Subgroups — #15881245

Order 3328.

898 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718845	C₂	1	2	yes	maximal
2	#13117757	C₂	1	2	yes	maximal
3	#13115575	C₂	1	2	yes	maximal
4	#13115092	C₂	1	4	yes	maximal
5	#13115000	C₂	1	1	yes	maximal, characteristic
6	#13114851	C₂	1	2	yes	maximal
7	#13114703	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₂×Q₈×Dic₁₃	C₂²	1	4	yes	—
12	C₄×C₁₃×Q₁₆	C₄	1	8	yes	complemented
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#103	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88075	C₂²	1	4	yes	—
17	#11718845	—	2	8	no	—
18	C₂×#88058	C₂²	1	2	yes	—
19	C₂×#88057	C₂²	1	2	yes	—
20	C₂×#88055	C₂²	1	2	yes	—
21	#11718646	—	2	4	no	—
22	#11718646	C₂²	1	4	yes	—
23	#11718597	—	2	2	no	—
24	#11718597	C₂²	1	2	yes	—
25	C₄×C₄⋊Dic₁₃	—	2	2	no	—
26	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
28	#11718142	C₂²	1	2	yes	—
29	#11718121	C₂²	1	4	yes	—
30	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
31	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
32	C₂×C₁₃×Q₁₆	D₄	1	4	yes	—
33	C₂×C₁₃×Q₁₆	Q₈	1	4	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₁₃	—	2	4	no	—
37	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
40	C₂×C₄²×C₁₃	—	2	2	no	—
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	Q₈×Dic₁₃	—	2	8	no	—
43	Q₈×Dic₁₃	—	4	4	no	—
44	C₂×C₄⋊Dic₁₃	—	2	4	no	—
45	C₂×C₄⋊Dic₁₃	—	2	4	no	—
46	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
47	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
48	C₂×C₄×Dic₁₃	—	2	4	no	—
49	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
50	C₂²×#1055	C₂³	1	2	yes	—
51	C₁₃×#107	C₂×C₄	1	4	yes	—
52	C₁₃×#103	C₂×C₄	1	8	yes	—
53	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
54	#88075	—	2	8	no	—
55	#88071	—	2	2	no	—
56	#88071	—	2	2	no	—
57	#88071	—	2	4	no	—
58	#88071	D₄	1	2	yes	—
59	#88058	—	4	2	no	—
60	#88057	—	2	4	no	—
61	#88055	—	2	4	no	—
62	C₄⋊#1055	—	2	2	no	—
63	#22675	—	13	1	no	maximal, Sylow
64	C₂×Q₈×C₁₃	—	2	2	no	—
65	C₂×Q₈×C₁₃	—	2	2	no	—
66	C₂×Q₈×C₁₃	—	2	2	no	—
67	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
68	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
69	C₂×Q₈×C₁₃	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₁₃	—	2	2	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₁₃	—	2	4	no	—
76	C₂²×Dic₁₃	C₂×D₄	1	4	yes	—
77	C₁₃×Q₁₆	C₄⋊C₄	1	16	yes	complemented
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
84	C₁₃×C₄⋊C₄	—	2	8	no	—
85	C₁₃×C₄⋊C₄	—	2	2	no	—
86	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
87	C₄²×C₁₃	—	2	8	no	—
88	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₁₃	#41	1	2	yes	—
90	C₄⋊Dic₁₃	—	2	4	no	—
91	C₄⋊Dic₁₃	—	2	4	no	—
92	C₄⋊Dic₁₃	—	4	2	no	—
93	C₄⋊Dic₁₃	—	4	8	no	—
94	C₄⋊Dic₁₃	—	4	8	no	—
95	C₄×Dic₁₃	—	2	2	no	—
96	C₄×Dic₁₃	—	2	2	no	—
97	C₄×Dic₁₃	—	4	8	no	—
98	C₂×#1055	—	2	2	no	—
99	C₂×#1055	—	2	2	no	—
100	C₂×#1055	—	2	2	no	—
101	C₂×#441	—	13	2	no	—
102	C₂×C₄×Q₁₆	D₁₃	1	1	yes	characteristic
103	#2351	—	13	2	no	—
104	#1943	—	13	2	no	—
105	#1863	—	13	4	no	—
106	#1846	—	13	1	no	—
107	#1805	—	13	2	no	—
108	#1770	—	13	1	no	—
109	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
110	Q₈×C₁₃	—	2	4	no	—
111	Q₈×C₁₃	—	2	4	no	—
112	Q₈×C₁₃	—	2	4	no	—
113	Q₈×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	—	4	4	no	—
121	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
129	C₂×Dic₁₃	—	2	4	no	—
130	C₂×Dic₁₃	—	2	4	no	—
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	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
137	#1055	—	4	2	no	—
138	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
139	C₂×C₄×Q₈	—	13	4	no	—
140	C₂×C₄×Q₈	C₂×D₁₃	1	2	yes	—
141	#441	—	26	8	no	—
142	C₄×Q₁₆	Dic₁₃	1	8	yes	complemented
143	C₂×#107	—	13	2	no	—
144	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
145	C₂×#106	—	13	2	no	—
146	C₂×C₄⋊C₈	—	13	1	no	—
147	C₂×#103	—	13	4	no	—
148	C₂×#103	C₂×D₁₃	1	2	yes	—
149	C₂×#97	—	13	2	no	—
150	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
151	#389	—	13	4	no	—
152	#389	—	26	4	no	—
153	#382	—	13	2	no	—
154	#382	—	26	2	no	—
155	C₄×C₄⋊C₄	—	13	1	no	—
156	C₄×C₄⋊C₄	—	26	2	no	—
157	#340	—	13	4	no	—
158	#336	—	13	2	no	—
159	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
160	C₂²×C₁₃	D₄²	1	1	yes	characteristic
161	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
162	C₂²×C₁₃	#575	1	2	yes	—
163	C₂²×C₁₃	#578	1	2	yes	—
164	C₄×C₁₃	—	2	8	no	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	4	4	no	—
168	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₄×C₁₃	#520	1	1	yes	characteristic
170	C₄×C₁₃	#547	1	1	yes	characteristic
171	C₄×C₁₃	#554	1	1	yes	characteristic
172	Dic₁₃	—	4	4	no	—
173	Dic₁₃	—	8	4	no	—
174	C₂²×Q₈	C₂²×D₁₃	1	2	yes	—
175	C₂×Q₁₆	Dic₂₆	1	4	yes	—
176	C₂×Q₁₆	D₅₂	1	4	yes	—
177	C₂×Q₁₆	C₂×Dic₁₃	1	4	yes	—
178	C₂²×C₈	—	13	2	no	—
179	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
181	C₄×Q₈	—	2	4	no	—
182	C₄×Q₈	—	26	8	no	—
183	C₄×Q₈	—	52	4	no	—
184	C₄×Q₈	C₂×Dic₁₃	1	8	yes	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	13	4	no	—
187	C₂×C₄⋊C₄	—	13	2	no	—
188	C₂×C₄⋊C₄	—	26	4	no	—
189	C₂×C₄⋊C₄	—	26	4	no	—
190	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	13	2	no	—
193	C₂×C₄²	—	26	4	no	—
194	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
195	#107	—	52	2	no	—
196	#107	C₂×Dic₁₃	1	4	yes	—
197	#106	—	26	4	no	—
198	C₄⋊C₈	—	26	2	no	—
199	#103	—	26	8	no	—
200	#103	C₂×Dic₁₃	1	8	yes	—
201	#97	—	26	4	no	—
202	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
203	#95	—	13	2	no	—
204	#95	—	26	4	no	—
205	#95	—	26	2	no	—
206	#95	—	26	2	no	—
207	C₂×C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic
208	C₂×C₁₃	#2943	1	1	yes	characteristic
209	C₂×C₁₃	#2950	1	1	yes	characteristic
210	C₂×C₁₃	#3277	1	2	yes	—
211	C₂×C₁₃	#3390	1	2	yes	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	C₂×Dic₂₆	1	4	yes	—
216	C₂×Q₈	C₂×D₅₂	1	4	yes	—
217	C₂×Q₈	C₂²×Dic₁₃	1	4	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	13	4	no	—
221	C₂²×C₄	—	26	4	no	—
222	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
223	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
224	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
225	Q₁₆	C₄⋊Dic₁₃	1	16	yes	complemented
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	26	2	no	—
229	C₂×C₈	—	26	2	no	—
230	C₂×C₈	—	26	2	no	—
231	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
232	C₂×C₈	C₂×D₅₂	1	2	yes	—
233	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
234	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
235	C₄⋊C₄	—	2	8	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	26	4	no	—
238	C₄⋊C₄	—	26	4	no	—
239	C₄⋊C₄	—	52	2	no	—
240	C₄⋊C₄	—	52	8	no	—
241	C₄⋊C₄	—	52	8	no	—
242	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
243	C₄²	—	2	8	no	—
244	C₄²	—	26	2	no	—
245	C₄²	—	26	2	no	—
246	C₄²	—	52	8	no	—
247	C₄²	#6129	1	2	yes	—
248	C₄²	C₂²×Dic₁₃	1	2	yes	—
249	C₁₃	#22675	1	1	yes	characteristic, complemented, Sylow
250	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
251	Q₈	—	2	4	no	—
252	Q₈	—	2	4	no	—
253	Q₈	—	2	4	no	—
254	Q₈	C₂×C₄⋊Dic₁₃	1	8	yes	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	26	4	no	—
263	C₂×C₄	—	26	4	no	—
264	C₂×C₄	—	26	4	no	—
265	C₂×C₄	—	52	4	no	—
266	C₂×C₄	—	52	4	no	—
267	C₂×C₄	—	52	4	no	—
268	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
269	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
270	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
272	C₂×C₄	C₂×#6129	1	1	yes	characteristic
273	C₂×C₄	#88256	1	1	yes	characteristic
274	C₂×C₄	#88257	1	1	yes	characteristic
275	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	52	2	no	—
278	C₈	C₂×C₄⋊Dic₁₃	1	4	yes	—
279	C₂²	D₄×Dic₂₆	1	1	yes	characteristic
280	C₂²	D₄×D₅₂	1	1	yes	characteristic
281	C₂²	#11719471	1	2	yes	—
282	C₂²	#11719473	1	2	yes	—
283	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
284	C₄	—	2	8	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	4	no	—
288	C₄	—	52	4	no	—
289	C₄	—	104	4	no	—
290	C₄	#11719248	1	1	yes	characteristic
291	C₄	#11719260	1	1	yes	characteristic
292	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
293	C₄	#11719532	1	1	yes	characteristic
294	C₂	D₄×C₄⋊Dic₁₃	1	1	yes	characteristic
295	C₂	#13123637	1	1	yes	characteristic
296	C₂	#13123704	1	1	yes	characteristic
297	C₂	#13126492	1	2	yes	—
298	C₂	#13127801	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×Q₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₁₃
hypocenter	C₁₃	#22675
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
12. C₄×C₁₃×Q₁₆	C₄	semidirect
77. C₁₃×Q₁₆	C₄⋊C₄	semidirect
142. C₄×Q₁₆	Dic₁₃	semidirect
225. Q₁₆	C₄⋊Dic₁₃	semidirect
249. C₁₃	#22675	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718845	2
#13117757	2
#13115575	2
#13115092	2
#13115000	2
#13114851	2
#13114703	2
C₂×C₄×C₁₃×Q₁₆	2
#22675	13