Subgroups — #16017570

Order 3328.

598 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	C₂×C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic
2	#13116990	C₂	1	4	yes	maximal, complemented
3	#13116980	C₂	1	2	yes	maximal, complemented
4	#13116970	C₂	1	2	yes	maximal, complemented
5	#13116625	C₂	1	2	yes	maximal, complemented
6	#13116592	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116551	C₂	1	2	yes	maximal
8	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#533	—	2	1	no	—
10	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₅₂	—	2	1	no	—
12	C₁₃×#500	C₂²	1	1	yes	characteristic, complemented
13	C₁₃×#500	C₂²	1	1	yes	characteristic, complemented
14	C₁₃×#499	—	2	1	no	—
15	C₁₃×#499	C₂²	1	2	yes	complemented
16	C₁₃×#498	—	2	1	no	—
17	C₁₃×#427	—	2	1	no	—
18	C₁₃×#427	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×#88061	—	2	1	no	—
21	C₂×#88058	C₂²	1	1	yes	characteristic
22	C₂×#88052	C₂²	1	2	yes	—
23	C₄×D₁₀₄	—	2	1	no	—
24	C₄×D₁₀₄	—	2	1	no	—
25	C₄×D₁₀₄	C₂²	1	2	yes	—
26	C₄×D₁₀₄	C₂²	1	2	yes	—
27	#11718083	C₂²	1	4	yes	—
28	#11718083	C₂²	1	4	yes	—
29	#11718082	C₂²	1	4	yes	complemented
30	#11718081	C₂²	1	4	yes	complemented
31	C₄⋊#6090	C₂²	1	2	yes	complemented
32	C₄⋊#6090	C₂²	1	2	yes	complemented
33	C₂²×D₅₂	—	2	1	no	—
34	C₂²×C₄×D₁₃	—	4	1	no	—
35	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₁₃×C₄⋊Q₈	—	2	1	no	—
38	C₁₃×C₄⋊Q₈	—	2	1	no	—
39	C₁₃×#125	—	2	2	no	—
40	C₁₃×C₂²⋊Q₈	—	4	1	no	—
41	C₁₃×C₂²⋊Q₈	—	4	1	no	—
42	C₄×Q₈×C₁₃	—	4	1	no	—
43	C₄×Q₈×C₁₃	—	4	1	no	—
44	C₁₃×#117	—	2	2	no	—
45	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
46	C₂×#6103	—	4	1	no	—
47	C₂×C₄⋊Dic₁₃	—	2	1	no	—
48	C₂×D₁₀₄	—	2	1	no	—
49	C₂×D₁₀₄	—	2	2	no	—
50	C₂×D₁₀₄	Q₈	1	4	yes	complemented
51	C₂×D₁₀₄	C₂³	1	2	yes	—
52	C₄×D₅₂	—	2	2	no	—
53	C₄×D₅₂	—	2	2	no	—
54	C₄×D₅₂	—	4	1	no	—
55	C₄×D₅₂	—	4	1	no	—
56	C₁₃×#107	—	2	1	no	—
57	C₁₃×#107	—	2	1	no	—
58	C₁₃×#107	C₂³	1	2	yes	—
59	C₁₃×#107	C₂³	1	2	yes	—
60	C₁₃×#106	—	2	2	no	—
61	C₁₃×#106	—	2	2	no	—
62	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	#88061	—	2	2	no	—
67	#88061	—	2	2	no	—
68	#88058	—	2	1	no	—
69	#88058	C₂³	1	2	yes	—
70	#88052	—	2	4	no	—
71	C₂×#6090	C₂³	1	4	yes	—
72	#33627	—	13	1	no	maximal, Sylow
73	C₂³×D₁₃	—	4	1	no	—
74	C₂×Q₈×C₁₃	—	4	1	no	—
75	C₂×Q₈×C₁₃	—	4	1	no	—
76	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×Dic₁₃	—	4	1	no	—
80	C₂×D₅₂	—	2	4	no	—
81	C₂×D₅₂	—	2	2	no	—
82	C₂×D₅₂	—	4	1	no	—
83	C₂×D₅₂	—	4	2	no	—
84	C₂×C₄×D₁₃	—	4	2	no	—
85	C₂×C₄×D₁₃	—	4	2	no	—
86	C₂×C₄×D₁₃	—	8	4	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₁₃	#41	1	2	yes	—
93	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₄⋊C₄	—	4	2	no	—
98	C₁₃×C₄⋊C₄	—	4	1	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₂²⋊C₄	—	4	2	no	—
101	C₄²×C₁₃	—	4	2	no	—
102	C₄²×C₁₃	D₈	1	1	yes	characteristic
103	C₄²×C₁₃	D₈	1	1	yes	characteristic
104	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
106	#6103	—	4	2	no	—
107	#6103	—	4	2	no	—
108	C₄⋊Dic₁₃	—	2	2	no	—
109	C₄⋊Dic₁₃	—	4	1	no	—
110	D₁₀₄	—	2	4	no	—
111	D₁₀₄	—	2	2	no	—
112	D₁₀₄	C₂×Q₈	1	4	yes	—
113	#6090	—	2	4	no	—
114	#3160	D₁₃	1	1	yes	characteristic, complemented
115	C₂×C₄×D₈	—	13	1	no	—
116	#2228	—	13	4	no	—
117	#2226	—	13	2	no	—
118	#2224	—	13	2	no	—
119	#2159	—	13	2	no	—
120	#2150	—	13	1	no	—
121	#2140	—	13	2	no	—
122	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
123	C₂²×D₁₃	—	4	2	no	—
124	C₂²×D₁₃	—	4	4	no	—
125	C₂²×D₁₃	—	8	2	no	—
126	C₂²×D₁₃	—	8	2	no	—
127	Q₈×C₁₃	—	8	2	no	—
128	Q₈×C₁₃	—	8	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	4	1	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#137	1	2	yes	—
140	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×Dic₁₃	—	4	2	no	—
142	C₂×Dic₁₃	—	8	2	no	—
143	D₅₂	—	2	4	no	—
144	D₅₂	—	4	2	no	—
145	D₅₂	—	4	4	no	—
146	C₄×D₁₃	—	8	4	no	—
147	C₄×D₁₃	—	8	4	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
151	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
152	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
153	C₂²×D₈	—	13	1	no	—
154	#533	—	2	1	no	—
155	C₂×C₄×D₄	—	26	1	no	—
156	C₂×M₃₂	—	13	2	no	—
157	#500	C₂×D₁₃	1	1	yes	characteristic, complemented
158	#500	C₂×D₁₃	1	1	yes	characteristic, complemented
159	#499	—	2	1	no	—
160	#499	C₂×D₁₃	1	2	yes	complemented
161	#498	—	2	1	no	—
162	C₄×D₈	—	13	2	no	—
163	C₄×D₈	—	13	2	no	—
164	C₄×D₈	—	26	1	no	—
165	C₄×D₈	—	26	1	no	—
166	#427	—	2	1	no	—
167	#427	C₂×D₁₃	1	2	yes	—
168	C₂×#107	—	13	1	no	—
169	C₂×#102	—	26	1	no	—
170	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
171	#367	—	13	4	no	—
172	#366	—	13	4	no	—
173	C₄⋊C₁₆	—	13	2	no	—
174	C₄⋊C₁₆	—	13	2	no	—
175	#357	—	13	4	no	—
176	#357	—	13	4	no	—
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	#474	1	2	yes	—
180	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
181	C₂×D₁₃	—	4	4	no	—
182	C₂×D₁₃	—	8	8	no	—
183	C₂×D₁₃	—	8	2	no	—
184	C₂×D₁₃	—	8	2	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	8	2	no	—
189	C₄×C₁₃	—	8	2	no	—
190	C₄×C₁₃	#474	1	2	yes	—
191	C₄×C₁₃	#509	1	2	yes	—
192	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
193	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
194	Dic₁₃	—	8	2	no	—
195	C₂²×D₄	—	26	1	no	—
196	C₂³×C₄	—	52	1	no	—
197	C₂×D₈	—	13	4	no	—
198	C₂×D₈	—	13	2	no	—
199	C₂×D₈	—	26	2	no	—
200	C₂×D₈	—	26	1	no	—
201	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
203	C₄⋊Q₈	—	2	1	no	—
204	C₄⋊Q₈	—	2	1	no	—
205	#125	—	2	2	no	—
206	C₂²⋊Q₈	—	4	1	no	—
207	C₂²⋊Q₈	—	4	1	no	—
208	C₄×Q₈	—	4	1	no	—
209	C₄×Q₈	—	4	1	no	—
210	C₄×D₄	—	26	2	no	—
211	C₄×D₄	—	26	2	no	—
212	C₄×D₄	—	52	1	no	—
213	C₄×D₄	—	52	1	no	—
214	#117	—	2	2	no	—
215	C₂×C₄⋊C₄	—	26	1	no	—
216	C₂×C₂²⋊C₄	—	52	1	no	—
217	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
218	M₃₂	—	26	4	no	—
219	C₂×C₁₆	—	13	4	no	—
220	#107	—	2	1	no	—
221	#107	—	2	1	no	—
222	#107	—	13	2	no	—
223	#107	—	26	1	no	—
224	#107	C₂²×D₁₃	1	2	yes	—
225	#107	C₂²×D₁₃	1	2	yes	—
226	#106	—	2	2	no	—
227	#106	—	2	2	no	—
228	#102	—	26	2	no	—
229	#102	—	26	2	no	—
230	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
231	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
233	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
234	C₂×C₁₃	—	2	2	no	—
235	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
236	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
237	C₂×C₁₃	#3414	1	1	yes	characteristic
238	D₁₃	—	8	4	no	—
239	C₂⁴	—	52	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×D₄	—	26	4	no	—
243	C₂×D₄	—	26	2	no	—
244	C₂×D₄	—	52	1	no	—
245	C₂×D₄	—	52	2	no	—
246	C₂²×C₄	—	52	1	no	—
247	C₂²×C₄	—	52	2	no	—
248	C₂²×C₄	—	52	2	no	—
249	C₂²×C₄	—	104	4	no	—
250	C₂²×C₄	#6104	1	1	yes	characteristic
251	C₂²×C₄	#6104	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
253	D₈	—	13	4	no	—
254	D₈	—	26	4	no	—
255	D₈	—	26	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	Q₈×D₁₃	1	2	yes	—
258	C₂×C₈	#6131	1	2	yes	—
259	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	26	2	no	—
270	C₄⋊C₄	—	52	1	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	52	2	no	—
273	C₂²⋊C₄	—	52	2	no	—
274	C₄²	—	4	2	no	—
275	C₄²	#6104	1	1	yes	characteristic
276	C₄²	#6104	1	1	yes	characteristic
277	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
278	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
279	C₁₆	—	26	4	no	—
280	C₁₃	#33627	1	1	yes	characteristic, complemented, Sylow
281	C₂³	—	52	2	no	—
282	C₂³	—	52	4	no	—
283	C₂³	—	104	2	no	—
284	C₂³	—	104	2	no	—
285	C₂³	C₂×#6104	1	1	yes	characteristic
286	Q₈	—	8	2	no	—
287	Q₈	—	8	2	no	—
288	D₄	—	26	4	no	—
289	D₄	—	52	2	no	—
290	D₄	—	52	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	52	2	no	—
296	C₂×C₄	—	104	4	no	—
297	C₂×C₄	—	104	2	no	—
298	C₂×C₄	—	104	4	no	—
299	C₂×C₄	C₂×#6104	1	1	yes	characteristic
300	C₂×C₄	C₂×#6104	1	1	yes	characteristic
301	C₂×C₄	C₂×#6104	1	1	yes	characteristic
302	C₂×C₄	C₂×#6104	1	1	yes	characteristic
303	C₂×C₄	C₂×#6104	1	1	yes	characteristic
304	C₂×C₄	#88196	1	2	yes	—
305	C₂×C₄	#88200	1	2	yes	—
306	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	#88200	1	2	yes	—
310	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
311	C₈	C₂×#6131	1	1	yes	characteristic
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	52	4	no	—
315	C₂²	—	104	8	no	—
316	C₂²	—	104	2	no	—
317	C₂²	—	104	2	no	—
318	C₂²	#11718748	1	2	yes	—
319	C₂²	C₂²×#6104	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	104	2	no	—
326	C₄	#11718748	1	2	yes	—
327	C₄	#11718854	1	2	yes	—
328	C₄	C₂×#88196	1	1	yes	characteristic
329	C₄	C₂×#88200	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	104	4	no	—
332	C₂	C₂×#11718748	1	1	yes	characteristic
333	C₂	C₂×#11718854	1	1	yes	characteristic
334	C₂	#13127970	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3160	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₀₄	C₂²
hypercenter	#3160	D₁₃
hypocenter	C₁₃	#33627
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #13116990	C₂	semidirect
3. #13116980	C₂	semidirect
4. #13116970	C₂	semidirect
5. #13116625	C₂	semidirect
6. #13116592	C₂	semidirect
8. C₁₃×#3160	C₂	semidirect
12. C₁₃×#500	C₂²	semidirect
13. C₁₃×#500	C₂²	semidirect
15. C₁₃×#499	C₂²	semidirect
29. #11718082	C₂²	semidirect
30. #11718081	C₂²	semidirect
31. C₄⋊#6090	C₂²	semidirect
32. C₄⋊#6090	C₂²	semidirect
50. C₂×D₁₀₄	Q₈	semidirect
114. #3160	D₁₃	semidirect
157. #500	C₂×D₁₃	semidirect
158. #500	C₂×D₁₃	semidirect
160. #499	C₂×D₁₃	semidirect
280. C₁₃	#33627	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₁₀₄	2
#13116990	2
#13116980	2
#13116970	2
#13116625	2
#13116592	2
#13116551	2
C₁₃×#3160	2
#33627	13