Subgroups — #15881235

Order 3328.

914 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718514	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
3	#13117748	C₂	1	2	yes	maximal
4	#13115572	C₂	1	2	yes	maximal
5	#13115101	C₂	1	4	yes	maximal
6	#13114838	C₂	1	2	yes	maximal
7	#13114703	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#1844	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
10	C₂×#88141	C₂²	1	4	yes	—
11	C₂×C₄×Dic₂₆	C₂²	1	2	yes	—
12	C₂×C₁₃×#106	C₂²	1	2	yes	—
13	C₂×C₁₃×#106	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#384	C₂²	1	2	yes	—
16	C₂×#88058	C₂²	1	1	yes	characteristic
17	C₂×#88056	C₂²	1	2	yes	—
18	C₂×#88055	C₂²	1	2	yes	—
19	#11718637	—	2	2	no	—
20	C₄⋊C₄×Dic₁₃	—	2	2	no	—
21	#11718634	C₂²	1	4	yes	—
22	#11718633	—	2	4	no	—
23	C₂×#88050	C₂²	1	4	yes	—
24	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
26	#11718514	—	2	8	no	—
27	C₄×Dic₅₂	C₄	1	8	yes	complemented
28	#11718142	C₂²	1	2	yes	—
29	#11718121	C₂²	1	4	yes	—
30	C₂²×Dic₂₆	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
38	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
39	C₂×#6101	—	2	2	no	—
40	C₂×#6101	—	2	4	no	—
41	C₂×#6101	—	2	4	no	—
42	C₂×C₄×Dic₁₃	—	2	2	no	—
43	C₂×C₄×Dic₁₃	—	2	4	no	—
44	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
45	C₂²×#1055	C₂³	1	2	yes	—
46	C₂×Dic₅₂	C₂×C₄	1	4	yes	—
47	C₂×Dic₅₂	D₄	1	4	yes	—
48	C₂×Dic₅₂	Q₈	1	4	yes	—
49	#88141	—	2	8	no	—
50	#88141	—	4	4	no	—
51	C₄×Dic₂₆	—	2	4	no	—
52	C₄×Dic₂₆	C₂×C₄	1	8	yes	—
53	C₁₃×#106	—	2	4	no	—
54	C₁₃×#106	—	4	2	no	—
55	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
56	#88071	—	2	4	no	—
57	#88071	D₄	1	2	yes	—
58	#88058	C₂×C₄	1	4	yes	—
59	#88056	C₂×C₄	1	8	yes	—
60	#88055	—	2	4	no	—
61	#88050	—	2	8	no	—
62	C₄⋊#1055	—	2	2	no	—
63	#22674	—	13	1	no	maximal, Sylow
64	C₂²×C₄×C₁₃	—	2	4	no	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₁₃	—	2	2	no	—
69	C₂²×Dic₁₃	—	2	2	no	—
70	C₂²×Dic₁₃	C₂×D₄	1	4	yes	—
71	C₂×Dic₂₆	—	2	2	no	—
72	C₂×Dic₂₆	—	2	2	no	—
73	C₂×Dic₂₆	—	2	2	no	—
74	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
75	C₂×Dic₂₆	C₂×D₄	1	4	yes	—
76	C₂×Dic₂₆	C₂×Q₈	1	4	yes	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
83	C₁₃×C₄⋊C₄	—	2	4	no	—
84	C₁₃×C₄⋊C₄	—	2	4	no	—
85	C₁₃×C₄⋊C₄	—	4	4	no	—
86	C₁₃×C₄⋊C₄	—	8	2	no	—
87	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₁₃	#41	1	2	yes	—
89	C₄⋊Dic₁₃	—	2	4	no	—
90	C₄⋊Dic₁₃	—	2	2	no	—
91	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
92	#6101	—	2	8	no	—
93	#6101	—	4	8	no	—
94	#6101	—	8	4	no	—
95	C₄×Dic₁₃	—	2	8	no	—
96	C₄×Dic₁₃	—	2	2	no	—
97	C₄×Dic₁₃	—	2	2	no	—
98	C₄×Dic₁₃	—	4	8	no	—
99	C₂×#1055	—	2	2	no	—
100	C₂×#1055	—	2	2	no	—
101	C₂×#1055	—	2	2	no	—
102	Dic₅₂	C₄⋊C₄	1	16	yes	complemented
103	C₂×#441	—	13	2	no	—
104	C₂×C₄×Q₁₆	—	13	1	no	—
105	Q₈×C₄⋊C₄	—	13	2	no	—
106	#1943	—	13	2	no	—
107	#1865	—	13	4	no	—
108	#1844	D₁₃	1	1	yes	characteristic
109	#1803	—	13	2	no	—
110	#1770	—	13	1	no	—
111	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	4	4	no	—
114	C₂×C₄×C₁₃	—	4	4	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₃	#137	1	4	yes	—
119	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
125	C₂×Dic₁₃	—	2	4	no	—
126	C₂×Dic₁₃	—	2	4	no	—
127	C₂×Dic₁₃	—	2	4	no	—
128	C₂×Dic₁₃	—	2	4	no	—
129	C₂×Dic₁₃	—	2	4	no	—
130	C₂×Dic₁₃	—	2	4	no	—
131	C₂×Dic₁₃	—	2	4	no	—
132	C₂×Dic₁₃	—	4	2	no	—
133	C₂×Dic₁₃	—	4	2	no	—
134	Dic₂₆	—	2	4	no	—
135	Dic₂₆	—	2	4	no	—
136	Dic₂₆	—	2	4	no	—
137	Dic₂₆	C₂×C₄⋊C₄	1	8	yes	—
138	C₈×C₁₃	—	2	2	no	—
139	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
140	#1055	—	4	2	no	—
141	C₂²×Q₁₆	—	13	1	no	—
142	C₂×C₄×Q₈	—	13	4	no	—
143	C₂×C₄×Q₈	—	13	2	no	—
144	#441	—	26	8	no	—
145	C₄×Q₁₆	—	13	8	no	—
146	C₂×#107	—	13	1	no	—
147	C₂×#106	C₂×D₁₃	1	2	yes	—
148	C₂×#106	C₂×D₁₃	1	2	yes	—
149	C₂×C₄⋊C₈	—	13	1	no	—
150	C₂×#103	—	13	2	no	—
151	C₂×#103	—	13	4	no	—
152	C₂×#97	—	13	2	no	—
153	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
154	#389	—	13	4	no	—
155	#389	—	26	4	no	—
156	#384	—	26	2	no	—
157	#384	C₂×D₁₃	1	2	yes	—
158	C₄×C₄⋊C₄	—	13	1	no	—
159	C₄×C₄⋊C₄	—	26	2	no	—
160	#340	—	13	4	no	—
161	#336	—	13	2	no	—
162	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂²×C₁₃	D₄²	1	1	yes	characteristic
164	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
165	C₂²×C₁₃	C₂×#137	1	2	yes	—
166	C₂²×C₁₃	#577	1	2	yes	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	8	4	no	—
169	C₄×C₁₃	#441	1	4	yes	complemented
170	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
171	C₄×C₁₃	#520	1	1	yes	characteristic
172	C₄×C₁₃	#547	1	1	yes	characteristic
173	C₄×C₁₃	#554	1	1	yes	characteristic
174	Dic₁₃	—	2	8	no	—
175	Dic₁₃	—	4	4	no	—
176	Dic₁₃	—	4	4	no	—
177	C₂²×Q₈	—	13	2	no	—
178	C₂×Q₁₆	—	13	4	no	—
179	C₂×Q₁₆	—	13	4	no	—
180	C₂×Q₁₆	—	13	4	no	—
181	C₂²×C₈	—	13	2	no	—
182	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
184	C₄×Q₈	—	13	8	no	—
185	C₄×Q₈	—	26	4	no	—
186	C₄×Q₈	—	26	8	no	—
187	C₄×Q₈	—	52	4	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	13	2	no	—
191	C₂×C₄⋊C₄	—	13	2	no	—
192	C₂×C₄⋊C₄	—	26	2	no	—
193	C₂×C₄⋊C₄	—	26	4	no	—
194	C₂×C₄⋊C₄	—	26	4	no	—
195	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
196	C₂×C₄²	—	13	2	no	—
197	C₂×C₄²	—	26	2	no	—
198	C₂×C₄²	—	26	4	no	—
199	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
200	#107	—	13	4	no	—
201	#106	—	2	4	no	—
202	#106	—	4	2	no	—
203	C₄⋊C₈	—	26	2	no	—
204	#103	—	13	8	no	—
205	#103	—	26	8	no	—
206	#97	—	26	4	no	—
207	C₄×C₈	C₄×D₁₃	1	4	yes	—
208	#95	—	13	2	no	—
209	#95	—	26	4	no	—
210	C₂×C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic
211	C₂×C₁₃	C₂×#441	1	1	yes	characteristic
212	C₂×C₁₃	#2952	1	1	yes	characteristic
213	C₂×C₁₃	#3278	1	2	yes	—
214	C₂×C₁₃	#3387	1	2	yes	—
215	C₂×Q₈	—	13	4	no	—
216	C₂×Q₈	—	13	4	no	—
217	C₂×Q₈	—	13	4	no	—
218	C₂×Q₈	—	26	2	no	—
219	C₂×Q₈	—	26	2	no	—
220	C₂×Q₈	—	26	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	13	4	no	—
223	C₂²×C₄	—	26	2	no	—
224	C₂²×C₄	—	26	2	no	—
225	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
226	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
228	Q₁₆	—	13	16	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	26	2	no	—
232	C₂×C₈	—	26	2	no	—
233	C₂×C₈	—	26	2	no	—
234	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
236	C₂×C₈	D₄×D₁₃	1	2	yes	—
237	C₂×C₈	Q₈×D₁₃	1	2	yes	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₄⋊C₄	—	8	2	no	—
242	C₄⋊C₄	—	13	4	no	—
243	C₄⋊C₄	—	26	8	no	—
244	C₄⋊C₄	—	26	4	no	—
245	C₄⋊C₄	—	26	2	no	—
246	C₄⋊C₄	—	52	8	no	—
247	C₄⋊C₄	—	104	4	no	—
248	C₄²	—	26	8	no	—
249	C₄²	—	26	2	no	—
250	C₄²	—	26	2	no	—
251	C₄²	—	52	8	no	—
252	C₄²	C₂×C₄×D₁₃	1	2	yes	—
253	C₄²	#6131	1	2	yes	—
254	C₁₃	#22674	1	1	yes	characteristic, complemented, Sylow
255	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
256	Q₈	—	13	8	no	—
257	Q₈	—	26	4	no	—
258	Q₈	—	26	4	no	—
259	Q₈	—	26	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	26	4	no	—
265	C₂×C₄	—	26	4	no	—
266	C₂×C₄	—	26	4	no	—
267	C₂×C₄	—	26	4	no	—
268	C₂×C₄	—	26	4	no	—
269	C₂×C₄	—	26	4	no	—
270	C₂×C₄	—	26	4	no	—
271	C₂×C₄	—	52	2	no	—
272	C₂×C₄	—	52	2	no	—
273	C₂×C₄	#88147	1	2	yes	—
274	C₂×C₄	#88147	1	2	yes	—
275	C₂×C₄	#88169	1	4	yes	—
276	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
278	C₂×C₄	#88251	1	1	yes	characteristic
279	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
280	C₂×C₄	C₂×#6131	1	1	yes	characteristic
281	C₂×C₄	#88254	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	52	2	no	—
284	C₈	C₄⋊C₄×D₁₃	1	4	yes	—
285	C₂²	C₂×#88147	1	1	yes	characteristic
286	C₂²	#11719422	1	1	yes	characteristic
287	C₂²	#11719432	1	1	yes	characteristic
288	C₂²	C₂×#88169	1	2	yes	—
289	C₂²	#11719467	1	2	yes	—
290	C₄	—	2	2	no	—
291	C₄	—	8	4	no	—
292	C₄	—	26	8	no	—
293	C₄	—	52	4	no	—
294	C₄	—	52	4	no	—
295	C₄	#11718514	1	4	yes	complemented
296	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
297	C₄	#11719209	1	1	yes	characteristic
298	C₄	#11719431	1	1	yes	characteristic
299	C₄	#11719435	1	1	yes	characteristic
300	C₂	#13117736	1	1	yes	characteristic
301	C₂	C₂×#11718514	1	1	yes	characteristic
302	C₂	#13123719	1	1	yes	characteristic
303	C₂	#13126501	1	2	yes	—
304	C₂	#13127773	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1844	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1844	D₁₃
hypocenter	C₁₃	#22674
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
27. C₄×Dic₅₂	C₄	semidirect
102. Dic₅₂	C₄⋊C₄	semidirect
169. C₄×C₁₃	#441	semidirect
254. C₁₃	#22674	semidirect
295. C₄	#11718514	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718514	2
C₂×C₄×Dic₅₂	2
#13117748	2
#13115572	2
#13115101	2
#13114838	2
#13114703	2
C₁₃×#1844	2
#22674	13