Subgroups — #15834382

Order 3328.

652 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123855	C₂	1	2	yes	maximal, complemented
2	C₄×#88173	C₂	1	1	yes	maximal, characteristic
3	#13113722	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113720	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113679	C₂	1	2	yes	maximal, complemented
6	#13113676	C₂	1	2	yes	maximal, complemented
7	Q₁₆×#1055	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113598	C₂	1	1	yes	maximal, characteristic, complemented
9	#13113269	C₂	1	2	yes	maximal, complemented
10	C₈×#6094	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88173	C₄	1	1	yes	characteristic
13	C₄×#6131	C₂²	1	2	yes	—
14	C₈×Q₈×C₁₃	C₂²	1	2	yes	—
15	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
16	Q₁₆×Dic₁₃	C₄	1	1	yes	characteristic
17	C₄×#6106	C₂²	1	2	yes	—
18	Q₈×#1055	C₂²	1	2	yes	—
19	#11718541	C₄	1	1	yes	characteristic
20	#11718533	C₄	1	1	yes	characteristic
21	#11718499	—	2	2	no	—
22	#11718499	C₂²	1	2	yes	—
23	#11718496	—	2	2	no	—
24	#11718496	C₂²	1	2	yes	—
25	#11718465	C₄	1	2	yes	—
26	#11718449	C₄	1	2	yes	—
27	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
28	#11718349	—	2	2	no	—
29	#11718349	C₂²	1	2	yes	—
30	C₄×#6094	—	2	2	no	—
31	C₄×#6094	C₂²	1	1	yes	characteristic
32	#11718346	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
34	C₁₃×#335	C₂²	1	1	yes	characteristic
35	C₁₃×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₈²×C₁₃	C₂²	1	1	yes	characteristic
37	Q₈⋊#1055	C₂²	1	2	yes	—
38	#11718075	C₂²	1	2	yes	—
39	#11718072	C₂²	1	2	yes	—
40	#11718051	C₂²	1	2	yes	—
41	#11718050	C₂²	1	1	yes	characteristic
42	#11718047	C₂²	1	1	yes	characteristic
43	#11718046	C₂²	1	1	yes	characteristic
44	C₈×#1055	C₂²	1	1	yes	characteristic
45	C₂×#6131	C₂×C₄	1	2	yes	—
46	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
47	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
48	Q₈×Dic₁₃	C₂×C₄	1	2	yes	—
49	C₂×#6106	C₂×C₄	1	2	yes	—
50	#88173	—	2	4	no	—
51	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
52	C₂×#6094	—	2	2	no	—
53	C₂×#6094	—	2	2	no	—
54	C₂×#6094	D₄	1	1	yes	characteristic
55	C₂×#6094	D₄	1	1	yes	characteristic
56	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
58	#88147	—	2	2	no	—
59	#88147	C₂×C₄	1	2	yes	—
60	#88146	—	2	2	no	—
61	#88146	C₂×C₄	1	2	yes	—
62	C₄×D₅₂	—	2	2	no	—
63	C₄×D₅₂	C₂³	1	2	yes	—
64	C₄²×D₁₃	—	2	2	no	—
65	C₄²×D₁₃	C₂³	1	1	yes	characteristic
66	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
67	C₁₃×C₄⋊C₈	—	2	2	no	—
68	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
69	C₁₃×#103	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₁₃	—	2	2	no	—
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	#88075	C₂×C₄	1	2	yes	—
75	#88061	C₂×C₄	1	2	yes	—
76	#88060	—	2	2	no	—
77	#88060	—	2	1	no	—
78	#88060	—	2	2	no	—
79	#88060	—	2	2	no	—
80	#88058	C₂×C₄	1	1	yes	characteristic
81	#88055	—	2	2	no	—
82	#88055	D₄	1	1	yes	characteristic
83	#88055	D₄	1	1	yes	characteristic
84	#88054	—	2	1	no	—
85	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
86	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
87	#88049	C₂×C₄	1	2	yes	—
88	#88048	C₂×C₄	1	2	yes	—
89	C₄⋊#1055	—	2	2	no	—
90	C₄⋊#1055	C₂³	1	2	yes	—
91	C₄×#1055	—	2	2	no	—
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	#18039	—	13	1	no	maximal, Sylow
96	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
97	#6131	—	2	4	no	—
98	#6131	—	4	2	no	—
99	C₂×D₅₂	—	2	2	no	—
100	C₂×D₅₂	C₂²×C₄	1	2	yes	—
101	C₂×C₄×D₁₃	—	2	2	no	—
102	C₂×C₄×D₁₃	—	2	2	no	—
103	C₂×C₄×D₁₃	—	2	2	no	—
104	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
107	C₁₃×Q₁₆	M₁₆	1	4	yes	complemented
108	C₂×C₈×C₁₃	—	2	2	no	—
109	C₂×C₈×C₁₃	—	2	1	no	—
110	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₁₃×C₄⋊C₄	—	2	2	no	—
115	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
116	C₄²×C₁₃	—	2	2	no	—
117	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
118	#6106	—	2	4	no	—
119	#6103	—	2	2	no	—
120	#6103	—	2	2	no	—
121	#6103	—	2	2	no	—
122	C₄⋊Dic₁₃	—	2	2	no	—
123	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
124	C₄×Dic₁₃	—	2	2	no	—
125	C₄×Dic₁₃	—	2	2	no	—
126	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
127	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
128	C₄×Dic₁₃	#41	1	1	yes	characteristic
129	C₂×#1055	—	2	2	no	—
130	C₂×#1055	—	2	1	no	—
131	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
132	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
133	C₂×#1055	C₂×D₄	1	1	yes	characteristic
134	C₂×#1055	C₂×D₄	1	1	yes	characteristic
135	D₁₀₄	—	2	2	no	—
136	#6094	—	2	2	no	—
137	#6094	—	2	2	no	—
138	#6094	—	4	4	no	—
139	#6094	—	4	4	no	—
140	C₈×D₁₃	—	2	2	no	—
141	C₈×D₁₃	—	4	1	no	—
142	#2971	—	13	2	no	—
143	C₄×#135	—	13	1	no	—
144	#1589	—	13	1	no	—
145	#1589	—	13	1	no	—
146	#1583	—	13	2	no	—
147	#1583	—	13	2	no	—
148	C₈×Q₁₆	—	13	1	no	—
149	C₈×Q₁₆	D₁₃	1	1	yes	characteristic, complemented
150	#1568	—	13	1	no	—
151	#1488	—	13	2	no	—
152	C₈×M₁₆	—	13	1	no	—
153	C₂²×D₁₃	—	2	2	no	—
154	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
155	Q₈×C₁₃	—	2	2	no	—
156	Q₈×C₁₃	C₂×M₁₆	1	4	yes	—
157	C₂×C₄×C₁₃	—	2	2	no	—
158	C₂×C₄×C₁₃	—	2	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	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	D₅₂	—	2	2	no	—
167	D₅₂	—	2	2	no	—
168	D₅₂	—	4	4	no	—
169	C₄×D₁₃	—	2	1	no	—
170	C₄×D₁₃	—	2	1	no	—
171	C₄×D₁₃	—	2	1	no	—
172	C₄×D₁₃	—	2	1	no	—
173	C₄×D₁₃	—	4	1	no	—
174	C₄×D₁₃	—	4	4	no	—
175	C₄×D₁₃	—	4	2	no	—
176	C₄×D₁₃	—	4	2	no	—
177	C₄×D₁₃	—	8	2	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	C₈×C₁₃	#135	1	2	yes	complemented
184	#1055	—	2	1	no	—
185	#1055	—	2	1	no	—
186	#1055	—	2	1	no	—
187	#1055	—	4	1	no	—
188	#1055	—	4	2	no	—
189	#1055	#135	1	2	yes	complemented
190	C₂×#135	—	13	1	no	—
191	C₄×#41	—	13	2	no	—
192	C₈×Q₈	—	13	2	no	—
193	C₈×Q₈	C₂×D₁₃	1	2	yes	—
194	C₄×Q₁₆	—	13	1	no	—
195	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
196	C₄×SD₁₆	—	13	2	no	—
197	C₄×SD₁₆	—	13	2	no	—
198	C₄×D₈	—	13	1	no	—
199	C₄×D₈	—	13	1	no	—
200	#436	—	13	2	no	—
201	#436	—	13	2	no	—
202	#436	—	26	2	no	—
203	#436	—	26	2	no	—
204	#432	—	13	2	no	—
205	#432	—	26	2	no	—
206	#431	—	13	1	no	—
207	#427	—	13	1	no	—
208	#416	—	13	2	no	—
209	C₄×M₁₆	—	13	1	no	—
210	C₄×M₁₆	—	26	2	no	—
211	C₂×C₄×C₈	—	13	1	no	—
212	#335	—	13	1	no	—
213	#335	C₂×D₁₃	1	1	yes	characteristic
214	#334	—	13	2	no	—
215	Q₈⋊C₈	—	13	2	no	—
216	Q₈⋊C₈	C₂×D₁₃	1	2	yes	—
217	D₄⋊C₈	—	13	2	no	—
218	D₄⋊C₈	—	13	2	no	—
219	#322	—	13	1	no	—
220	#322	—	13	1	no	—
221	C₈²	—	13	1	no	—
222	C₈²	C₂×D₁₃	1	1	yes	characteristic
223	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
224	C₂×D₁₃	—	2	1	no	—
225	C₂×D₁₃	—	2	1	no	—
226	C₂×D₁₃	—	2	1	no	—
227	C₂×D₁₃	—	4	2	no	—
228	C₂×D₁₃	—	4	2	no	—
229	C₂×D₁₃	—	4	2	no	—
230	C₄×C₁₃	—	2	4	no	—
231	C₄×C₁₃	—	2	1	no	—
232	C₄×C₁₃	—	4	2	no	—
233	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
234	C₄×C₁₃	#568	1	1	yes	characteristic
235	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
236	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
237	Dic₁₃	—	2	1	no	—
238	Dic₁₃	—	2	1	no	—
239	Dic₁₃	—	2	1	no	—
240	Dic₁₃	—	4	2	no	—
241	C₂×#41	—	13	2	no	—
242	#135	—	26	4	no	—
243	C₂×Q₁₆	C₄×D₁₃	1	1	yes	characteristic
244	C₂×SD₁₆	—	13	2	no	—
245	C₂×D₈	—	13	1	no	—
246	C₂×M₁₆	—	13	1	no	—
247	C₂×M₁₆	—	13	1	no	—
248	C₂×M₁₆	—	26	2	no	—
249	C₂×M₁₆	—	26	2	no	—
250	C₂²×C₈	—	13	1	no	—
251	C₂²×C₈	—	13	1	no	—
252	C₄×Q₈	—	13	2	no	—
253	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
254	C₄×D₄	—	13	2	no	—
255	C₄×D₄	—	13	2	no	—
256	C₄×D₄	—	26	2	no	—
257	C₄×D₄	—	26	2	no	—
258	#117	—	13	2	no	—
259	#117	—	26	2	no	—
260	C₂×C₄²	—	13	1	no	—
261	C₂×C₄²	—	26	2	no	—
262	#107	—	13	1	no	—
263	#107	C₄×D₁₃	1	1	yes	characteristic
264	#106	—	13	2	no	—
265	C₄⋊C₈	—	2	2	no	—
266	C₄⋊C₈	—	13	2	no	—
267	C₄⋊C₈	—	26	2	no	—
268	C₄⋊C₈	—	26	1	no	—
269	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
270	#103	—	13	2	no	—
271	#103	C₄×D₁₃	1	2	yes	—
272	#102	—	13	2	no	—
273	#102	—	13	2	no	—
274	C₂²⋊C₈	—	26	2	no	—
275	C₂²⋊C₈	—	26	1	no	—
276	C₂²⋊C₈	—	26	2	no	—
277	C₂²⋊C₈	—	26	2	no	—
278	#97	—	13	1	no	—
279	#97	—	13	1	no	—
280	#97	—	26	2	no	—
281	C₄×C₈	—	2	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	—	26	2	no	—
288	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
291	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
292	C₂×C₁₃	C₄×#135	1	1	yes	characteristic
293	C₂×C₁₃	#2957	1	1	yes	characteristic
294	D₁₃	—	4	1	no	—
295	D₁₃	—	8	2	no	—
296	#41	—	26	4	no	—
297	#41	—	52	2	no	—
298	C₂×Q₈	C₂×C₄×D₁₃	1	2	yes	—
299	C₂×D₄	—	13	2	no	—
300	C₂×D₄	—	26	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	C₂²×C₄	—	26	2	no	—
307	Q₁₆	#6094	1	4	yes	complemented
308	SD₁₆	—	26	4	no	—
309	D₈	—	26	2	no	—
310	M₁₆	—	26	2	no	—
311	M₁₆	—	26	2	no	—
312	M₁₆	—	52	4	no	—
313	M₁₆	—	52	4	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	13	1	no	—
317	C₂×C₈	—	13	1	no	—
318	C₂×C₈	—	13	1	no	—
319	C₂×C₈	—	13	1	no	—
320	C₂×C₈	—	26	2	no	—
321	C₂×C₈	—	26	2	no	—
322	C₂×C₈	—	26	1	no	—
323	C₂×C₈	—	52	1	no	—
324	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
325	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
326	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
327	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	13	2	no	—
330	C₄⋊C₄	—	26	2	no	—
331	C₄⋊C₄	C₂×C₄×D₁₃	1	2	yes	—
332	C₂²⋊C₄	—	26	2	no	—
333	C₂²⋊C₄	—	26	2	no	—
334	C₂²⋊C₄	—	26	2	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	13	1	no	—
337	C₄²	—	13	1	no	—
338	C₄²	—	13	1	no	—
339	C₄²	—	26	2	no	—
340	C₄²	—	26	2	no	—
341	C₄²	C₂³×D₁₃	1	1	yes	characteristic
342	C₁₃	#18039	1	1	yes	characteristic, complemented, Sylow
343	C₂³	—	13	1	no	—
344	C₂³	—	26	2	no	—
345	Q₈	—	2	2	no	—
346	Q₈	C₂×#6094	1	4	yes	—
347	D₄	—	26	2	no	—
348	D₄	—	26	2	no	—
349	D₄	—	52	4	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	13	1	no	—
353	C₂×C₄	—	13	1	no	—
354	C₂×C₄	—	13	1	no	—
355	C₂×C₄	—	26	1	no	—
356	C₂×C₄	—	26	1	no	—
357	C₂×C₄	—	26	1	no	—
358	C₂×C₄	—	26	1	no	—
359	C₂×C₄	—	26	2	no	—
360	C₂×C₄	—	52	1	no	—
361	C₂×C₄	—	52	4	no	—
362	C₂×C₄	—	52	2	no	—
363	C₂×C₄	—	52	2	no	—
364	C₂×C₄	—	104	2	no	—
365	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
366	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
367	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
368	C₈	—	2	1	no	—
369	C₈	—	2	1	no	—
370	C₈	—	4	1	no	—
371	C₈	—	4	2	no	—
372	C₈	—	13	2	no	—
373	C₈	—	26	1	no	—
374	C₈	—	26	1	no	—
375	C₈	—	26	1	no	—
376	C₈	—	52	1	no	—
377	C₈	—	52	2	no	—
378	C₈	C₂×#6094	1	2	yes	—
379	C₈	#88173	1	2	yes	complemented
380	C₂²	—	26	1	no	—
381	C₂²	—	26	1	no	—
382	C₂²	—	26	1	no	—
383	C₂²	—	52	2	no	—
384	C₂²	—	52	2	no	—
385	C₂²	—	52	2	no	—
386	C₂²	C₄×D₄×D₁₃	1	1	yes	characteristic
387	C₄	—	2	1	no	—
388	C₄	—	2	4	no	—
389	C₄	—	4	2	no	—
390	C₄	—	26	1	no	—
391	C₄	—	26	1	no	—
392	C₄	—	26	1	no	—
393	C₄	—	52	2	no	—
394	C₄	C₂²×#6094	1	1	yes	characteristic
395	C₄	#11719455	1	1	yes	characteristic
396	C₄	C₂×#88173	1	1	yes	characteristic
397	C₄	D₁₃×#135	1	1	yes	characteristic
398	C₂	—	52	1	no	—
399	C₂	—	104	2	no	—
400	C₂	D₄×#6094	1	1	yes	characteristic
401	C₂	C₄×#88173	1	1	yes	characteristic
402	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×Q₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×#88173	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₈×Q₁₆	D₁₃
hypocenter	C₁₃	#18039
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #13123855	C₂	semidirect
3. #13113722	C₂	semidirect
4. #13113720	C₂	semidirect
5. #13113679	C₂	semidirect
6. #13113676	C₂	semidirect
7. Q₁₆×#1055	C₂	semidirect
8. #13113598	C₂	semidirect
9. #13113269	C₂	semidirect
10. C₈×#6094	C₂	semidirect
11. C₈×C₁₃×Q₁₆	C₂	semidirect
107. C₁₃×Q₁₆	M₁₆	semidirect
149. C₈×Q₁₆	D₁₃	semidirect
183. C₈×C₁₃	#135	semidirect
189. #1055	#135	semidirect
307. Q₁₆	#6094	semidirect
342. C₁₃	#18039	semidirect
379. C₈	#88173	semidirect

Maximal subgroups

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

subgroup	index
#13123855	2
C₄×#88173	2
#13113722	2
#13113720	2
#13113679	2
#13113676	2
Q₁₆×#1055	2
#13113598	2
#13113269	2
C₈×#6094	2
C₈×C₁₃×Q₁₆	2
#18039	13