Subgroups — #15750893

Order 3328.

684 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88128	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
3	#13112656	C₂	1	4	yes	maximal, complemented
4	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#88128	C₄	1	1	yes	characteristic
6	C₂×C₄×D₅₂	—	2	1	no	—
7	C₂×C₄×D₅₂	—	2	1	no	—
8	C₂×C₄×D₅₂	C₄	1	2	yes	—
9	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
10	C₂×C₄×C₈×C₁₃	—	2	1	no	—
11	C₄³×C₁₃	C₂²	1	1	yes	characteristic
12	#11718600	—	2	1	no	—
13	#11718600	—	2	1	no	—
14	#11718600	C₂²	1	2	yes	—
15	#11718596	C₄	1	1	yes	characteristic
16	C₂×C₄⋊#1055	—	2	1	no	—
17	C₂×C₄×#1055	—	2	1	no	—
18	C₁₃×#336	C₂²	1	2	yes	—
19	#11718116	C₂²	1	2	yes	—
20	C₂²×D₅₂	—	2	1	no	—
21	C₂²×D₅₂	—	2	1	no	—
22	C₂²×D₅₂	C₂×C₄	1	2	yes	—
23	C₂²×C₄×D₁₃	—	2	2	no	—
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×#6103	—	2	2	no	—
33	C₂×#6103	—	2	2	no	—
34	C₂×#6103	—	2	2	no	—
35	C₂×C₄⋊Dic₁₃	—	2	1	no	—
36	C₂×C₄⋊Dic₁₃	—	2	1	no	—
37	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
38	C₂²×#1055	—	2	2	no	—
39	#88128	—	2	2	no	—
40	#88128	C₈	1	4	yes	complemented
41	C₄×D₅₂	—	4	4	no	—
42	C₄×D₅₂	—	4	2	no	—
43	C₄×D₅₂	—	4	2	no	—
44	C₄×D₅₂	—	4	2	no	—
45	C₄×D₅₂	—	4	2	no	—
46	C₁₃×C₄⋊C₈	—	2	4	no	—
47	C₄×C₈×C₁₃	—	4	2	no	—
48	C₄⋊#1055	—	2	4	no	—
49	C₄×#1055	—	4	2	no	—
50	#7569	—	13	1	no	maximal, Sylow
51	C₂³×D₁₃	—	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×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₁₃	—	2	2	no	—
60	C₂×D₅₂	—	2	4	no	—
61	C₂×D₅₂	—	2	4	no	—
62	C₂×D₅₂	—	2	2	no	—
63	C₂×D₅₂	—	4	2	no	—
64	C₂×D₅₂	—	4	2	no	—
65	C₂×D₅₂	—	4	2	no	—
66	C₂×D₅₂	—	4	2	no	—
67	C₂×D₅₂	—	4	2	no	—
68	C₂×D₅₂	—	4	2	no	—
69	C₂×D₅₂	C₂×C₈	1	4	yes	—
70	C₂×C₄×D₁₃	—	4	4	no	—
71	C₂×C₄×D₁₃	—	4	4	no	—
72	C₂×C₄×D₁₃	—	4	4	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	4	2	no	—
75	C₂×C₈×C₁₃	—	4	2	no	—
76	C₄²×C₁₃	—	2	2	no	—
77	C₄²×C₁₃	—	2	2	no	—
78	C₄²×C₁₃	—	2	2	no	—
79	C₄²×C₁₃	—	4	1	no	—
80	C₄²×C₁₃	C₂×C₈	1	2	yes	—
81	C₄²×C₁₃	M₁₆	1	2	yes	—
82	C₄²×C₁₃	D₈	1	2	yes	—
83	C₄²×C₁₃	D₈	1	2	yes	—
84	C₄²×C₁₃	SD₁₆	1	2	yes	—
85	C₄²×C₁₃	SD₁₆	1	2	yes	—
86	#6103	—	4	4	no	—
87	#6103	—	4	2	no	—
88	#6103	—	4	2	no	—
89	#6103	—	4	2	no	—
90	#6103	—	4	2	no	—
91	C₄⋊Dic₁₃	—	4	2	no	—
92	C₄⋊Dic₁₃	—	4	2	no	—
93	C₄⋊Dic₁₃	—	4	2	no	—
94	C₂×#1055	—	2	4	no	—
95	C₂×#1055	—	4	2	no	—
96	C₂×#1055	—	4	2	no	—
97	C₄×#127	—	13	1	no	—
98	C₄×C₄⋊C₈	—	13	1	no	—
99	C₄×C₄⋊C₈	D₁₃	1	1	yes	characteristic, complemented
100	#1270	—	13	4	no	—
101	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂²×D₁₃	—	2	4	no	—
103	C₂²×D₁₃	—	4	2	no	—
104	C₂²×D₁₃	—	4	2	no	—
105	C₂²×D₁₃	—	4	2	no	—
106	C₂²×D₁₃	—	4	2	no	—
107	C₂²×D₁₃	—	4	2	no	—
108	C₂²×D₁₃	—	4	2	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	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
119	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
120	C₂×C₄×C₁₃	#100	1	2	yes	—
121	C₂×C₄×C₁₃	#102	1	2	yes	—
122	C₂×C₄×C₁₃	#102	1	2	yes	—
123	C₂×C₄×C₁₃	#102	1	2	yes	—
124	C₂×C₄×C₁₃	#102	1	2	yes	—
125	C₂×C₄×C₁₃	#104	1	2	yes	—
126	C₂×C₄×C₁₃	#104	1	2	yes	—
127	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×Dic₁₃	—	4	2	no	—
134	C₂×Dic₁₃	—	4	2	no	—
135	C₂×Dic₁₃	—	4	2	no	—
136	D₅₂	—	4	4	no	—
137	D₅₂	—	4	4	no	—
138	D₅₂	—	4	8	no	—
139	D₅₂	—	4	8	no	—
140	C₄×D₁₃	—	8	8	no	—
141	C₈×C₁₃	—	4	4	no	—
142	#1055	—	4	4	no	—
143	C₂×#127	—	13	1	no	—
144	C₂×C₄×D₄	—	13	2	no	—
145	C₂×C₄×D₄	—	26	1	no	—
146	C₂×C₄×D₄	—	26	1	no	—
147	C₂×C₄⋊C₈	—	2	1	no	—
148	C₂×C₄⋊C₈	—	26	1	no	—
149	C₂×C₄×C₈	—	2	1	no	—
150	C₂×C₄×C₈	—	26	1	no	—
151	#390	—	13	2	no	—
152	#390	—	26	1	no	—
153	#390	—	26	1	no	—
154	#384	—	13	1	no	—
155	C₄³	C₂×D₁₃	1	1	yes	characteristic
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	C₂×#100	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
162	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
163	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
164	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
165	C₂×D₁₃	—	4	4	no	—
166	C₂×D₁₃	—	4	4	no	—
167	C₂×D₁₃	—	4	4	no	—
168	C₂×D₁₃	—	8	2	no	—
169	C₂×D₁₃	—	8	2	no	—
170	C₂×D₁₃	—	8	2	no	—
171	C₂×D₁₃	—	8	2	no	—
172	C₄×C₁₃	—	2	1	no	—
173	C₄×C₁₃	—	2	1	no	—
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	—	2	1	no	—
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	2	1	no	—
178	C₄×C₁₃	—	4	1	no	—
179	C₄×C₁₃	D₄⋊C₈	1	4	yes	—
180	C₄×C₁₃	D₄⋊C₈	1	4	yes	—
181	C₄×C₁₃	#331	1	4	yes	—
182	Dic₁₃	—	8	2	no	—
183	C₂²×D₄	—	13	2	no	—
184	C₂²×D₄	—	26	1	no	—
185	C₂²×D₄	—	26	1	no	—
186	C₂³×C₄	—	26	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	26	2	no	—
189	#127	—	13	4	no	—
190	#127	—	26	2	no	—
191	C₄×D₄	—	52	4	no	—
192	C₄×D₄	—	52	2	no	—
193	C₄×D₄	—	52	2	no	—
194	C₄×D₄	—	52	2	no	—
195	C₄×D₄	—	52	2	no	—
196	C₂×C₄⋊C₄	—	13	2	no	—
197	C₂×C₄⋊C₄	—	26	1	no	—
198	C₂×C₄⋊C₄	—	26	1	no	—
199	C₂×C₂²⋊C₄	—	26	2	no	—
200	C₂×C₂²⋊C₄	—	26	2	no	—
201	C₂×C₂²⋊C₄	—	26	2	no	—
202	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
203	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
204	C₂×C₄²	D₅₂	1	1	yes	characteristic
205	C₂×C₄²	D₅₂	1	1	yes	characteristic
206	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
207	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
208	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
209	C₄⋊C₈	—	2	4	no	—
210	C₄⋊C₈	—	26	4	no	—
211	C₄×C₈	—	4	2	no	—
212	C₄×C₈	—	52	2	no	—
213	C₂×C₁₃	#1460	1	1	yes	characteristic
214	C₂×C₁₃	C₂×D₄⋊C₈	1	1	yes	characteristic
215	C₂×C₁₃	C₂×D₄⋊C₈	1	1	yes	characteristic
216	C₂×C₁₃	#1502	1	1	yes	characteristic
217	C₂×C₁₃	#1502	1	1	yes	characteristic
218	C₂×C₁₃	#1525	1	1	yes	characteristic
219	C₂×C₁₃	C₂×#331	1	1	yes	characteristic
220	D₁₃	—	8	4	no	—
221	C₂⁴	—	26	2	no	—
222	C₂×D₄	—	13	4	no	—
223	C₂×D₄	—	26	2	no	—
224	C₂×D₄	—	26	4	no	—
225	C₂×D₄	—	26	4	no	—
226	C₂×D₄	—	52	2	no	—
227	C₂×D₄	—	52	2	no	—
228	C₂×D₄	—	52	2	no	—
229	C₂×D₄	—	52	2	no	—
230	C₂×D₄	—	52	2	no	—
231	C₂×D₄	—	52	2	no	—
232	C₂²×C₄	—	26	2	no	—
233	C₂²×C₄	—	52	4	no	—
234	C₂²×C₄	—	52	4	no	—
235	C₂²×C₄	—	52	4	no	—
236	C₂²×C₄	#6103	1	1	yes	characteristic
237	C₂²×C₄	#6103	1	1	yes	characteristic
238	C₂²×C₄	#6103	1	1	yes	characteristic
239	C₂²×C₄	#6103	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	26	4	no	—
247	C₂×C₈	—	52	2	no	—
248	C₂×C₈	—	52	2	no	—
249	C₄⋊C₄	—	52	2	no	—
250	C₄⋊C₄	—	52	2	no	—
251	C₄⋊C₄	—	52	2	no	—
252	C₂²⋊C₄	—	52	4	no	—
253	C₂²⋊C₄	—	52	2	no	—
254	C₂²⋊C₄	—	52	2	no	—
255	C₂²⋊C₄	—	52	2	no	—
256	C₂²⋊C₄	—	52	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	4	1	no	—
261	C₄²	C₈×D₁₃	1	2	yes	—
262	C₄²	#6094	1	2	yes	—
263	C₄²	#6095	1	2	yes	—
264	C₄²	D₁₀₄	1	2	yes	—
265	C₄²	#6104	1	2	yes	—
266	C₄²	#6106	1	2	yes	—
267	C₁₃	#7569	1	1	yes	characteristic, complemented, Sylow
268	C₂³	—	26	4	no	—
269	C₂³	—	52	2	no	—
270	C₂³	—	52	2	no	—
271	C₂³	—	52	2	no	—
272	C₂³	—	52	2	no	—
273	C₂³	—	52	2	no	—
274	C₂³	—	52	2	no	—
275	C₂³	C₂×#6103	1	1	yes	characteristic
276	D₄	—	52	4	no	—
277	D₄	—	52	4	no	—
278	D₄	—	52	8	no	—
279	D₄	—	52	8	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	52	2	no	—
290	C₂×C₄	—	52	2	no	—
291	C₂×C₄	—	52	2	no	—
292	C₂×C₄	—	104	8	no	—
293	C₂×C₄	#88045	1	2	yes	—
294	C₂×C₄	#88049	1	2	yes	—
295	C₂×C₄	#88049	1	2	yes	—
296	C₂×C₄	#88060	1	2	yes	—
297	C₂×C₄	#88060	1	2	yes	—
298	C₂×C₄	#88061	1	2	yes	—
299	C₂×C₄	#88061	1	2	yes	—
300	C₂×C₄	#88063	1	2	yes	—
301	C₂×C₄	#88065	1	2	yes	—
302	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
303	C₂×C₄	C₂×#6094	1	1	yes	characteristic
304	C₂×C₄	C₂×#6095	1	1	yes	characteristic
305	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
306	C₂×C₄	C₂×#6104	1	1	yes	characteristic
307	C₂×C₄	C₂×#6106	1	1	yes	characteristic
308	C₈	—	4	4	no	—
309	C₈	—	52	4	no	—
310	C₂²	—	52	4	no	—
311	C₂²	—	52	4	no	—
312	C₂²	—	52	4	no	—
313	C₂²	—	104	2	no	—
314	C₂²	—	104	2	no	—
315	C₂²	—	104	2	no	—
316	C₂²	—	104	2	no	—
317	C₂²	C₂×#88045	1	1	yes	characteristic
318	C₂²	C₂×#88049	1	1	yes	characteristic
319	C₂²	#11718667	1	1	yes	characteristic
320	C₂²	C₂×#88060	1	1	yes	characteristic
321	C₂²	C₂×#88061	1	1	yes	characteristic
322	C₂²	C₂×#88063	1	1	yes	characteristic
323	C₂²	C₂×#88065	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	104	2	no	—
332	C₄	#11718051	1	4	yes	—
333	C₄	#11718075	1	4	yes	—
334	C₄	#11718077	1	4	yes	—
335	C₂	—	104	4	no	—
336	C₂	#13113156	1	1	yes	characteristic
337	C₂	C₂×#11718075	1	1	yes	characteristic
338	C₂	C₂×#11718051	1	1	yes	characteristic
339	C₂	#13113330	1	1	yes	characteristic
340	C₂	#13113332	1	1	yes	characteristic
341	C₂	#13113427	1	1	yes	characteristic
342	C₂	C₂×#11718077	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#6103
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×C₄⋊C₈	C₂
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂×#88128	C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#7569
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊#1055	C₂	semidirect
3. #13112656	C₂	semidirect
4. C₄×C₁₃×C₄⋊C₈	C₂	semidirect
40. #88128	C₈	semidirect
99. C₄×C₄⋊C₈	D₁₃	semidirect
267. C₁₃	#7569	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88128	2
C₄×C₄⋊#1055	2
#13112656	2
C₄×C₁₃×C₄⋊C₈	2
#7569	13