Subgroups — #16795205

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131642	C₂	1	2	yes	maximal, complemented
2	#13130434	C₂	1	2	yes	maximal, complemented
3	#13130429	C₂	1	2	yes	maximal, complemented
4	#13130426	C₂	1	4	yes	maximal, complemented
5	#13130384	C₂	1	1	yes	maximal, characteristic, complemented
6	#13130130	C₂	1	2	yes	maximal, complemented
7	#13130030	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#88173	C₂	1	1	yes	maximal, characteristic
9	C₂×#88173	C₄	1	1	yes	characteristic
10	C₄×#6131	C₂²	1	2	yes	—
11	Q₈×#1057	C₂²	1	4	yes	—
12	#11719073	—	2	4	no	—
13	#11719073	C₂²	1	4	yes	—
14	#11719051	—	2	2	no	—
15	#11719051	C₂²	1	2	yes	—
16	C₄×D₁₃⋊C₈	—	2	2	no	—
17	C₄×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
20	Q₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
21	C₄×#6106	C₄	1	2	yes	—
22	#11718541	C₄	1	1	yes	characteristic
23	#11718533	C₂²	1	1	yes	characteristic
24	#11718465	C₄	1	2	yes	—
25	#11718449	C₂²	1	2	yes	—
26	C₄×D₁₀₄	C₄	1	1	yes	characteristic
27	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
28	Q₈⋊#1057	C₂²	1	4	yes	—
29	#11718256	C₂²	1	2	yes	—
30	#11718253	C₂²	1	2	yes	—
31	#11718251	C₂²	1	2	yes	—
32	#11718251	C₂²	1	2	yes	—
33	#11718243	C₂²	1	4	yes	—
34	C₂×#6131	C₂×C₄	1	2	yes	—
35	C₂×D₁₃⋊C₈	—	2	1	no	—
36	C₂×D₁₃⋊C₈	—	2	4	no	—
37	C₂×D₁₃⋊C₈	D₄	1	2	yes	—
38	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
40	Q₈×Dic₁₃	C₂³	1	2	yes	—
41	C₂×#6106	C₂×C₄	1	2	yes	—
42	#88173	C₈	1	8	yes	complemented
43	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
46	#88147	—	2	2	no	—
47	#88147	C₂³	1	2	yes	—
48	#88146	—	2	2	no	—
49	#88146	C₂×C₄	1	2	yes	—
50	C₄×D₅₂	—	2	2	no	—
51	C₄×D₅₂	C₂×C₄	1	2	yes	—
52	C₄²×D₁₃	—	2	2	no	—
53	C₄²×D₁₃	C₂³	1	1	yes	characteristic
54	#88111	—	2	2	no	—
55	#88111	—	2	4	no	—
56	C₄⋊#1057	—	2	4	no	—
57	C₄⋊#1057	C₂³	1	4	yes	—
58	C₄×#1057	—	2	4	no	—
59	C₄×#1057	—	2	2	no	—
60	C₄×#1057	—	2	1	no	—
61	C₄×#1057	D₄	1	2	yes	—
62	C₄×#1057	C₂³	1	2	yes	—
63	C₄×#1057	C₂³	1	2	yes	—
64	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
65	C₁₃×#103	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
67	#88075	C₂×C₄	1	2	yes	—
68	#88061	C₂×C₄	1	2	yes	—
69	#88058	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
71	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
72	#88049	C₂×C₄	1	2	yes	—
73	#88048	C₂×C₄	1	2	yes	—
74	C₄×#1055	C₂×C₄	1	1	yes	characteristic
75	#17849	—	13	1	no	maximal, Sylow
76	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
77	#6131	—	2	4	no	—
78	#6131	C₂×C₈	1	8	yes	complemented
79	C₂×D₅₂	—	2	2	no	—
80	C₂×D₅₂	C₂²×C₄	1	2	yes	—
81	C₂×C₄×D₁₃	—	2	2	no	—
82	C₂×C₄×D₁₃	—	2	2	no	—
83	C₂×C₄×D₁₃	—	2	2	no	—
84	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
87	C₂×#1057	—	2	4	no	—
88	C₂×#1057	—	2	2	no	—
89	C₂×#1057	C₂×D₄	1	4	yes	—
90	D₁₃⋊C₈	—	4	2	no	—
91	D₁₃⋊C₈	—	4	8	no	—
92	D₁₃⋊C₈	—	4	2	no	—
93	C₁₃×Q₁₆	C₂×C₈	1	4	yes	complemented
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
98	C₄²×C₁₃	—	2	2	no	—
99	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	#6106	C₂×C₈	1	8	yes	complemented
101	#6103	—	2	2	no	—
102	#6103	—	2	2	no	—
103	#6103	—	2	2	no	—
104	C₄⋊Dic₁₃	—	2	2	no	—
105	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
106	C₄×Dic₁₃	—	2	2	no	—
107	C₄×Dic₁₃	—	2	2	no	—
108	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄×Dic₁₃	#41	1	1	yes	characteristic
110	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
111	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
112	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
113	D₁₀₄	C₂×C₈	1	4	yes	complemented
114	C₈×D₁₃	—	2	2	no	—
115	C₈×D₁₃	C₂×C₈	1	4	yes	complemented
116	C₈×#41	—	13	2	no	—
117	C₄×#135	—	13	1	no	—
118	#1580	—	13	2	no	—
119	#1579	—	13	2	no	—
120	SD₁₆⋊C₈	—	13	4	no	—
121	#1564	—	13	1	no	—
122	#1476	—	13	2	no	—
123	C₂×#322	—	13	1	no	—
124	C₂²×D₁₃	—	2	2	no	—
125	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
126	Q₈×C₁₃	—	2	2	no	—
127	Q₈×C₁₃	C₂²×C₈	1	4	yes	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×Dic₁₃	—	2	2	no	—
134	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
135	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
137	D₅₂	—	2	8	no	—
138	D₅₂	—	2	2	no	—
139	D₅₂	C₂²×C₈	1	4	yes	—
140	C₄×D₁₃	—	2	2	no	—
141	C₄×D₁₃	—	2	2	no	—
142	C₄×D₁₃	—	2	8	no	—
143	C₄×D₁₃	—	4	4	no	—
144	C₄×D₁₃	—	4	4	no	—
145	C₄×D₁₃	C₂²×C₈	1	2	yes	—
146	C₄×D₁₃	#131	1	2	yes	—
147	#1057	—	2	4	no	—
148	#1057	—	4	2	no	—
149	#1057	—	4	4	no	—
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	C₂²×C₈	1	2	yes	—
152	#1055	—	2	1	no	—
153	#1055	C₂²×C₈	1	2	yes	—
154	C₂×#135	—	13	1	no	—
155	C₄×#41	—	13	2	no	—
156	C₈×Q₈	—	13	4	no	—
157	C₄×Q₁₆	—	13	1	no	—
158	C₄×Q₁₆	#261	1	1	yes	characteristic
159	C₄×SD₁₆	—	13	2	no	—
160	C₄×SD₁₆	—	13	2	no	—
161	C₄×D₈	—	13	1	no	—
162	C₄×D₈	—	13	1	no	—
163	C₈×D₄	—	13	4	no	—
164	C₈×D₄	—	26	4	no	—
165	#431	—	13	2	no	—
166	#431	—	26	2	no	—
167	#427	—	13	1	no	—
168	#416	—	13	2	no	—
169	C₂×C₄×C₈	—	13	1	no	—
170	C₂×C₄×C₈	—	13	1	no	—
171	C₂×C₄×C₈	—	13	1	no	—
172	C₂×C₄×C₈	—	26	2	no	—
173	#335	—	13	2	no	—
174	#334	—	13	2	no	—
175	Q₈⋊C₈	—	13	4	no	—
176	D₄⋊C₈	—	13	4	no	—
177	#322	—	13	2	no	—
178	#322	—	13	2	no	—
179	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
180	C₂×D₁₃	—	2	1	no	—
181	C₂×D₁₃	—	2	1	no	—
182	C₂×D₁₃	—	2	4	no	—
183	C₂×D₁₃	—	4	2	no	—
184	C₂×D₁₃	—	4	2	no	—
185	C₂×D₁₃	C₈×D₄	1	2	yes	—
186	C₄×C₁₃	—	2	4	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	4	2	no	—
189	C₄×C₁₃	#444	1	2	yes	—
190	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
191	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
192	Dic₁₃	—	2	1	no	—
193	Dic₁₃	—	4	2	no	—
194	Dic₁₃	C₈×D₄	1	2	yes	—
195	Dic₁₃	#575	1	2	yes	—
196	C₂×#41	—	13	2	no	—
197	#135	—	13	8	no	—
198	C₂×Q₁₆	C₂×#261	1	1	yes	characteristic
199	C₂×SD₁₆	—	13	2	no	—
200	C₂×D₈	—	13	1	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	C₂²×C₈	—	26	1	no	—
206	C₄×Q₈	—	13	2	no	—
207	C₄×Q₈	C₂×#261	1	2	yes	—
208	C₄×D₄	—	13	2	no	—
209	C₄×D₄	—	13	2	no	—
210	C₄×D₄	—	26	2	no	—
211	C₄×D₄	—	26	2	no	—
212	#117	—	13	2	no	—
213	#117	—	26	2	no	—
214	C₂×C₄²	—	13	1	no	—
215	C₂×C₄²	—	26	2	no	—
216	#107	—	13	1	no	—
217	#107	C₂×#261	1	1	yes	characteristic
218	#106	—	13	2	no	—
219	C₄⋊C₈	—	13	4	no	—
220	C₄⋊C₈	—	26	4	no	—
221	#103	—	13	2	no	—
222	#103	C₂×#261	1	2	yes	—
223	#102	—	13	2	no	—
224	#102	—	13	2	no	—
225	C₂²⋊C₈	—	26	4	no	—
226	C₂²⋊C₈	—	26	2	no	—
227	C₄×C₈	—	13	2	no	—
228	C₄×C₈	—	13	2	no	—
229	C₄×C₈	—	13	1	no	—
230	C₄×C₈	—	13	1	no	—
231	C₄×C₈	—	13	2	no	—
232	C₄×C₈	—	13	1	no	—
233	C₄×C₈	—	26	1	no	—
234	C₄×C₈	—	26	2	no	—
235	C₄×C₈	—	26	4	no	—
236	C₄×C₈	C₂×#261	1	1	yes	characteristic
237	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
238	C₂×C₁₃	#2943	1	1	yes	characteristic
239	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
240	D₁₃	—	2	2	no	—
241	D₁₃	—	4	4	no	—
242	#41	—	13	8	no	—
243	#41	—	26	4	no	—
244	C₂×Q₈	C₂²×#261	1	2	yes	—
245	C₂×D₄	—	13	2	no	—
246	C₂×D₄	—	26	2	no	—
247	C₂²×C₄	—	13	1	no	—
248	C₂²×C₄	—	13	1	no	—
249	C₂²×C₄	—	13	1	no	—
250	C₂²×C₄	—	26	2	no	—
251	C₂²×C₄	—	26	2	no	—
252	C₂²×C₄	—	26	2	no	—
253	Q₁₆	D₁₃⋊C₈	1	4	yes	complemented
254	SD₁₆	—	13	8	no	—
255	D₈	—	13	4	no	—
256	C₂×C₈	—	13	1	no	—
257	C₂×C₈	—	13	4	no	—
258	C₂×C₈	—	13	4	no	—
259	C₂×C₈	—	13	1	no	—
260	C₂×C₈	—	26	2	no	—
261	C₂×C₈	—	26	2	no	—
262	C₂×C₈	—	26	4	no	—
263	C₂×C₈	—	52	2	no	—
264	C₂×C₈	—	52	8	no	—
265	C₂×C₈	—	52	2	no	—
266	C₂×C₈	C₂²×#261	1	1	yes	characteristic
267	C₂×C₈	C₂²×#261	1	1	yes	characteristic
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	13	2	no	—
270	C₄⋊C₄	—	26	2	no	—
271	C₄⋊C₄	C₂²×#261	1	2	yes	—
272	C₂²⋊C₄	—	26	2	no	—
273	C₂²⋊C₄	—	26	2	no	—
274	C₂²⋊C₄	—	26	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	13	1	no	—
277	C₄²	—	13	1	no	—
278	C₄²	—	13	1	no	—
279	C₄²	—	26	2	no	—
280	C₄²	—	26	2	no	—
281	C₄²	C₂²×#261	1	1	yes	characteristic
282	C₁₃	#17849	1	1	yes	characteristic, complemented, Sylow
283	C₂³	—	13	1	no	—
284	C₂³	—	26	2	no	—
285	Q₈	—	2	2	no	—
286	Q₈	C₂×D₁₃⋊C₈	1	4	yes	—
287	D₄	—	13	4	no	—
288	D₄	—	26	8	no	—
289	D₄	—	26	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	13	1	no	—
293	C₂×C₄	—	13	1	no	—
294	C₂×C₄	—	13	2	no	—
295	C₂×C₄	—	13	2	no	—
296	C₂×C₄	—	13	1	no	—
297	C₂×C₄	—	26	2	no	—
298	C₂×C₄	—	26	8	no	—
299	C₂×C₄	—	26	2	no	—
300	C₂×C₄	—	26	2	no	—
301	C₂×C₄	—	52	4	no	—
302	C₂×C₄	—	52	4	no	—
303	C₂×C₄	D₄×#261	1	1	yes	characteristic
304	C₂×C₄	D₄×#261	1	1	yes	characteristic
305	C₂×C₄	C₂³×#261	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	13	2	no	—
308	C₈	—	26	4	no	—
309	C₈	—	26	1	no	—
310	C₈	—	52	4	no	—
311	C₈	—	52	2	no	—
312	C₈	C₂×D₁₃⋊C₈	1	2	yes	—
313	C₂²	—	13	2	no	—
314	C₂²	—	26	1	no	—
315	C₂²	—	26	1	no	—
316	C₂²	—	26	4	no	—
317	C₂²	—	52	2	no	—
318	C₂²	—	52	2	no	—
319	C₂²	C₂×D₄×#261	1	1	yes	characteristic
320	C₄	—	2	4	no	—
321	C₄	—	2	1	no	—
322	C₄	—	4	2	no	—
323	C₄	—	13	2	no	—
324	C₄	—	13	2	no	—
325	C₄	—	26	1	no	—
326	C₄	—	52	2	no	—
327	C₄	#11719112	1	2	yes	—
328	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
329	C₄	D₁₃⋊#131	1	1	yes	characteristic
330	C₂	—	26	2	no	—
331	C₂	—	52	4	no	—
332	C₂	D₄×D₁₃⋊C₈	1	1	yes	characteristic
333	C₂	#13131590	1	1	yes	characteristic
334	C₂	C₂×#11719112	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₄×C₁₃×Q₁₆	C₄
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×#88173	C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₄×Q₁₆	#261
hypocenter	C₁₃	#17849
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #13131642	C₂	semidirect
2. #13130434	C₂	semidirect
3. #13130429	C₂	semidirect
4. #13130426	C₂	semidirect
5. #13130384	C₂	semidirect
6. #13130130	C₂	semidirect
7. #13130030	C₂	semidirect
42. #88173	C₈	semidirect
78. #6131	C₂×C₈	semidirect
93. C₁₃×Q₁₆	C₂×C₈	semidirect
100. #6106	C₂×C₈	semidirect
113. D₁₀₄	C₂×C₈	semidirect
115. C₈×D₁₃	C₂×C₈	semidirect
253. Q₁₆	D₁₃⋊C₈	semidirect
282. C₁₃	#17849	semidirect

Maximal subgroups

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

subgroup	index
#13131642	2
#13130434	2
#13130429	2
#13130426	2
#13130384	2
#13130130	2
#13130030	2
C₄×#88173	2
#17849	13