Subgroups — #16799518

Order 3328.

780 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131513	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#6118	C₂	1	1	yes	maximal, characteristic
3	#13130951	C₂	1	4	yes	maximal, complemented
4	#13130789	C₂	1	4	yes	maximal
5	C₄×#88103	C₂	1	4	yes	maximal, complemented
6	D₁₃×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×#6118	C₂²	1	2	yes	—
8	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
9	#11719130	—	2	1	no	—
10	#11719127	C₂²	1	2	yes	—
11	C₂×#88110	—	2	1	no	—
12	#11719119	—	2	1	no	—
13	C₂×C₄×#1057	—	2	1	no	—
14	C₂×#88103	C₂²	1	8	yes	—
15	#11719054	—	2	1	no	—
16	#11719052	C₂²	1	2	yes	—
17	#11719051	—	2	1	no	—
18	C₄×#6118	—	2	1	no	—
19	C₄×#6118	—	2	1	no	—
20	C₄×#6118	C₂²	1	2	yes	—
21	C₄×#6118	C₂²	1	2	yes	—
22	C₄×#6118	C₂²	1	2	yes	—
23	C₁₃×#431	C₄	1	1	yes	characteristic
24	#11718586	C₄	1	1	yes	characteristic
25	#11718494	C₄	1	2	yes	—
26	D₁₃×C₄⋊C₈	C₂²	1	2	yes	—
27	D₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
28	#11718379	C₄	1	2	yes	—
29	#11718346	C₄	1	2	yes	—
30	C₄×C₈×D₁₃	C₂²	1	2	yes	—
31	#11718268	C₂²	1	8	yes	—
32	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
33	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
34	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
35	C₂²×#1057	—	2	2	no	—
36	C₂×#6118	—	2	2	no	—
37	C₂×#6118	—	2	8	no	—
38	C₂×#6118	C₂³	1	4	yes	—
39	C₂×#6118	C₂³	1	4	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
43	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
44	C₂×C₈×D₁₃	C₂³	1	4	yes	—
45	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×D₁₃	D₄	1	1	yes	characteristic
48	C₄²×D₁₃	D₄	1	1	yes	characteristic
49	C₄²×D₁₃	Q₈	1	1	yes	characteristic
50	C₄²×D₁₃	Q₈	1	1	yes	characteristic
51	C₄²×D₁₃	C₂³	1	1	yes	characteristic
52	C₄²×D₁₃	C₂³	1	1	yes	characteristic
53	C₂²⋊#1057	—	2	2	no	—
54	#88111	—	2	2	no	—
55	#88110	—	2	2	no	—
56	#88110	—	2	2	no	—
57	#88110	—	2	2	no	—
58	C₄⋊#1057	—	2	2	no	—
59	C₄⋊#1057	—	2	2	no	—
60	C₄×#1057	—	2	2	no	—
61	C₄×#1057	—	2	2	no	—
62	C₄×#1057	—	2	2	no	—
63	#88103	—	2	16	no	—
64	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
65	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
67	C₂²⋊#1055	C₂×C₄	1	2	yes	—
68	#88060	C₂×C₄	1	4	yes	—
69	#88054	C₂×C₄	1	4	yes	—
70	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
71	C₄⋊#1055	C₂×C₄	1	2	yes	—
72	C₄×#1055	C₂×C₄	1	2	yes	—
73	#21370	—	13	1	no	maximal, Sylow
74	C₂³×D₁₃	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×D₁₃	—	2	4	no	—
82	C₂×C₄×D₁₃	—	2	4	no	—
83	C₂×C₄×D₁₃	—	4	2	no	—
84	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
91	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
96	C₂×#1057	—	2	8	no	—
97	C₂×#1057	—	2	4	no	—
98	C₂×#1057	—	2	4	no	—
99	#6118	—	2	8	no	—
100	#6118	—	2	4	no	—
101	#6118	—	2	8	no	—
102	#6118	#41	1	8	yes	—
103	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
104	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
109	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
110	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
111	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
112	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
113	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
114	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
117	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
118	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
119	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
120	C₂×#1055	C₂²×C₄	1	4	yes	—
121	C₈×D₁₃	—	2	8	no	—
122	#2920	—	13	1	no	—
123	C₂×#431	—	13	1	no	—
124	C₂×C₄×M₁₆	—	13	1	no	—
125	#1950	—	13	4	no	—
126	#1855	—	13	4	no	—
127	C₄×#108	—	13	4	no	—
128	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×D₁₃	—	2	2	no	—
130	C₂²×D₁₃	—	2	2	no	—
131	C₂²×D₁₃	—	4	1	no	—
132	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×D₁₃	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	#108	1	4	yes	complemented
137	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×Dic₁₃	—	2	2	no	—
144	C₂×Dic₁₃	—	2	2	no	—
145	C₂×Dic₁₃	—	2	2	no	—
146	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
152	C₄×D₁₃	—	2	2	no	—
153	C₄×D₁₃	—	2	2	no	—
154	C₄×D₁₃	—	2	2	no	—
155	C₄×D₁₃	—	2	2	no	—
156	C₄×D₁₃	—	2	2	no	—
157	C₄×D₁₃	—	2	2	no	—
158	C₄×D₁₃	—	2	2	no	—
159	C₄×D₁₃	—	2	2	no	—
160	C₄×D₁₃	—	2	2	no	—
161	C₄×D₁₃	—	4	1	no	—
162	C₄×D₁₃	—	4	1	no	—
163	C₄×D₁₃	C₂×#41	1	4	yes	—
164	#1057	—	2	8	no	—
165	#1057	—	4	4	no	—
166	C₈×C₁₃	—	2	4	no	—
167	#1055	—	2	4	no	—
168	C₂²×M₁₆	—	13	2	no	—
169	C₂²×C₄²	—	13	1	no	—
170	#432	—	26	1	no	—
171	#432	—	26	1	no	—
172	#431	—	13	1	no	—
173	#431	—	13	2	no	—
174	#431	—	13	2	no	—
175	#431	—	13	2	no	—
176	#431	—	26	1	no	—
177	#431	—	26	1	no	—
178	#431	#261	1	1	yes	characteristic
179	C₂×#108	—	13	8	no	—
180	#423	—	13	2	no	—
181	C₂×C₄⋊C₈	—	13	2	no	—
182	#407	—	13	2	no	—
183	C₂×C₂²⋊C₈	—	13	2	no	—
184	C₄×M₁₆	—	13	2	no	—
185	C₄×M₁₆	—	13	2	no	—
186	C₄×M₁₆	—	13	2	no	—
187	C₄×M₁₆	—	26	1	no	—
188	C₄×M₁₆	—	26	1	no	—
189	C₂×#97	—	26	1	no	—
190	C₂×C₄×C₈	—	13	2	no	—
191	C₂×C₄×C₈	—	26	1	no	—
192	#341	—	13	8	no	—
193	C₂²×C₁₃	—	2	1	no	—
194	C₂²×C₁₃	—	2	1	no	—
195	C₂²×C₁₃	C₂×#108	1	2	yes	—
196	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
197	C₂×D₁₃	—	2	1	no	—
198	C₂×D₁₃	—	2	1	no	—
199	C₂×D₁₃	—	2	1	no	—
200	C₂×D₁₃	—	2	1	no	—
201	C₂×D₁₃	—	2	1	no	—
202	C₂×D₁₃	—	4	4	no	—
203	C₂×D₁₃	#533	1	2	yes	—
204	C₄×C₁₃	—	2	1	no	—
205	C₄×C₁₃	—	2	1	no	—
206	C₄×C₁₃	—	2	1	no	—
207	C₄×C₁₃	C₂×#108	1	2	yes	—
208	C₄×C₁₃	C₂×#108	1	2	yes	—
209	C₄×C₁₃	C₄×#41	1	2	yes	—
210	Dic₁₃	—	2	1	no	—
211	Dic₁₃	—	2	1	no	—
212	Dic₁₃	—	2	1	no	—
213	Dic₁₃	—	2	1	no	—
214	Dic₁₃	#430	1	2	yes	—
215	Dic₁₃	#525	1	2	yes	—
216	C₂³×C₄	—	13	1	no	—
217	C₂³×C₄	—	13	1	no	—
218	C₂³×C₄	—	13	1	no	—
219	C₂×M₁₆	—	13	4	no	—
220	C₂×M₁₆	—	13	4	no	—
221	C₂×M₁₆	—	26	8	no	—
222	C₂×M₁₆	—	26	2	no	—
223	C₂²×C₈	—	13	4	no	—
224	C₂²×C₈	—	26	2	no	—
225	C₂×C₄²	—	13	1	no	—
226	C₂×C₄²	—	13	1	no	—
227	C₂×C₄²	—	13	1	no	—
228	C₂×C₄²	—	13	1	no	—
229	C₂×C₄²	—	13	1	no	—
230	C₂×C₄²	—	13	1	no	—
231	C₂×C₄²	—	13	1	no	—
232	C₂×C₄²	—	13	1	no	—
233	C₂×C₄²	—	13	1	no	—
234	C₂×C₄²	—	13	1	no	—
235	C₂×C₄²	—	13	1	no	—
236	C₂×C₄²	C₂×#261	1	1	yes	characteristic
237	#108	—	26	16	no	—
238	C₄⋊C₈	—	13	4	no	—
239	C₄⋊C₈	—	13	2	no	—
240	C₄⋊C₈	—	26	2	no	—
241	C₄⋊C₈	—	26	2	no	—
242	C₄⋊C₈	C₂×#261	1	2	yes	—
243	C₂²⋊C₈	—	13	2	no	—
244	C₂²⋊C₈	—	13	4	no	—
245	C₂²⋊C₈	—	26	2	no	—
246	C₂²⋊C₈	—	26	2	no	—
247	C₂²⋊C₈	C₂×#261	1	2	yes	—
248	#97	—	26	2	no	—
249	#97	—	26	2	no	—
250	#97	—	26	2	no	—
251	C₄×C₈	—	13	4	no	—
252	C₄×C₈	—	13	2	no	—
253	C₄×C₈	—	26	2	no	—
254	C₄×C₈	—	26	2	no	—
255	C₄×C₈	—	26	2	no	—
256	C₄×C₈	C₂×#261	1	2	yes	—
257	C₂×C₁₃	—	2	2	no	—
258	C₂×C₁₃	#2296	1	1	yes	characteristic
259	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
260	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
261	D₁₃	—	2	2	no	—
262	D₁₃	—	4	1	no	—
263	C₂⁴	—	13	1	no	—
264	C₂²×C₄	—	13	1	no	—
265	C₂²×C₄	—	13	1	no	—
266	C₂²×C₄	—	13	1	no	—
267	C₂²×C₄	—	13	1	no	—
268	C₂²×C₄	—	13	1	no	—
269	C₂²×C₄	—	13	1	no	—
270	C₂²×C₄	—	13	1	no	—
271	C₂²×C₄	—	13	1	no	—
272	C₂²×C₄	—	13	1	no	—
273	C₂²×C₄	—	13	1	no	—
274	C₂²×C₄	—	13	1	no	—
275	C₂²×C₄	—	13	1	no	—
276	C₂²×C₄	—	13	1	no	—
277	C₂²×C₄	—	13	1	no	—
278	C₂²×C₄	—	13	1	no	—
279	C₂²×C₄	—	26	4	no	—
280	C₂²×C₄	—	26	4	no	—
281	C₂²×C₄	—	52	2	no	—
282	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
283	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
284	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
285	M₁₆	—	13	8	no	—
286	M₁₆	—	26	4	no	—
287	M₁₆	—	26	8	no	—
288	M₁₆	—	26	8	no	—
289	C₂×C₈	—	13	4	no	—
290	C₂×C₈	—	26	8	no	—
291	C₂×C₈	—	26	4	no	—
292	C₂×C₈	—	26	8	no	—
293	C₂×C₈	—	26	4	no	—
294	C₂×C₈	C₂²×#261	1	4	yes	—
295	C₄²	—	13	1	no	—
296	C₄²	—	13	1	no	—
297	C₄²	—	13	1	no	—
298	C₄²	—	13	1	no	—
299	C₄²	—	13	1	no	—
300	C₄²	—	13	1	no	—
301	C₄²	—	13	1	no	—
302	C₄²	—	13	1	no	—
303	C₄²	—	13	1	no	—
304	C₄²	—	13	1	no	—
305	C₄²	—	13	1	no	—
306	C₄²	—	13	1	no	—
307	C₄²	C₄⋊#261	1	1	yes	characteristic
308	C₄²	C₄⋊#261	1	1	yes	characteristic
309	C₄²	C₂²×#261	1	1	yes	characteristic
310	C₄²	C₂²×#261	1	1	yes	characteristic
311	C₁₃	#21370	1	1	yes	characteristic, complemented, Sylow
312	C₂³	—	13	1	no	—
313	C₂³	—	13	1	no	—
314	C₂³	—	26	2	no	—
315	C₂³	—	26	2	no	—
316	C₂³	—	52	1	no	—
317	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	13	1	no	—
321	C₂×C₄	—	13	1	no	—
322	C₂×C₄	—	13	4	no	—
323	C₂×C₄	—	13	1	no	—
324	C₂×C₄	—	13	1	no	—
325	C₂×C₄	—	13	1	no	—
326	C₂×C₄	—	13	1	no	—
327	C₂×C₄	—	26	2	no	—
328	C₂×C₄	—	26	2	no	—
329	C₂×C₄	—	26	2	no	—
330	C₂×C₄	—	26	2	no	—
331	C₂×C₄	—	26	2	no	—
332	C₂×C₄	—	26	2	no	—
333	C₂×C₄	—	26	2	no	—
334	C₂×C₄	—	26	2	no	—
335	C₂×C₄	—	26	2	no	—
336	C₂×C₄	—	26	2	no	—
337	C₂×C₄	—	26	2	no	—
338	C₂×C₄	—	26	2	no	—
339	C₂×C₄	—	52	1	no	—
340	C₂×C₄	—	52	1	no	—
341	C₂×C₄	#88103	1	4	yes	complemented
342	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
343	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
344	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
345	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
346	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
347	C₂×C₄	C₂³×#261	1	1	yes	characteristic
348	C₈	—	2	4	no	—
349	C₈	—	26	8	no	—
350	C₈	—	26	4	no	—
351	C₈	—	52	4	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	13	2	no	—
355	C₂²	—	26	1	no	—
356	C₂²	—	26	1	no	—
357	C₂²	—	26	1	no	—
358	C₂²	—	26	1	no	—
359	C₂²	—	26	1	no	—
360	C₂²	—	52	4	no	—
361	C₂²	C₂×#88103	1	2	yes	—
362	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	13	2	no	—
367	C₄	—	13	2	no	—
368	C₄	—	26	1	no	—
369	C₄	—	26	1	no	—
370	C₄	—	26	1	no	—
371	C₄	—	26	1	no	—
372	C₄	C₂×#88103	1	2	yes	—
373	C₄	C₂×#88103	1	2	yes	—
374	C₄	#41×#261	1	2	yes	—
375	C₂	—	2	2	no	—
376	C₂	—	26	2	no	—
377	C₂	—	52	1	no	—
378	C₂	#13131296	1	1	yes	characteristic
379	C₂	C₂²×#88103	1	1	yes	characteristic
380	C₂	D₁₃⋊#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#431	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₁₃	C₂×Q₈
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	#431	#261
hypocenter	C₁₃	#21370
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
3. #13130951	C₂	semidirect
5. C₄×#88103	C₂	semidirect
136. C₂×C₄×C₁₃	#108	semidirect
311. C₁₃	#21370	semidirect
341. C₂×C₄	#88103	semidirect

Maximal subgroups

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

subgroup	index
#13131513	2
C₂×C₄×#6118	2
#13130951	2
#13130789	2
C₄×#88103	2
D₁₃×#431	2
#21370	13