Subgroups — #15753988

Order 3328.

482 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123549	C₂	1	1	yes	maximal, characteristic
2	#13113486	C₂	1	1	yes	maximal, characteristic
3	#13113330	C₂	1	1	yes	maximal, characteristic
4	#13113119	C₂	1	1	yes	maximal, characteristic, complemented
5	#13112739	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112733	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88155	—	2	1	no	—
9	#11719182	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	—	2	1	no	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	#11718771	—	2	1	no	—
15	#11718586	C₂²	1	1	yes	characteristic
16	#11718586	C₂²	1	1	yes	characteristic
17	#11718585	C₂²	1	1	yes	characteristic
18	#11718349	—	2	1	no	—
19	#11718349	C₄	1	2	yes	complemented
20	C₈×D₅₂	—	2	1	no	—
21	C₈×D₅₂	C₄	1	2	yes	complemented
22	C₁₃×#336	—	2	1	no	—
23	#11718077	C₄	1	2	yes	complemented
24	#11718074	C₄	1	2	yes	complemented
25	#11718047	—	2	1	no	—
26	C₈×#1055	—	2	1	no	—
27	#11718043	C₄	1	4	yes	—
28	C₂×#6127	—	2	1	no	—
29	C₂²×C₈×C₁₃	—	2	1	no	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
33	C₂×C₄²×C₁₃	—	2	1	no	—
34	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
35	#88184	—	2	1	no	—
36	#88184	C₂×C₄	1	2	yes	—
37	C₂×#6099	—	2	1	no	—
38	#88155	—	4	2	no	—
39	#88155	—	4	2	no	—
40	C₂×#6094	—	2	2	no	—
41	C₂×C₈×D₁₃	—	2	2	no	—
42	#88129	—	2	1	no	—
43	#88128	C₂×C₄	1	1	yes	characteristic
44	C₄×D₅₂	—	2	1	no	—
45	C₄×D₅₂	C₂×C₄	1	2	yes	—
46	#88123	C₂×C₄	1	1	yes	characteristic
47	C₁₃×C₄⋊C₈	—	2	1	no	—
48	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₁₃	—	4	2	no	—
50	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
54	C₁₃×#95	C₂×C₄	1	2	yes	—
55	C₂²⋊#1055	—	2	1	no	—
56	C₂²⋊#1055	C₂×C₄	1	2	yes	—
57	#88060	—	2	2	no	—
58	#88060	—	2	2	no	—
59	C₄⋊#1055	—	2	1	no	—
60	C₄⋊#1055	—	2	1	no	—
61	C₄⋊#1055	C₂×C₄	1	2	yes	—
62	C₄⋊#1055	C₂×C₄	1	2	yes	—
63	C₄×#1055	—	2	1	no	—
64	C₄×#1055	—	2	1	no	—
65	#8251	—	13	1	no	maximal, Sylow
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
70	C₂×C₂²⋊D₁₃	—	2	2	no	—
71	#6127	—	4	1	no	—
72	#6127	—	4	1	no	—
73	#6127	—	4	1	no	—
74	#6127	—	4	1	no	—
75	C₂×D₅₂	—	2	1	no	—
76	C₂×D₅₂	—	2	1	no	—
77	C₂×D₅₂	C₄²	1	2	yes	—
78	C₂×C₄×D₁₃	—	2	2	no	—
79	C₂×Dic₂₆	—	2	1	no	—
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	—	2	1	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	4	1	no	—
84	C₂×C₈×C₁₃	—	4	1	no	—
85	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	D₈	1	2	yes	—
90	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
91	C₁₃×C₄⋊C₄	—	2	4	no	—
92	C₄²×C₁₃	—	4	1	no	—
93	C₄²×C₁₃	—	4	1	no	—
94	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₃	#41	1	1	yes	characteristic
98	#6103	—	2	2	no	—
99	#6103	—	2	2	no	—
100	C₄⋊Dic₁₃	—	2	1	no	—
101	C₄⋊Dic₁₃	C₄²	1	2	yes	—
102	#6099	—	4	1	no	—
103	#6099	—	4	1	no	—
104	C₂×#1055	—	2	2	no	—
105	C₂×#1055	—	2	2	no	—
106	C₂×#1055	—	2	2	no	—
107	#6094	—	4	4	no	—
108	C₈×D₁₃	—	4	4	no	—
109	#2931	—	13	1	no	—
110	C₈×C₄⋊C₄	D₁₃	1	1	yes	characteristic, complemented
111	#1540	—	13	1	no	—
112	#1502	—	13	1	no	—
113	#1450	—	13	1	no	—
114	#1303	—	13	1	no	—
115	#1301	—	13	1	no	—
116	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	—	4	1	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
133	C₂²⋊D₁₃	—	4	2	no	—
134	C₂²⋊D₁₃	—	4	2	no	—
135	C₂×Dic₁₃	—	2	2	no	—
136	D₅₂	—	4	2	no	—
137	D₅₂	—	4	1	no	—
138	D₅₂	—	4	1	no	—
139	D₅₂	—	4	2	no	—
140	C₄×D₁₃	—	4	2	no	—
141	C₄×D₁₃	—	4	2	no	—
142	Dic₂₆	—	4	1	no	—
143	Dic₂₆	—	4	1	no	—
144	C₈×C₁₃	—	2	1	no	—
145	C₈×C₁₃	—	2	1	no	—
146	C₈×C₁₃	—	4	2	no	—
147	C₈×C₁₃	#102	1	2	yes	—
148	C₈×C₁₃	#102	1	2	yes	—
149	#1055	—	4	2	no	—
150	#1055	—	4	2	no	—
151	#1055	—	4	2	no	—
152	C₂×#131	—	26	1	no	—
153	#532	—	13	1	no	—
154	#436	—	13	2	no	—
155	#436	—	26	1	no	—
156	C₈×D₄	—	13	2	no	—
157	C₈×D₄	—	26	1	no	—
158	#431	—	13	1	no	—
159	#431	—	13	1	no	—
160	#423	—	13	1	no	—
161	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
162	#408	—	26	1	no	—
163	C₂×C₄×C₈	—	2	1	no	—
164	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
165	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
166	#336	—	2	1	no	—
167	#332	—	13	2	no	—
168	#331	—	13	2	no	—
169	#327	—	13	4	no	—
170	#322	—	26	1	no	—
171	C₈²	—	26	1	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
175	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
176	C₂²×C₁₃	C₄×SD₁₆	1	1	yes	characteristic
177	C₂×D₁₃	—	4	2	no	—
178	C₂×D₁₃	—	4	2	no	—
179	C₂×D₁₃	—	4	2	no	—
180	C₄×C₁₃	—	2	1	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	4	4	no	—
184	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
185	C₄×C₁₃	#416	1	1	yes	characteristic
186	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
187	C₄×C₁₃	C₄×SD₁₆	1	1	yes	characteristic
188	C₄×C₁₃	#443	1	2	yes	—
189	Dic₁₃	—	4	2	no	—
190	C₂×#41	—	26	1	no	—
191	#131	—	52	2	no	—
192	#131	—	52	2	no	—
193	C₂×M₁₆	—	26	1	no	—
194	C₂×M₁₆	—	26	2	no	—
195	C₂²×C₈	—	2	1	no	—
196	C₂²×C₈	—	26	2	no	—
197	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
199	C₄⋊Q₈	—	13	1	no	—
200	#127	—	13	1	no	—
201	#124	—	26	1	no	—
202	#121	—	13	2	no	—
203	#121	—	26	1	no	—
204	C₄×D₄	—	13	2	no	—
205	C₄×D₄	—	26	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	13	2	no	—
211	C₄⋊C₈	—	13	2	no	—
212	C₄⋊C₈	—	26	1	no	—
213	C₄⋊C₈	—	26	1	no	—
214	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
215	C₂²⋊C₈	—	13	2	no	—
216	C₂²⋊C₈	—	26	1	no	—
217	C₂²⋊C₈	—	26	2	no	—
218	C₂²⋊C₈	—	26	2	no	—
219	C₄×C₈	—	4	2	no	—
220	C₄×C₈	—	26	1	no	—
221	C₄×C₈	—	26	1	no	—
222	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
223	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
224	C₄×C₈	D₅₂	1	1	yes	characteristic
225	C₄×C₈	D₅₂	1	1	yes	characteristic
226	#95	C₄×D₁₃	1	2	yes	—
227	C₂×C₁₃	—	2	1	no	—
228	C₂×C₁₃	—	2	1	no	—
229	C₂×C₁₃	#1757	1	1	yes	characteristic
230	C₂×C₁₃	C₄×#102	1	1	yes	characteristic
231	C₂×C₁₃	#1764	1	1	yes	characteristic
232	D₁₃	—	8	2	no	—
233	#41	—	52	1	no	—
234	#41	—	52	1	no	—
235	#41	—	52	1	no	—
236	#41	—	52	1	no	—
237	C₂×Q₈	—	26	1	no	—
238	C₂×D₄	—	13	2	no	—
239	C₂×D₄	—	26	2	no	—
240	C₂×D₄	—	26	1	no	—
241	C₂×D₄	—	26	1	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	26	2	no	—
244	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
245	C₂²×C₄	#6127	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
247	M₁₆	—	52	1	no	—
248	M₁₆	—	52	1	no	—
249	M₁₆	—	52	4	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	26	2	no	—
256	C₂×C₈	—	26	2	no	—
257	C₂×C₈	—	26	2	no	—
258	C₂×C₈	—	52	4	no	—
259	C₂×C₈	#6103	1	1	yes	characteristic
260	C₂×C₈	#6103	1	1	yes	characteristic
261	C₂×C₈	#6103	1	1	yes	characteristic
262	C₂×C₈	#6103	1	1	yes	characteristic
263	C₂×C₈	#6104	1	2	yes	—
264	C₂×C₈	#6106	1	2	yes	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	13	2	no	—
267	C₄⋊C₄	—	26	1	no	—
268	C₂²⋊C₄	—	26	2	no	—
269	C₂²⋊C₄	—	26	2	no	—
270	C₄²	—	4	1	no	—
271	C₄²	—	4	1	no	—
272	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
274	C₄²	C₂×D₅₂	1	1	yes	characteristic
275	C₄²	#6127	1	1	yes	characteristic
276	C₁₃	#8251	1	1	yes	characteristic, complemented, Sylow
277	C₂³	—	26	2	no	—
278	C₂³	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
279	Q₈	—	52	1	no	—
280	Q₈	—	52	1	no	—
281	D₄	—	52	2	no	—
282	D₄	—	52	2	no	—
283	D₄	—	52	2	no	—
284	D₄	—	52	1	no	—
285	D₄	—	52	1	no	—
286	D₄	—	52	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	26	2	no	—
293	C₂×C₄	—	52	2	no	—
294	C₂×C₄	—	52	2	no	—
295	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
296	C₂×C₄	#88126	1	1	yes	characteristic
297	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
298	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
299	C₂×C₄	C₂×#6103	1	1	yes	characteristic
300	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
301	C₂×C₄	C₂×#6104	1	1	yes	characteristic
302	C₂×C₄	C₂×#6106	1	1	yes	characteristic
303	C₂×C₄	#88204	1	1	yes	characteristic
304	C₂×C₄	#88204	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	4	2	no	—
308	C₈	—	52	2	no	—
309	C₈	—	52	2	no	—
310	C₈	—	52	2	no	—
311	C₈	#88049	1	2	yes	—
312	C₈	#88049	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	52	2	no	—
316	C₂²	—	52	2	no	—
317	C₂²	—	52	2	no	—
318	C₂²	C₄×#6103	1	1	yes	characteristic
319	C₂²	C₄×#6104	1	1	yes	characteristic
320	C₂²	C₄×#6106	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	4	no	—
325	C₄	—	52	2	no	—
326	C₄	#11718361	1	2	yes	—
327	C₄	C₂×#88049	1	1	yes	characteristic
328	C₄	#11718663	1	1	yes	characteristic
329	C₄	C₄×#6104	1	1	yes	characteristic
330	C₄	C₄×#6106	1	1	yes	characteristic
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	104	2	no	—
334	C₂	#13114637	1	1	yes	characteristic
335	C₂	C₄×#88049	1	1	yes	characteristic
336	C₂	#13114673	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#6103
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₈×C₁₃×C₄⋊C₄	C₂
socle	C₂²×C₁₃	C₄×C₂²⋊C₄
prime radical	#11719182	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₈×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#8251
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #13113119	C₂	semidirect
5. #13112739	C₂	semidirect
6. #13112733	C₂	semidirect
7. C₈×C₁₃×C₄⋊C₄	C₂	semidirect
19. #11718349	C₄	semidirect
21. C₈×D₅₂	C₄	semidirect
23. #11718077	C₄	semidirect
24. #11718074	C₄	semidirect
110. C₈×C₄⋊C₄	D₁₃	semidirect
276. C₁₃	#8251	semidirect

Maximal subgroups

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

subgroup	index
#13123549	2
#13113486	2
#13113330	2
#13113119	2
#13112739	2
#13112733	2
C₈×C₁₃×C₄⋊C₄	2
#8251	13