Subgroups — #16785048

Order 3328.

566 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130768	C₂	1	1	yes	maximal, characteristic
2	C₄²×#1057	C₂	1	1	yes	maximal, characteristic
3	#13129672	C₂	1	4	yes	maximal
4	#13117403	C₂	1	1	yes	maximal, characteristic
5	C₂×#88141	—	2	1	no	—
6	C₂×#88123	C₄	1	1	yes	characteristic
7	C₂×C₄⋊#1057	—	2	1	no	—
8	C₂×C₄⋊#1057	—	2	1	no	—
9	C₂×C₄⋊#1057	C₂²	1	2	yes	—
10	C₂×C₄×#1057	—	2	1	no	—
11	C₂×C₄×#1057	—	2	1	no	—
12	C₂×C₄×#1057	C₂²	1	2	yes	—
13	C₁₃×#384	C₄	1	1	yes	characteristic
14	C₄⋊C₄×Dic₁₃	—	2	1	no	—
15	C₄⋊C₄×Dic₁₃	C₂²	1	2	yes	—
16	#11718635	—	2	1	no	—
17	#11718634	—	2	1	no	—
18	#11718634	C₄	1	2	yes	—
19	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂²×Dic₂₆	—	2	1	no	—
21	C₂²×#1057	—	2	2	no	—
22	C₂²×#1057	—	2	2	no	—
23	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
25	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₃	—	2	1	no	—
28	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
29	C₂×#6101	—	2	2	no	—
30	C₂×C₄×Dic₁₃	—	2	2	no	—
31	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
34	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
35	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
37	#88141	—	4	2	no	—
38	#88141	—	4	2	no	—
39	#88123	—	2	2	no	—
40	#88123	C₈	1	4	yes	complemented
41	C₄⋊#1057	—	2	4	no	—
42	C₄⋊#1057	—	4	2	no	—
43	C₄⋊#1057	—	4	2	no	—
44	C₄×#1057	—	2	4	no	—
45	C₄×#1057	—	2	4	no	—
46	C₄×#1057	—	2	2	no	—
47	C₄×#1057	Q₈	1	4	yes	complemented
48	#88071	—	2	2	no	—
49	#88071	—	2	2	no	—
50	#9870	—	13	1	no	maximal, Sylow
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₁₃	—	2	2	no	—
56	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×Dic₂₆	—	2	4	no	—
61	C₂×Dic₂₆	—	4	2	no	—
62	C₂×Dic₂₆	—	4	2	no	—
63	C₂×#1057	—	2	4	no	—
64	C₂×#1057	—	2	4	no	—
65	C₂×#1057	—	2	4	no	—
66	C₂×#1057	—	4	2	no	—
67	C₂×#1057	—	4	2	no	—
68	C₂×#1057	—	4	2	no	—
69	C₁₃×C₄⋊C₄	—	2	4	no	—
70	C₁₃×C₄⋊C₄	—	4	2	no	—
71	C₁₃×C₄⋊C₄	—	4	2	no	—
72	C₄²×C₁₃	C₂×C₈	1	2	yes	—
73	C₄²×C₁₃	M₁₆	1	2	yes	—
74	C₄⋊Dic₁₃	—	2	4	no	—
75	C₄⋊Dic₁₃	—	2	2	no	—
76	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
77	#6101	—	4	4	no	—
78	C₄×Dic₁₃	—	2	2	no	—
79	C₄×Dic₁₃	—	2	2	no	—
80	C₄×Dic₁₃	—	2	2	no	—
81	C₄×Dic₁₃	—	4	2	no	—
82	C₄×Dic₁₃	—	4	2	no	—
83	C₄×Dic₁₃	D₈	1	2	yes	—
84	C₄×Dic₁₃	SD₁₆	1	2	yes	—
85	C₄×Dic₁₃	SD₁₆	1	2	yes	—
86	C₄×Dic₁₃	Q₁₆	1	2	yes	—
87	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
88	C₄×Dic₁₃	#41	1	2	yes	—
89	C₄×C₄⋊Q₈	—	13	1	no	—
90	#1842	—	13	1	no	—
91	C₄²×C₈	—	13	1	no	—
92	#1279	—	13	4	no	—
93	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
96	C₂×C₄×C₁₃	—	4	2	no	—
97	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
98	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
99	C₂×C₄×C₁₃	#102	1	2	yes	—
100	C₂×C₄×C₁₃	#103	1	2	yes	—
101	C₂×C₄×C₁₃	#104	1	2	yes	—
102	C₂×C₄×C₁₃	#104	1	2	yes	—
103	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#131	1	2	yes	—
107	C₂×Dic₁₃	—	2	1	no	—
108	C₂×Dic₁₃	—	2	1	no	—
109	C₂×Dic₁₃	—	2	1	no	—
110	C₂×Dic₁₃	—	2	1	no	—
111	C₂×Dic₁₃	—	2	1	no	—
112	C₂×Dic₁₃	—	2	1	no	—
113	C₂×Dic₁₃	—	2	4	no	—
114	C₂×Dic₁₃	—	4	2	no	—
115	C₂×Dic₁₃	—	4	2	no	—
116	C₂×Dic₁₃	#102	1	2	yes	—
117	C₂×Dic₁₃	#103	1	2	yes	—
118	C₂×Dic₁₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×Dic₁₃	#124	1	1	yes	characteristic
120	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
121	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
122	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
125	Dic₂₆	—	4	8	no	—
126	#1057	—	2	8	no	—
127	#1057	—	8	2	no	—
128	C₂×C₄⋊Q₈	—	13	1	no	—
129	C₂×C₄×Q₈	—	26	1	no	—
130	C₂×C₄⋊C₈	—	13	2	no	—
131	C₂×C₄⋊C₈	—	26	1	no	—
132	C₂×C₄⋊C₈	—	26	1	no	—
133	C₂×C₄×C₈	—	13	2	no	—
134	C₂×C₄×C₈	—	26	1	no	—
135	C₂×C₄×C₈	—	26	1	no	—
136	#391	—	26	1	no	—
137	#389	—	13	2	no	—
138	#389	—	26	1	no	—
139	#384	#261	1	1	yes	characteristic
140	C₄×C₄⋊C₄	—	13	2	no	—
141	C₄×C₄⋊C₄	—	26	1	no	—
142	C₄³	—	13	1	no	—
143	C₂²×C₁₃	#391	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₂²×C₁₃	#408	1	1	yes	characteristic
146	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
147	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
148	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
149	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
150	C₄×C₁₃	—	8	2	no	—
151	C₄×C₁₃	D₄⋊C₈	1	4	yes	—
152	C₄×C₁₃	Q₈⋊C₈	1	4	yes	—
153	C₄×C₁₃	C₈×Q₈	1	2	yes	—
154	C₄×C₁₃	#446	1	2	yes	—
155	Dic₁₃	—	2	2	no	—
156	Dic₁₃	—	2	2	no	—
157	Dic₁₃	—	2	2	no	—
158	Dic₁₃	—	4	4	no	—
159	Dic₁₃	#486	1	1	yes	characteristic
160	Dic₁₃	#487	1	1	yes	characteristic
161	Dic₁₃	#498	1	1	yes	characteristic
162	Dic₁₃	#500	1	1	yes	characteristic
163	C₂²×Q₈	—	26	1	no	—
164	C₂²×C₈	—	26	2	no	—
165	C₂²×C₈	—	26	2	no	—
166	C₄⋊Q₈	—	13	4	no	—
167	C₄⋊Q₈	—	26	2	no	—
168	C₄×Q₈	—	52	2	no	—
169	C₄×Q₈	—	52	2	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	—	13	2	no	—
173	C₂×C₄⋊C₄	—	26	1	no	—
174	C₂×C₄⋊C₄	—	26	2	no	—
175	C₂×C₄⋊C₄	C₂×#261	1	2	yes	—
176	C₂×C₄²	—	13	1	no	—
177	C₂×C₄²	—	13	1	no	—
178	C₂×C₄²	—	13	1	no	—
179	C₂×C₄²	—	13	1	no	—
180	C₂×C₄²	—	13	1	no	—
181	C₂×C₄²	—	13	1	no	—
182	C₂×C₄²	—	26	2	no	—
183	C₂×C₄²	C₂×#261	1	1	yes	characteristic
184	C₄⋊C₈	—	26	4	no	—
185	C₄⋊C₈	—	52	2	no	—
186	C₄⋊C₈	—	52	2	no	—
187	C₄×C₈	—	13	4	no	—
188	C₄×C₈	—	26	2	no	—
189	C₄×C₈	—	26	4	no	—
190	C₄×C₈	—	26	4	no	—
191	#95	—	26	2	no	—
192	#95	—	26	2	no	—
193	C₂×C₁₃	C₂×D₄⋊C₈	1	1	yes	characteristic
194	C₂×C₁₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
195	C₂×C₁₃	#1492	1	1	yes	characteristic
196	C₂×C₁₃	#1493	1	1	yes	characteristic
197	C₂×C₁₃	#1984	1	1	yes	characteristic
198	C₂×C₁₃	#1990	1	1	yes	characteristic
199	C₂×C₁₃	#1994	1	1	yes	characteristic
200	C₂×Q₈	—	26	4	no	—
201	C₂×Q₈	—	52	2	no	—
202	C₂×Q₈	—	52	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	13	1	no	—
205	C₂²×C₄	—	13	1	no	—
206	C₂²×C₄	—	13	1	no	—
207	C₂²×C₄	—	13	1	no	—
208	C₂²×C₄	—	26	2	no	—
209	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
211	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
212	C₂×C₈	—	26	4	no	—
213	C₂×C₈	—	26	4	no	—
214	C₂×C₈	—	26	4	no	—
215	C₂×C₈	—	52	2	no	—
216	C₂×C₈	—	52	2	no	—
217	C₂×C₈	—	52	2	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	13	4	no	—
222	C₄⋊C₄	—	26	4	no	—
223	C₄⋊C₄	—	26	2	no	—
224	C₄⋊C₄	—	52	4	no	—
225	C₄²	—	13	2	no	—
226	C₄²	—	13	2	no	—
227	C₄²	—	13	2	no	—
228	C₄²	—	13	2	no	—
229	C₄²	—	13	2	no	—
230	C₄²	—	13	2	no	—
231	C₄²	—	26	2	no	—
232	C₄²	—	26	2	no	—
233	C₄²	—	26	2	no	—
234	C₄²	—	52	2	no	—
235	C₄²	—	52	2	no	—
236	C₄²	D₁₃⋊C₈	1	2	yes	—
237	C₄²	#6118	1	2	yes	—
238	C₁₃	#9870	1	1	yes	characteristic, complemented, Sylow
239	C₂³	C₂×C₂²⋊#261	1	1	yes	characteristic
240	Q₈	—	52	8	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	13	1	no	—
245	C₂×C₄	—	13	1	no	—
246	C₂×C₄	—	13	1	no	—
247	C₂×C₄	—	13	1	no	—
248	C₂×C₄	—	13	1	no	—
249	C₂×C₄	—	13	1	no	—
250	C₂×C₄	—	13	2	no	—
251	C₂×C₄	—	13	2	no	—
252	C₂×C₄	—	13	2	no	—
253	C₂×C₄	—	26	1	no	—
254	C₂×C₄	—	26	1	no	—
255	C₂×C₄	—	26	1	no	—
256	C₂×C₄	—	26	1	no	—
257	C₂×C₄	—	26	1	no	—
258	C₂×C₄	—	26	1	no	—
259	C₂×C₄	—	26	4	no	—
260	C₂×C₄	—	52	2	no	—
261	C₂×C₄	—	52	2	no	—
262	C₂×C₄	#88111	1	2	yes	—
263	C₂×C₄	#88111	1	2	yes	—
264	C₂×C₄	#88115	1	2	yes	—
265	C₂×C₄	#88116	1	2	yes	—
266	C₂×C₄	#88117	1	2	yes	—
267	C₂×C₄	#88118	1	2	yes	—
268	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
269	C₂×C₄	C₂×#6118	1	1	yes	characteristic
270	C₂×C₄	#88238	1	2	yes	—
271	C₂×C₄	Q₈×#261	1	2	yes	—
272	C₈	—	26	8	no	—
273	C₈	—	104	2	no	—
274	C₂²	C₂×#88111	1	1	yes	characteristic
275	C₂²	C₂×#88115	1	1	yes	characteristic
276	C₂²	C₂×#88116	1	1	yes	characteristic
277	C₂²	#11719147	1	1	yes	characteristic
278	C₂²	C₂×#88117	1	1	yes	characteristic
279	C₂²	C₂×#88118	1	1	yes	characteristic
280	C₂²	(C₂×#261)⋊Q₈	1	1	yes	characteristic
281	C₄	—	8	2	no	—
282	C₄	—	13	1	no	—
283	C₄	—	13	1	no	—
284	C₄	—	13	1	no	—
285	C₄	—	13	1	no	—
286	C₄	—	26	2	no	—
287	C₄	—	26	2	no	—
288	C₄	—	26	2	no	—
289	C₄	—	52	4	no	—
290	C₄	#11718243	1	4	yes	—
291	C₄	#11718244	1	4	yes	—
292	C₄	#11719074	1	2	yes	—
293	C₄	#11719080	1	2	yes	—
294	C₂	C₂×#11718243	1	1	yes	characteristic
295	C₂	C₂×#11718244	1	1	yes	characteristic
296	C₂	#13130182	1	1	yes	characteristic
297	C₂	#13130185	1	1	yes	characteristic
298	C₂	#13131006	1	1	yes	characteristic
299	C₂	#13131017	1	1	yes	characteristic
300	C₂	#13131021	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#384	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	#384	#261
hypocenter	C₁₃	#9870
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

Complements

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

N	H ≅ G/N	type
40. #88123	C₈	semidirect
47. C₄×#1057	Q₈	semidirect
238. C₁₃	#9870	semidirect

Maximal subgroups

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

subgroup	index
#13130768	2
C₄²×#1057	2
#13129672	2
#13117403	2
#9870	13