Subgroups — #15867333

Order 3328.

722 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88158	C₂	1	1	yes	maximal, characteristic
2	#13123485	C₂	1	2	yes	maximal, complemented
3	#13115729	C₂	1	2	yes	maximal
4	#13115589	C₂	1	2	yes	maximal, complemented
5	#13115540	C₂	1	4	yes	maximal, complemented
6	C₄×#88059	C₂	1	1	yes	maximal, characteristic, complemented
7	#13114647	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88158	C₂²	1	1	yes	characteristic
10	C₄×#6127	C₂²	1	2	yes	—
11	C₂×C₁₃×#97	C₂²	1	2	yes	—
12	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#376	C₂²	1	1	yes	characteristic
16	#11718774	C₂²	1	2	yes	—
17	#11718771	C₂²	1	4	yes	—
18	C₂×#88059	C₂²	1	2	yes	—
19	#11718767	C₂²	1	1	yes	characteristic
20	#11718763	C₂²	1	4	yes	—
21	C₂×#88045	C₂²	1	4	yes	—
22	C₄×#6099	C₂²	1	2	yes	—
23	#11718370	—	2	4	no	—
24	#11718366	—	2	4	no	—
25	C₄×Dic₅₂	C₄	1	2	yes	complemented
26	C₄×D₁₀₄	C₄	1	2	yes	complemented
27	C₄×#6095	C₄	1	4	yes	complemented
28	#11718144	C₂²	1	2	yes	—
29	#11718115	C₂²	1	4	yes	—
30	C₂×#6127	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₄×C₂²⋊D₁₃	—	2	4	no	—
37	#88180	C₂³	1	2	yes	—
38	C₂×#6099	C₂³	1	4	yes	—
39	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
40	#88158	—	2	4	no	—
41	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
42	C₂×#6095	C₂×C₄	1	2	yes	—
43	C₄×D₅₂	C₂×C₄	1	4	yes	—
44	#88126	—	2	2	no	—
45	C₄²×D₁₃	—	2	2	no	—
46	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
47	C₁₃×#97	—	2	4	no	—
48	C₁₃×#97	D₄	1	2	yes	—
49	C₁₃×#97	D₄	1	2	yes	—
50	C₄×C₈×C₁₃	—	2	4	no	—
51	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
53	C₁₃×#95	—	2	2	no	—
54	#88061	C₂×C₄	1	4	yes	—
55	#88060	—	2	8	no	—
56	#88059	—	2	4	no	—
57	#88058	C₂×C₄	1	2	yes	—
58	#88057	C₂×C₄	1	2	yes	—
59	#88056	C₂×C₄	1	4	yes	—
60	#88054	—	2	8	no	—
61	#88045	—	2	8	no	—
62	#88043	—	2	2	no	—
63	C₄×#1055	—	2	2	no	—
64	#21295	—	13	1	no	maximal, Sylow
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	#41	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₂×C₂²⋊D₁₃	—	2	2	no	—
69	#6127	—	2	4	no	—
70	#6127	—	2	4	no	—
71	C₂×D₅₂	C₂²×C₄	1	2	yes	—
72	C₂×C₄×D₁₃	—	2	2	no	—
73	C₂×C₄×D₁₃	—	2	4	no	—
74	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
82	C₂×C₈×C₁₃	#41	1	4	yes	—
83	C₄²×C₁₃	—	2	2	no	—
84	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
85	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₃	#41	1	4	yes	—
88	C₂²⋊Dic₁₃	—	2	2	no	—
89	#6103	—	2	4	no	—
90	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
91	#6101	—	2	4	no	—
92	C₄×Dic₁₃	—	2	4	no	—
93	C₄×Dic₁₃	—	2	2	no	—
94	#6099	—	2	4	no	—
95	#6099	—	2	4	no	—
96	C₂×#1055	—	2	4	no	—
97	Dic₅₂	—	2	2	no	—
98	D₁₀₄	—	2	2	no	—
99	#6095	—	2	4	no	—
100	C₄×#135	—	13	1	no	—
101	#2923	—	13	2	no	—
102	#1975	—	13	2	no	—
103	#1946	—	13	2	no	—
104	#1937	—	13	4	no	—
105	C₄×#108	—	13	1	no	—
106	#1759	—	13	2	no	—
107	#1726	D₁₃	1	1	yes	characteristic, complemented
108	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
109	C₂²×D₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
119	C₂²⋊D₁₃	—	4	4	no	—
120	C₂×Dic₁₃	—	2	2	no	—
121	C₂×Dic₁₃	—	2	4	no	—
122	D₅₂	—	2	2	no	—
123	D₅₂	—	2	2	no	—
124	C₄×D₁₃	—	4	4	no	—
125	C₄×D₁₃	—	8	4	no	—
126	Dic₂₆	—	2	2	no	—
127	Dic₂₆	—	2	2	no	—
128	C₈×C₁₃	—	2	2	no	—
129	C₈×C₁₃	—	2	4	no	—
130	C₈×C₁₃	#117	1	2	yes	—
131	C₈×C₁₃	#117	1	2	yes	—
132	#1055	—	4	4	no	—
133	C₂×#135	—	13	1	no	—
134	C₄×#41	—	13	2	no	—
135	C₄×Q₁₆	—	13	2	no	—
136	C₄×SD₁₆	—	13	4	no	—
137	C₄×D₈	—	13	2	no	—
138	#433	—	26	4	no	—
139	#432	—	26	4	no	—
140	C₂×#108	—	13	2	no	—
141	#427	—	13	1	no	—
142	#424	—	13	4	no	—
143	C₂×#104	—	13	4	no	—
144	#416	—	13	2	no	—
145	#408	—	13	4	no	—
146	C₄×M₁₆	—	13	2	no	—
147	C₂×#97	C₂×D₁₃	1	2	yes	—
148	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
149	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
150	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
151	#376	C₂×D₁₃	1	1	yes	characteristic
152	#341	—	13	2	no	—
153	#339	—	13	4	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂²×C₁₃	#529	1	2	yes	—
157	C₂×D₁₃	—	4	4	no	—
158	C₂×D₁₃	—	4	2	no	—
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	—	2	2	no	—
161	C₄×C₁₃	—	4	4	no	—
162	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
163	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
164	C₄×C₁₃	#532	1	2	yes	—
165	Dic₁₃	—	4	4	no	—
166	Dic₁₃	—	4	2	no	—
167	C₂×#41	—	13	2	no	—
168	#135	—	26	4	no	—
169	C₂×Q₁₆	—	13	1	no	—
170	C₂×SD₁₆	—	13	2	no	—
171	C₂×D₈	—	13	1	no	—
172	C₂×M₁₆	—	13	4	no	—
173	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
175	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
176	C₄×Q₈	—	13	4	no	—
177	C₄×D₄	—	13	4	no	—
178	C₄×D₄	—	26	4	no	—
179	#117	—	13	2	no	—
180	#117	—	26	2	no	—
181	C₂×C₄²	—	26	2	no	—
182	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
183	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
184	#108	—	26	4	no	—
185	#107	—	13	2	no	—
186	#106	—	13	2	no	—
187	C₄⋊C₈	—	26	8	no	—
188	#104	—	26	8	no	—
189	#103	—	13	4	no	—
190	#102	—	13	4	no	—
191	C₂²⋊C₈	—	26	8	no	—
192	#97	—	2	4	no	—
193	#97	—	26	2	no	—
194	#97	D₅₂	1	2	yes	—
195	#97	D₅₂	1	2	yes	—
196	C₄×C₈	—	2	4	no	—
197	C₄×C₈	—	26	2	no	—
198	C₄×C₈	C₄×D₁₃	1	2	yes	—
199	C₄×C₈	C₄×D₁₃	1	2	yes	—
200	#95	—	2	2	no	—
201	C₂×C₁₃	—	2	2	no	—
202	C₂×C₁₃	#2293	1	1	yes	characteristic
203	C₂×C₁₃	#2957	1	2	yes	—
204	D₁₃	—	8	2	no	—
205	#41	—	26	4	no	—
206	#41	—	26	4	no	—
207	C₂×Q₈	—	13	2	no	—
208	C₂×D₄	—	13	2	no	—
209	C₂×D₄	—	26	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	26	2	no	—
212	C₂²×C₄	—	26	4	no	—
213	C₂²×C₄	#6127	1	2	yes	—
214	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
215	Q₁₆	—	26	2	no	—
216	SD₁₆	—	26	4	no	—
217	D₈	—	26	2	no	—
218	M₁₆	—	26	4	no	—
219	M₁₆	—	26	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	26	4	no	—
224	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
225	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
227	C₂×C₈	C₂×D₅₂	1	4	yes	—
228	C₂×C₈	#6127	1	4	yes	—
229	C₄⋊C₄	—	13	4	no	—
230	C₄⋊C₄	—	26	4	no	—
231	C₂²⋊C₄	—	26	2	no	—
232	C₂²⋊C₄	—	26	4	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	26	4	no	—
235	C₄²	—	26	2	no	—
236	C₄²	C₂×C₄×D₁₃	1	2	yes	—
237	C₄²	C₂×D₅₂	1	1	yes	characteristic
238	C₄²	C₂×D₅₂	1	1	yes	characteristic
239	C₄²	#6127	1	4	yes	—
240	C₁₃	#21295	1	1	yes	characteristic, complemented, Sylow
241	C₂³	—	26	2	no	—
242	C₂³	C₂×#6127	1	1	yes	characteristic
243	Q₈	—	26	2	no	—
244	Q₈	—	26	2	no	—
245	D₄	—	26	2	no	—
246	D₄	—	26	2	no	—
247	D₄	—	52	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	26	4	no	—
252	C₂×C₄	—	26	2	no	—
253	C₂×C₄	—	52	4	no	—
254	C₂×C₄	—	104	4	no	—
255	C₂×C₄	C₄×D₅₂	1	2	yes	—
256	C₂×C₄	C₄×D₅₂	1	2	yes	—
257	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
258	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
259	C₂×C₄	C₂×#6127	1	2	yes	—
260	C₂×C₄	C₂×#6127	1	2	yes	—
261	C₈	—	2	2	no	—
262	C₈	—	2	4	no	—
263	C₈	—	52	4	no	—
264	C₈	#88126	1	2	yes	—
265	C₈	#88126	1	2	yes	—
266	C₂²	—	2	2	no	—
267	C₂²	—	52	2	no	—
268	C₂²	—	52	4	no	—
269	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
270	C₂²	#11719184	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	4	4	no	—
274	C₄	—	52	2	no	—
275	C₄	—	52	4	no	—
276	C₄	C₂×#88126	1	1	yes	characteristic
277	C₄	C₄×#6127	1	1	yes	characteristic
278	C₄	#11719182	1	2	yes	—
279	C₂	—	2	2	no	—
280	C₂	—	104	2	no	—
281	C₂	#13117300	1	1	yes	characteristic
282	C₂	#13123754	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1726	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×#88158	C₂²
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1726	D₁₃
hypocenter	C₁₃	#21295
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726

Complements

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

N	H ≅ G/N	type
2. #13123485	C₂	semidirect
4. #13115589	C₂	semidirect
5. #13115540	C₂	semidirect
6. C₄×#88059	C₂	semidirect
7. #13114647	C₂	semidirect
8. C₁₃×#1726	C₂	semidirect
25. C₄×Dic₅₂	C₄	semidirect
26. C₄×D₁₀₄	C₄	semidirect
27. C₄×#6095	C₄	semidirect
107. #1726	D₁₃	semidirect
240. C₁₃	#21295	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88158	2
#13123485	2
#13115729	2
#13115589	2
#13115540	2
C₄×#88059	2
#13114647	2
C₁₃×#1726	2
#21295	13