Subgroups — #15867146

Order 3328.

822 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
2	#13117313	C₂	1	2	yes	maximal
3	#13115697	C₂	1	2	yes	maximal
4	#13115569	C₂	1	2	yes	maximal
5	#13115512	C₂	1	4	yes	maximal
6	C₄×#88058	C₂	1	1	yes	maximal, characteristic
7	C₄×#88056	C₂	1	2	yes	maximal
8	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₂₆	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₄³×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88058	C₂²	1	2	yes	—
17	C₂×#88058	C₂²	1	1	yes	characteristic
18	C₂×#88056	C₂²	1	4	yes	—
19	C₂×#88056	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
21	#11718594	—	2	4	no	—
22	#11718593	—	2	2	no	—
23	C₄×#6101	—	2	2	no	—
24	#11718590	C₂²	1	4	yes	—
25	#11718589	C₂²	1	4	yes	—
26	C₄×Dic₅₂	C₄	1	8	yes	complemented
27	#11718143	C₂²	1	4	yes	—
28	#11718143	C₂²	1	2	yes	—
29	C₂²×Dic₂₆	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	D₄	1	2	yes	—
35	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
39	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
40	C₂×#6101	—	2	2	no	—
41	C₂×#6101	—	2	4	no	—
42	C₂×C₄×Dic₁₃	—	2	2	no	—
43	C₂×C₄×Dic₁₃	—	2	2	no	—
44	C₂×Dic₅₂	—	2	4	no	—
45	C₂×Dic₅₂	C₂×C₄	1	4	yes	—
46	C₄×Dic₂₆	—	2	4	no	—
47	C₄×Dic₂₆	C₂×C₄	1	8	yes	—
48	C₄×C₈×C₁₃	—	2	4	no	—
49	C₄×C₈×C₁₃	—	2	4	no	—
50	C₄×C₈×C₁₃	—	2	2	no	—
51	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
52	#88071	—	2	4	no	—
53	#88071	—	2	4	no	—
54	#88071	—	2	4	no	—
55	#88071	—	2	2	no	—
56	#88058	—	2	4	no	—
57	#88058	C₂×C₄	1	4	yes	—
58	#88056	—	2	8	no	—
59	#88056	C₂×C₄	1	8	yes	—
60	#21276	—	13	1	no	maximal, Sylow
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₁₃	—	2	2	no	—
67	C₂²×Dic₁₃	—	2	2	no	—
68	C₂²×Dic₁₃	—	2	4	no	—
69	C₂×Dic₂₆	—	2	4	no	—
70	C₂×Dic₂₆	—	2	4	no	—
71	C₂×Dic₂₆	—	2	2	no	—
72	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	#41	1	4	yes	—
79	C₂×C₈×C₁₃	#41	1	4	yes	—
80	C₄²×C₁₃	—	2	4	no	—
81	C₄²×C₁₃	—	2	2	no	—
82	C₄²×C₁₃	—	2	2	no	—
83	C₄²×C₁₃	—	2	2	no	—
84	C₄²×C₁₃	Q₁₆	1	4	yes	complemented
85	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
86	C₄²×C₁₃	#41	1	2	yes	—
87	C₄⋊Dic₁₃	—	2	2	no	—
88	C₄⋊Dic₁₃	—	2	4	no	—
89	C₄⋊Dic₁₃	—	2	4	no	—
90	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
91	#6101	—	2	8	no	—
92	#6101	—	4	8	no	—
93	C₄×Dic₁₃	—	2	8	no	—
94	C₄×Dic₁₃	—	4	4	no	—
95	Dic₅₂	—	2	8	no	—
96	C₂×C₄×Q₁₆	—	13	1	no	—
97	#2295	—	13	2	no	—
98	#1969	—	13	2	no	—
99	#1942	—	13	2	no	—
100	#1933	—	13	4	no	—
101	C₄×#107	—	13	1	no	—
102	C₄×#103	—	13	2	no	—
103	C₄²×C₈	D₁₃	1	1	yes	characteristic
104	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	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	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
113	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
114	C₂×C₄×C₁₃	#135	1	4	yes	—
115	C₂×C₄×C₁₃	#135	1	2	yes	—
116	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
120	C₂×Dic₁₃	—	2	4	no	—
121	C₂×Dic₁₃	—	2	4	no	—
122	C₂×Dic₁₃	—	4	4	no	—
123	C₂×Dic₁₃	—	4	4	no	—
124	C₂×Dic₁₃	—	4	4	no	—
125	C₂×Dic₁₃	—	4	4	no	—
126	C₂×Dic₁₃	—	4	4	no	—
127	Dic₂₆	—	2	4	no	—
128	Dic₂₆	—	2	4	no	—
129	Dic₂₆	—	4	4	no	—
130	C₈×C₁₃	—	2	2	no	—
131	C₈×C₁₃	—	2	4	no	—
132	C₈×C₁₃	#117	1	4	yes	—
133	C₂²×Q₁₆	—	13	1	no	—
134	C₂×C₄×Q₈	—	13	2	no	—
135	C₄×Q₁₆	—	13	8	no	—
136	C₂×#107	—	13	1	no	—
137	C₂×#107	—	13	2	no	—
138	C₂×#103	—	13	2	no	—
139	C₂×#103	—	13	4	no	—
140	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
141	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
142	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
144	#391	—	13	4	no	—
145	#389	—	13	4	no	—
146	#387	—	26	4	no	—
147	C₄×C₄⋊C₄	—	13	2	no	—
148	C₄×C₄⋊C₄	—	26	2	no	—
149	#376	—	26	2	no	—
150	C₄³	C₂×D₁₃	1	1	yes	characteristic
151	#340	—	13	2	no	—
152	#340	—	13	4	no	—
153	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
154	C₂²×C₁₃	#532	1	2	yes	—
155	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×#135	1	2	yes	—
158	C₄×C₁₃	—	2	4	no	—
159	C₄×C₁₃	—	2	4	no	—
160	C₄×C₁₃	—	2	2	no	—
161	C₄×C₁₃	C₄×Q₁₆	1	4	yes	—
162	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
163	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
164	C₄×C₁₃	#529	1	2	yes	—
165	Dic₁₃	—	4	4	no	—
166	Dic₁₃	—	4	4	no	—
167	Dic₁₃	—	8	4	no	—
168	C₂²×Q₈	—	13	2	no	—
169	C₂×Q₁₆	—	13	4	no	—
170	C₂×Q₁₆	—	26	4	no	—
171	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
173	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
174	C₄×Q₈	—	13	8	no	—
175	C₄×Q₈	—	26	4	no	—
176	C₂×C₄⋊C₄	—	13	2	no	—
177	C₂×C₄⋊C₄	—	13	4	no	—
178	C₂×C₄⋊C₄	—	26	2	no	—
179	C₂×C₄⋊C₄	—	26	4	no	—
180	C₂×C₄²	—	26	2	no	—
181	C₂×C₄²	—	26	2	no	—
182	C₂×C₄²	D₅₂	1	2	yes	—
183	C₂×C₄²	D₅₂	1	1	yes	characteristic
184	C₂×C₄²	D₅₂	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
186	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
187	#107	—	13	4	no	—
188	#107	—	26	4	no	—
189	#103	—	13	8	no	—
190	#103	—	26	8	no	—
191	C₄×C₈	—	2	4	no	—
192	C₄×C₈	—	2	4	no	—
193	C₄×C₈	—	2	2	no	—
194	C₄×C₈	C₄×D₁₃	1	4	yes	—
195	#95	—	26	4	no	—
196	#95	—	26	4	no	—
197	#95	—	26	2	no	—
198	#95	—	26	4	no	—
199	C₂×C₁₃	#2293	1	1	yes	characteristic
200	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
201	C₂×C₁₃	C₄×#135	1	1	yes	characteristic
202	C₂×C₁₃	#3168	1	2	yes	—
203	C₂×C₁₃	#3170	1	2	yes	—
204	C₂×Q₈	—	13	4	no	—
205	C₂×Q₈	—	26	2	no	—
206	C₂×Q₈	—	26	4	no	—
207	C₂×Q₈	—	26	4	no	—
208	C₂²×C₄	—	26	2	no	—
209	C₂²×C₄	—	26	2	no	—
210	C₂²×C₄	—	26	4	no	—
211	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₅₂	1	2	yes	—
213	C₂²×C₄	C₂×D₅₂	1	2	yes	—
214	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
215	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
216	Q₁₆	—	26	8	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
221	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
222	C₂×C₈	#6127	1	4	yes	—
223	C₂×C₈	#6127	1	4	yes	—
224	C₄⋊C₄	—	13	4	no	—
225	C₄⋊C₄	—	26	8	no	—
226	C₄⋊C₄	—	26	2	no	—
227	C₄⋊C₄	—	26	4	no	—
228	C₄⋊C₄	—	26	4	no	—
229	C₄⋊C₄	—	52	8	no	—
230	C₄²	—	2	4	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	26	8	no	—
235	C₄²	—	52	4	no	—
236	C₄²	Dic₅₂	1	4	yes	complemented
237	C₄²	C₂×C₄×D₁₃	1	2	yes	—
238	C₄²	#6127	1	2	yes	—
239	C₁₃	#21276	1	1	yes	characteristic, complemented, Sylow
240	C₂³	C₂²×D₅₂	1	1	yes	characteristic
241	Q₈	—	26	4	no	—
242	Q₈	—	26	4	no	—
243	Q₈	—	52	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	26	4	no	—
250	C₂×C₄	—	26	4	no	—
251	C₂×C₄	—	52	4	no	—
252	C₂×C₄	—	52	4	no	—
253	C₂×C₄	—	52	4	no	—
254	C₂×C₄	—	52	4	no	—
255	C₂×C₄	—	52	4	no	—
256	C₂×C₄	C₄×D₅₂	1	2	yes	—
257	C₂×C₄	C₄×D₅₂	1	2	yes	—
258	C₂×C₄	#88158	1	4	yes	—
259	C₂×C₄	#88158	1	2	yes	—
260	C₂×C₄	C₂×Dic₅₂	1	4	yes	—
261	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
262	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
263	C₂×C₄	C₂×#6127	1	2	yes	—
264	C₂×C₄	C₂×#6127	1	2	yes	—
265	C₂×C₄	C₂×#6127	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	2	4	no	—
268	C₈	#88126	1	4	yes	—
269	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
270	C₂²	#11719182	1	2	yes	—
271	C₂²	C₂×#88158	1	1	yes	characteristic
272	C₂²	C₂×#88158	1	2	yes	—
273	C₂²	C₂²×Dic₅₂	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	4	no	—
277	C₄	—	52	4	no	—
278	C₄	—	52	4	no	—
279	C₄	—	104	4	no	—
280	C₄	C₄×Dic₅₂	1	4	yes	—
281	C₄	C₂×#88126	1	1	yes	characteristic
282	C₄	C₄×#6127	1	1	yes	characteristic
283	C₄	#11719184	1	2	yes	—
284	C₂	#13117300	1	1	yes	characteristic
285	C₂	C₂×C₄×Dic₅₂	1	1	yes	characteristic
286	C₂	C₄×#88158	1	1	yes	characteristic
287	C₂	#13125406	1	2	yes	—
288	C₂	#13125422	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄²×C₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄²×C₈	D₁₃
hypocenter	C₁₃	#21276
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
26. C₄×Dic₅₂	C₄	semidirect
84. C₄²×C₁₃	Q₁₆	semidirect
236. C₄²	Dic₅₂	semidirect
239. C₁₃	#21276	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₅₂	2
#13117313	2
#13115697	2
#13115569	2
#13115512	2
C₄×#88058	2
C₄×#88056	2
C₄²×C₈×C₁₃	2
#21276	13