Subgroups — #15834660

Order 3328.

718 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123873	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic
3	#13113749	C₂	1	2	yes	maximal, complemented
4	#13113664	C₂	1	4	yes	maximal, complemented
5	#13113659	C₂	1	2	yes	maximal, complemented
6	#13113278	C₂	1	2	yes	maximal, complemented
7	#13113125	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₁₀₄	C₄	1	1	yes	characteristic
10	C₂×C₄×D₅₂	C₂²	1	2	yes	—
11	C₁₃×#432	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88061	C₄	1	2	yes	—
14	C₂×#88058	C₄	1	1	yes	characteristic
15	#11718586	C₂²	1	1	yes	characteristic
16	C₄×#6099	C₂²	1	1	yes	characteristic
17	#11718499	C₂²	1	4	yes	complemented
18	#11718495	C₂²	1	4	yes	complemented
19	#11718387	—	2	2	no	—
20	D₁₃×C₂²⋊C₈	—	2	2	no	—
21	#11718367	—	2	4	no	—
22	C₄×D₁₀₄	C₄	1	2	yes	—
23	C₄×D₁₀₄	C₄	1	1	yes	characteristic
24	C₄×D₁₀₄	C₄	1	1	yes	characteristic
25	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
26	C₄×D₁₀₄	C₂²	1	2	yes	—
27	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
28	C₁₃×#335	C₂²	1	2	yes	complemented
29	C₁₃×#334	C₂²	1	2	yes	complemented
30	#11718075	C₂²	1	4	yes	complemented
31	#11718075	C₂²	1	4	yes	complemented
32	#11718047	C₂²	1	2	yes	complemented
33	#11718046	C₂²	1	2	yes	complemented
34	C₂²×D₅₂	C₂×C₄	1	2	yes	—
35	C₂²×C₄×D₁₃	—	2	2	no	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×#6103	—	2	2	no	—
40	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
41	C₂×#6099	—	2	1	no	—
42	C₂×D₁₀₄	—	2	4	no	—
43	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
44	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
45	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
46	C₂×#6094	—	2	4	no	—
47	C₂×C₈×D₁₃	—	2	4	no	—
48	C₄×D₅₂	—	2	2	no	—
49	C₄×D₅₂	—	2	2	no	—
50	C₄×D₅₂	C₂×C₄	1	2	yes	—
51	C₄×D₅₂	C₂×C₄	1	2	yes	—
52	C₄×D₅₂	C₂³	1	2	yes	—
53	C₄×D₅₂	C₂³	1	2	yes	—
54	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
55	C₁₃×C₂²⋊C₈	—	2	2	no	—
56	C₁₃×#97	—	2	2	no	—
57	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₂²⋊#1055	D₄	1	2	yes	—
62	#88061	C₂×C₄	1	2	yes	—
63	#88061	C₂×C₄	1	2	yes	—
64	#88061	C₂×C₄	1	2	yes	—
65	#88061	C₂×C₄	1	2	yes	—
66	#88060	—	2	4	no	—
67	#88060	—	2	4	no	—
68	#88058	C₂×C₄	1	2	yes	—
69	#88058	C₂×C₄	1	1	yes	characteristic
70	#88058	C₂×C₄	1	1	yes	characteristic
71	C₄⋊#1055	D₄	1	2	yes	—
72	#88043	—	2	1	no	—
73	C₄×#1055	C₂³	1	2	yes	—
74	C₄×#1055	C₂³	1	2	yes	—
75	#18064	—	13	1	no	maximal, Sylow
76	C₂³×D₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
80	C₂²×Dic₁₃	—	2	2	no	—
81	C₂×D₅₂	—	2	4	no	—
82	C₂×D₅₂	—	2	2	no	—
83	C₂×D₅₂	—	4	2	no	—
84	C₂×D₅₂	C₂²×C₄	1	2	yes	—
85	C₂×D₅₂	C₂²×C₄	1	2	yes	—
86	C₂×C₄×D₁₃	—	2	2	no	—
87	C₂×C₄×D₁₃	—	2	2	no	—
88	C₂×C₄×D₁₃	—	2	2	no	—
89	C₂×C₄×D₁₃	—	2	2	no	—
90	C₂×C₄×D₁₃	—	8	4	no	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	2	4	no	—
93	C₂×C₈×C₁₃	—	4	1	no	—
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₁₃	#41	1	1	yes	characteristic
101	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
102	#6103	—	2	2	no	—
103	#6103	—	2	2	no	—
104	#6103	—	2	2	no	—
105	#6103	—	2	2	no	—
106	C₄⋊Dic₁₃	—	2	2	no	—
107	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
108	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
109	#6099	—	4	2	no	—
110	C₂×#1055	—	2	2	no	—
111	C₂×#1055	C₂×D₄	1	4	yes	—
112	D₁₀₄	—	2	4	no	—
113	D₁₀₄	—	2	2	no	—
114	D₁₀₄	M₁₆	1	4	yes	complemented
115	#6094	—	8	4	no	—
116	C₈×D₁₃	—	8	4	no	—
117	#2974	—	13	2	no	—
118	C₂×C₄×D₈	—	13	1	no	—
119	#1594	—	13	2	no	—
120	#1581	—	13	4	no	—
121	#1580	—	13	2	no	—
122	#1568	D₁₃	1	1	yes	characteristic, complemented
123	#1490	—	13	2	no	—
124	#1451	—	13	1	no	—
125	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
126	C₂²×D₁₃	—	2	2	no	—
127	C₂²×D₁₃	—	2	2	no	—
128	C₂²×D₁₃	—	4	4	no	—
129	C₂²×D₁₃	—	4	4	no	—
130	C₂²×D₁₃	—	8	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
141	C₂×Dic₁₃	—	2	2	no	—
142	C₂×Dic₁₃	—	2	2	no	—
143	C₂×Dic₁₃	—	8	2	no	—
144	D₅₂	—	2	2	no	—
145	D₅₂	—	2	2	no	—
146	D₅₂	—	2	2	no	—
147	D₅₂	—	4	4	no	—
148	D₅₂	C₂×M₁₆	1	4	yes	—
149	C₄×D₁₃	—	4	4	no	—
150	C₄×D₁₃	—	4	4	no	—
151	C₄×D₁₃	—	8	2	no	—
152	C₄×D₁₃	—	8	2	no	—
153	C₈×C₁₃	—	2	1	no	—
154	C₈×C₁₃	—	2	1	no	—
155	C₈×C₁₃	—	2	1	no	—
156	C₈×C₁₃	—	4	4	no	—
157	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
158	#1055	—	2	4	no	—
159	#1055	—	4	2	no	—
160	C₂²×D₈	—	13	1	no	—
161	C₂×C₄×D₄	—	13	2	no	—
162	C₄×D₈	—	13	1	no	—
163	C₄×D₈	—	13	2	no	—
164	C₄×D₈	—	13	1	no	—
165	C₄×D₈	—	13	1	no	—
166	C₄×D₈	—	13	2	no	—
167	C₄×D₈	—	13	1	no	—
168	#436	—	13	4	no	—
169	#435	—	26	4	no	—
170	C₈×D₄	—	13	4	no	—
171	#432	C₂×D₁₃	1	2	yes	—
172	#431	—	13	1	no	—
173	C₂×#107	—	13	1	no	—
174	C₂×#102	—	13	2	no	—
175	#407	—	26	2	no	—
176	C₂×C₂²⋊C₈	—	26	2	no	—
177	C₄×M₁₆	—	13	1	no	—
178	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
179	#335	C₂×D₁₃	1	2	yes	complemented
180	#334	C₂×D₁₃	1	2	yes	complemented
181	D₄⋊C₈	—	13	4	no	—
182	D₄⋊C₈	—	13	4	no	—
183	#322	—	13	2	no	—
184	#322	—	13	2	no	—
185	C₂²×C₁₃	—	2	1	no	—
186	C₂²×C₁₃	—	2	1	no	—
187	C₂²×C₁₃	#444	1	2	yes	—
188	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
189	C₂×D₁₃	—	2	4	no	—
190	C₂×D₁₃	—	4	2	no	—
191	C₂×D₁₃	—	4	2	no	—
192	C₂×D₁₃	—	4	2	no	—
193	C₂×D₁₃	—	4	2	no	—
194	C₂×D₁₃	—	4	2	no	—
195	C₂×D₁₃	—	8	8	no	—
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	1	no	—
198	C₄×C₁₃	—	2	1	no	—
199	C₄×C₁₃	—	2	1	no	—
200	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
201	C₄×C₁₃	#568	1	1	yes	characteristic
202	C₄×C₁₃	#576	1	2	yes	—
203	Dic₁₃	—	4	2	no	—
204	Dic₁₃	—	4	2	no	—
205	C₂²×D₄	—	13	2	no	—
206	C₂³×C₄	—	26	2	no	—
207	C₂×D₈	—	13	1	no	—
208	C₂×D₈	—	13	2	no	—
209	C₂×D₈	—	13	1	no	—
210	C₂×D₈	—	26	4	no	—
211	C₂×M₁₆	—	26	4	no	—
212	C₂×M₁₆	—	26	1	no	—
213	C₂²×C₈	—	26	4	no	—
214	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
215	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
216	C₄×D₄	—	13	2	no	—
217	C₄×D₄	—	13	2	no	—
218	C₄×D₄	—	13	2	no	—
219	C₄×D₄	—	13	2	no	—
220	C₄×D₄	—	26	2	no	—
221	C₄×D₄	—	26	2	no	—
222	C₂×C₄⋊C₄	—	13	2	no	—
223	C₂×C₂²⋊C₄	—	26	2	no	—
224	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
225	#107	—	13	2	no	—
226	#107	—	13	1	no	—
227	#107	—	13	1	no	—
228	C₄⋊C₈	—	13	2	no	—
229	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
230	#102	—	13	2	no	—
231	#102	—	13	2	no	—
232	#102	—	13	2	no	—
233	#102	—	13	2	no	—
234	C₂²⋊C₈	—	2	2	no	—
235	C₂²⋊C₈	—	13	2	no	—
236	C₂²⋊C₈	—	26	4	no	—
237	C₂²⋊C₈	—	26	4	no	—
238	#97	—	2	2	no	—
239	#97	—	26	1	no	—
240	C₄×C₈	—	13	2	no	—
241	C₄×C₈	—	13	2	no	—
242	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
243	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
246	C₂×C₁₃	—	2	2	no	—
247	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
248	C₂×C₁₃	#2948	1	1	yes	characteristic
249	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
250	D₁₃	—	4	4	no	—
251	D₁₃	—	8	2	no	—
252	C₂⁴	—	26	2	no	—
253	C₂×D₄	—	13	2	no	—
254	C₂×D₄	—	13	2	no	—
255	C₂×D₄	—	26	4	no	—
256	C₂×D₄	—	26	2	no	—
257	C₂×D₄	—	52	2	no	—
258	C₂²×C₄	—	26	2	no	—
259	C₂²×C₄	—	26	2	no	—
260	C₂²×C₄	—	26	2	no	—
261	C₂²×C₄	—	26	2	no	—
262	C₂²×C₄	—	26	2	no	—
263	C₂²×C₄	—	104	4	no	—
264	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
265	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
266	C₂²×C₄	#6131	1	1	yes	characteristic
267	D₈	—	13	4	no	—
268	D₈	—	26	4	no	—
269	D₈	—	26	2	no	—
270	M₁₆	—	52	2	no	—
271	M₁₆	—	104	4	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	—	13	4	no	—
276	C₂×C₈	—	26	2	no	—
277	C₂×C₈	—	104	4	no	—
278	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
279	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
280	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
281	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
282	C₄⋊C₄	—	13	2	no	—
283	C₄⋊C₄	—	13	2	no	—
284	C₄⋊C₄	—	26	2	no	—
285	C₂²⋊C₄	—	26	2	no	—
286	C₂²⋊C₄	—	26	2	no	—
287	C₂²⋊C₄	—	26	2	no	—
288	C₂²⋊C₄	—	26	2	no	—
289	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
290	C₄²	D₄×D₁₃	1	1	yes	characteristic
291	C₄²	#6131	1	1	yes	characteristic
292	C₄²	C₂³×D₁₃	1	1	yes	characteristic
293	C₁₃	#18064	1	1	yes	characteristic, complemented, Sylow
294	C₂³	—	26	2	no	—
295	C₂³	—	26	2	no	—
296	C₂³	—	52	4	no	—
297	C₂³	—	52	4	no	—
298	C₂³	—	104	2	no	—
299	C₂³	#88147	1	1	yes	characteristic
300	D₄	—	13	4	no	—
301	D₄	—	26	2	no	—
302	D₄	—	26	2	no	—
303	D₄	—	26	2	no	—
304	D₄	—	52	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	26	2	no	—
310	C₂×C₄	—	26	2	no	—
311	C₂×C₄	—	52	4	no	—
312	C₂×C₄	—	52	4	no	—
313	C₂×C₄	—	104	2	no	—
314	C₂×C₄	—	104	2	no	—
315	C₂×C₄	—	104	2	no	—
316	C₂×C₄	#88147	1	1	yes	characteristic
317	C₂×C₄	#88147	1	1	yes	characteristic
318	C₂×C₄	#88147	1	1	yes	characteristic
319	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
320	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
321	C₂×C₄	C₂×#6131	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	4	4	no	—
326	C₈	—	26	4	no	—
327	C₈	—	52	2	no	—
328	C₈	M₁₆×D₁₃	1	2	yes	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	26	4	no	—
332	C₂²	—	52	2	no	—
333	C₂²	—	52	2	no	—
334	C₂²	—	52	2	no	—
335	C₂²	—	52	2	no	—
336	C₂²	—	52	2	no	—
337	C₂²	—	104	8	no	—
338	C₂²	#11718555	1	2	yes	—
339	C₂²	C₂×#88147	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	52	2	no	—
345	C₄	—	52	2	no	—
346	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
347	C₄	#11719450	1	1	yes	characteristic
348	C₄	#11719474	1	2	yes	—
349	C₂	—	2	2	no	—
350	C₂	—	52	4	no	—
351	C₂	—	104	2	no	—
352	C₂	#13123571	1	1	yes	characteristic
353	C₂	#13123687	1	1	yes	characteristic
354	C₂	C₂×#11718555	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×#1568	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂²×D₁₀₄	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1568	D₁₃
hypocenter	C₁₃	#18064
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
1. #13123873	C₂	semidirect
3. #13113749	C₂	semidirect
4. #13113664	C₂	semidirect
5. #13113659	C₂	semidirect
6. #13113278	C₂	semidirect
7. #13113125	C₂	semidirect
8. C₁₃×#1568	C₂	semidirect
17. #11718499	C₂²	semidirect
18. #11718495	C₂²	semidirect
28. C₁₃×#335	C₂²	semidirect
29. C₁₃×#334	C₂²	semidirect
30. #11718075	C₂²	semidirect
31. #11718075	C₂²	semidirect
32. #11718047	C₂²	semidirect
33. #11718046	C₂²	semidirect
114. D₁₀₄	M₁₆	semidirect
122. #1568	D₁₃	semidirect
179. #335	C₂×D₁₃	semidirect
180. #334	C₂×D₁₃	semidirect
293. C₁₃	#18064	semidirect

Maximal subgroups

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

subgroup	index
#13123873	2
C₂×C₄×D₁₀₄	2
#13113749	2
#13113664	2
#13113659	2
#13113278	2
#13113125	2
C₁₃×#1568	2
#18064	13