Subgroups — #15834699

Order 3328.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125404	C₂	1	1	yes	maximal, characteristic
2	#13123837	C₂	1	2	yes	maximal, complemented
3	#13113734	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113721	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113682	C₂	1	2	yes	maximal, complemented
6	#13113672	C₂	1	2	yes	maximal, complemented
7	#13113642	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113583	C₂	1	1	yes	maximal, characteristic, complemented
9	#13113241	C₂	1	2	yes	maximal, complemented
10	C₈×#6094	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719262	C₂²	1	2	yes	—
13	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
14	C₈×D₄×C₁₃	C₂²	1	2	yes	—
15	#11718818	C₄	1	1	yes	characteristic
16	#11718816	C₄	1	1	yes	characteristic
17	#11718772	—	2	2	no	—
18	#11718762	—	2	2	no	—
19	C₄×#6105	C₂²	1	2	yes	—
20	#11718673	C₂²	1	2	yes	—
21	#11718548	C₄	1	1	yes	characteristic
22	#11718539	C₄	1	1	yes	characteristic
23	#11718494	C₂²	1	2	yes	—
24	#11718491	C₂²	1	2	yes	—
25	#11718439	C₄	1	2	yes	—
26	#11718429	C₄	1	2	yes	—
27	#11718390	—	2	2	no	—
28	#11718381	—	2	2	no	—
29	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
30	C₄×#6094	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	#11718072	C₂²	1	2	yes	—
40	#11718050	C₂²	1	1	yes	characteristic
41	#11718048	C₂²	1	2	yes	—
42	#11718047	C₂²	1	1	yes	characteristic
43	#11718046	C₂²	1	1	yes	characteristic
44	C₈×#1055	C₂²	1	1	yes	characteristic
45	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	—	2	2	no	—
47	C₄×D₄×C₁₃	C₂³	1	2	yes	—
48	#88192	C₂×C₄	1	2	yes	—
49	#88190	C₂×C₄	1	2	yes	—
50	C₂×#6105	C₂×C₄	1	2	yes	—
51	C₄×C₂²⋊D₁₃	—	2	2	no	—
52	#88180	—	2	2	no	—
53	C₂×#6099	—	2	2	no	—
54	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
55	C₂×#6094	D₄	1	1	yes	characteristic
56	C₂×#6094	D₄	1	1	yes	characteristic
57	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
59	#88151	C₂×C₄	1	2	yes	—
60	#88144	C₂×C₄	1	2	yes	—
61	#88137	—	2	2	no	—
62	#88133	—	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	2	no	—
75	#88060	—	2	1	no	—
76	#88058	C₂×C₄	1	1	yes	characteristic
77	#88056	C₂×C₄	1	2	yes	—
78	#88055	—	2	2	no	—
79	#88055	D₄	1	1	yes	characteristic
80	#88055	D₄	1	1	yes	characteristic
81	#88054	—	2	2	no	—
82	#88054	—	2	1	no	—
83	#88054	—	2	2	no	—
84	C₈×Dic₁₃	—	2	2	no	—
85	C₈×Dic₁₃	—	2	2	no	—
86	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
87	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
88	#88050	C₂×C₄	1	2	yes	—
89	#88048	C₂×C₄	1	2	yes	—
90	C₄⋊#1055	C₂³	1	2	yes	—
91	C₄×#1055	C₂³	1	1	yes	characteristic
92	C₄×#1055	C₂³	1	1	yes	characteristic
93	C₄×#1055	C₂³	1	1	yes	characteristic
94	#18069	—	13	1	no	maximal, Sylow
95	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₁₃	—	2	2	no	—
97	C₂×C₂²⋊D₁₃	—	2	2	no	—
98	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
101	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
102	C₁₃×D₈	—	2	2	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	2	1	no	—
105	C₂×C₈×C₁₃	—	4	4	no	—
106	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
111	C₁₃×C₂²⋊C₄	—	2	2	no	—
112	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
113	C₂²⋊Dic₁₃	—	2	2	no	—
114	C₂²⋊Dic₁₃	—	2	2	no	—
115	C₂²⋊Dic₁₃	—	2	2	no	—
116	#6105	—	4	2	no	—
117	#6103	—	2	2	no	—
118	C₄⋊Dic₁₃	—	2	2	no	—
119	C₄⋊Dic₁₃	—	2	2	no	—
120	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
121	#6101	—	2	2	no	—
122	#6101	—	2	2	no	—
123	C₄×Dic₁₃	—	2	2	no	—
124	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
125	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
126	C₄×Dic₁₃	#41	1	1	yes	characteristic
127	#6099	—	4	4	no	—
128	C₂×#1055	—	2	2	no	—
129	C₂×#1055	—	2	1	no	—
130	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
131	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
132	C₂×#1055	C₂×D₄	1	1	yes	characteristic
133	C₂×#1055	C₂×D₄	1	1	yes	characteristic
134	Dic₅₂	—	2	2	no	—
135	#6094	—	2	2	no	—
136	#6094	—	2	2	no	—
137	C₈×D₁₃	—	2	2	no	—
138	C₈×D₁₃	—	4	1	no	—
139	#3168	—	13	1	no	—
140	#2969	—	13	2	no	—
141	#1591	—	13	1	no	—
142	#1589	—	13	1	no	—
143	#1583	—	13	2	no	—
144	#1582	—	13	2	no	—
145	C₈×Q₁₆	—	13	1	no	—
146	C₈×D₈	D₁₃	1	1	yes	characteristic, complemented
147	#1564	—	13	1	no	—
148	#1482	—	13	2	no	—
149	C₈×M₁₆	—	13	1	no	—
150	C₂³×C₁₃	—	2	2	no	—
151	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
152	D₄×C₁₃	—	2	2	no	—
153	D₄×C₁₃	—	2	2	no	—
154	C₂×C₄×C₁₃	—	2	2	no	—
155	C₂×C₄×C₁₃	—	4	2	no	—
156	C₂×C₄×C₁₃	—	4	2	no	—
157	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
160	C₂²⋊D₁₃	—	8	2	no	—
161	C₂×Dic₁₃	—	2	2	no	—
162	C₂×Dic₁₃	—	2	2	no	—
163	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
164	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
165	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
166	C₄×D₁₃	—	2	1	no	—
167	C₄×D₁₃	—	2	1	no	—
168	C₄×D₁₃	—	2	1	no	—
169	C₄×D₁₃	—	2	1	no	—
170	C₄×D₁₃	—	4	1	no	—
171	Dic₂₆	—	2	2	no	—
172	Dic₂₆	—	2	2	no	—
173	C₈×C₁₃	—	2	1	no	—
174	C₈×C₁₃	—	2	1	no	—
175	C₈×C₁₃	—	4	1	no	—
176	C₈×C₁₃	—	4	2	no	—
177	C₈×C₁₃	#131	1	2	yes	—
178	C₈×C₁₃	#135	1	2	yes	—
179	#1055	—	2	1	no	—
180	#1055	—	2	1	no	—
181	#1055	—	2	1	no	—
182	#1055	—	4	1	no	—
183	#1055	—	4	2	no	—
184	#1055	#135	1	2	yes	—
185	#529	—	13	2	no	—
186	#499	—	13	1	no	—
187	#495	—	13	1	no	—
188	#488	—	13	2	no	—
189	#467	—	13	1	no	—
190	#466	—	13	1	no	—
191	#465	—	13	2	no	—
192	#446	—	13	2	no	—
193	C₈×Q₈	—	13	2	no	—
194	C₄×Q₁₆	—	13	1	no	—
195	C₄×SD₁₆	—	13	2	no	—
196	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
197	#436	—	13	2	no	—
198	#436	—	26	2	no	—
199	#435	—	26	2	no	—
200	C₈×D₄	C₂×D₁₃	1	2	yes	—
201	#433	—	26	2	no	—
202	#431	—	13	1	no	—
203	#431	—	13	2	no	—
204	#405	—	26	2	no	—
205	C₄×M₁₆	—	13	1	no	—
206	C₂×C₄×C₈	—	13	1	no	—
207	#335	—	13	1	no	—
208	#335	C₂×D₁₃	1	1	yes	characteristic
209	#334	—	13	2	no	—
210	Q₈⋊C₈	—	13	2	no	—
211	Q₈⋊C₈	—	13	2	no	—
212	D₄⋊C₈	—	13	2	no	—
213	D₄⋊C₈	C₂×D₁₃	1	2	yes	—
214	#322	—	13	1	no	—
215	#322	—	13	1	no	—
216	C₈²	—	13	1	no	—
217	C₈²	C₂×D₁₃	1	1	yes	characteristic
218	C₂²×C₁₃	—	4	2	no	—
219	C₂²×C₁₃	—	4	2	no	—
220	C₂²×C₁₃	—	4	2	no	—
221	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂×D₁₃	—	2	1	no	—
223	C₂×D₁₃	—	2	1	no	—
224	C₂×D₁₃	—	2	1	no	—
225	C₄×C₁₃	—	2	1	no	—
226	C₄×C₁₃	—	4	2	no	—
227	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
228	C₄×C₁₃	#568	1	1	yes	characteristic
229	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
230	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
231	Dic₁₃	—	2	1	no	—
232	Dic₁₃	—	2	1	no	—
233	Dic₁₃	—	2	1	no	—
234	Dic₁₃	—	4	2	no	—
235	Dic₁₃	—	4	2	no	—
236	C₂×Q₁₆	—	13	1	no	—
237	C₂×SD₁₆	—	13	2	no	—
238	C₂×D₈	C₄×D₁₃	1	1	yes	characteristic
239	C₂×M₁₆	—	13	1	no	—
240	C₂×M₁₆	—	13	1	no	—
241	C₂×M₁₆	—	26	2	no	—
242	C₂²×C₈	—	2	2	no	—
243	C₂²×C₈	—	13	1	no	—
244	C₂²×C₈	—	13	1	no	—
245	#126	—	26	2	no	—
246	#125	—	13	2	no	—
247	#124	—	13	2	no	—
248	#123	—	26	2	no	—
249	C₂²⋊Q₈	—	13	2	no	—
250	#121	—	13	2	no	—
251	C₄×Q₈	—	13	2	no	—
252	C₄×D₄	—	26	2	no	—
253	C₄×D₄	C₂²×D₁₃	1	2	yes	—
254	#117	—	26	2	no	—
255	C₂×C₄²	—	13	1	no	—
256	#107	—	13	1	no	—
257	#107	C₄×D₁₃	1	1	yes	characteristic
258	#106	—	13	2	no	—
259	C₄⋊C₈	—	13	2	no	—
260	C₄⋊C₈	—	26	1	no	—
261	C₄⋊C₈	—	26	2	no	—
262	C₄⋊C₈	—	26	2	no	—
263	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
264	#103	—	13	2	no	—
265	#103	—	13	2	no	—
266	#102	—	13	2	no	—
267	#102	C₄×D₁₃	1	2	yes	—
268	C₂²⋊C₈	—	2	2	no	—
269	C₂²⋊C₈	—	26	2	no	—
270	C₂²⋊C₈	—	26	1	no	—
271	C₂²⋊C₈	—	26	2	no	—
272	#97	—	13	1	no	—
273	#97	—	13	1	no	—
274	#97	—	26	2	no	—
275	C₄×C₈	—	13	1	no	—
276	C₄×C₈	—	13	1	no	—
277	C₄×C₈	—	13	1	no	—
278	C₄×C₈	—	13	1	no	—
279	C₄×C₈	—	13	1	no	—
280	C₄×C₈	—	26	2	no	—
281	C₄×C₈	—	26	2	no	—
282	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
285	C₂×C₁₃	—	8	2	no	—
286	C₂×C₁₃	#2935	1	1	yes	characteristic
287	C₂×C₁₃	C₄×#135	1	1	yes	characteristic
288	C₂×C₁₃	#2957	1	1	yes	characteristic
289	D₁₃	—	4	1	no	—
290	C₂×Q₈	—	13	2	no	—
291	C₂×D₄	—	26	2	no	—
292	C₂×D₄	C₂×C₄×D₁₃	1	2	yes	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	13	1	no	—
295	C₂²×C₄	—	13	1	no	—
296	C₂²×C₄	—	13	1	no	—
297	Q₁₆	—	26	2	no	—
298	SD₁₆	—	52	2	no	—
299	D₈	—	2	2	no	—
300	M₁₆	—	26	2	no	—
301	M₁₆	—	26	2	no	—
302	M₁₆	—	52	4	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	4	4	no	—
306	C₂×C₈	—	13	1	no	—
307	C₂×C₈	—	13	1	no	—
308	C₂×C₈	—	13	1	no	—
309	C₂×C₈	—	13	1	no	—
310	C₂×C₈	—	26	2	no	—
311	C₂×C₈	—	26	2	no	—
312	C₂×C₈	—	26	1	no	—
313	C₂×C₈	—	52	1	no	—
314	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
315	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
316	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
317	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
318	C₄⋊C₄	—	13	2	no	—
319	C₄⋊C₄	—	26	2	no	—
320	C₄⋊C₄	—	26	2	no	—
321	C₄⋊C₄	—	26	2	no	—
322	C₄⋊C₄	—	26	2	no	—
323	C₄⋊C₄	C₂×C₄×D₁₃	1	2	yes	—
324	C₂²⋊C₄	—	2	2	no	—
325	C₂²⋊C₄	—	26	2	no	—
326	C₂²⋊C₄	—	26	2	no	—
327	C₂²⋊C₄	—	26	2	no	—
328	C₂²⋊C₄	—	26	2	no	—
329	C₄²	—	13	1	no	—
330	C₄²	—	13	1	no	—
331	C₄²	—	13	1	no	—
332	C₄²	—	26	2	no	—
333	C₄²	C₂³×D₁₃	1	1	yes	characteristic
334	C₁₃	#18069	1	1	yes	characteristic, complemented, Sylow
335	C₂³	—	2	2	no	—
336	C₂³	—	13	1	no	—
337	Q₈	—	26	2	no	—
338	Q₈	—	26	2	no	—
339	D₄	—	2	2	no	—
340	D₄	—	2	2	no	—
341	D₄	—	104	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	13	1	no	—
346	C₂×C₄	—	13	1	no	—
347	C₂×C₄	—	13	1	no	—
348	C₂×C₄	—	26	1	no	—
349	C₂×C₄	—	26	1	no	—
350	C₂×C₄	—	26	1	no	—
351	C₂×C₄	—	26	1	no	—
352	C₂×C₄	—	26	2	no	—
353	C₂×C₄	—	26	2	no	—
354	C₂×C₄	—	52	1	no	—
355	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
356	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
357	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	4	1	no	—
361	C₈	—	4	2	no	—
362	C₈	—	13	2	no	—
363	C₈	—	26	1	no	—
364	C₈	—	26	1	no	—
365	C₈	—	26	1	no	—
366	C₈	—	52	1	no	—
367	C₈	—	52	2	no	—
368	C₈	#88155	1	2	yes	—
369	C₈	#88166	1	2	yes	—
370	C₂²	—	4	2	no	—
371	C₂²	—	4	2	no	—
372	C₂²	—	4	2	no	—
373	C₂²	—	26	1	no	—
374	C₂²	—	26	1	no	—
375	C₂²	—	26	1	no	—
376	C₂²	C₄×D₄×D₁₃	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	4	2	no	—
379	C₄	—	26	1	no	—
380	C₄	—	26	1	no	—
381	C₄	—	26	1	no	—
382	C₄	—	52	2	no	—
383	C₄	—	52	2	no	—
384	C₄	C₂×#88155	1	1	yes	characteristic
385	C₄	#11719455	1	1	yes	characteristic
386	C₄	C₂×#88166	1	1	yes	characteristic
387	C₄	D₁₃×#135	1	1	yes	characteristic
388	C₂	—	8	2	no	—
389	C₂	—	52	1	no	—
390	C₂	#13123575	1	1	yes	characteristic
391	C₂	C₄×#88166	1	1	yes	characteristic
392	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	C₈×C₁₃×D₈	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11718818	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₈×D₈	D₁₃
hypocenter	C₁₃	#18069
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×D₈

Complements

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

N	H ≅ G/N	type
2. #13123837	C₂	semidirect
3. #13113734	C₂	semidirect
4. #13113721	C₂	semidirect
5. #13113682	C₂	semidirect
6. #13113672	C₂	semidirect
7. #13113642	C₂	semidirect
8. #13113583	C₂	semidirect
9. #13113241	C₂	semidirect
10. C₈×#6094	C₂	semidirect
11. C₈×C₁₃×D₈	C₂	semidirect
146. C₈×D₈	D₁₃	semidirect
334. C₁₃	#18069	semidirect

Maximal subgroups

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

subgroup	index
#13125404	2
#13123837	2
#13113734	2
#13113721	2
#13113682	2
#13113672	2
#13113642	2
#13113583	2
#13113241	2
C₈×#6094	2
C₈×C₁₃×D₈	2
#18069	13