Subgroups — #15779865

Order 3328.

574 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718777	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88058	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718095	C₂	1	2	yes	maximal, complemented
4	#13112928	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊Dic₁₃	—	2	1	no	—
8	C₂×#88178	—	2	1	no	—
9	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
11	#11718777	—	2	2	no	—
12	#11718777	C₄	1	4	yes	complemented
13	C₂×#88058	—	2	4	no	—
14	C₂×#88058	—	2	1	no	—
15	C₂×#88058	C₂²	1	1	yes	characteristic
16	C₂×#88058	C₂²	1	2	yes	—
17	C₂×#88056	—	2	1	no	—
18	C₁₃×C₂²⋊C₁₆	—	2	2	no	—
19	#11718095	—	2	4	no	—
20	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
21	C₂³×Dic₁₃	—	4	1	no	—
22	C₂²×Dic₂₆	—	2	1	no	—
23	C₂²×C₈×C₁₃	—	2	2	no	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	D₄	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₂²⋊Dic₁₃	—	4	1	no	—
30	C₂×C₄⋊Dic₁₃	—	2	4	no	—
31	C₂×C₄⋊Dic₁₃	—	2	1	no	—
32	C₂×C₄⋊Dic₁₃	—	2	2	no	—
33	C₂×C₄⋊Dic₁₃	—	4	2	no	—
34	C₂×C₄⋊Dic₁₃	—	4	1	no	—
35	#88178	—	2	4	no	—
36	#88178	—	4	2	no	—
37	C₂×#6101	—	4	1	no	—
38	C₂×Dic₅₂	—	2	2	no	—
39	C₂×Dic₅₂	—	2	2	no	—
40	C₂×Dic₅₂	—	2	1	no	—
41	C₂×Dic₅₂	C₂×C₄	1	2	yes	complemented
42	C₂×C₁₃×C₁₆	—	2	2	no	—
43	C₂×C₁₃×C₁₆	—	2	2	no	—
44	C₂×C₁₃×C₁₆	—	2	2	no	—
45	#88058	—	2	4	no	—
46	#88058	—	2	2	no	—
47	#88058	—	2	1	no	—
48	#88058	C₂×C₄	1	2	yes	complemented
49	#88058	D₄	1	4	yes	—
50	#88056	—	2	4	no	—
51	#12993	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₁₃	—	4	4	no	—
62	C₂²×Dic₁₃	—	4	1	no	—
63	C₂²×Dic₁₃	—	4	1	no	—
64	C₂²×Dic₁₃	—	4	2	no	—
65	C₂²×Dic₁₃	—	8	2	no	—
66	C₂²×Dic₁₃	—	8	2	no	—
67	C₂×Dic₂₆	—	2	2	no	—
68	C₂×Dic₂₆	—	4	1	no	—
69	C₂×Dic₂₆	—	4	1	no	—
70	C₂×Dic₂₆	—	4	1	no	—
71	C₂×Dic₂₆	—	4	1	no	—
72	C₂×Dic₂₆	—	4	1	no	—
73	C₂×C₈×C₁₃	—	2	1	no	—
74	C₂×C₈×C₁₃	—	2	1	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	#41	1	2	yes	—
82	C₂²⋊Dic₁₃	—	4	4	no	—
83	C₄⋊Dic₁₃	—	2	2	no	—
84	C₄⋊Dic₁₃	—	2	4	no	—
85	C₄⋊Dic₁₃	—	4	4	no	—
86	C₄⋊Dic₁₃	—	4	1	no	—
87	C₄⋊Dic₁₃	—	4	2	no	—
88	#6101	—	4	4	no	—
89	Dic₅₂	—	2	4	no	—
90	Dic₅₂	—	4	2	no	—
91	C₁₃×C₁₆	—	4	2	no	—
92	C₂×#467	—	13	1	no	—
93	C₂²×#107	—	13	1	no	—
94	C₂×#358	—	13	2	no	—
95	C₂×C₂²⋊C₁₆	D₁₃	1	1	yes	characteristic
96	#1380	—	13	2	no	—
97	C₂³×C₁₃	—	2	1	no	—
98	C₂³×C₁₃	—	2	1	no	—
99	C₂³×C₁₃	—	2	1	no	—
100	C₂³×C₁₃	—	2	1	no	—
101	C₂³×C₁₃	#102	1	2	yes	—
102	C₂³×C₁₃	SD₃₂	1	2	yes	—
103	C₂³×C₁₃	Q₃₂	1	2	yes	—
104	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	#102	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
118	C₂×Dic₁₃	—	4	2	no	—
119	C₂×Dic₁₃	—	4	2	no	—
120	C₂×Dic₁₃	—	4	4	no	—
121	C₂×Dic₁₃	—	8	8	no	—
122	C₂×Dic₁₃	—	8	1	no	—
123	C₂×Dic₁₃	—	8	1	no	—
124	C₂×Dic₁₃	—	8	2	no	—
125	C₂×Dic₁₃	—	8	1	no	—
126	C₂×Dic₁₃	—	8	2	no	—
127	C₂×Dic₁₃	—	8	1	no	—
128	Dic₂₆	—	4	2	no	—
129	Dic₂₆	—	4	2	no	—
130	Dic₂₆	—	8	1	no	—
131	Dic₂₆	—	8	1	no	—
132	C₈×C₁₃	—	2	2	no	—
133	C₈×C₁₃	C₄×D₄	1	2	yes	—
134	C₈×C₁₃	#121	1	2	yes	—
135	C₂²×Q₁₆	—	13	1	no	—
136	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
137	C₂×C₂²⋊Q₈	—	26	1	no	—
138	C₂²×C₄⋊C₄	—	26	1	no	—
139	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
140	#467	—	13	4	no	—
141	#467	—	26	2	no	—
142	C₂×#107	—	13	2	no	—
143	C₂×#107	—	13	1	no	—
144	C₂×#107	—	26	4	no	—
145	C₂×#107	—	26	1	no	—
146	C₂×#103	—	26	1	no	—
147	#358	—	26	4	no	—
148	C₂²⋊C₁₆	—	2	2	no	—
149	C₂²×C₁₃	—	2	2	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	#358	1	4	yes	complemented
156	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
157	C₂²×C₁₃	#447	1	1	yes	characteristic
158	C₂²×C₁₃	#480	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
160	C₂²×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
161	C₂²×C₁₃	#509	1	1	yes	characteristic
162	C₂²×C₁₃	#510	1	1	yes	characteristic
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	#386	1	1	yes	characteristic
165	C₄×C₁₃	#417	1	1	yes	characteristic
166	C₄×C₁₃	#448	1	1	yes	characteristic
167	C₄×C₁₃	#481	1	1	yes	characteristic
168	Dic₁₃	—	8	2	no	—
169	Dic₁₃	—	8	2	no	—
170	Dic₁₃	—	8	4	no	—
171	C₂²×Q₈	—	26	1	no	—
172	C₂³×C₄	—	52	1	no	—
173	C₂³×C₄	D₅₂	1	1	yes	characteristic
174	C₂×Q₁₆	—	13	2	no	—
175	C₂×Q₁₆	—	26	2	no	—
176	C₂×Q₁₆	—	26	2	no	—
177	C₂×Q₁₆	—	26	1	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	C₄×D₁₃	1	2	yes	—
181	C₂²×C₈	D₅₂	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
183	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
184	C₂²⋊Q₈	—	26	4	no	—
185	C₂²⋊Q₈	—	52	2	no	—
186	C₂×C₄⋊C₄	—	26	4	no	—
187	C₂×C₄⋊C₄	—	26	2	no	—
188	C₂×C₄⋊C₄	—	26	1	no	—
189	C₂×C₄⋊C₄	—	52	2	no	—
190	C₂×C₄⋊C₄	—	52	1	no	—
191	C₂×C₄⋊C₄	—	52	1	no	—
192	C₂×C₂²⋊C₄	—	52	1	no	—
193	C₂×C₁₆	—	2	2	no	—
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	—	2	2	no	—
196	#107	—	13	2	no	—
197	#107	—	13	4	no	—
198	#107	—	26	4	no	—
199	#107	—	26	2	no	—
200	#107	—	26	1	no	—
201	#103	—	26	4	no	—
202	C₂×C₁₃	—	2	4	no	—
203	C₂×C₁₃	#1874	1	1	yes	characteristic
204	C₂×C₁₃	C₂×#358	1	1	yes	characteristic
205	C₂×C₁₃	#2139	1	1	yes	characteristic
206	C₂×C₁₃	#2187	1	1	yes	characteristic
207	C₂×C₁₃	#2188	1	1	yes	characteristic
208	C₂×C₁₃	#2235	1	1	yes	characteristic
209	C₂×C₁₃	#2236	1	1	yes	characteristic
210	C₂⁴	D₁₀₄	1	1	yes	characteristic
211	C₂×Q₈	—	26	2	no	—
212	C₂×Q₈	—	52	1	no	—
213	C₂×Q₈	—	52	1	no	—
214	C₂×Q₈	—	52	1	no	—
215	C₂×Q₈	—	52	1	no	—
216	C₂×Q₈	—	52	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	52	4	no	—
222	C₂²×C₄	—	52	1	no	—
223	C₂²×C₄	—	52	1	no	—
224	C₂²×C₄	—	52	2	no	—
225	C₂²×C₄	—	104	2	no	—
226	C₂²×C₄	—	104	2	no	—
227	C₂²×C₄	#6095	1	2	yes	—
228	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
229	C₂²×C₄	#6103	1	2	yes	—
230	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
231	Q₁₆	—	26	4	no	—
232	Q₁₆	—	52	2	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	#6103	1	2	yes	—
238	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
239	C₂×C₈	D₄×D₁₃	1	2	yes	—
240	C₂×C₈	#6129	1	2	yes	—
241	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
242	C₄⋊C₄	—	26	4	no	—
243	C₄⋊C₄	—	26	2	no	—
244	C₄⋊C₄	—	52	4	no	—
245	C₄⋊C₄	—	52	1	no	—
246	C₄⋊C₄	—	52	2	no	—
247	C₄⋊C₄	—	52	4	no	—
248	C₂²⋊C₄	—	52	4	no	—
249	C₁₆	—	4	2	no	—
250	C₁₃	#12993	1	1	yes	characteristic, complemented, Sylow
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	#88040	1	2	yes	—
256	C₂³	Dic₁₀₄	1	2	yes	—
257	C₂³	#88061	1	2	yes	—
258	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
259	Q₈	—	52	2	no	—
260	Q₈	—	52	2	no	—
261	Q₈	—	104	1	no	—
262	Q₈	—	104	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	52	2	no	—
270	C₂×C₄	—	52	2	no	—
271	C₂×C₄	—	52	4	no	—
272	C₂×C₄	—	104	8	no	—
273	C₂×C₄	—	104	2	no	—
274	C₂×C₄	—	104	1	no	—
275	C₂×C₄	—	104	2	no	—
276	C₂×C₄	—	104	1	no	—
277	C₂×C₄	—	104	1	no	—
278	C₂×C₄	—	104	1	no	—
279	C₂×C₄	#88061	1	2	yes	—
280	C₂×C₄	#88136	1	1	yes	characteristic
281	C₂×C₄	#88140	1	1	yes	characteristic
282	C₂×C₄	C₂×#6095	1	1	yes	characteristic
283	C₂×C₄	#88162	1	1	yes	characteristic
284	C₂×C₄	#88163	1	1	yes	characteristic
285	C₂×C₄	C₂×#6103	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	#88135	1	2	yes	—
288	C₈	#88193	1	2	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	#11718095	1	4	yes	complemented
296	C₂²	#11718393	1	1	yes	characteristic
297	C₂²	#11718397	1	1	yes	characteristic
298	C₂²	C₂×#88040	1	1	yes	characteristic
299	C₂²	C₂×Dic₁₀₄	1	1	yes	characteristic
300	C₂²	#11718569	1	1	yes	characteristic
301	C₂²	#11718570	1	1	yes	characteristic
302	C₂²	C₂×#88061	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	104	2	no	—
305	C₄	—	104	4	no	—
306	C₄	—	104	2	no	—
307	C₄	#11718396	1	1	yes	characteristic
308	C₄	#11718399	1	1	yes	characteristic
309	C₄	#11718619	1	1	yes	characteristic
310	C₄	#11718794	1	1	yes	characteristic
311	C₂	—	2	4	no	—
312	C₂	#13115151	1	1	yes	characteristic
313	C₂	C₂×#11718095	1	1	yes	characteristic
314	C₂	#13116549	1	1	yes	characteristic
315	C₂	#13116767	1	1	yes	characteristic
316	C₂	#13116772	1	1	yes	characteristic
317	C₂	#13117034	1	1	yes	characteristic
318	C₂	#13117039	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×C₂²⋊C₁₆	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂⁴×C₁₃	D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₁₃
hypocenter	C₁₃	#12993
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#11718095	C₂	semidirect
4. #13112928	C₂	semidirect
12. #11718777	C₄	semidirect
41. C₂×Dic₅₂	C₂×C₄	semidirect
48. #88058	C₂×C₄	semidirect
155. C₂²×C₁₃	#358	semidirect
250. C₁₃	#12993	semidirect
295. C₂²	#11718095	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718777	2
C₂²×#88058	2
C₂×#11718095	2
#13112928	2
C₂×C₁₃×C₂²⋊C₁₆	2
#12993	13