Subgroups — #16789356

Order 3328.

474 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃⋊(C₂²⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718216	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×#261	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₂²⋊#261	C₂	1	1	yes	maximal, characteristic
5	(C₂×#261)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×#261)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	C₄×C₂²⋊#261	C₄	1	1	yes	characteristic
10	D₁₃⋊(C₂²⋊C₈)	C₄	1	1	yes	characteristic
11	C₂×C₈×#261	C₂²	1	1	yes	characteristic
12	C₂×C₈×#261	C₂²	1	1	yes	characteristic
13	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic, complemented
16	C₂×#88052	C₄	1	1	yes	characteristic, complemented
17	D₁₃×M₃₂	—	2	2	no	—
18	D₁₃×M₃₂	—	2	2	no	—
19	C₂×#88038	C₄	1	1	yes	characteristic, complemented
20	C₂×#88038	C₄	1	1	yes	characteristic, complemented
21	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
23	C₂²⋊#6092	C₄	1	2	yes	complemented
24	(C₂×#261)⋊C₈	C₄	1	1	yes	characteristic
25	(C₂×#261)⋊C₈	C₄	1	1	yes	characteristic
26	#11718259	C₄	1	2	yes	complemented
27	#11718216	—	2	2	no	—
28	C₁₆×#261	—	2	2	no	—
29	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₂²⋊#261	C₈	1	2	yes	—
31	C₂×C₄×#261	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×#261	C₂×C₄	1	1	yes	characteristic
33	C₂×D₁₃⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×D₁₃⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₁₃	—	2	1	no	—
38	C₂×C₈×D₁₃	—	2	1	no	—
39	C₂×C₈×D₁₃	—	2	1	no	—
40	C₂×C₈×D₁₃	—	2	1	no	—
41	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
45	C₂²⋊#1057	C₈	1	2	yes	—
46	(C₂×#261)⋊C₄	C₈	1	2	yes	—
47	#88111	C₈	1	2	yes	—
48	C₂×#6092	C₂×C₄	1	2	yes	—
49	C₂×#6092	C₂×C₄	1	2	yes	—
50	C₈×#261	—	2	1	no	—
51	C₈×#261	—	2	1	no	—
52	C₈×#261	—	2	1	no	—
53	C₈×#261	—	2	1	no	—
54	D₁₃⋊C₁₆	—	2	1	no	—
55	D₁₃⋊C₁₆	—	2	1	no	—
56	D₁₃⋊C₁₆	—	2	1	no	—
57	D₁₃⋊C₁₆	—	2	1	no	—
58	C₁₃×M₃₂	—	2	2	no	—
59	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
60	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
61	#88052	—	2	2	no	—
62	C₂×#6090	C₂×C₄	1	1	yes	characteristic
63	C₂×#6090	C₂×C₄	1	1	yes	characteristic
64	#88038	—	2	2	no	—
65	#88038	—	2	2	no	—
66	C₁₆×D₁₃	—	2	1	no	—
67	C₁₆×D₁₃	—	2	1	no	—
68	C₁₆×D₁₃	D₄	1	2	yes	—
69	C₁₆×D₁₃	D₄	1	2	yes	—
70	#12612	—	13	1	no	maximal, Sylow
71	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×#261	C₂×C₈	1	2	yes	—
73	C₂²×#261	C₂×C₈	1	2	yes	—
74	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
75	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₁₃	—	2	1	no	—
77	C₂×C₄×D₁₃	—	2	1	no	—
78	C₂×C₄×D₁₃	—	2	1	no	—
79	C₂×C₄×D₁₃	—	2	1	no	—
80	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
81	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²⋊#261	—	2	4	no	—
85	C₂×#1057	C₂×C₈	1	2	yes	—
86	C₂×#1057	C₂×C₈	1	2	yes	—
87	C₄×#261	—	2	1	no	—
88	C₄×#261	—	2	1	no	—
89	C₄×#261	—	2	1	no	—
90	C₄×#261	—	2	1	no	—
91	D₁₃⋊C₈	—	2	1	no	—
92	D₁₃⋊C₈	—	2	1	no	—
93	D₁₃⋊C₈	—	2	1	no	—
94	D₁₃⋊C₈	—	2	1	no	—
95	C₂×C₈×C₁₃	—	2	1	no	—
96	C₂×C₈×C₁₃	—	2	1	no	—
97	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×#1055	—	2	1	no	—
100	C₂×#1055	—	2	1	no	—
101	C₂×#1055	C₄²	1	1	yes	characteristic
102	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
103	C₈×D₁₃	—	2	1	no	—
104	C₈×D₁₃	—	2	1	no	—
105	C₈×D₁₃	—	2	1	no	—
106	C₈×D₁₃	—	2	1	no	—
107	C₈×D₁₃	—	2	1	no	—
108	C₈×D₁₃	—	2	1	no	—
109	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
110	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
111	C₈×D₁₃	#41	1	1	yes	characteristic
112	C₈×D₁₃	#41	1	1	yes	characteristic
113	#6092	—	2	2	no	—
114	#6092	—	2	2	no	—
115	C₁₃×C₁₆	—	2	1	no	—
116	C₁₃×C₁₆	C₂²⋊C₄	1	2	yes	complemented
117	#6090	—	2	1	no	—
118	#6090	C₂²⋊C₄	1	2	yes	complemented
119	C₂²×M₃₂	—	13	1	no	—
120	C₂×C₂²⋊C₁₆	—	13	1	no	—
121	C₂×#346	—	13	1	no	—
122	C₂×C₄×C₁₆	—	13	1	no	—
123	C₈×C₂²⋊C₄	—	13	1	no	—
124	#1374	—	13	1	no	—
125	#1374	—	13	1	no	—
126	C₂³×C₁₃	C₄×C₈	1	1	yes	characteristic
127	C₂²×D₁₃	—	2	1	no	—
128	C₂²×D₁₃	—	2	1	no	—
129	C₂²×D₁₃	—	2	2	no	—
130	C₂²×D₁₃	—	2	2	no	—
131	C₂²×D₁₃	C₄×C₈	1	1	yes	characteristic
132	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×#261	—	2	2	no	—
134	C₂×#261	—	2	2	no	—
135	C₂×#261	—	2	2	no	—
136	C₂×#261	—	2	2	no	—
137	C₂×#261	—	2	2	no	—
138	C₂×#261	M₃₂	1	4	yes	complemented
139	C₂×C₄×C₁₃	—	2	1	no	—
140	C₂×C₄×C₁₃	—	2	1	no	—
141	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
143	C₂×Dic₁₃	—	2	1	no	—
144	C₂×Dic₁₃	—	2	1	no	—
145	C₂×Dic₁₃	C₄×C₈	1	1	yes	characteristic
146	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
147	C₄×D₁₃	—	2	1	no	—
148	C₄×D₁₃	—	2	1	no	—
149	C₄×D₁₃	—	2	1	no	—
150	C₄×D₁₃	—	2	1	no	—
151	C₄×D₁₃	—	2	1	no	—
152	C₄×D₁₃	—	2	1	no	—
153	C₄×D₁₃	C₄×D₄	1	1	yes	characteristic
154	C₄×D₁₃	C₄×D₄	1	1	yes	characteristic
155	C₄×D₁₃	#131	1	1	yes	characteristic
156	C₄×D₁₃	#131	1	1	yes	characteristic
157	#1057	—	2	2	no	—
158	#1057	—	2	2	no	—
159	C₈×C₁₃	—	2	1	no	—
160	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₈×C₁₃	#117	1	1	yes	characteristic
162	#1055	—	2	1	no	—
163	#1055	C₄×D₄	1	1	yes	characteristic
164	#1055	C₄×D₄	1	1	yes	characteristic
165	C₂³×C₈	—	13	1	no	—
166	C₂×M₃₂	—	13	1	no	—
167	C₂×M₃₂	—	13	1	no	—
168	C₂×M₃₂	—	13	1	no	—
169	C₂×M₃₂	—	26	2	no	—
170	C₂×M₃₂	—	26	2	no	—
171	C₂×M₃₂	#261	1	1	yes	characteristic, complemented
172	C₂²×C₁₆	—	13	1	no	—
173	C₂²×C₁₆	—	13	1	no	—
174	C₂²×C₁₆	—	13	1	no	—
175	C₂²×C₁₆	—	13	1	no	—
176	C₂×C₂²⋊C₈	—	13	1	no	—
177	C₂×C₄×C₈	—	13	1	no	—
178	C₂×C₄×C₈	—	13	1	no	—
179	C₄×C₂²⋊C₄	—	13	1	no	—
180	C₂²⋊C₁₆	—	13	2	no	—
181	C₂²⋊C₁₆	—	13	2	no	—
182	#346	—	26	2	no	—
183	C₄×C₁₆	—	26	2	no	—
184	#336	—	13	1	no	—
185	#336	—	13	1	no	—
186	C₂²×C₁₃	—	2	1	no	—
187	C₂²×C₁₃	—	2	1	no	—
188	C₂²×C₁₃	—	2	1	no	—
189	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
190	C₂×D₁₃	—	2	1	no	—
191	C₂×D₁₃	—	2	1	no	—
192	C₂×D₁₃	—	2	1	no	—
193	C₂×D₁₃	—	4	2	no	—
194	C₂×D₁₃	—	4	2	no	—
195	C₂×D₁₃	C₈×D₄	1	1	yes	characteristic
196	C₂×D₁₃	C₈×D₄	1	1	yes	characteristic
197	C₂×D₁₃	C₂×M₃₂	1	2	yes	—
198	C₂×D₁₃	#504	1	2	yes	—
199	#261	—	2	4	no	—
200	#261	—	4	2	no	—
201	C₄×C₁₃	—	2	1	no	—
202	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
203	C₄×C₁₃	#405	1	1	yes	characteristic
204	Dic₁₃	—	2	1	no	—
205	Dic₁₃	C₈×D₄	1	1	yes	characteristic
206	Dic₁₃	C₈×D₄	1	1	yes	characteristic
207	C₂³×C₄	—	13	1	no	—
208	C₂²×C₈	—	13	1	no	—
209	C₂²×C₈	—	13	1	no	—
210	C₂²×C₈	—	13	1	no	—
211	C₂²×C₈	—	13	1	no	—
212	C₂²×C₈	—	13	1	no	—
213	C₂²×C₈	—	13	1	no	—
214	C₂²×C₈	—	13	1	no	—
215	C₂²×C₈	—	26	1	no	—
216	C₂²×C₈	—	26	1	no	—
217	C₂²×C₈	—	26	1	no	—
218	C₂²×C₈	—	26	1	no	—
219	C₂²×C₈	C₂×#261	1	1	yes	characteristic
220	C₂×C₂²⋊C₄	—	13	2	no	—
221	C₂×C₄²	—	13	1	no	—
222	C₂×C₄²	—	13	1	no	—
223	M₃₂	—	2	2	no	—
224	M₃₂	—	26	2	no	—
225	M₃₂	—	26	2	no	—
226	M₃₂	—	26	2	no	—
227	C₂×C₁₆	—	13	1	no	—
228	C₂×C₁₆	—	13	2	no	—
229	C₂×C₁₆	—	13	2	no	—
230	C₂×C₁₆	—	13	2	no	—
231	C₂×C₁₆	—	13	2	no	—
232	C₂×C₁₆	—	13	1	no	—
233	C₂×C₁₆	—	26	1	no	—
234	C₂×C₁₆	—	26	1	no	—
235	C₂×C₁₆	—	26	1	no	—
236	C₂×C₁₆	—	26	1	no	—
237	C₂×C₁₆	—	26	1	no	—
238	C₂×C₁₆	—	26	1	no	—
239	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
240	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
241	C₂²⋊C₈	—	13	2	no	—
242	C₂²⋊C₈	—	13	2	no	—
243	C₄×C₈	—	26	1	no	—
244	C₄×C₈	—	26	1	no	—
245	C₄×C₈	—	26	1	no	—
246	C₄×C₈	—	26	1	no	—
247	#95	—	13	2	no	—
248	C₂×C₁₃	—	4	1	no	—
249	C₂×C₁₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
250	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
251	C₂×C₁₃	#2108	1	1	yes	characteristic
252	D₁₃	—	4	1	no	—
253	D₁₃	#2168	1	2	yes	—
254	D₁₃	#2169	1	2	yes	—
255	C₂⁴	—	13	1	no	—
256	C₂²×C₄	—	13	1	no	—
257	C₂²×C₄	—	13	2	no	—
258	C₂²×C₄	—	13	1	no	—
259	C₂²×C₄	—	13	1	no	—
260	C₂²×C₄	—	13	1	no	—
261	C₂²×C₄	—	13	1	no	—
262	C₂²×C₄	—	13	2	no	—
263	C₂²×C₄	—	26	1	no	—
264	C₂²×C₄	—	26	1	no	—
265	C₂²×C₄	—	26	1	no	—
266	C₂²×C₄	—	26	1	no	—
267	C₂²×C₄	C₄×#261	1	1	yes	characteristic
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	13	1	no	—
271	C₂×C₈	—	13	1	no	—
272	C₂×C₈	—	13	1	no	—
273	C₂×C₈	—	13	1	no	—
274	C₂×C₈	—	13	1	no	—
275	C₂×C₈	—	13	2	no	—
276	C₂×C₈	—	13	2	no	—
277	C₂×C₈	—	13	1	no	—
278	C₂×C₈	—	26	1	no	—
279	C₂×C₈	—	26	1	no	—
280	C₂×C₈	—	26	1	no	—
281	C₂×C₈	—	26	1	no	—
282	C₂×C₈	—	26	1	no	—
283	C₂×C₈	—	26	1	no	—
284	C₂×C₈	—	26	1	no	—
285	C₂×C₈	—	26	1	no	—
286	C₂×C₈	—	26	1	no	—
287	C₂×C₈	—	26	1	no	—
288	C₂×C₈	—	26	1	no	—
289	C₂×C₈	—	26	1	no	—
290	C₂×C₈	C₄×#261	1	1	yes	characteristic
291	C₂×C₈	C₂²×#261	1	1	yes	characteristic
292	C₂²⋊C₄	—	26	4	no	—
293	C₄²	—	26	1	no	—
294	C₄²	—	26	1	no	—
295	C₄²	—	26	1	no	—
296	C₄²	—	26	1	no	—
297	C₁₆	—	2	1	no	—
298	C₁₆	—	13	2	no	—
299	C₁₆	—	26	2	no	—
300	C₁₆	—	26	2	no	—
301	C₁₆	—	26	1	no	—
302	C₁₆	C₂²⋊#261	1	2	yes	complemented
303	C₁₃	#12612	1	1	yes	characteristic, complemented, Sylow
304	C₂³	—	13	1	no	—
305	C₂³	—	13	1	no	—
306	C₂³	—	26	2	no	—
307	C₂³	—	26	1	no	—
308	C₂³	—	26	1	no	—
309	C₂³	—	26	2	no	—
310	C₂³	C₈×#261	1	1	yes	characteristic
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	13	1	no	—
314	C₂×C₄	—	13	1	no	—
315	C₂×C₄	—	13	1	no	—
316	C₂×C₄	—	13	1	no	—
317	C₂×C₄	—	13	1	no	—
318	C₂×C₄	—	13	4	no	—
319	C₂×C₄	—	13	1	no	—
320	C₂×C₄	—	26	1	no	—
321	C₂×C₄	—	26	1	no	—
322	C₂×C₄	—	26	1	no	—
323	C₂×C₄	—	26	1	no	—
324	C₂×C₄	—	26	1	no	—
325	C₂×C₄	—	26	1	no	—
326	C₂×C₄	—	26	2	no	—
327	C₂×C₄	—	26	2	no	—
328	C₂×C₄	—	26	1	no	—
329	C₂×C₄	—	26	1	no	—
330	C₂×C₄	—	26	2	no	—
331	C₂×C₄	—	26	2	no	—
332	C₂×C₄	—	26	2	no	—
333	C₂×C₄	C₈×#261	1	1	yes	characteristic
334	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	13	1	no	—
337	C₈	—	13	1	no	—
338	C₈	—	26	2	no	—
339	C₈	—	26	2	no	—
340	C₈	—	26	1	no	—
341	C₈	#88237	1	1	yes	characteristic
342	C₈	C₂×C₂²⋊#261	1	1	yes	characteristic
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	13	2	no	—
347	C₂²	—	13	2	no	—
348	C₂²	—	13	1	no	—
349	C₂²	—	13	1	no	—
350	C₂²	—	26	1	no	—
351	C₂²	—	26	1	no	—
352	C₂²	—	26	1	no	—
353	C₂²	—	52	2	no	—
354	C₂²	—	52	2	no	—
355	C₂²	C₂×C₈×#261	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	13	1	no	—
358	C₄	—	13	1	no	—
359	C₄	—	26	4	no	—
360	C₄	—	26	1	no	—
361	C₄	—	52	2	no	—
362	C₄	#11719089	1	1	yes	characteristic
363	C₄	C₄×C₂²⋊#261	1	1	yes	characteristic
364	C₂	—	4	1	no	—
365	C₂	—	13	2	no	—
366	C₂	—	13	2	no	—
367	C₂	—	52	1	no	—
368	C₂	C₈×C₂²⋊#261	1	1	yes	characteristic
369	C₂	M₃₂×#261	1	1	yes	characteristic
370	C₂	#13131048	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#261
commutator, socle	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂×C₁₃×M₃₂	C₄
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂×C₈×D₁₃	C₂³
hypercenter	C₂×M₃₂	#261
hypocenter	C₁₃	#12612
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#11718216	C₂	semidirect
3. C₂×C₁₆×#261	C₂	semidirect
5. (C₂×#261)⋊C₁₆	C₂	semidirect
6. (C₂×#261)⋊C₁₆	C₂	semidirect
15. C₂×C₁₃×M₃₂	C₄	semidirect
16. C₂×#88052	C₄	semidirect
19. C₂×#88038	C₄	semidirect
20. C₂×#88038	C₄	semidirect
23. C₂²⋊#6092	C₄	semidirect
26. #11718259	C₄	semidirect
116. C₁₃×C₁₆	C₂²⋊C₄	semidirect
118. #6090	C₂²⋊C₄	semidirect
138. C₂×#261	M₃₂	semidirect
171. C₂×M₃₂	#261	semidirect
302. C₁₆	C₂²⋊#261	semidirect
303. C₁₃	#12612	semidirect

Maximal subgroups

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

subgroup	index
D₁₃⋊(C₂²⋊C₁₆)	2
C₂×#11718216	2
C₂×C₁₆×#261	2
C₈×C₂²⋊#261	2
(C₂×#261)⋊C₁₆	2
C₂×D₁₃×M₃₂	2
#12612	13