
56	C₄⋊#1057	C₂³	1	2	yes	—
57	C₄⋊#1057	C₂³	1	2	yes	—
58	C₄⋊#1057	C₂³	1	4	yes	—
59	C₄×#1057	—	2	2	no	—
60	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
61	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
64	#88060	C₂×C₄	1	2	yes	—
65	#88060	C₂×C₄	1	2	yes	—
66	#88060	C₂×C₄	1	2	yes	—
67	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
68	C₈×Dic₁₃	C₂³	1	2	yes	—
69	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
70	C₄⋊#1055	C₂×C₄	1	2	yes	—
71	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
72	C₄×#1055	C₂×C₄	1	2	yes	—
73	C₄×#1055	C₂×C₄	1	1	yes	characteristic
74	#17836	—	13	1	no	maximal, Sylow
75	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
76	#6131	—	2	4	no	—
77	C₂×D₅₂	C₄⋊C₄	1	2	yes	—
78	C₂×D₅₂	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
82	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×D₁₃	C₂×Q₈	1	2	yes	—
84	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
85	C₂×#1057	—	2	2	no	—
86	C₂×#1057	—	2	2	no	—
87	C₂×#1057	—	2	4	no	—
88	#6118	—	8	2	no	—
89	D₁₃⋊C₈	—	8	4	no	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
93	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
94	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	#6103	C₄⋊C₄	1	2	yes	—
98	#6103	C₄⋊C₄	1	2	yes	—
99	#6103	C₄⋊C₄	1	2	yes	—
100	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
101	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
103	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
104	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
105	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
106	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
107	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
108	C₂×#1055	C₂²×C₄	1	2	yes	—
109	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
110	C₈×D₁₃	—	2	4	no	—
111	C₈×D₁₃	—	2	2	no	—
112	C₈×D₁₃	—	4	2	no	—
113	C₈×D₁₃	—	4	1	no	—
114	#2990	—	13	2	no	—
115	C₈×#41	—	13	1	no	—
116	#1574	—	13	2	no	—
117	#1573	—	13	4	no	—
118	#1568	—	13	2	no	—
119	#1567	—	13	1	no	—
120	#1564	—	13	1	no	—
121	C₂×#335	—	13	2	no	—
122	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
124	Q₈×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
134	D₅₂	—	2	4	no	—
135	D₅₂	#108	1	4	yes	—
136	C₄×D₁₃	—	2	4	no	—
137	C₄×D₁₃	—	2	2	no	—
138	C₄×D₁₃	—	2	1	no	—
139	C₄×D₁₃	—	2	2	no	—
140	C₄×D₁₃	—	2	1	no	—
141	C₄×D₁₃	—	2	2	no	—
142	C₄×D₁₃	—	2	2	no	—
143	C₄×D₁₃	—	2	1	no	—
144	C₄×D₁₃	—	2	2	no	—
145	C₄×D₁₃	—	2	1	no	—
146	#1057	—	4	2	no	—
147	#1057	—	4	4	no	—
148	#1057	—	4	2	no	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	2	2	no	—
151	C₈×C₁₃	—	2	1	no	—
152	#1055	—	2	1	no	—
153	#1055	—	2	2	no	—
154	#1055	#131	1	2	yes	—
155	C₄×#41	—	13	1	no	—
156	#446	—	26	2	no	—
157	C₈×Q₈	—	13	1	no	—
158	C₈×Q₈	#261	1	1	yes	characteristic
159	#435	—	26	4	no	—
160	C₈×D₄	—	13	2	no	—
161	C₈×D₄	—	13	1	no	—
162	C₈×D₄	—	13	2	no	—
163	C₈×D₄	—	13	1	no	—
164	C₈×D₄	—	26	2	no	—
165	#432	—	13	4	no	—
166	#431	—	13	2	no	—
167	#431	—	13	1	no	—
168	#431	—	13	2	no	—
169	#423	—	13	2	no	—
170	C₂×C₄⋊C₈	—	13	4	no	—
171	C₂×C₄×C₈	—	13	2	no	—
172	C₂×C₄×C₈	—	13	1	no	—
173	#335	—	13	1	no	—
174	#335	—	13	2	no	—
175	#335	—	13	4	no	—
176	#335	—	13	1	no	—
177	#335	—	13	2	no	—
178	#334	—	13	2	no	—
179	#334	—	13	4	no	—
180	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
181	C₂×D₁₃	—	2	2	no	—
182	C₂×D₁₃	—	2	2	no	—
183	C₂×D₁₃	—	2	1	no	—
184	C₂×D₁₃	—	2	1	no	—
185	C₂×D₁₃	—	2	1	no	—
186	C₂×D₁₃	C₂×#108	1	4	yes	—
187	C₄×C₁₃	—	2	2	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	2	no	—
190	C₄×C₁₃	C₂×#108	1	2	yes	—
191	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
192	C₄×C₁₃	#568	1	1	yes	characteristic
193	Dic₁₃	—	2	2	no	—
194	Dic₁₃	—	2	1	no	—
195	Dic₁₃	—	2	1	no	—
196	Dic₁₃	#576	1	1	yes	characteristic
197	Dic₁₃	#578	1	1	yes	characteristic
198	C₂×#41	—	13	1	no	—
199	C₂×M₁₆	—	26	2	no	—
200	C₂²×C₈	—	13	2	no	—
201	C₂²×C₈	—	13	1	no	—
202	C₂²×C₈	—	13	2	no	—
203	C₂²×C₈	—	13	1	no	—
204	C₂²×C₈	—	26	4	no	—
205	C₄×Q₈	—	13	1	no	—
206	C₄×Q₈	C₂×#261	1	1	yes	characteristic
207	C₄×D₄	—	13	1	no	—
208	C₄×D₄	—	13	2	no	—
209	C₄×D₄	—	13	1	no	—
210	C₄×D₄	—	13	2	no	—
211	#117	—	13	1	no	—
212	#117	—	13	2	no	—
213	C₂×C₄²	—	13	2	no	—
214	C₂×C₄²	—	13	1	no	—
215	C₄⋊C₈	—	13	2	no	—
216	C₄⋊C₈	—	13	2	no	—
217	C₄⋊C₈	—	13	4	no	—
218	C₄⋊C₈	—	13	2	no	—
219	C₄⋊C₈	—	13	1	no	—
220	C₄⋊C₈	—	26	2	no	—
221	C₄⋊C₈	—	26	4	no	—
222	C₄⋊C₈	C₂×#261	1	2	yes	—
223	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
224	C₂²⋊C₈	—	13	2	no	—
225	C₂²⋊C₈	—	13	2	no	—
226	C₂²⋊C₈	—	13	2	no	—
227	C₂²⋊C₈	—	26	2	no	—
228	C₂²⋊C₈	—	26	4	no	—
229	#97	—	26	4	no	—
230	C₄×C₈	—	13	1	no	—
231	C₄×C₈	—	13	2	no	—
232	C₄×C₈	—	13	1	no	—
233	C₄×C₈	—	13	2	no	—
234	C₄×C₈	—	13	1	no	—
235	C₄×C₈	—	26	2	no	—
236	C₄×C₈	C₂×#261	1	2	yes	—
237	C₄×C₈	C₂×#261	1	1	yes	characteristic
238	C₂×C₁₃	#2906	1	1	yes	characteristic
239	C₂×C₁₃	#2913	1	1	yes	characteristic
240	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
241	D₁₃	—	2	4	no	—
242	D₁₃	—	4	1	no	—
243	#41	—	26	4	no	—
244	C₂×Q₈	C₄⋊#261	1	1	yes	characteristic
245	C₂×D₄	—	13	1	no	—
246	C₂×D₄	—	13	2	no	—
247	C₂²×C₄	—	13	2	no	—
248	C₂²×C₄	—	13	1	no	—
249	C₂²×C₄	—	13	2	no	—
250	C₂²×C₄	—	13	1	no	—
251	C₂²×C₄	—	13	2	no	—
252	C₂²×C₄	—	13	1	no	—
253	M₁₆	—	104	2	no	—
254	C₂×C₈	—	13	1	no	—
255	C₂×C₈	—	13	1	no	—
256	C₂×C₈	—	13	2	no	—
257	C₂×C₈	—	26	4	no	—
258	C₂×C₈	—	26	2	no	—
259	C₂×C₈	—	26	2	no	—
260	C₂×C₈	—	26	2	no	—
261	C₂×C₈	—	26	4	no	—
262	C₂×C₈	—	52	2	no	—
263	C₂×C₈	—	52	1	no	—
264	C₂×C₈	—	104	4	no	—
265	C₂×C₈	C₂²×#261	1	1	yes	characteristic
266	C₂×C₈	C₂²×#261	1	1	yes	characteristic
267	C₂×C₈	C₂²×#261	1	2	yes	—
268	C₄⋊C₄	—	13	2	no	—
269	C₄⋊C₄	—	13	1	no	—
270	C₄⋊C₄	C₄⋊#261	1	2	yes	—
271	C₄⋊C₄	C₄⋊#261	1	1	yes	characteristic
272	C₂²⋊C₄	—	13	2	no	—
273	C₂²⋊C₄	—	13	2	no	—
274	C₂²⋊C₄	—	13	2	no	—
275	C₄²	—	13	1	no	—
276	C₄²	—	13	2	no	—
277	C₄²	—	13	2	no	—
278	C₄²	—	13	1	no	—
279	C₄²	—	13	1	no	—
280	C₄²	C₂²×#261	1	2	yes	—
281	C₄²	C₂²×#261	1	1	yes	characteristic
282	C₁₃	#17836	1	1	yes	characteristic, complemented, Sylow
283	C₂³	—	13	2	no	—
284	C₂³	—	13	1	no	—
285	Q₈	—	2	2	no	—
286	D₄	—	13	4	no	—
287	D₄	—	26	4	no	—
288	C₂×C₄	—	13	1	no	—
289	C₂×C₄	—	13	1	no	—
290	C₂×C₄	—	13	1	no	—
291	C₂×C₄	—	13	2	no	—
292	C₂×C₄	—	26	2	no	—
293	C₂×C₄	—	26	1	no	—
294	C₂×C₄	—	26	2	no	—
295	C₂×C₄	—	26	1	no	—
296	C₂×C₄	—	26	4	no	—
297	C₂×C₄	—	26	2	no	—
298	C₂×C₄	—	26	2	no	—
299	C₂×C₄	—	26	1	no	—
300	C₂×C₄	—	26	2	no	—
301	C₂×C₄	—	26	1	no	—
302	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
303	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
305	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
306	C₂×C₄	C₂³×#261	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	2	no	—
310	C₈	—	13	2	no	—
311	C₈	—	26	1	no	—
312	C₈	—	26	2	no	—
313	C₈	—	52	2	no	—
314	C₈	—	52	4	no	—
315	C₈	—	52	2	no	—
316	C₂²	—	13	4	no	—
317	C₂²	—	26	2	no	—
318	C₂²	—	26	2	no	—
319	C₂²	—	26	1	no	—
320	C₂²	—	26	1	no	—
321	C₂²	—	26	1	no	—
322	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	2	no	—
326	C₄	—	13	1	no	—
327	C₄	—	13	1	no	—
328	C₄	—	26	1	no	—
329	C₄	—	26	2	no	—
330	C₄	—	26	1	no	—
331	C₄	D₁₃⋊#108	1	2	yes	—
332	C₄	#11719625	1	1	yes	characteristic
333	C₄	D₁₃⋊#131	1	1	yes	characteristic
334	C₂	—	26	4	no	—
335	C₂	—	52	1	no	—
336	C₂	#13131484	1	1	yes	characteristic
337	C₂	#13131491	1	1	yes	characteristic
338	C₂	C₂×D₁₃⋊#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₈×Q₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂×#6131	Q₈
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₈×Q₈	#261
hypocenter	C₁₃	#17836
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #13130414	C₂	semidirect
3. #13130410	C₂	semidirect
4. #13130399	C₂	semidirect
5. #13130395	C₂	semidirect
6. #13130384	C₂	semidirect
7. D₁₃⋊#335	C₂	semidirect
282. C₁₃	#17836	semidirect

Maximal subgroups

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

subgroup	index
#13131766	2
#13130414	2
#13130410	2
#13130399	2
#13130395	2
#13130384	2
D₁₃⋊#335	2
C₈×#6131	2
#17836	13