Subgroups — #15750927

Order 3328.

444 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13117313	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	#13112661	C₂	1	2	yes	maximal
4	#13112658	C₂	1	2	yes	maximal
5	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₂₆	C₄	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
8	C₄³×C₁₃	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
10	#11718594	—	2	1	no	—
11	#11718594	—	2	1	no	—
12	#11718593	—	2	1	no	—
13	C₄×#6101	—	2	1	no	—
14	#11718590	C₄	1	1	yes	characteristic
15	#11718590	C₂²	1	1	yes	characteristic
16	#11718589	C₄	1	1	yes	characteristic
17	#11718589	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊#1055	—	2	1	no	—
19	C₂×C₄×#1055	—	2	1	no	—
20	C₁₃×#336	—	2	1	no	—
21	C₁₃×#336	C₂²	1	2	yes	—
22	#11718116	C₂²	1	2	yes	—
23	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×#6101	—	2	1	no	—
36	C₂×#6101	—	2	1	no	—
37	C₂×#6101	—	2	1	no	—
38	C₂×C₄×Dic₁₃	—	2	1	no	—
39	C₂×C₄×Dic₁₃	—	2	1	no	—
40	C₂²×#1055	—	2	2	no	—
41	C₄×Dic₂₆	—	2	2	no	—
42	C₄×Dic₂₆	C₈	1	4	yes	complemented
43	C₁₃×C₄⋊C₈	—	4	2	no	—
44	#88071	—	2	1	no	—
45	#88071	—	2	1	no	—
46	#88071	—	2	1	no	—
47	#88071	—	2	1	no	—
48	#88071	—	2	1	no	—
49	#88071	—	2	1	no	—
50	#88071	—	2	1	no	—
51	C₄⋊#1055	—	2	4	no	—
52	C₄×#1055	—	4	2	no	—
53	#7577	—	13	1	no	maximal, Sylow
54	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₁₃	—	2	1	no	—
62	C₂²×Dic₁₃	—	2	1	no	—
63	C₂²×Dic₁₃	—	2	1	no	—
64	C₂²×Dic₁₃	—	2	1	no	—
65	C₂×Dic₂₆	—	2	1	no	—
66	C₂×Dic₂₆	—	4	1	no	—
67	C₂×Dic₂₆	—	4	1	no	—
68	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₁₃	—	4	2	no	—
70	C₂×C₈×C₁₃	—	4	2	no	—
71	C₂×C₈×C₁₃	—	4	2	no	—
72	C₄²×C₁₃	—	2	2	no	—
73	C₄²×C₁₃	—	2	1	no	—
74	C₄²×C₁₃	—	2	1	no	—
75	C₄²×C₁₃	—	4	1	no	—
76	C₄²×C₁₃	—	4	1	no	—
77	C₄²×C₁₃	—	4	1	no	—
78	C₄²×C₁₃	C₂×C₈	1	2	yes	—
79	C₄²×C₁₃	M₁₆	1	2	yes	—
80	C₄²×C₁₃	SD₁₆	1	2	yes	—
81	C₄²×C₁₃	Q₁₆	1	2	yes	—
82	C₄⋊Dic₁₃	—	2	1	no	—
83	C₄⋊Dic₁₃	—	4	1	no	—
84	C₄⋊Dic₁₃	—	4	1	no	—
85	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
86	#6101	—	2	4	no	—
87	#6101	—	4	2	no	—
88	#6101	—	4	1	no	—
89	#6101	—	4	1	no	—
90	C₄×Dic₁₃	—	2	4	no	—
91	C₄×Dic₁₃	—	4	2	no	—
92	C₂×#1055	—	2	4	no	—
93	C₂×#1055	—	4	2	no	—
94	C₂×#1055	—	4	2	no	—
95	#2295	—	13	1	no	—
96	#1831	D₁₃	1	1	yes	characteristic
97	C₄×C₄⋊C₈	—	13	1	no	—
98	#1272	—	13	2	no	—
99	#1271	—	13	2	no	—
100	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
115	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
116	C₂×C₄×C₁₃	#103	1	2	yes	—
117	C₂×C₄×C₁₃	#103	1	2	yes	—
118	C₂×C₄×C₁₃	#104	1	2	yes	—
119	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
123	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
125	C₂×Dic₁₃	—	2	2	no	—
126	C₂×Dic₁₃	—	2	2	no	—
127	C₂×Dic₁₃	—	4	1	no	—
128	C₂×Dic₁₃	—	4	1	no	—
129	C₂×Dic₁₃	—	4	1	no	—
130	C₂×Dic₁₃	—	4	1	no	—
131	C₂×Dic₁₃	—	4	1	no	—
132	C₂×Dic₁₃	—	4	1	no	—
133	C₂×Dic₁₃	—	4	1	no	—
134	C₂×Dic₁₃	—	4	1	no	—
135	C₂×Dic₁₃	—	4	1	no	—
136	C₂×Dic₁₃	—	4	1	no	—
137	Dic₂₆	—	4	2	no	—
138	Dic₂₆	—	4	2	no	—
139	C₈×C₁₃	—	8	2	no	—
140	#1055	—	4	4	no	—
141	C₂×C₄×Q₈	—	13	1	no	—
142	C₂×C₄⋊C₈	—	2	1	no	—
143	C₂×C₄⋊C₈	—	26	1	no	—
144	C₂×C₄×C₈	—	26	1	no	—
145	#391	—	13	1	no	—
146	#391	—	13	1	no	—
147	#389	—	13	1	no	—
148	#389	—	13	1	no	—
149	#387	—	26	1	no	—
150	#387	—	26	1	no	—
151	C₄×C₄⋊C₄	—	13	1	no	—
152	C₄×C₄⋊C₄	—	26	1	no	—
153	#376	—	26	1	no	—
154	C₄³	C₂×D₁₃	1	1	yes	characteristic
155	#336	—	2	1	no	—
156	#336	—	13	2	no	—
157	#336	C₂×D₁₃	1	2	yes	—
158	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
159	C₂²×C₁₃	#410	1	1	yes	characteristic
160	C₂²×C₁₃	#413	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
162	C₂²×C₁₃	#417	1	1	yes	characteristic
163	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
164	C₂²×C₁₃	#421	1	1	yes	characteristic
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	2	1	no	—
170	C₄×C₁₃	—	2	1	no	—
171	C₄×C₁₃	—	2	1	no	—
172	C₄×C₁₃	—	2	1	no	—
173	C₄×C₁₃	—	4	1	no	—
174	C₄×C₁₃	Q₈⋊C₈	1	4	yes	—
175	Dic₁₃	—	4	2	no	—
176	Dic₁₃	—	4	2	no	—
177	Dic₁₃	—	8	1	no	—
178	Dic₁₃	—	8	1	no	—
179	C₂²×Q₈	—	13	1	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	26	2	no	—
182	C₄×Q₈	—	13	4	no	—
183	C₄×Q₈	—	26	2	no	—
184	C₂×C₄⋊C₄	—	13	1	no	—
185	C₂×C₄⋊C₄	—	13	1	no	—
186	C₂×C₄⋊C₄	—	13	1	no	—
187	C₂×C₄⋊C₄	—	26	1	no	—
188	C₂×C₄⋊C₄	—	26	1	no	—
189	C₂×C₄⋊C₄	—	26	1	no	—
190	C₂×C₄²	—	26	1	no	—
191	C₂×C₄²	—	26	1	no	—
192	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
193	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
194	C₂×C₄²	D₅₂	1	1	yes	characteristic
195	C₂×C₄²	D₅₂	1	1	yes	characteristic
196	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
197	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
199	C₄⋊C₈	—	4	2	no	—
200	C₄⋊C₈	—	26	4	no	—
201	C₄×C₈	—	52	2	no	—
202	#95	—	26	1	no	—
203	#95	—	26	1	no	—
204	#95	—	26	1	no	—
205	#95	—	26	1	no	—
206	#95	—	26	1	no	—
207	#95	—	26	1	no	—
208	#95	—	26	1	no	—
209	C₂×C₁₃	#1458	1	1	yes	characteristic
210	C₂×C₁₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
211	C₂×C₁₃	#1477	1	1	yes	characteristic
212	C₂×C₁₃	#1509	1	1	yes	characteristic
213	C₂×C₁₃	#1518	1	1	yes	characteristic
214	C₂×C₁₃	#1529	1	1	yes	characteristic
215	C₂×C₁₃	#1550	1	1	yes	characteristic
216	C₂×Q₈	—	13	2	no	—
217	C₂×Q₈	—	26	1	no	—
218	C₂×Q₈	—	52	1	no	—
219	C₂×Q₈	—	52	1	no	—
220	C₂²×C₄	—	26	1	no	—
221	C₂²×C₄	—	26	1	no	—
222	C₂²×C₄	—	26	1	no	—
223	C₂²×C₄	—	26	1	no	—
224	C₂²×C₄	#6103	1	1	yes	characteristic
225	C₂²×C₄	#6103	1	1	yes	characteristic
226	C₂²×C₄	#6103	1	1	yes	characteristic
227	C₂²×C₄	#6103	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	26	4	no	—
235	C₂×C₈	—	52	2	no	—
236	C₂×C₈	—	52	2	no	—
237	C₄⋊C₄	—	13	2	no	—
238	C₄⋊C₄	—	26	1	no	—
239	C₄⋊C₄	—	26	4	no	—
240	C₄⋊C₄	—	52	1	no	—
241	C₄⋊C₄	—	52	1	no	—
242	C₄⋊C₄	—	52	2	no	—
243	C₄⋊C₄	—	52	1	no	—
244	C₄⋊C₄	—	52	1	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	4	1	no	—
249	C₄²	—	4	1	no	—
250	C₄²	—	4	1	no	—
251	C₄²	—	26	4	no	—
252	C₄²	—	52	2	no	—
253	C₄²	C₈×D₁₃	1	2	yes	—
254	C₄²	#6094	1	2	yes	—
255	C₄²	#6105	1	2	yes	—
256	C₄²	#6107	1	2	yes	—
257	C₁₃	#7577	1	1	yes	characteristic, complemented, Sylow
258	C₂³	C₂×#6103	1	1	yes	characteristic
259	Q₈	—	52	2	no	—
260	Q₈	—	52	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	26	2	no	—
275	C₂×C₄	—	26	2	no	—
276	C₂×C₄	—	52	1	no	—
277	C₂×C₄	—	52	1	no	—
278	C₂×C₄	—	52	1	no	—
279	C₂×C₄	—	52	1	no	—
280	C₂×C₄	—	52	1	no	—
281	C₂×C₄	—	52	1	no	—
282	C₂×C₄	—	52	1	no	—
283	C₂×C₄	—	52	1	no	—
284	C₂×C₄	—	52	1	no	—
285	C₂×C₄	—	52	1	no	—
286	C₂×C₄	#88050	1	2	yes	—
287	C₂×C₄	#88050	1	2	yes	—
288	C₂×C₄	#88060	1	2	yes	—
289	C₂×C₄	#88060	1	2	yes	—
290	C₂×C₄	#88065	1	2	yes	—
291	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
292	C₂×C₄	C₂×#6094	1	1	yes	characteristic
293	C₂×C₄	#88162	1	1	yes	characteristic
294	C₂×C₄	#88163	1	1	yes	characteristic
295	C₂×C₄	C₂×#6105	1	1	yes	characteristic
296	C₂×C₄	C₂×#6107	1	1	yes	characteristic
297	C₈	—	8	2	no	—
298	C₈	—	52	4	no	—
299	C₂²	C₂×#88050	1	1	yes	characteristic
300	C₂²	#11718666	1	1	yes	characteristic
301	C₂²	#11718667	1	1	yes	characteristic
302	C₂²	C₂×#88060	1	1	yes	characteristic
303	C₂²	#11718793	1	1	yes	characteristic
304	C₂²	#11718794	1	1	yes	characteristic
305	C₂²	C₂×#88065	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	4	1	no	—
315	C₄	—	52	2	no	—
316	C₄	—	52	2	no	—
317	C₄	—	104	1	no	—
318	C₄	—	104	1	no	—
319	C₄	#11718076	1	4	yes	—
320	C₂	#13113148	1	1	yes	characteristic
321	C₂	C₂×#11718076	1	1	yes	characteristic
322	C₂	#13113224	1	1	yes	characteristic
323	C₂	#13113362	1	1	yes	characteristic
324	C₂	#13113399	1	1	yes	characteristic
325	C₂	#13113444	1	1	yes	characteristic
326	C₂	#13113527	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6103
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1831	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1831	D₁₃
hypocenter	C₁₃	#7577
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
42. C₄×Dic₂₆	C₈	semidirect
257. C₁₃	#7577	semidirect

Maximal subgroups

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

subgroup	index
#13117313	2
C₄×C₄⋊#1055	2
#13112661	2
#13112658	2
C₁₃×#1831	2
#7577	13