Subgroups — #16016131

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃×M₃₂	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₅₂	C₂	1	1	yes	maximal, characteristic
3	#13116667	C₂	1	2	yes	maximal, complemented
4	#13116660	C₂	1	4	yes	maximal, complemented
5	#13116655	C₂	1	2	yes	maximal, complemented
6	#13116446	C₂	1	2	yes	maximal, complemented
7	C₄×#88052	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
10	C₂×C₄×D₅₂	C₄	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88060	C₄	1	2	yes	—
14	C₂×#88052	C₂²	1	1	yes	characteristic
15	C₂×#88052	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
17	D₁₃×M₃₂	—	2	8	no	—
18	C₂×#88038	C₂²	1	4	yes	complemented
19	C₂×C₁₆×D₁₃	C₂²	1	4	yes	complemented
20	C₈×D₅₂	C₄	1	2	yes	—
21	C₈×D₅₂	C₄	1	1	yes	characteristic
22	C₈×D₅₂	C₄	1	1	yes	characteristic
23	C₈×D₅₂	C₂²	1	1	yes	characteristic
24	C₈×D₅₂	C₂²	1	2	yes	—
25	C₈×D₅₂	C₂²	1	1	yes	characteristic
26	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
27	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
28	#11718099	C₂²	1	4	yes	complemented
29	#11718099	C₂²	1	4	yes	complemented
30	#11718053	C₂²	1	2	yes	complemented
31	C₄×#6090	C₂²	1	2	yes	complemented
32	C₂²×D₅₂	C₈	1	1	yes	characteristic
33	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×#6103	C₈	1	2	yes	—
38	C₂×C₄⋊Dic₁₃	C₈	1	1	yes	characteristic
39	C₂²×#1055	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₁₃	—	2	4	no	—
41	C₂×C₈×D₁₃	—	2	4	no	—
42	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₁₃	C₂³	1	2	yes	—
45	C₂×C₈×D₁₃	C₂³	1	2	yes	—
46	C₄×D₅₂	C₈	1	2	yes	—
47	C₄×D₅₂	C₈	1	1	yes	characteristic
48	C₄×D₅₂	C₈	1	1	yes	characteristic
49	C₄×D₅₂	C₂×C₄	1	2	yes	—
50	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
51	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
52	C₁₃×M₃₂	—	2	4	no	—
53	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
54	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	#88060	C₂×C₄	1	2	yes	—
59	#88060	C₂×C₄	1	2	yes	—
60	#88060	C₂×C₄	1	2	yes	—
61	#88060	C₂×C₄	1	2	yes	—
62	#88052	—	2	2	no	—
63	#88052	D₄	1	4	yes	complemented
64	C₂×#6090	C₂³	1	2	yes	—
65	C₂×#6090	C₂³	1	2	yes	—
66	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
67	C₄⋊#1055	C₂×C₄	1	2	yes	—
68	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
69	#88038	—	2	8	no	—
70	C₁₆×D₁₃	—	2	8	no	—
71	D₄×M₃₂	—	13	1	no	maximal, Sylow
72	C₂³×D₁₃	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
77	C₂×D₅₂	—	2	4	no	—
78	C₂×D₅₂	C₂×C₈	1	2	yes	—
79	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
80	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₁₃	—	2	4	no	—
82	C₂×C₄×D₁₃	—	2	4	no	—
83	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
84	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
85	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₃	#41	1	2	yes	—
93	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
95	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	#6103	C₂×C₈	1	2	yes	—
99	#6103	C₂×C₈	1	2	yes	—
100	#6103	C₂×C₈	1	2	yes	—
101	#6103	C₂×C₈	1	2	yes	—
102	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
103	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
104	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
105	C₂×#1055	—	2	4	no	—
106	C₂×#1055	C₂²×C₄	1	2	yes	—
107	C₂×#1055	C₂²×C₄	1	2	yes	—
108	C₈×D₁₃	—	2	4	no	—
109	C₈×D₁₃	—	2	4	no	—
110	C₈×D₁₃	—	2	2	no	—
111	C₈×D₁₃	—	2	2	no	—
112	C₈×D₁₃	—	2	2	no	—
113	C₈×D₁₃	—	2	2	no	—
114	C₁₃×C₁₆	—	2	4	no	—
115	#6090	—	2	2	no	—
116	#6090	C₂×D₄	1	4	yes	complemented
117	C₂²×M₃₂	—	13	2	no	—
118	C₂×C₈×D₄	—	13	1	no	—
119	#2169	—	13	2	no	—
120	#2168	—	13	4	no	—
121	D₄×C₁₆	—	13	2	no	—
122	#2150	D₁₃	1	1	yes	characteristic, complemented
123	#2112	—	13	2	no	—
124	C₄×M₃₂	—	13	1	no	—
125	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
126	C₂²×D₁₃	—	2	4	no	—
127	C₂²×D₁₃	—	2	4	no	—
128	C₂²×D₁₃	—	4	2	no	—
129	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
130	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	4	1	no	—
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#131	1	2	yes	—
140	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
141	C₂×Dic₁₃	—	2	4	no	—
142	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
143	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
144	D₅₂	—	2	4	no	—
145	D₅₂	—	2	2	no	—
146	D₅₂	M₃₂	1	4	yes	complemented
147	C₄×D₁₃	—	2	2	no	—
148	C₄×D₁₃	—	2	2	no	—
149	C₄×D₁₃	—	2	2	no	—
150	C₄×D₁₃	—	2	2	no	—
151	C₄×D₁₃	—	2	4	no	—
152	C₄×D₁₃	—	2	4	no	—
153	C₈×C₁₃	—	2	1	no	—
154	C₈×C₁₃	—	2	1	no	—
155	C₈×C₁₃	C₄×D₄	1	2	yes	—
156	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
157	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
158	#1055	—	2	2	no	—
159	#1055	—	2	2	no	—
160	C₂³×C₈	—	13	2	no	—
161	C₂×C₄×D₄	—	13	1	no	—
162	C₂×M₃₂	—	13	1	no	—
163	C₂×M₃₂	—	13	1	no	—
164	C₂×M₃₂	—	13	4	no	—
165	C₂×M₃₂	—	26	8	no	—
166	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
167	C₂²×C₁₆	—	13	4	no	—
168	C₈×D₄	—	13	2	no	—
169	C₈×D₄	—	13	1	no	—
170	C₈×D₄	—	13	1	no	—
171	C₈×D₄	—	13	1	no	—
172	C₈×D₄	—	13	2	no	—
173	C₈×D₄	—	13	1	no	—
174	C₂×C₄⋊C₈	—	13	1	no	—
175	C₂×C₂²⋊C₈	—	13	2	no	—
176	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
177	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	complemented
178	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	complemented
179	C₂²⋊C₁₆	—	13	4	no	—
180	C₂²⋊C₁₆	—	13	4	no	—
181	#346	—	13	2	no	—
182	C₄×C₁₆	—	13	2	no	—
183	C₂²×C₁₃	—	2	1	no	—
184	C₂²×C₁₃	—	2	1	no	—
185	C₂²×C₁₃	C₈×D₄	1	2	yes	—
186	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
187	C₂×D₁₃	—	2	2	no	—
188	C₂×D₁₃	—	2	2	no	—
189	C₂×D₁₃	—	2	2	no	—
190	C₂×D₁₃	—	2	2	no	—
191	C₂×D₁₃	—	2	2	no	—
192	C₂×D₁₃	—	4	8	no	—
193	C₂×D₁₃	C₂×M₃₂	1	4	yes	complemented
194	C₄×C₁₃	—	2	1	no	—
195	C₄×C₁₃	—	2	1	no	—
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	C₈×D₄	1	2	yes	—
198	C₄×C₁₃	C₂×M₃₂	1	2	yes	complemented
199	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
200	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
201	Dic₁₃	—	2	2	no	—
202	Dic₁₃	—	2	2	no	—
203	C₂²×D₄	—	13	1	no	—
204	C₂³×C₄	—	13	2	no	—
205	C₂²×C₈	—	13	2	no	—
206	C₂²×C₈	—	13	2	no	—
207	C₂²×C₈	—	13	2	no	—
208	C₂²×C₈	—	13	2	no	—
209	C₂²×C₈	—	13	2	no	—
210	C₂²×C₈	—	26	4	no	—
211	C₂²×C₈	—	26	4	no	—
212	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
214	C₄×D₄	—	13	1	no	—
215	C₄×D₄	—	13	2	no	—
216	C₄×D₄	—	13	1	no	—
217	C₄×D₄	—	13	1	no	—
218	C₄×D₄	—	13	2	no	—
219	C₄×D₄	—	13	1	no	—
220	C₂×C₄⋊C₄	—	13	1	no	—
221	C₂×C₂²⋊C₄	—	13	2	no	—
222	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
223	M₃₂	—	2	4	no	—
224	M₃₂	—	13	4	no	—
225	M₃₂	—	26	8	no	—
226	M₃₂	—	26	2	no	—
227	C₂×C₁₆	—	13	2	no	—
228	C₂×C₁₆	—	13	2	no	—
229	C₂×C₁₆	—	26	8	no	—
230	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
231	C₄⋊C₈	—	13	1	no	—
232	C₄⋊C₈	—	13	2	no	—
233	C₄⋊C₈	—	13	1	no	—
234	C₂²⋊C₈	—	13	2	no	—
235	C₂²⋊C₈	—	13	2	no	—
236	C₂²⋊C₈	—	13	2	no	—
237	C₂²⋊C₈	—	13	2	no	—
238	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
239	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
242	C₂×C₁₃	—	2	2	no	—
243	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
244	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
245	C₂×C₁₃	#3407	1	1	yes	characteristic
246	D₁₃	—	2	4	no	—
247	D₁₃	—	4	2	no	—
248	C₂⁴	—	13	2	no	—
249	C₂×D₄	—	13	1	no	—
250	C₂×D₄	—	13	2	no	—
251	C₂×D₄	—	13	1	no	—
252	C₂×D₄	—	26	4	no	—
253	C₂²×C₄	—	13	2	no	—
254	C₂²×C₄	—	13	2	no	—
255	C₂²×C₄	—	13	2	no	—
256	C₂²×C₄	—	13	2	no	—
257	C₂²×C₄	—	13	2	no	—
258	C₂²×C₄	—	26	4	no	—
259	C₂²×C₄	—	26	4	no	—
260	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
261	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	13	2	no	—
265	C₂×C₈	—	13	2	no	—
266	C₂×C₈	—	26	4	no	—
267	C₂×C₈	—	26	4	no	—
268	C₂×C₈	—	26	4	no	—
269	C₂×C₈	—	26	2	no	—
270	C₂×C₈	—	26	2	no	—
271	C₂×C₈	—	26	2	no	—
272	C₂×C₈	—	26	2	no	—
273	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
274	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
275	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
276	C₂×C₈	D₄×D₁₃	1	2	yes	—
277	C₂×C₈	#6131	1	2	yes	—
278	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
279	C₄⋊C₄	—	13	2	no	—
280	C₄⋊C₄	—	13	1	no	—
281	C₄⋊C₄	—	13	1	no	—
282	C₂²⋊C₄	—	13	2	no	—
283	C₂²⋊C₄	—	13	2	no	—
284	C₂²⋊C₄	—	13	2	no	—
285	C₂²⋊C₄	—	13	2	no	—
286	C₄²	C₈×D₁₃	1	1	yes	characteristic
287	C₄²	C₈×D₁₃	1	1	yes	characteristic
288	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
289	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
290	C₁₆	—	2	4	no	—
291	C₁₆	—	13	4	no	—
292	C₁₆	—	26	2	no	—
293	C₁₃	D₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
294	C₂³	—	13	2	no	—
295	C₂³	—	13	2	no	—
296	C₂³	—	26	4	no	—
297	C₂³	—	26	4	no	—
298	C₂³	—	52	2	no	—
299	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
300	D₄	—	13	4	no	—
301	D₄	—	26	4	no	—
302	D₄	—	26	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	13	2	no	—
306	C₂×C₄	—	13	2	no	—
307	C₂×C₄	—	26	4	no	—
308	C₂×C₄	—	26	2	no	—
309	C₂×C₄	—	26	2	no	—
310	C₂×C₄	—	26	2	no	—
311	C₂×C₄	—	26	2	no	—
312	C₂×C₄	—	26	4	no	—
313	C₂×C₄	—	26	4	no	—
314	C₂×C₄	#88147	1	2	yes	—
315	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
316	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
317	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
318	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
319	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
320	C₂×C₄	#88161	1	2	yes	—
321	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	26	2	no	—
325	C₈	—	26	2	no	—
326	C₈	#88147	1	2	yes	—
327	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
328	C₈	C₂×#6131	1	1	yes	characteristic
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	13	4	no	—
332	C₂²	—	26	2	no	—
333	C₂²	—	26	2	no	—
334	C₂²	—	26	2	no	—
335	C₂²	—	26	2	no	—
336	C₂²	—	26	2	no	—
337	C₂²	—	52	8	no	—
338	C₂²	#11718495	1	2	yes	—
339	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	26	2	no	—
344	C₄	—	26	2	no	—
345	C₄	#11718495	1	2	yes	—
346	C₄	D₁₃×M₃₂	1	2	yes	complemented
347	C₄	C₂×#88147	1	1	yes	characteristic
348	C₄	C₂×#88161	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	26	4	no	—
351	C₂	—	52	2	no	—
352	C₂	C₂×#11718495	1	1	yes	characteristic
353	C₂	C₂×D₁₃×M₃₂	1	1	yes	characteristic
354	C₂	#13127938	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#2150	C₂
prime radical	C₂²×D₅₂	C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2150	D₁₃
hypocenter	C₁₃	D₄×M₃₂
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₃×M₃₂	C₂	semidirect
3. #13116667	C₂	semidirect
4. #13116660	C₂	semidirect
5. #13116655	C₂	semidirect
6. #13116446	C₂	semidirect
7. C₄×#88052	C₂	semidirect
8. C₁₃×#2150	C₂	semidirect
18. C₂×#88038	C₂²	semidirect
19. C₂×C₁₆×D₁₃	C₂²	semidirect
26. C₁₃×C₄⋊C₁₆	C₂²	semidirect
27. C₁₃×C₄⋊C₁₆	C₂²	semidirect
28. #11718099	C₂²	semidirect
29. #11718099	C₂²	semidirect
30. #11718053	C₂²	semidirect
31. C₄×#6090	C₂²	semidirect
63. #88052	D₄	semidirect
116. #6090	C₂×D₄	semidirect
122. #2150	D₁₃	semidirect
146. D₅₂	M₃₂	semidirect
177. C₄⋊C₁₆	C₂×D₁₃	semidirect
178. C₄⋊C₁₆	C₂×D₁₃	semidirect
193. C₂×D₁₃	C₂×M₃₂	semidirect
198. C₄×C₁₃	C₂×M₃₂	semidirect
293. C₁₃	D₄×M₃₂	semidirect
346. C₄	D₁₃×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₃×M₃₂	2
C₂×C₈×D₅₂	2
#13116667	2
#13116660	2
#13116655	2
#13116446	2
C₄×#88052	2
C₁₃×#2150	2
D₄×M₃₂	13