Subgroups — #15832766

Order 3328.

646 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123951	C₂	1	2	yes	maximal, complemented
2	C₄×#88166	C₂	1	1	yes	maximal, characteristic
3	#13113738	C₂	1	2	yes	maximal, complemented
4	#13113735	C₂	1	2	yes	maximal, complemented
5	#13113734	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113732	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113721	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
9	#13113220	C₂	1	2	yes	maximal, complemented
10	#13113118	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#1589	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88166	C₄	1	1	yes	characteristic
13	C₄×#6129	C₂²	1	2	yes	—
14	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
15	C₁₃×#436	C₂²	1	2	yes	—
16	D₈×Dic₁₃	C₄	1	1	yes	characteristic
17	#11718788	—	2	2	no	—
18	#11718779	—	2	2	no	—
19	C₄×#6105	C₂²	1	2	yes	—
20	#11718673	C₂²	1	2	yes	—
21	#11718541	C₄	1	1	yes	characteristic
22	#11718534	C₄	1	1	yes	characteristic
23	D₁₃×C₄⋊C₈	C₂²	1	2	yes	—
24	#11718492	C₂²	1	2	yes	—
25	#11718433	C₄	1	2	yes	—
26	#11718419	C₄	1	2	yes	—
27	#11718389	—	2	2	no	—
28	#11718380	—	2	2	no	—
29	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
30	#11718346	C₂²	1	1	yes	characteristic
31	#11718346	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
33	#11718340	C₂²	1	2	yes	—
34	C₁₃×#335	C₂²	1	1	yes	characteristic
35	C₁₃×D₄⋊C₈	C₂²	1	2	yes	—
36	C₈²×C₁₃	C₂²	1	1	yes	characteristic
37	D₄⋊#1055	C₂²	1	2	yes	—
38	#11718076	C₂²	1	2	yes	—
39	#11718071	C₂²	1	2	yes	—
40	#11718050	C₂²	1	1	yes	characteristic
41	#11718048	C₂²	1	2	yes	—
42	#11718046	C₂²	1	1	yes	characteristic
43	#11718046	C₂²	1	1	yes	characteristic
44	C₈×#1055	C₂²	1	1	yes	characteristic
45	C₂×#6129	C₂×C₄	1	2	yes	—
46	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₃×M₁₆	—	2	2	no	—
48	C₄×D₄×C₁₃	C₂³	1	2	yes	—
49	D₄×Dic₁₃	C₂×C₄	1	2	yes	—
50	C₂×#6105	C₂×C₄	1	2	yes	—
51	C₄×C₂²⋊D₁₃	—	2	2	no	—
52	C₂×C₄×Dic₁₃	—	2	2	no	—
53	C₂×#6099	—	2	2	no	—
54	#88166	—	2	4	no	—
55	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
56	C₂×C₈×D₁₃	—	2	2	no	—
57	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
59	#88146	C₂×C₄	1	2	yes	—
60	#88141	C₂×C₄	1	2	yes	—
61	#88135	—	2	2	no	—
62	#88131	—	2	2	no	—
63	C₄²×D₁₃	C₂³	1	1	yes	characteristic
64	C₄×Dic₂₆	C₂³	1	2	yes	—
65	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
66	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
67	C₁₃×#102	C₂×C₄	1	2	yes	—
68	C₁₃×C₂²⋊C₈	—	2	2	no	—
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	#88072	C₂×C₄	1	2	yes	—
73	C₂²⋊#1055	—	2	2	no	—
74	#88060	—	2	1	no	—
75	#88060	D₄	1	1	yes	characteristic
76	#88060	D₄	1	1	yes	characteristic
77	#88058	C₂×C₄	1	1	yes	characteristic
78	#88056	C₂×C₄	1	2	yes	—
79	#88055	—	2	2	no	—
80	#88055	—	2	2	no	—
81	#88054	—	2	2	no	—
82	#88054	—	2	2	no	—
83	#88054	—	2	1	no	—
84	#88054	—	2	2	no	—
85	#88054	D₄	1	1	yes	characteristic
86	#88054	D₄	1	1	yes	characteristic
87	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
88	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
89	#88050	C₂×C₄	1	2	yes	—
90	#88048	C₂×C₄	1	2	yes	—
91	C₄⋊#1055	C₂³	1	2	yes	—
92	C₄×#1055	C₂³	1	1	yes	characteristic
93	C₄×#1055	C₂³	1	1	yes	characteristic
94	C₄×#1055	C₂³	1	1	yes	characteristic
95	#17898	—	13	1	no	maximal, Sylow
96	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₁₃	—	2	2	no	—
98	C₂×C₂²⋊D₁₃	—	2	2	no	—
99	C₂²×Dic₁₃	—	2	2	no	—
100	C₂²×Dic₁₃	—	2	2	no	—
101	#6129	—	2	4	no	—
102	#6129	—	2	4	no	—
103	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
106	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
107	C₁₃×D₈	—	2	2	no	—
108	C₁₃×M₁₆	—	8	2	no	—
109	C₂×C₈×C₁₃	—	2	2	no	—
110	C₂×C₈×C₁₃	—	2	1	no	—
111	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
116	C₁₃×C₂²⋊C₄	—	2	2	no	—
117	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
118	C₂²⋊Dic₁₃	—	2	2	no	—
119	#6105	—	2	4	no	—
120	#6103	—	2	2	no	—
121	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
122	#6101	—	2	2	no	—
123	#6101	—	2	2	no	—
124	C₄×Dic₁₃	—	2	2	no	—
125	C₄×Dic₁₃	—	2	2	no	—
126	C₄×Dic₁₃	—	2	2	no	—
127	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
128	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
129	C₄×Dic₁₃	#41	1	1	yes	characteristic
130	#6099	—	8	2	no	—
131	C₂×#1055	—	2	2	no	—
132	C₂×#1055	—	2	1	no	—
133	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
134	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
135	C₂×#1055	C₂×D₄	1	1	yes	characteristic
136	C₂×#1055	C₂×D₄	1	1	yes	characteristic
137	Dic₅₂	—	2	2	no	—
138	C₈×D₁₃	—	2	2	no	—
139	C₈×D₁₃	—	8	2	no	—
140	C₈×D₁₃	M₁₆	1	4	yes	complemented
141	#2989	—	13	2	no	—
142	C₄×#135	—	13	1	no	—
143	#1592	—	13	2	no	—
144	#1592	—	13	2	no	—
145	#1591	—	13	1	no	—
146	#1591	—	13	1	no	—
147	#1589	—	13	1	no	—
148	#1589	D₁₃	1	1	yes	characteristic, complemented
149	C₂×#335	—	13	1	no	—
150	#1477	—	13	2	no	—
151	#1450	—	13	1	no	—
152	C₂³×C₁₃	—	2	2	no	—
153	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
154	D₄×C₁₃	—	2	2	no	—
155	D₄×C₁₃	—	2	2	no	—
156	C₂×C₄×C₁₃	—	2	2	no	—
157	C₂×C₄×C₁₃	—	4	2	no	—
158	C₂×C₄×C₁₃	—	4	2	no	—
159	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
162	C₂²⋊D₁₃	—	4	4	no	—
163	C₂×Dic₁₃	—	2	2	no	—
164	C₂×Dic₁₃	—	2	2	no	—
165	C₂×Dic₁₃	—	4	4	no	—
166	C₂×Dic₁₃	—	8	2	no	—
167	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
168	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
169	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
170	C₄×D₁₃	—	2	1	no	—
171	C₄×D₁₃	—	2	1	no	—
172	C₄×D₁₃	—	4	1	no	—
173	C₄×D₁₃	C₂×M₁₆	1	2	yes	—
174	C₄×D₁₃	#131	1	2	yes	—
175	Dic₂₆	—	2	2	no	—
176	Dic₂₆	—	2	2	no	—
177	C₈×C₁₃	—	2	1	no	—
178	C₈×C₁₃	—	2	1	no	—
179	C₈×C₁₃	—	2	1	no	—
180	C₈×C₁₃	—	4	1	no	—
181	C₈×C₁₃	—	4	2	no	—
182	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
183	#1055	—	2	1	no	—
184	#1055	—	2	1	no	—
185	#1055	—	2	1	no	—
186	#1055	—	4	1	no	—
187	#1055	—	4	2	no	—
188	#1055	C₂×M₁₆	1	2	yes	—
189	C₂×#135	—	13	1	no	—
190	C₄×#41	—	13	2	no	—
191	#446	—	13	2	no	—
192	#446	—	13	2	no	—
193	C₄×Q₁₆	—	13	1	no	—
194	C₄×Q₁₆	—	13	1	no	—
195	C₄×SD₁₆	—	13	2	no	—
196	C₄×SD₁₆	—	13	2	no	—
197	C₄×D₈	—	13	1	no	—
198	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
199	#436	—	13	2	no	—
200	#436	C₂×D₁₃	1	2	yes	—
201	#435	—	26	2	no	—
202	#435	—	26	2	no	—
203	#432	—	26	2	no	—
204	#431	—	13	1	no	—
205	#431	—	13	1	no	—
206	#427	—	13	1	no	—
207	#423	—	26	2	no	—
208	C₂×C₄⋊C₈	—	13	2	no	—
209	#416	—	13	2	no	—
210	C₂×C₄×C₈	—	13	1	no	—
211	#335	—	13	2	no	—
212	#335	—	13	1	no	—
213	#335	C₂×D₁₃	1	1	yes	characteristic
214	Q₈⋊C₈	—	13	2	no	—
215	Q₈⋊C₈	—	13	2	no	—
216	D₄⋊C₈	—	13	2	no	—
217	D₄⋊C₈	C₂×D₁₃	1	2	yes	—
218	#322	—	13	1	no	—
219	#322	—	13	1	no	—
220	C₈²	—	13	1	no	—
221	C₈²	C₂×D₁₃	1	1	yes	characteristic
222	C₂²×C₁₃	—	4	2	no	—
223	C₂²×C₁₃	—	4	2	no	—
224	C₂²×C₁₃	—	4	2	no	—
225	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
226	C₂×D₁₃	—	2	1	no	—
227	C₂×D₁₃	—	2	1	no	—
228	C₂×D₁₃	#436	1	2	yes	—
229	C₄×C₁₃	—	2	1	no	—
230	C₄×C₁₃	—	4	2	no	—
231	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
232	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
233	C₄×C₁₃	#576	1	1	yes	characteristic
234	C₄×C₁₃	#578	1	1	yes	characteristic
235	Dic₁₃	—	2	1	no	—
236	Dic₁₃	—	2	1	no	—
237	Dic₁₃	—	4	2	no	—
238	Dic₁₃	—	4	2	no	—
239	Dic₁₃	#436	1	2	yes	—
240	C₂×#41	—	13	2	no	—
241	#135	—	26	4	no	—
242	C₂×Q₁₆	—	13	1	no	—
243	C₂×SD₁₆	—	13	2	no	—
244	C₂×D₈	C₄×D₁₃	1	1	yes	characteristic
245	C₂×M₁₆	—	2	2	no	—
246	C₂×M₁₆	—	26	2	no	—
247	C₂²×C₈	—	13	1	no	—
248	C₂²×C₈	—	13	1	no	—
249	C₂²×C₈	—	26	2	no	—
250	C₄×Q₈	—	13	2	no	—
251	C₄×Q₈	—	13	2	no	—
252	C₄×D₄	—	13	2	no	—
253	C₄×D₄	—	26	2	no	—
254	C₄×D₄	—	26	2	no	—
255	C₄×D₄	C₂²×D₁₃	1	2	yes	—
256	#117	—	13	2	no	—
257	#117	—	26	2	no	—
258	C₂×C₄²	—	13	1	no	—
259	C₂×C₄²	—	26	2	no	—
260	#107	—	13	1	no	—
261	#107	C₄×D₁₃	1	1	yes	characteristic
262	#106	—	13	2	no	—
263	C₄⋊C₈	—	13	1	no	—
264	C₄⋊C₈	—	13	1	no	—
265	C₄⋊C₈	—	13	2	no	—
266	C₄⋊C₈	—	26	2	no	—
267	C₄⋊C₈	—	26	1	no	—
268	C₄⋊C₈	—	26	2	no	—
269	C₄⋊C₈	—	26	2	no	—
270	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
271	#103	—	13	2	no	—
272	#103	—	13	2	no	—
273	#102	—	13	2	no	—
274	#102	C₄×D₁₃	1	2	yes	—
275	C₂²⋊C₈	—	2	2	no	—
276	C₂²⋊C₈	—	13	1	no	—
277	C₂²⋊C₈	—	13	1	no	—
278	C₂²⋊C₈	—	26	1	no	—
279	C₂²⋊C₈	—	26	2	no	—
280	#97	—	26	2	no	—
281	#97	—	26	2	no	—
282	C₄×C₈	—	13	1	no	—
283	C₄×C₈	—	13	1	no	—
284	C₄×C₈	—	13	1	no	—
285	C₄×C₈	—	13	1	no	—
286	C₄×C₈	—	13	1	no	—
287	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
290	C₂×C₁₃	—	8	2	no	—
291	C₂×C₁₃	C₂×#436	1	1	yes	characteristic
292	C₂×C₁₃	#2951	1	1	yes	characteristic
293	C₂×C₁₃	#2957	1	1	yes	characteristic
294	D₁₃	—	2	2	no	—
295	#41	—	26	4	no	—
296	#41	—	26	4	no	—
297	C₂×Q₈	—	13	2	no	—
298	C₂×D₄	—	26	2	no	—
299	C₂×D₄	C₂×C₄×D₁₃	1	2	yes	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	13	1	no	—
302	C₂²×C₄	—	13	1	no	—
303	C₂²×C₄	—	13	1	no	—
304	C₂²×C₄	—	26	2	no	—
305	C₂²×C₄	—	26	2	no	—
306	Q₁₆	—	26	2	no	—
307	SD₁₆	—	26	4	no	—
308	D₈	—	2	2	no	—
309	M₁₆	—	8	2	no	—
310	M₁₆	—	104	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	13	1	no	—
314	C₂×C₈	—	13	4	no	—
315	C₂×C₈	—	13	1	no	—
316	C₂×C₈	—	13	1	no	—
317	C₂×C₈	—	13	1	no	—
318	C₂×C₈	—	26	2	no	—
319	C₂×C₈	—	26	2	no	—
320	C₂×C₈	—	26	1	no	—
321	C₂×C₈	—	104	2	no	—
322	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
323	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
324	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
325	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
326	C₄⋊C₄	—	13	2	no	—
327	C₄⋊C₄	—	26	2	no	—
328	C₄⋊C₄	—	26	2	no	—
329	C₄⋊C₄	C₂×C₄×D₁₃	1	2	yes	—
330	C₂²⋊C₄	—	2	2	no	—
331	C₂²⋊C₄	—	26	2	no	—
332	C₂²⋊C₄	—	26	2	no	—
333	C₄²	—	13	1	no	—
334	C₄²	—	13	1	no	—
335	C₄²	—	13	1	no	—
336	C₄²	—	26	2	no	—
337	C₄²	—	26	2	no	—
338	C₄²	—	26	2	no	—
339	C₄²	C₂³×D₁₃	1	1	yes	characteristic
340	C₁₃	#17898	1	1	yes	characteristic, complemented, Sylow
341	C₂³	—	2	2	no	—
342	C₂³	—	13	1	no	—
343	Q₈	—	26	2	no	—
344	Q₈	—	26	2	no	—
345	D₄	—	2	2	no	—
346	D₄	—	2	2	no	—
347	D₄	—	52	4	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	13	1	no	—
352	C₂×C₄	—	13	2	no	—
353	C₂×C₄	—	13	2	no	—
354	C₂×C₄	—	13	1	no	—
355	C₂×C₄	—	13	1	no	—
356	C₂×C₄	—	26	1	no	—
357	C₂×C₄	—	26	1	no	—
358	C₂×C₄	—	26	2	no	—
359	C₂×C₄	—	26	2	no	—
360	C₂×C₄	—	52	1	no	—
361	C₂×C₄	—	52	4	no	—
362	C₂×C₄	—	104	2	no	—
363	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
364	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
365	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	2	1	no	—
368	C₈	—	2	1	no	—
369	C₈	—	4	1	no	—
370	C₈	—	4	2	no	—
371	C₈	—	13	2	no	—
372	C₈	—	26	1	no	—
373	C₈	—	26	1	no	—
374	C₈	—	26	1	no	—
375	C₈	—	52	1	no	—
376	C₈	—	52	2	no	—
377	C₈	M₁₆×D₁₃	1	2	yes	—
378	C₂²	—	4	2	no	—
379	C₂²	—	4	2	no	—
380	C₂²	—	4	2	no	—
381	C₂²	—	13	2	no	—
382	C₂²	—	26	1	no	—
383	C₂²	—	26	1	no	—
384	C₂²	C₄×D₄×D₁₃	1	1	yes	characteristic
385	C₄	—	2	1	no	—
386	C₄	—	4	2	no	—
387	C₄	—	13	2	no	—
388	C₄	—	26	1	no	—
389	C₄	—	26	1	no	—
390	C₄	—	52	2	no	—
391	C₄	—	52	2	no	—
392	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
393	C₄	D₁₃×#131	1	1	yes	characteristic
394	C₄	#11719462	1	1	yes	characteristic
395	C₄	#11719476	1	1	yes	characteristic
396	C₂	—	8	2	no	—
397	C₂	—	26	2	no	—
398	C₂	D₁₃×#436	1	1	yes	characteristic
399	C₂	#13123706	1	1	yes	characteristic
400	C₂	#13123748	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1589	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×#88166	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1589	D₁₃
hypocenter	C₁₃	#17898
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1589

Complements

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

N	H ≅ G/N	type
1. #13123951	C₂	semidirect
3. #13113738	C₂	semidirect
4. #13113735	C₂	semidirect
5. #13113734	C₂	semidirect
6. #13113732	C₂	semidirect
7. #13113721	C₂	semidirect
8. D₁₃×#335	C₂	semidirect
9. #13113220	C₂	semidirect
10. #13113118	C₂	semidirect
11. C₁₃×#1589	C₂	semidirect
140. C₈×D₁₃	M₁₆	semidirect
148. #1589	D₁₃	semidirect
340. C₁₃	#17898	semidirect

Maximal subgroups

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

subgroup	index
#13123951	2
C₄×#88166	2
#13113738	2
#13113735	2
#13113734	2
#13113732	2
#13113721	2
D₁₃×#335	2
#13113220	2
#13113118	2
C₁₃×#1589	2
#17898	13