Subgroups — #16798330

Order 3328.

810 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131513	C₂	1	2	yes	maximal
2	#13130777	C₂	1	2	yes	maximal, complemented
3	#13130760	C₂	1	2	yes	maximal, complemented
4	#13130716	C₂	1	4	yes	maximal
5	C₄×#88104	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
8	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
9	#11719130	—	2	2	no	—
10	#11719127	C₂²	1	4	yes	—
11	#11719119	—	2	2	no	—
12	C₂×#88104	C₂²	1	8	yes	—
13	#11719054	—	2	2	no	—
14	#11719052	C₂²	1	4	yes	—
15	#11719051	—	2	2	no	—
16	C₄×#6118	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
19	C₂×C₄×#1055	C₄	1	1	yes	characteristic
20	C₄×C₈×D₁₃	C₄	1	1	yes	characteristic
21	C₄×C₈×D₁₃	C₄	1	2	yes	—
22	C₄×C₈×D₁₃	C₄	1	1	yes	characteristic
23	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
24	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
25	C₄×C₈×D₁₃	C₂²	1	2	yes	—
26	#11718268	C₂²	1	8	yes	—
27	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
28	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
29	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
30	C₂×#6118	C₂³	1	8	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
35	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
36	C₂²×#1055	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₁₃	—	2	8	no	—
38	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₁₃	C₂³	1	2	yes	—
41	C₂×C₈×D₁₃	C₂³	1	2	yes	—
42	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
44	C₄²×D₁₃	D₄	1	1	yes	characteristic
45	C₄²×D₁₃	D₄	1	1	yes	characteristic
46	C₄²×D₁₃	Q₈	1	1	yes	characteristic
47	C₄²×D₁₃	Q₈	1	1	yes	characteristic
48	C₄²×D₁₃	C₂³	1	1	yes	characteristic
49	C₄²×D₁₃	C₂³	1	1	yes	characteristic
50	C₂²⋊#1057	—	2	4	no	—
51	#88111	—	2	4	no	—
52	#88110	—	2	4	no	—
53	C₄⋊#1057	—	2	4	no	—
54	C₄⋊#1057	—	2	4	no	—
55	C₄×#1057	—	2	4	no	—
56	#88104	—	2	16	no	—
57	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
60	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
61	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
62	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
63	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
64	C₄×#1055	C₂×C₄	1	1	yes	characteristic
65	C₄×#1055	C₂×C₄	1	2	yes	—
66	C₄×#1055	C₂×C₄	1	1	yes	characteristic
67	#20826	—	13	1	no	maximal, Sylow
68	C₂³×D₁₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×D₁₃	—	2	4	no	—
76	C₂×C₄×D₁₃	—	2	4	no	—
77	C₂×C₄×D₁₃	—	2	4	no	—
78	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
90	C₂×#1057	—	2	8	no	—
91	#6118	—	2	16	no	—
92	C₂×C₈×C₁₃	—	2	4	no	—
93	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
103	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
104	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
105	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×#1055	—	2	4	no	—
112	C₂×#1055	C₂²×C₄	1	2	yes	—
113	C₂×#1055	C₂²×C₄	1	2	yes	—
114	C₈×D₁₃	—	2	4	no	—
115	C₈×D₁₃	—	2	4	no	—
116	C₈×D₁₃	—	2	4	no	—
117	C₈×D₁₃	#41	1	8	yes	—
118	#2920	—	13	2	no	—
119	C₂²×C₄×C₈	—	13	1	no	—
120	#1848	—	13	2	no	—
121	#1837	—	13	2	no	—
122	#1810	—	13	4	no	—
123	C₄×#108	—	13	4	no	—
124	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×D₁₃	—	2	2	no	—
126	C₂²×D₁₃	—	2	2	no	—
127	C₂²×D₁₃	—	2	2	no	—
128	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×D₁₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	#108	1	4	yes	complemented
133	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×Dic₁₃	—	2	2	no	—
140	C₂×Dic₁₃	—	2	2	no	—
141	C₂×Dic₁₃	—	2	2	no	—
142	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₄×D₁₃	—	2	2	no	—
149	C₄×D₁₃	—	2	2	no	—
150	C₄×D₁₃	—	2	2	no	—
151	C₄×D₁₃	—	2	2	no	—
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	#108	1	4	yes	complemented
159	C₄×D₁₃	C₂×#41	1	4	yes	—
160	#1057	—	4	8	no	—
161	C₈×C₁₃	—	2	2	no	—
162	C₈×C₁₃	#117	1	4	yes	—
163	#1055	—	2	2	no	—
164	#1055	#117	1	4	yes	—
165	C₂³×C₈	—	13	2	no	—
166	C₂²×C₄²	—	13	1	no	—
167	#432	—	26	2	no	—
168	#432	—	26	2	no	—
169	#431	—	26	2	no	—
170	#431	—	26	2	no	—
171	C₂×#108	—	13	8	no	—
172	#423	—	13	4	no	—
173	#407	—	13	4	no	—
174	C₄×M₁₆	—	13	4	no	—
175	C₂×C₄×C₈	—	13	2	no	—
176	C₂×C₄×C₈	—	13	1	no	—
177	C₂×C₄×C₈	—	13	1	no	—
178	C₂×C₄×C₈	—	13	1	no	—
179	C₂×C₄×C₈	—	13	1	no	—
180	C₂×C₄×C₈	—	13	2	no	—
181	C₂×C₄×C₈	—	13	1	no	—
182	C₂×C₄×C₈	—	13	2	no	—
183	C₂×C₄×C₈	#261	1	1	yes	characteristic
184	#341	—	13	8	no	—
185	C₂²×C₁₃	—	2	1	no	—
186	C₂²×C₁₃	—	2	1	no	—
187	C₂²×C₁₃	C₂×#108	1	2	yes	—
188	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
189	C₂×D₁₃	—	2	8	no	—
190	C₂×D₁₃	—	2	1	no	—
191	C₂×D₁₃	—	2	1	no	—
192	C₂×D₁₃	—	2	1	no	—
193	C₂×D₁₃	—	2	1	no	—
194	C₂×D₁₃	C₂×#108	1	2	yes	—
195	C₂×D₁₃	#533	1	2	yes	—
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	1	no	—
198	C₄×C₁₃	C₂×#108	1	2	yes	—
199	C₄×C₁₃	C₂×#108	1	2	yes	—
200	C₄×C₁₃	C₂×#108	1	2	yes	—
201	C₄×C₁₃	C₂×#117	1	2	yes	—
202	Dic₁₃	—	2	1	no	—
203	Dic₁₃	—	2	1	no	—
204	Dic₁₃	—	2	1	no	—
205	Dic₁₃	—	2	1	no	—
206	Dic₁₃	C₂×#108	1	2	yes	—
207	Dic₁₃	#532	1	2	yes	—
208	C₂³×C₄	—	13	1	no	—
209	C₂³×C₄	—	13	1	no	—
210	C₂³×C₄	—	13	1	no	—
211	C₂×M₁₆	—	13	8	no	—
212	C₂²×C₈	—	13	2	no	—
213	C₂²×C₈	—	13	2	no	—
214	C₂²×C₈	—	13	2	no	—
215	C₂²×C₈	—	13	2	no	—
216	C₂²×C₈	—	13	2	no	—
217	C₂²×C₈	—	26	8	no	—
218	C₂²×C₈	C₂×#261	1	2	yes	—
219	C₂×C₄²	—	13	1	no	—
220	C₂×C₄²	—	13	1	no	—
221	C₂×C₄²	—	13	1	no	—
222	C₂×C₄²	—	13	1	no	—
223	C₂×C₄²	—	13	1	no	—
224	C₂×C₄²	—	13	1	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	C₂×#261	1	1	yes	characteristic
231	#108	—	26	16	no	—
232	C₄⋊C₈	—	26	4	no	—
233	C₄⋊C₈	—	26	4	no	—
234	C₂²⋊C₈	—	26	4	no	—
235	C₂²⋊C₈	—	26	4	no	—
236	#97	—	26	4	no	—
237	C₄×C₈	—	13	2	no	—
238	C₄×C₈	—	13	2	no	—
239	C₄×C₈	—	13	2	no	—
240	C₄×C₈	—	13	2	no	—
241	C₄×C₈	—	13	1	no	—
242	C₄×C₈	—	13	1	no	—
243	C₄×C₈	—	13	2	no	—
244	C₄×C₈	—	26	4	no	—
245	C₄×C₈	C₂×#261	1	1	yes	characteristic
246	C₄×C₈	C₂×#261	1	1	yes	characteristic
247	C₄×C₈	C₂×#261	1	2	yes	—
248	C₂×C₁₃	—	2	2	no	—
249	C₂×C₁₃	#2285	1	1	yes	characteristic
250	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
251	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
252	D₁₃	—	2	2	no	—
253	D₁₃	—	2	2	no	—
254	C₂⁴	—	13	1	no	—
255	C₂²×C₄	—	13	1	no	—
256	C₂²×C₄	—	13	1	no	—
257	C₂²×C₄	—	13	1	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	1	no	—
263	C₂²×C₄	—	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	—	26	4	no	—
271	C₂²×C₄	—	26	4	no	—
272	C₂²×C₄	—	26	4	no	—
273	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
274	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
275	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
276	M₁₆	—	26	16	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	13	2	no	—
279	C₂×C₈	—	13	8	no	—
280	C₂×C₈	—	13	2	no	—
281	C₂×C₈	—	26	4	no	—
282	C₂×C₈	—	26	4	no	—
283	C₂×C₈	—	26	4	no	—
284	C₂×C₈	—	26	4	no	—
285	C₂×C₈	—	26	8	no	—
286	C₂×C₈	C₂²×#261	1	2	yes	—
287	C₂×C₈	C₂²×#261	1	2	yes	—
288	C₄²	—	13	1	no	—
289	C₄²	—	13	1	no	—
290	C₄²	—	13	1	no	—
291	C₄²	—	13	1	no	—
292	C₄²	—	13	1	no	—
293	C₄²	—	13	1	no	—
294	C₄²	—	13	1	no	—
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	C₄⋊#261	1	1	yes	characteristic
301	C₄²	C₄⋊#261	1	1	yes	characteristic
302	C₄²	C₂²×#261	1	1	yes	characteristic
303	C₄²	C₂²×#261	1	1	yes	characteristic
304	C₁₃	#20826	1	1	yes	characteristic, complemented, Sylow
305	C₂³	—	13	1	no	—
306	C₂³	—	13	1	no	—
307	C₂³	—	26	2	no	—
308	C₂³	—	26	2	no	—
309	C₂³	—	26	2	no	—
310	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	13	1	no	—
314	C₂×C₄	—	13	1	no	—
315	C₂×C₄	—	13	4	no	—
316	C₂×C₄	—	13	4	no	—
317	C₂×C₄	—	13	1	no	—
318	C₂×C₄	—	13	1	no	—
319	C₂×C₄	—	13	1	no	—
320	C₂×C₄	—	13	1	no	—
321	C₂×C₄	—	26	2	no	—
322	C₂×C₄	—	26	2	no	—
323	C₂×C₄	—	26	2	no	—
324	C₂×C₄	—	26	2	no	—
325	C₂×C₄	—	26	2	no	—
326	C₂×C₄	—	26	2	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	#88104	1	4	yes	complemented
335	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
336	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
337	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
338	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
339	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
340	C₂×C₄	C₂³×#261	1	1	yes	characteristic
341	C₈	—	2	2	no	—
342	C₈	—	13	4	no	—
343	C₈	—	26	2	no	—
344	C₈	—	52	8	no	—
345	C₈	#88237	1	4	yes	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	13	2	no	—
349	C₂²	—	13	2	no	—
350	C₂²	—	26	8	no	—
351	C₂²	—	26	1	no	—
352	C₂²	—	26	1	no	—
353	C₂²	—	26	1	no	—
354	C₂²	—	26	1	no	—
355	C₂²	C₂×#88104	1	2	yes	—
356	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	13	2	no	—
360	C₄	—	13	2	no	—
361	C₄	—	26	1	no	—
362	C₄	—	26	1	no	—
363	C₄	—	26	1	no	—
364	C₄	—	26	1	no	—
365	C₄	C₂×#88104	1	2	yes	—
366	C₄	C₂×#88104	1	2	yes	—
367	C₄	D₁₃⋊#108	1	2	yes	—
368	C₄	C₂×#88237	1	2	yes	—
369	C₂	—	2	2	no	—
370	C₂	—	26	2	no	—
371	C₂	—	26	2	no	—
372	C₂	#13131285	1	1	yes	characteristic
373	C₂	C₂×D₁₃⋊#108	1	1	yes	characteristic
374	C₂	C₂²×#88104	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	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₁₃	C₂×Q₈
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#20826
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13130777	C₂	semidirect
3. #13130760	C₂	semidirect
5. C₄×#88104	C₂	semidirect
132. C₂×C₄×C₁₃	#108	semidirect
158. C₄×D₁₃	#108	semidirect
304. C₁₃	#20826	semidirect
334. C₂×C₄	#88104	semidirect

Maximal subgroups

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

subgroup	index
#13131513	2
#13130777	2
#13130760	2
#13130716	2
C₄×#88104	2
C₂×C₄×C₈×D₁₃	2
#20826	13