Subgroups — #15881219

Order 3328.

1050 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718372	C₂	1	2	yes	maximal
2	C₂×C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic
3	#13117747	C₂	1	2	yes	maximal, complemented
4	#13115574	C₂	1	2	yes	maximal, complemented
5	#13115086	C₂	1	4	yes	maximal, complemented
6	#13115000	C₂	1	1	yes	maximal, characteristic, complemented
7	#13114847	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₅₂	C₂²	1	4	yes	—
11	C₂×C₄×D₅₂	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#97	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×#88061	C₂²	1	4	yes	—
17	C₂×#88061	C₂²	1	2	yes	—
18	C₂×#88058	C₂²	1	2	yes	—
19	C₂×#88058	C₂²	1	1	yes	characteristic
20	C₂×#88057	C₂²	1	2	yes	—
21	#11718650	—	2	4	no	—
22	#11718597	C₂²	1	2	yes	—
23	#11718589	C₂²	1	4	yes	—
24	#11718372	—	2	8	no	—
25	C₄×D₁₀₄	C₄	1	8	yes	complemented
26	#11718328	—	2	2	no	—
27	D₁₃×#95	—	2	2	no	—
28	C₁₃×#336	C₂²	1	2	yes	—
29	#11718143	C₂²	1	4	yes	—
30	C₂²×D₅₂	C₂³	1	2	yes	—
31	C₂²×C₄×D₁₃	—	2	2	no	—
32	C₂²×C₄×D₁₃	—	2	4	no	—
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×#6103	—	2	2	no	—
40	C₂×#6103	—	2	4	no	—
41	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
42	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
43	C₂×#6101	—	2	4	no	—
44	C₂×D₁₀₄	C₂×C₄	1	4	yes	—
45	C₂×D₁₀₄	D₄	1	4	yes	—
46	C₂×D₁₀₄	Q₈	1	4	yes	—
47	C₄×D₅₂	—	2	4	no	—
48	C₄×D₅₂	—	2	8	no	—
49	C₄×D₅₂	—	4	4	no	—
50	C₄×D₅₂	C₂×C₄	1	8	yes	complemented
51	C₁₃×C₄⋊C₈	—	2	2	no	—
52	C₁₃×#97	—	2	4	no	—
53	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	complemented
54	C₁₃×#95	D₄	1	2	yes	—
55	#88071	—	2	2	no	—
56	#88071	—	2	4	no	—
57	#88071	—	2	2	no	—
58	#88061	—	2	8	no	—
59	#88061	C₂×C₄	1	8	yes	complemented
60	#88058	—	4	2	no	—
61	#88058	C₂×C₄	1	4	yes	complemented
62	#88057	—	2	4	no	—
63	#22672	—	13	1	no	maximal, Sylow
64	C₂³×D₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₁₃	—	2	2	no	—
70	C₂²×Dic₁₃	—	2	4	no	—
71	C₂×D₅₂	—	2	2	no	—
72	C₂×D₅₂	—	2	2	no	—
73	C₂×D₅₂	—	2	2	no	—
74	C₂×D₅₂	C₂²×C₄	1	4	yes	—
75	C₂×D₅₂	C₂×D₄	1	4	yes	—
76	C₂×D₅₂	C₂×Q₈	1	4	yes	—
77	C₂×C₄×D₁₃	—	2	8	no	—
78	C₂×C₄×D₁₃	—	4	8	no	—
79	C₂×C₄×D₁₃	—	4	8	no	—
80	C₂×C₄×D₁₃	—	8	4	no	—
81	C₂×C₄×D₁₃	—	8	4	no	—
82	C₂×C₈×C₁₃	—	2	1	no	—
83	C₂×C₈×C₁₃	—	2	1	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
91	C₁₃×C₄⋊C₄	—	4	2	no	—
92	C₄²×C₁₃	—	2	2	no	—
93	C₄²×C₁₃	—	2	2	no	—
94	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₃	#41	1	2	yes	—
96	#6103	—	2	8	no	—
97	#6103	—	4	8	no	—
98	C₄⋊Dic₁₃	—	2	2	no	—
99	C₄⋊Dic₁₃	—	2	4	no	—
100	C₄⋊Dic₁₃	—	2	4	no	—
101	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
102	#6101	—	8	4	no	—
103	D₁₀₄	C₄⋊C₄	1	16	yes	complemented
104	C₂×#442	—	13	2	no	—
105	C₂×C₄×D₈	—	13	1	no	—
106	#2349	—	13	2	no	—
107	#1943	—	13	2	no	—
108	#1862	—	13	4	no	—
109	#1846	—	13	1	no	—
110	#1804	—	13	2	no	—
111	#1770	D₁₃	1	1	yes	characteristic, complemented
112	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₂²×D₁₃	—	2	4	no	—
114	C₂²×D₁₃	—	2	4	no	—
115	C₂²×D₁₃	—	2	4	no	—
116	C₂²×D₁₃	—	4	4	no	—
117	C₂²×D₁₃	—	4	4	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	4	no	—
121	C₂×C₄×C₁₃	—	2	4	no	—
122	C₂×C₄×C₁₃	—	2	4	no	—
123	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
131	C₂×Dic₁₃	—	2	4	no	—
132	C₂×Dic₁₃	—	4	4	no	—
133	C₂×Dic₁₃	—	4	4	no	—
134	C₂×Dic₁₃	—	4	4	no	—
135	C₂×Dic₁₃	—	4	4	no	—
136	D₅₂	—	2	4	no	—
137	D₅₂	—	2	4	no	—
138	D₅₂	—	2	4	no	—
139	D₅₂	C₂×C₄⋊C₄	1	8	yes	—
140	C₄×D₁₃	—	8	8	no	—
141	C₄×D₁₃	—	8	16	no	—
142	C₈×C₁₃	—	2	2	no	—
143	C₈×C₁₃	—	4	2	no	—
144	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
145	C₂²×D₈	—	13	1	no	—
146	C₂×C₄×D₄	—	13	2	no	—
147	C₂×C₄×D₄	—	13	4	no	—
148	#442	—	26	8	no	—
149	C₄×D₈	—	13	8	no	—
150	C₂×#107	—	13	1	no	—
151	C₂×#107	—	13	2	no	—
152	C₂×#106	—	13	2	no	—
153	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
154	C₂×#102	—	13	2	no	—
155	C₂×#102	—	13	4	no	—
156	C₂×#97	C₂×D₁₃	1	2	yes	—
157	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
158	#389	—	13	4	no	—
159	#385	—	26	4	no	—
160	#382	—	13	2	no	—
161	#380	—	26	2	no	—
162	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
163	C₂×#95	—	26	2	no	—
164	#340	—	13	4	no	—
165	#336	C₂×D₁₃	1	2	yes	—
166	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₁₃	D₄²	1	1	yes	characteristic
168	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
169	C₂²×C₁₃	#575	1	2	yes	—
170	C₂²×C₁₃	#576	1	2	yes	—
171	C₂×D₁₃	—	2	8	no	—
172	C₂×D₁₃	—	4	8	no	—
173	C₂×D₁₃	—	4	4	no	—
174	C₂×D₁₃	—	4	4	no	—
175	C₂×D₁₃	—	4	8	no	—
176	C₄×C₁₃	—	2	2	no	—
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	—	4	4	no	—
179	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
180	C₄×C₁₃	#520	1	1	yes	characteristic
181	C₄×C₁₃	#547	1	1	yes	characteristic
182	C₄×C₁₃	#554	1	1	yes	characteristic
183	Dic₁₃	—	4	4	no	—
184	Dic₁₃	—	8	4	no	—
185	C₂²×D₄	—	13	2	no	—
186	C₂³×C₄	—	26	2	no	—
187	C₂³×C₄	—	26	4	no	—
188	C₂×D₈	—	13	4	no	—
189	C₂×D₈	—	13	4	no	—
190	C₂×D₈	—	13	4	no	—
191	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
192	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
194	C₄×D₄	—	13	8	no	—
195	C₄×D₄	—	26	8	no	—
196	C₄×D₄	—	26	4	no	—
197	C₄×D₄	—	52	4	no	—
198	C₂×C₄⋊C₄	—	13	2	no	—
199	C₂×C₄⋊C₄	—	13	4	no	—
200	C₂×C₄⋊C₄	—	26	4	no	—
201	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
202	C₂×C₂²⋊C₄	—	26	2	no	—
203	C₂×C₂²⋊C₄	—	26	4	no	—
204	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
205	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
206	#107	—	13	4	no	—
207	#107	—	52	2	no	—
208	#106	—	26	4	no	—
209	C₄⋊C₈	—	2	2	no	—
210	#102	—	13	8	no	—
211	#102	—	26	8	no	—
212	#97	—	2	4	no	—
213	C₄×C₈	C₄×D₁₃	1	4	yes	complemented
214	#95	—	26	2	no	—
215	#95	—	26	4	no	—
216	#95	—	26	2	no	—
217	#95	D₅₂	1	2	yes	—
218	C₂×C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic
219	C₂×C₁₃	#2943	1	1	yes	characteristic
220	C₂×C₁₃	#2948	1	1	yes	characteristic
221	C₂×C₁₃	#3273	1	2	yes	—
222	C₂×C₁₃	#3389	1	2	yes	—
223	D₁₃	—	4	8	no	—
224	C₂⁴	—	26	2	no	—
225	C₂×D₄	—	13	4	no	—
226	C₂×D₄	—	13	4	no	—
227	C₂×D₄	—	13	4	no	—
228	C₂×D₄	—	26	2	no	—
229	C₂×D₄	—	26	2	no	—
230	C₂×D₄	—	26	2	no	—
231	C₂²×C₄	—	26	2	no	—
232	C₂²×C₄	—	26	8	no	—
233	C₂²×C₄	—	26	4	no	—
234	C₂²×C₄	—	52	8	no	—
235	C₂²×C₄	—	52	8	no	—
236	C₂²×C₄	—	104	4	no	—
237	C₂²×C₄	—	104	4	no	—
238	C₂²×C₄	C₂×D₅₂	1	4	yes	—
239	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
241	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
242	D₈	—	13	16	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
249	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
250	C₂×C₈	D₄×D₁₃	1	2	yes	—
251	C₂×C₈	Q₈×D₁₃	1	2	yes	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	13	4	no	—
254	C₄⋊C₄	—	26	2	no	—
255	C₄⋊C₄	—	26	4	no	—
256	C₄⋊C₄	—	26	4	no	—
257	C₄⋊C₄	—	104	4	no	—
258	C₂²⋊C₄	—	26	8	no	—
259	C₂²⋊C₄	—	52	8	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	C₂×C₄×D₁₃	1	2	yes	—
263	C₄²	#6127	1	2	yes	—
264	C₁₃	#22672	1	1	yes	characteristic, complemented, Sylow
265	C₂³	—	26	4	no	—
266	C₂³	—	26	4	no	—
267	C₂³	—	26	4	no	—
268	C₂³	—	52	4	no	—
269	C₂³	—	52	4	no	—
270	C₂³	C₂²×D₅₂	1	1	yes	characteristic
271	D₄	—	13	8	no	—
272	D₄	—	26	4	no	—
273	D₄	—	26	4	no	—
274	D₄	—	26	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	26	4	no	—
281	C₂×C₄	—	52	4	no	—
282	C₂×C₄	—	52	4	no	—
283	C₂×C₄	—	52	4	no	—
284	C₂×C₄	—	52	4	no	—
285	C₂×C₄	—	104	8	no	—
286	C₂×C₄	—	104	16	no	—
287	C₂×C₄	C₄×D₅₂	1	2	yes	—
288	C₂×C₄	C₄×D₅₂	1	2	yes	—
289	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
290	C₂×C₄	C₂×#6127	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
292	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
293	C₂×C₄	#88256	1	1	yes	characteristic
294	C₂×C₄	#88257	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	4	2	no	—
297	C₈	C₄⋊C₄×D₁₃	1	4	yes	—
298	C₂²	—	26	8	no	—
299	C₂²	—	52	8	no	—
300	C₂²	—	52	4	no	—
301	C₂²	—	52	4	no	—
302	C₂²	—	52	8	no	—
303	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
304	C₂²	D₄×D₅₂	1	1	yes	characteristic
305	C₂²	Q₈×D₅₂	1	1	yes	characteristic
306	C₂²	#11719453	1	2	yes	—
307	C₂²	#11719457	1	2	yes	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	4	4	no	—
311	C₄	—	52	4	no	—
312	C₄	—	104	4	no	—
313	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
314	C₄	#11719228	1	1	yes	characteristic
315	C₄	#11719256	1	1	yes	characteristic
316	C₄	#11719284	1	1	yes	characteristic
317	C₂	—	52	8	no	—
318	C₂	C₄⋊C₄×D₅₂	1	1	yes	characteristic
319	C₂	#13123638	1	1	yes	characteristic
320	C₂	#13123689	1	1	yes	characteristic
321	C₂	#13126448	1	2	yes	—
322	C₂	#13127796	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1770	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂²×D₁₀₄	C₂²
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1770	D₁₃
hypocenter	C₁₃	#22672
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ #1770

Complements

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

N	H ≅ G/N	type
3. #13117747	C₂	semidirect
4. #13115574	C₂	semidirect
5. #13115086	C₂	semidirect
6. #13115000	C₂	semidirect
7. #13114847	C₂	semidirect
8. C₁₃×#1770	C₂	semidirect
25. C₄×D₁₀₄	C₄	semidirect
50. C₄×D₅₂	C₂×C₄	semidirect
53. C₄×C₈×C₁₃	C₂×C₄	semidirect
59. #88061	C₂×C₄	semidirect
61. #88058	C₂×C₄	semidirect
103. D₁₀₄	C₄⋊C₄	semidirect
111. #1770	D₁₃	semidirect
213. C₄×C₈	C₄×D₁₃	semidirect
264. C₁₃	#22672	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718372	2
C₂×C₄×D₁₀₄	2
#13117747	2
#13115574	2
#13115086	2
#13115000	2
#13114847	2
C₁₃×#1770	2
#22672	13