Subgroups — #15881192

Order 3328.

862 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×D₅₂	C₂	1	2	yes	maximal
2	C₂×#11718349	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718554	C₂	1	2	yes	maximal, complemented
4	#13115611	C₂	1	2	yes	maximal, complemented
5	#13115042	C₂	1	4	yes	maximal, complemented
6	#13115008	C₂	1	1	yes	maximal, characteristic, complemented
7	#13114875	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×M₁₆×D₁₃	C₂²	1	4	yes	—
10	C₂²×#6094	C₂²	1	2	yes	—
11	C₂×C₄×D₅₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718787	C₂²	1	4	yes	—
17	C₂×#88062	C₂²	1	4	yes	—
18	C₂×#88060	C₂²	1	2	yes	—
19	C₂×#88059	C₂²	1	2	yes	—
20	C₂×#88059	C₂²	1	2	yes	—
21	#11718585	C₂²	1	2	yes	—
22	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
23	#11718554	—	2	8	no	—
24	#11718367	—	2	4	no	—
25	#11718349	—	2	2	no	—
26	#11718349	C₄	1	4	yes	—
27	#11718349	C₄	1	4	yes	—
28	C₈×D₅₂	—	2	2	no	—
29	C₁₃×#341	C₂²	1	2	yes	—
30	#11718150	C₂²	1	4	yes	—
31	C₂²×D₅₂	Q₈	1	1	yes	characteristic
32	C₂²×C₄×D₁₃	C₂³	1	2	yes	—
33	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
34	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×#6103	Q₈	1	2	yes	—
39	C₂×C₄⋊Dic₁₃	Q₈	1	1	yes	characteristic
40	C₂×#6099	C₂³	1	4	yes	—
41	C₂²×#1055	C₂³	1	2	yes	—
42	M₁₆×D₁₃	—	2	8	no	—
43	M₁₆×D₁₃	—	4	4	no	—
44	C₂×#6094	—	2	4	no	—
45	C₂×#6094	—	2	4	no	—
46	C₂×#6094	—	2	4	no	—
47	C₂×#6094	C₂×C₄	1	8	yes	—
48	C₂×C₈×D₁₃	—	2	4	no	—
49	C₄×D₅₂	C₂×C₄	1	2	yes	—
50	C₄×D₅₂	C₂×C₄	1	2	yes	—
51	C₄×D₅₂	D₄	1	2	yes	—
52	C₄×D₅₂	D₄	1	2	yes	—
53	C₁₃×#108	—	2	2	no	—
54	C₁₃×#108	D₄	1	4	yes	complemented
55	C₁₃×#97	—	2	2	no	—
56	C₄×C₈×C₁₃	—	2	2	no	—
57	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
59	#88062	—	2	8	no	—
60	#88060	—	2	4	no	—
61	#88060	—	2	4	no	—
62	#88060	C₂×C₄	1	8	yes	—
63	#88059	—	2	4	no	—
64	#88059	—	4	2	no	—
65	C₄⋊#1055	—	2	2	no	—
66	C₄⋊#1055	—	2	2	no	—
67	C₄⋊#1055	C₂×C₄	1	2	yes	—
68	C₄⋊#1055	C₂×C₄	1	2	yes	—
69	#22669	—	13	1	no	maximal, Sylow
70	C₂³×D₁₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
75	C₂×D₅₂	—	4	2	no	—
76	C₂×D₅₂	C₄⋊C₄	1	2	yes	—
77	C₂×D₅₂	C₄⋊C₄	1	2	yes	—
78	C₂×C₄×D₁₃	—	2	2	no	—
79	C₂×C₄×D₁₃	—	2	2	no	—
80	C₂×C₄×D₁₃	—	2	2	no	—
81	C₂×C₄×D₁₃	—	2	2	no	—
82	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
83	C₂×C₄×D₁₃	C₂×D₄	1	4	yes	—
84	C₁₃×M₁₆	—	2	2	no	—
85	C₁₃×M₁₆	—	2	2	no	—
86	C₁₃×M₁₆	—	2	2	no	—
87	C₁₃×M₁₆	C₂×D₄	1	4	yes	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	1	no	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₁₃	#41	1	2	yes	—
96	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
100	#6103	C₄⋊C₄	1	4	yes	—
101	#6103	C₄⋊C₄	1	4	yes	—
102	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
103	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
104	#6099	—	2	4	no	—
105	#6099	—	2	4	no	—
106	C₂×#1055	—	2	2	no	—
107	C₂×#1055	—	2	2	no	—
108	C₂×#1055	—	2	4	no	—
109	C₂×#1055	C₂²×C₄	1	4	yes	—
110	#6094	—	2	16	no	—
111	#6094	—	4	8	no	—
112	C₈×D₁₃	—	4	8	no	—
113	D₄×M₁₆	—	13	2	no	—
114	C₂×#436	—	13	1	no	—
115	C₂×#430	—	13	2	no	—
116	#1950	—	13	2	no	—
117	#1855	—	13	4	no	—
118	#1848	—	13	1	no	—
119	#1811	—	13	2	no	—
120	C₄×#108	D₁₃	1	1	yes	characteristic, complemented
121	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
122	C₂²×D₁₃	—	2	2	no	—
123	C₂²×D₁₃	—	2	2	no	—
124	C₂²×D₁₃	—	4	2	no	—
125	C₂²×D₁₃	—	4	2	no	—
126	C₂²×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	4	1	no	—
129	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
139	C₂×Dic₁₃	—	2	2	no	—
140	C₂×Dic₁₃	—	2	2	no	—
141	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
142	D₅₂	—	2	4	no	—
143	D₅₂	—	4	2	no	—
144	C₄×D₁₃	—	2	4	no	—
145	C₄×D₁₃	—	2	4	no	—
146	C₄×D₁₃	—	2	4	no	—
147	C₄×D₁₃	—	2	4	no	—
148	C₈×C₁₃	—	2	2	no	—
149	C₈×C₁₃	—	2	4	no	—
150	C₈×C₁₃	—	4	2	no	—
151	C₈×C₁₃	C₄×D₄	1	4	yes	—
152	#1055	—	2	4	no	—
153	#1055	—	4	4	no	—
154	C₂²×M₁₆	—	13	2	no	—
155	C₂²×M₁₆	—	13	4	no	—
156	C₂×C₄×D₄	—	13	1	no	—
157	#436	—	13	4	no	—
158	#436	—	13	4	no	—
159	#436	—	26	2	no	—
160	#435	—	26	4	no	—
161	C₈×D₄	—	26	2	no	—
162	#430	—	26	8	no	—
163	C₂×#108	—	13	4	no	—
164	C₂×#108	—	13	2	no	—
165	C₂×#108	—	13	2	no	—
166	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
167	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
168	#423	—	13	2	no	—
169	C₂×C₄⋊C₈	—	13	1	no	—
170	#407	—	13	4	no	—
171	C₂×C₂²⋊C₈	—	13	2	no	—
172	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
173	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
174	#341	—	13	4	no	—
175	#341	C₂×D₁₃	1	2	yes	—
176	C₂²×C₁₃	—	2	1	no	—
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
179	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
180	C₂²×C₁₃	#554	1	1	yes	characteristic
181	C₂×D₁₃	—	2	4	no	—
182	C₂×D₁₃	—	2	4	no	—
183	C₂×D₁₃	—	2	4	no	—
184	C₂×D₁₃	—	8	2	no	—
185	C₂×D₁₃	—	8	2	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	#430	1	2	yes	complemented
190	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
191	C₄×C₁₃	#520	1	1	yes	characteristic
192	C₄×C₁₃	D₄²	1	1	yes	characteristic
193	C₄×C₁₃	#547	1	1	yes	characteristic
194	Dic₁₃	—	2	4	no	—
195	C₂²×D₄	—	13	1	no	—
196	C₂³×C₄	—	13	2	no	—
197	C₂×M₁₆	—	13	8	no	—
198	C₂×M₁₆	—	13	4	no	—
199	C₂×M₁₆	—	26	4	no	—
200	C₂×M₁₆	—	26	4	no	—
201	C₂×M₁₆	—	26	4	no	—
202	C₂×M₁₆	—	26	8	no	—
203	C₂×M₁₆	—	52	4	no	—
204	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
205	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
206	C₂²×C₈	—	13	2	no	—
207	C₂²×C₈	—	26	4	no	—
208	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
210	C₄×D₄	—	13	2	no	—
211	C₄×D₄	—	13	2	no	—
212	C₄×D₄	—	13	2	no	—
213	C₄×D₄	—	13	2	no	—
214	C₂×C₄⋊C₄	—	13	1	no	—
215	C₂×C₂²⋊C₄	—	13	2	no	—
216	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
217	#108	—	2	2	no	—
218	#108	—	26	8	no	—
219	#108	—	26	4	no	—
220	#108	—	52	2	no	—
221	#108	D₅₂	1	4	yes	complemented
222	C₄⋊C₈	—	13	2	no	—
223	C₄⋊C₈	—	13	2	no	—
224	C₄⋊C₈	—	26	2	no	—
225	C₄⋊C₈	—	26	2	no	—
226	C₂²⋊C₈	—	13	8	no	—
227	C₂²⋊C₈	—	26	4	no	—
228	C₂²⋊C₈	—	26	4	no	—
229	#97	—	2	2	no	—
230	C₄×C₈	—	2	2	no	—
231	C₄×C₈	C₄×D₁₃	1	2	yes	—
232	C₄×C₈	C₄×D₁₃	1	2	yes	—
233	C₂×C₁₃	—	2	1	no	—
234	C₂×C₁₃	—	2	1	no	—
235	C₂×C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
237	C₂×C₁₃	#2916	1	1	yes	characteristic
238	D₁₃	—	4	4	no	—
239	C₂⁴	—	13	2	no	—
240	C₂×D₄	—	13	2	no	—
241	C₂×D₄	—	13	2	no	—
242	C₂×D₄	—	52	2	no	—
243	C₂²×C₄	—	13	2	no	—
244	C₂²×C₄	—	13	4	no	—
245	C₂²×C₄	—	13	4	no	—
246	C₂²×C₄	—	26	2	no	—
247	C₂²×C₄	—	26	2	no	—
248	C₂²×C₄	—	26	2	no	—
249	C₂²×C₄	—	26	2	no	—
250	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
251	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
252	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
253	M₁₆	—	2	2	no	—
254	M₁₆	—	2	2	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	26	16	no	—
257	M₁₆	—	26	4	no	—
258	M₁₆	—	26	4	no	—
259	M₁₆	—	52	8	no	—
260	M₁₆	C₂×D₅₂	1	4	yes	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	13	4	no	—
266	C₂×C₈	—	26	2	no	—
267	C₂×C₈	—	26	2	no	—
268	C₂×C₈	—	26	4	no	—
269	C₂×C₈	—	52	8	no	—
270	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
271	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
272	C₂×C₈	C₂×D₅₂	1	2	yes	—
273	C₂×C₈	#6127	1	2	yes	—
274	C₄⋊C₄	—	13	2	no	—
275	C₄⋊C₄	—	13	2	no	—
276	C₂²⋊C₄	—	13	4	no	—
277	C₂²⋊C₄	—	13	4	no	—
278	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
279	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
280	C₄²	D₄×D₁₃	1	1	yes	characteristic
281	C₄²	D₄×D₁₃	1	1	yes	characteristic
282	C₁₃	#22669	1	1	yes	characteristic, complemented, Sylow
283	C₂³	—	13	4	no	—
284	C₂³	—	26	2	no	—
285	C₂³	—	26	2	no	—
286	C₂³	—	52	2	no	—
287	C₂³	—	52	2	no	—
288	C₂³	C₂×Q₈×D₁₃	1	1	yes	characteristic
289	D₄	—	26	4	no	—
290	D₄	—	52	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	13	4	no	—
294	C₂×C₄	—	26	2	no	—
295	C₂×C₄	—	26	2	no	—
296	C₂×C₄	—	26	4	no	—
297	C₂×C₄	—	26	4	no	—
298	C₂×C₄	—	26	4	no	—
299	C₂×C₄	—	26	4	no	—
300	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
301	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
302	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
303	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
306	C₂×C₄	C₂×#6127	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
308	C₂×C₄	#88256	1	1	yes	characteristic
309	C₂×C₄	#88257	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	2	4	no	—
312	C₈	—	4	2	no	—
313	C₈	—	26	4	no	—
314	C₈	—	52	4	no	—
315	C₈	C₄×D₅₂	1	4	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	26	4	no	—
319	C₂²	—	26	4	no	—
320	C₂²	—	26	4	no	—
321	C₂²	—	104	2	no	—
322	C₂²	—	104	2	no	—
323	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
324	C₂²	Q₈×D₅₂	1	1	yes	characteristic
325	C₂²	#11719284	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	26	4	no	—
330	C₄	#11718554	1	2	yes	complemented
331	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
332	C₄	#11719228	1	1	yes	characteristic
333	C₄	D₄×D₅₂	1	1	yes	characteristic
334	C₄	#11719256	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	52	4	no	—
338	C₂	C₄⋊C₄×D₅₂	1	1	yes	characteristic
339	C₂	C₂×#11718554	1	1	yes	characteristic
340	C₂	#13123444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#108	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂²×D₅₂	Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#108	D₁₃
hypocenter	C₁₃	#22669
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
3. C₂×#11718554	C₂	semidirect
4. #13115611	C₂	semidirect
5. #13115042	C₂	semidirect
6. #13115008	C₂	semidirect
7. #13114875	C₂	semidirect
8. C₄×C₁₃×#108	C₂	semidirect
54. C₁₃×#108	D₄	semidirect
120. C₄×#108	D₁₃	semidirect
189. C₄×C₁₃	#430	semidirect
221. #108	D₅₂	semidirect
282. C₁₃	#22669	semidirect
330. C₄	#11718554	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×D₅₂	2
C₂×#11718349	2
C₂×#11718554	2
#13115611	2
#13115042	2
#13115008	2
#13114875	2
C₄×C₁₃×#108	2
#22669	13