Subgroups — #16807314

Order 3328.

662 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131215	C₂	1	2	yes	maximal, complemented
2	#13131214	C₂	1	4	yes	maximal, complemented
3	D₄×#6092	C₂	1	2	yes	maximal, complemented
4	#13131196	C₂	1	1	yes	maximal, characteristic, complemented
5	#13131195	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718259	C₂	1	2	yes	maximal, complemented
7	D₁₃⋊(C₂²⋊C₁₆)	C₂	1	2	yes	maximal, complemented
8	C₈×D₄×D₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
10	C₄×D₄×D₁₃	C₄	1	1	yes	characteristic
11	C₂²×#6092	C₂²	1	4	yes	complemented
12	C₂×D₁₃⋊C₁₆	C₂²	1	4	yes	complemented
13	C₈×D₄×C₁₃	C₄	1	1	yes	characteristic
14	C₈×C₂²⋊D₁₃	C₄	1	2	yes	—
15	D₄×#1055	C₂²	1	1	yes	characteristic
16	#11718495	C₂²	1	1	yes	characteristic
17	D₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
18	#11718386	C₂²	1	2	yes	—
19	D₁₃×C₂²⋊C₈	C₄	1	2	yes	—
20	C₈×D₅₂	C₄	1	1	yes	characteristic
21	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
22	C₂²⋊#6092	C₂²	1	4	yes	complemented
23	#11718260	C₂²	1	2	yes	complemented
24	#11718259	C₂²	1	1	yes	characteristic
25	#11718259	C₂²	1	2	yes	—
26	#11718259	C₂²	1	1	yes	characteristic
27	#11718259	C₂²	1	4	yes	complemented
28	#11718257	C₂²	1	2	yes	complemented
29	C₄⋊#6092	C₂²	1	2	yes	complemented
30	C₄×#6092	C₂²	1	2	yes	complemented
31	C₂×D₄×D₁₃	C₈	1	1	yes	characteristic
32	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
34	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
35	D₄×Dic₁₃	C₈	1	1	yes	characteristic
36	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
37	C₂²×#1055	C₂³	1	2	yes	—
38	C₂×C₈×D₁₃	—	4	4	no	—
39	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₈×D₁₃	C₂³	1	2	yes	—
44	#88147	C₈	1	1	yes	characteristic
45	C₄⋊C₄×D₁₃	C₈	1	1	yes	characteristic
46	#88135	C₈	1	2	yes	—
47	C₂²⋊C₄×D₁₃	C₈	1	2	yes	—
48	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
49	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
50	C₂×#6092	—	4	4	no	—
51	C₂×#6092	C₂³	1	2	yes	—
52	C₂×#6092	C₂³	1	2	yes	—
53	C₂×#6092	C₂³	1	4	yes	—
54	D₁₃⋊C₁₆	—	4	4	no	—
55	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
58	C₂²⋊#1055	C₂×C₄	1	2	yes	—
59	#88060	C₂×C₄	1	2	yes	—
60	#88060	C₂×C₄	1	2	yes	—
61	#88054	C₂×C₄	1	2	yes	—
62	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
63	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
64	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
65	C₄×#1055	C₂³	1	1	yes	characteristic
66	#33486	—	13	1	no	maximal, Sylow
67	C₂³×D₁₃	C₂×C₈	1	2	yes	—
68	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
71	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
72	D₄×D₁₃	C₁₆	1	8	yes	complemented
73	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
74	C₂×C₄×D₁₃	—	4	2	no	—
75	C₂×C₄×D₁₃	—	4	2	no	—
76	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
78	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
79	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	—	4	2	no	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
87	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
88	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
90	#6103	C₂×C₈	1	2	yes	—
91	#6103	C₂×C₈	1	2	yes	—
92	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
93	#6101	C₂×C₈	1	2	yes	—
94	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
95	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
96	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×#1055	—	2	4	no	—
98	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
99	C₂×#1055	C₂²×C₄	1	2	yes	—
100	C₂×#1055	C₂⁴	1	1	yes	characteristic
101	C₈×D₁₃	—	2	4	no	—
102	C₈×D₁₃	—	2	2	no	—
103	C₈×D₁₃	—	4	2	no	—
104	C₈×D₁₃	—	4	2	no	—
105	C₈×D₁₃	—	4	1	no	—
106	#6092	—	2	2	no	—
107	#6092	—	2	4	no	—
108	#6092	—	2	2	no	—
109	C₂×C₈×D₄	—	13	1	no	—
110	D₄×C₁₆	—	13	2	no	—
111	D₄×C₁₆	—	13	4	no	—
112	D₄×C₁₆	—	13	2	no	—
113	#2162	—	13	1	no	—
114	#2162	—	13	1	no	—
115	C₂×C₂²⋊C₁₆	—	13	2	no	—
116	C₂×C₂²⋊C₁₆	—	13	2	no	—
117	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
118	C₂²×D₁₃	—	4	2	no	—
119	C₂²×D₁₃	—	4	2	no	—
120	C₂²×D₁₃	C₂×C₁₆	1	8	yes	complemented
121	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
122	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
123	D₄×C₁₃	C₂×C₁₆	1	4	yes	complemented
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
130	C₂²⋊D₁₃	C₂×C₁₆	1	8	yes	complemented
131	C₂×Dic₁₃	—	4	2	no	—
132	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
135	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
136	D₅₂	C₂×C₁₆	1	4	yes	complemented
137	C₄×D₁₃	—	2	1	no	—
138	C₄×D₁₃	—	2	2	no	—
139	C₄×D₁₃	—	2	1	no	—
140	C₄×D₁₃	—	2	2	no	—
141	C₄×D₁₃	—	4	2	no	—
142	C₄×D₁₃	—	4	2	no	—
143	C₄×D₁₃	—	4	1	no	—
144	C₄×D₁₃	C₂×C₁₆	1	4	yes	complemented
145	C₈×C₁₃	—	2	1	no	—
146	C₈×C₁₃	—	2	2	no	—
147	C₈×C₁₃	—	2	1	no	—
148	#1055	—	2	2	no	—
149	#1055	—	2	1	no	—
150	#1055	H(5,𝔽₂)	1	2	yes	—
151	C₂³×C₈	—	13	2	no	—
152	C₂×C₄×D₄	—	13	1	no	—
153	C₂²×C₁₆	—	13	4	no	—
154	C₂²×C₁₆	—	13	4	no	—
155	C₈×D₄	—	13	2	no	—
156	C₈×D₄	—	13	1	no	—
157	C₈×D₄	—	13	1	no	—
158	C₈×D₄	—	13	2	no	—
159	C₈×D₄	—	13	1	no	—
160	C₈×D₄	#261	1	1	yes	characteristic
161	C₂×C₄⋊C₈	—	13	1	no	—
162	C₂×C₂²⋊C₈	—	13	2	no	—
163	C₂×C₄×C₈	—	13	1	no	—
164	C₄⋊C₁₆	—	13	2	no	—
165	C₄⋊C₁₆	—	13	2	no	—
166	C₂²⋊C₁₆	—	13	1	no	—
167	C₂²⋊C₁₆	—	13	2	no	—
168	C₂²⋊C₁₆	—	13	1	no	—
169	C₂²⋊C₁₆	—	13	4	no	—
170	C₂²⋊C₁₆	—	13	4	no	—
171	C₄×C₁₆	—	13	2	no	—
172	C₄×C₁₆	—	13	2	no	—
173	C₂²×C₁₃	—	2	2	no	—
174	C₂²×C₁₃	—	2	2	no	—
175	C₂²×C₁₃	C₂²×C₁₆	1	4	yes	—
176	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
177	C₂×D₁₃	—	2	1	no	—
178	C₂×D₁₃	—	2	2	no	—
179	C₂×D₁₃	—	2	1	no	—
180	C₂×D₁₃	—	2	2	no	—
181	C₂×D₁₃	—	4	8	no	—
182	C₂×D₁₃	C₂²×C₁₆	1	2	yes	—
183	C₂×D₁₃	C₂²×C₁₆	1	4	yes	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	C₂²×C₁₆	1	2	yes	—
187	C₄×C₁₃	#518	1	1	yes	characteristic
188	C₄×C₁₃	#568	1	1	yes	characteristic
189	Dic₁₃	—	2	1	no	—
190	Dic₁₃	—	2	1	no	—
191	Dic₁₃	—	2	2	no	—
192	Dic₁₃	C₂²×C₁₆	1	2	yes	—
193	C₂²×D₄	—	13	1	no	—
194	C₂³×C₄	—	13	2	no	—
195	C₂²×C₈	—	13	2	no	—
196	C₂²×C₈	—	13	1	no	—
197	C₂²×C₈	—	13	2	no	—
198	C₂²×C₈	—	13	2	no	—
199	C₂²×C₈	—	13	2	no	—
200	C₂²×C₈	—	13	1	no	—
201	C₂²×C₈	—	52	4	no	—
202	C₂²×C₈	C₂×#261	1	2	yes	—
203	C₄×D₄	—	13	1	no	—
204	C₄×D₄	—	13	2	no	—
205	C₄×D₄	—	13	1	no	—
206	C₄×D₄	—	13	2	no	—
207	C₄×D₄	—	13	1	no	—
208	C₄×D₄	C₂×#261	1	1	yes	characteristic
209	C₂×C₄⋊C₄	—	13	1	no	—
210	C₂×C₂²⋊C₄	—	13	2	no	—
211	C₂×C₄²	—	13	1	no	—
212	C₂×C₁₆	—	13	2	no	—
213	C₂×C₁₆	—	13	2	no	—
214	C₂×C₁₆	—	13	4	no	—
215	C₂×C₁₆	—	52	4	no	—
216	C₂×C₁₆	—	52	4	no	—
217	C₄⋊C₈	—	13	2	no	—
218	C₄⋊C₈	—	13	1	no	—
219	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
220	C₂²⋊C₈	—	13	2	no	—
221	C₂²⋊C₈	—	13	2	no	—
222	C₂²⋊C₈	—	13	2	no	—
223	C₂²⋊C₈	C₂×#261	1	2	yes	—
224	C₄×C₈	—	13	1	no	—
225	C₄×C₈	—	13	1	no	—
226	C₄×C₈	—	13	1	no	—
227	C₄×C₈	C₂×#261	1	1	yes	characteristic
228	C₂×C₁₃	—	2	4	no	—
229	C₂×C₁₃	#2972	1	1	yes	characteristic
230	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
231	C₂×C₁₃	#3407	1	1	yes	characteristic
232	D₁₃	—	2	2	no	—
233	D₁₃	—	2	4	no	—
234	C₂⁴	—	13	2	no	—
235	C₂×D₄	—	13	2	no	—
236	C₂×D₄	—	13	8	no	—
237	C₂×D₄	—	13	1	no	—
238	C₂×D₄	D₁₃⋊C₈	1	1	yes	characteristic
239	C₂²×C₄	—	13	1	no	—
240	C₂²×C₄	—	13	2	no	—
241	C₂²×C₄	—	13	2	no	—
242	C₂²×C₄	—	13	1	no	—
243	C₂²×C₄	—	13	1	no	—
244	C₂²×C₄	—	13	2	no	—
245	C₂²×C₄	—	13	2	no	—
246	C₂²×C₄	—	52	2	no	—
247	C₂²×C₄	—	52	2	no	—
248	C₂²×C₄	C₂²×#261	1	2	yes	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	13	1	no	—
251	C₂×C₈	—	13	1	no	—
252	C₂×C₈	—	13	2	no	—
253	C₂×C₈	—	26	4	no	—
254	C₂×C₈	—	26	2	no	—
255	C₂×C₈	—	26	4	no	—
256	C₂×C₈	—	52	2	no	—
257	C₂×C₈	—	52	2	no	—
258	C₂×C₈	—	52	1	no	—
259	C₂×C₈	C₂²×#261	1	1	yes	characteristic
260	C₂×C₈	C₂²×#261	1	1	yes	characteristic
261	C₂×C₈	C₂²×#261	1	2	yes	—
262	C₄⋊C₄	—	13	2	no	—
263	C₄⋊C₄	—	13	1	no	—
264	C₄⋊C₄	D₁₃⋊C₈	1	1	yes	characteristic
265	C₂²⋊C₄	—	13	2	no	—
266	C₂²⋊C₄	—	13	2	no	—
267	C₂²⋊C₄	—	13	2	no	—
268	C₂²⋊C₄	D₁₃⋊C₈	1	2	yes	—
269	C₄²	—	13	1	no	—
270	C₄²	—	13	1	no	—
271	C₄²	—	13	1	no	—
272	C₄²	C₂²×#261	1	1	yes	characteristic
273	C₁₆	—	26	2	no	—
274	C₁₆	—	26	4	no	—
275	C₁₆	—	26	2	no	—
276	C₁₃	#33486	1	1	yes	characteristic, complemented, Sylow
277	C₂³	—	13	2	no	—
278	C₂³	—	13	1	no	—
279	C₂³	—	13	8	no	—
280	C₂³	—	52	2	no	—
281	C₂³	—	52	2	no	—
282	C₂³	C₂×D₁₃⋊C₈	1	2	yes	—
283	D₄	—	13	8	no	—
284	D₄	—	13	4	no	—
285	D₄	D₁₃⋊C₁₆	1	4	yes	complemented
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	13	1	no	—
289	C₂×C₄	—	13	4	no	—
290	C₂×C₄	—	13	1	no	—
291	C₂×C₄	—	13	1	no	—
292	C₂×C₄	—	13	2	no	—
293	C₂×C₄	—	26	1	no	—
294	C₂×C₄	—	26	2	no	—
295	C₂×C₄	—	26	1	no	—
296	C₂×C₄	—	26	2	no	—
297	C₂×C₄	—	52	1	no	—
298	C₂×C₄	—	52	2	no	—
299	C₂×C₄	—	52	2	no	—
300	C₂×C₄	—	52	2	no	—
301	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
302	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
303	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
304	C₂×C₄	C₂³×#261	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	—	13	2	no	—
309	C₈	—	26	2	no	—
310	C₈	—	26	1	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	13	2	no	—
314	C₂²	—	13	4	no	—
315	C₂²	—	26	1	no	—
316	C₂²	—	26	2	no	—
317	C₂²	—	26	1	no	—
318	C₂²	—	26	2	no	—
319	C₂²	—	52	8	no	—
320	C₂²	C₂×D₁₃⋊C₁₆	1	4	yes	—
321	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	2	no	—
324	C₄	—	13	2	no	—
325	C₄	—	26	1	no	—
326	C₄	—	26	2	no	—
327	C₄	—	26	1	no	—
328	C₄	C₂×D₁₃⋊C₁₆	1	2	yes	—
329	C₄	#11719623	1	1	yes	characteristic
330	C₄	#11719629	1	1	yes	characteristic
331	C₂	—	2	4	no	—
332	C₂	—	26	2	no	—
333	C₂	—	26	4	no	—
334	C₂	#13131684	1	1	yes	characteristic
335	C₂	C₂²×D₁₃⋊C₁₆	1	1	yes	characteristic
336	C₂	#13131804	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator	C₂×C₁₃	C₂³×C₁₆
Fitting	C₈×D₄×C₁₃	C₄
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂×D₄×D₁₃	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₈×D₄	#261
hypocenter	C₁₃	#33486
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #13131215	C₂	semidirect
2. #13131214	C₂	semidirect
3. D₄×#6092	C₂	semidirect
4. #13131196	C₂	semidirect
5. #13131195	C₂	semidirect
6. C₂×#11718259	C₂	semidirect
7. D₁₃⋊(C₂²⋊C₁₆)	C₂	semidirect
11. C₂²×#6092	C₂²	semidirect
12. C₂×D₁₃⋊C₁₆	C₂²	semidirect
22. C₂²⋊#6092	C₂²	semidirect
23. #11718260	C₂²	semidirect
27. #11718259	C₂²	semidirect
28. #11718257	C₂²	semidirect
29. C₄⋊#6092	C₂²	semidirect
30. C₄×#6092	C₂²	semidirect
72. D₄×D₁₃	C₁₆	semidirect
120. C₂²×D₁₃	C₂×C₁₆	semidirect
123. D₄×C₁₃	C₂×C₁₆	semidirect
130. C₂²⋊D₁₃	C₂×C₁₆	semidirect
136. D₅₂	C₂×C₁₆	semidirect
144. C₄×D₁₃	C₂×C₁₆	semidirect
276. C₁₃	#33486	semidirect
285. D₄	D₁₃⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#13131215	2
#13131214	2
D₄×#6092	2
#13131196	2
#13131195	2
C₂×#11718259	2
D₁₃⋊(C₂²⋊C₁₆)	2
C₈×D₄×D₁₃	2
#33486	13