Subgroups — #15834609

Order 3328.

626 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123923	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
3	#13113734	C₂	1	2	yes	maximal, complemented
4	#13113709	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₅₂	C₂	1	2	yes	maximal, complemented
6	#13113599	C₂	1	1	yes	maximal, characteristic
7	#13113284	C₂	1	2	yes	maximal
8	C₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₅₂	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₂₆	C₂²	1	2	yes	—
11	C₁₃×#431	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88058	C₄	1	1	yes	characteristic
15	C₂×#88056	C₄	1	2	yes	—
16	#11718587	C₂²	1	2	yes	—
17	#11718489	—	2	4	no	—
18	#11718380	—	2	2	no	—
19	#11718379	—	2	2	no	—
20	C₄×Dic₅₂	C₄	1	2	yes	—
21	C₄×Dic₅₂	C₄	1	1	yes	characteristic
22	C₄×Dic₅₂	C₄	1	1	yes	characteristic
23	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
24	C₄×Dic₅₂	C₂²	1	2	yes	—
25	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
26	#11718340	C₂²	1	4	yes	—
27	C₈×Dic₂₆	C₂²	1	4	yes	—
28	C₁₃×#322	C₂²	1	2	yes	—
29	C₈²×C₁₃	C₂²	1	2	yes	—
30	#11718050	C₂²	1	2	yes	—
31	#11718049	C₂²	1	2	yes	—
32	#11718048	C₂²	1	4	yes	—
33	#11718048	C₂²	1	4	yes	—
34	C₂²×Dic₂₆	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×M₁₆	—	2	1	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×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
40	C₂×#6101	—	2	2	no	—
41	C₂×C₄×Dic₁₃	—	2	2	no	—
42	C₂×Dic₅₂	—	2	4	no	—
43	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
44	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
45	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
46	C₄×Dic₂₆	—	2	2	no	—
47	C₄×Dic₂₆	—	2	2	no	—
48	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
49	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
50	C₄×Dic₂₆	C₂³	1	2	yes	—
51	C₄×Dic₂₆	C₂³	1	2	yes	—
52	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
53	C₁₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
54	C₁₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
55	C₁₃×#97	—	2	1	no	—
56	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	2	yes	—
59	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	2	yes	—
62	C₂²⋊#1055	—	2	2	no	—
63	#88058	C₂×C₄	1	2	yes	—
64	#88058	C₂×C₄	1	1	yes	characteristic
65	#88058	C₂×C₄	1	1	yes	characteristic
66	#88056	C₂×C₄	1	2	yes	—
67	#88056	C₂×C₄	1	2	yes	—
68	#88056	C₂×C₄	1	2	yes	—
69	#88056	C₂×C₄	1	2	yes	—
70	#88055	—	2	4	no	—
71	#88054	—	2	4	no	—
72	#88054	—	2	4	no	—
73	C₈×Dic₁₃	—	2	4	no	—
74	C₄⋊#1055	C₂³	1	4	yes	—
75	#88043	—	2	2	no	—
76	#18058	—	13	1	no	maximal, Sylow
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	C₂²×Dic₁₃	—	2	2	no	—
82	C₂×Dic₂₆	—	2	4	no	—
83	C₂×Dic₂₆	—	2	2	no	—
84	C₂×Dic₂₆	—	4	2	no	—
85	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
86	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
87	C₁₃×M₁₆	—	2	4	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	4	1	no	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
97	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₃	#41	1	1	yes	characteristic
100	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
101	C₄⋊Dic₁₃	—	2	2	no	—
102	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
103	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
104	#6101	—	2	2	no	—
105	#6101	—	2	2	no	—
106	#6101	—	2	2	no	—
107	#6101	—	2	2	no	—
108	C₄×Dic₁₃	—	2	2	no	—
109	C₄×Dic₁₃	—	2	2	no	—
110	C₄×Dic₁₃	—	2	2	no	—
111	C₄×Dic₁₃	—	2	2	no	—
112	C₂×#1055	—	2	4	no	—
113	Dic₅₂	—	2	4	no	—
114	Dic₅₂	—	2	2	no	—
115	Dic₅₂	M₁₆	1	4	yes	complemented
116	#2984	—	13	2	no	—
117	C₂×C₄×Q₁₆	—	13	1	no	—
118	#1591	—	13	2	no	—
119	#1587	—	13	4	no	—
120	C₈×Q₁₆	—	13	2	no	—
121	#1568	—	13	1	no	—
122	#1491	—	13	2	no	—
123	C₈×M₁₆	D₁₃	1	1	yes	characteristic
124	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
135	C₂×Dic₁₃	—	2	2	no	—
136	C₂×Dic₁₃	—	2	2	no	—
137	C₂×Dic₁₃	—	2	2	no	—
138	C₂×Dic₁₃	—	2	2	no	—
139	C₂×Dic₁₃	—	4	4	no	—
140	C₂×Dic₁₃	—	4	4	no	—
141	Dic₂₆	—	2	2	no	—
142	Dic₂₆	—	2	2	no	—
143	Dic₂₆	—	2	2	no	—
144	Dic₂₆	—	4	4	no	—
145	Dic₂₆	C₂×M₁₆	1	4	yes	—
146	C₈×C₁₃	—	2	4	no	—
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	2	2	no	—
151	C₈×C₁₃	—	2	2	no	—
152	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
153	#1055	—	4	4	no	—
154	C₂²×Q₁₆	—	13	1	no	—
155	C₂×C₄×Q₈	—	13	2	no	—
156	#446	—	13	4	no	—
157	#446	—	26	4	no	—
158	C₈×Q₈	—	13	4	no	—
159	C₄×Q₁₆	—	13	1	no	—
160	C₄×Q₁₆	—	13	2	no	—
161	C₄×Q₁₆	—	13	1	no	—
162	C₄×Q₁₆	—	13	1	no	—
163	C₄×Q₁₆	—	13	2	no	—
164	C₄×Q₁₆	—	13	1	no	—
165	#432	—	13	2	no	—
166	#432	—	26	2	no	—
167	#431	—	26	2	no	—
168	#431	C₂×D₁₃	1	1	yes	characteristic
169	C₂×#107	—	13	1	no	—
170	C₂×#103	—	13	2	no	—
171	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
173	#335	—	13	2	no	—
174	#334	—	13	2	no	—
175	Q₈⋊C₈	—	13	4	no	—
176	Q₈⋊C₈	—	13	4	no	—
177	#322	C₂×D₁₃	1	2	yes	—
178	C₈²	C₂×D₁₃	1	2	yes	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	#443	1	2	yes	—
182	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
188	C₄×C₁₃	#568	1	1	yes	characteristic
189	C₄×C₁₃	#578	1	2	yes	—
190	Dic₁₃	—	2	4	no	—
191	Dic₁₃	—	4	2	no	—
192	Dic₁₃	—	4	2	no	—
193	Dic₁₃	—	4	2	no	—
194	C₂²×Q₈	—	13	2	no	—
195	C₂×Q₁₆	—	13	1	no	—
196	C₂×Q₁₆	—	13	2	no	—
197	C₂×Q₁₆	—	13	1	no	—
198	C₂×Q₁₆	—	26	4	no	—
199	C₂×M₁₆	—	2	1	no	—
200	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
201	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
202	C₄×Q₈	—	13	2	no	—
203	C₄×Q₈	—	13	2	no	—
204	C₄×Q₈	—	13	2	no	—
205	C₄×Q₈	—	13	2	no	—
206	C₄×Q₈	—	26	2	no	—
207	C₄×Q₈	—	26	2	no	—
208	C₂×C₄⋊C₄	—	13	2	no	—
209	C₂×C₄⋊C₄	—	26	2	no	—
210	C₂×C₄²	—	26	2	no	—
211	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
212	#107	—	13	2	no	—
213	#107	—	13	1	no	—
214	#107	—	13	1	no	—
215	C₄⋊C₈	—	13	4	no	—
216	C₄⋊C₈	—	26	4	no	—
217	C₄⋊C₈	—	26	4	no	—
218	C₄⋊C₈	D₅₂	1	2	yes	—
219	#103	—	13	2	no	—
220	#103	—	13	2	no	—
221	#103	—	13	2	no	—
222	#103	—	13	2	no	—
223	C₂²⋊C₈	—	26	2	no	—
224	C₂²⋊C₈	D₅₂	1	1	yes	characteristic
225	C₂²⋊C₈	D₅₂	1	1	yes	characteristic
226	#97	—	2	1	no	—
227	#97	—	26	4	no	—
228	#97	—	26	2	no	—
229	C₄×C₈	—	26	4	no	—
230	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
231	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₁₃	1	2	yes	—
233	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₁₃	1	2	yes	—
236	C₂×C₁₃	—	2	2	no	—
237	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
238	C₂×C₁₃	#2950	1	1	yes	characteristic
239	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
240	C₂×Q₈	—	13	2	no	—
241	C₂×Q₈	—	13	2	no	—
242	C₂×Q₈	—	26	4	no	—
243	C₂×Q₈	—	26	2	no	—
244	C₂×Q₈	—	52	2	no	—
245	C₂²×C₄	—	26	2	no	—
246	C₂²×C₄	—	26	2	no	—
247	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
249	C₂²×C₄	#6127	1	1	yes	characteristic
250	Q₁₆	—	13	4	no	—
251	Q₁₆	—	26	4	no	—
252	Q₁₆	—	26	2	no	—
253	M₁₆	—	2	4	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	26	4	no	—
258	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂×D₅₂	1	2	yes	—
263	C₂×C₈	C₂×D₅₂	1	2	yes	—
264	C₄⋊C₄	—	13	2	no	—
265	C₄⋊C₄	—	13	2	no	—
266	C₄⋊C₄	—	26	2	no	—
267	C₄⋊C₄	—	26	2	no	—
268	C₄⋊C₄	—	26	2	no	—
269	C₄⋊C₄	—	26	2	no	—
270	C₄⋊C₄	—	26	2	no	—
271	C₄²	—	26	2	no	—
272	C₄²	—	26	2	no	—
273	C₄²	—	26	2	no	—
274	C₄²	—	26	2	no	—
275	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
276	C₄²	C₂×D₅₂	1	1	yes	characteristic
277	C₄²	#6127	1	1	yes	characteristic
278	C₄²	C₂³×D₁₃	1	1	yes	characteristic
279	C₁₃	#18058	1	1	yes	characteristic, complemented, Sylow
280	C₂³	C₄×D₅₂	1	1	yes	characteristic
281	Q₈	—	13	4	no	—
282	Q₈	—	26	2	no	—
283	Q₈	—	26	2	no	—
284	Q₈	—	26	2	no	—
285	Q₈	—	52	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	26	2	no	—
291	C₂×C₄	—	26	2	no	—
292	C₂×C₄	—	26	2	no	—
293	C₂×C₄	—	26	2	no	—
294	C₂×C₄	—	52	4	no	—
295	C₂×C₄	—	52	4	no	—
296	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
297	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
298	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
301	C₂×C₄	C₂×#6127	1	1	yes	characteristic
302	C₈	—	2	4	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	2	no	—
307	C₈	—	2	2	no	—
308	C₈	—	52	4	no	—
309	C₈	M₁₆×D₁₃	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	#11718361	1	2	yes	—
313	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	26	4	no	—
319	C₄	—	52	2	no	—
320	C₄	—	52	2	no	—
321	C₄	—	52	2	no	—
322	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
323	C₄	#11719450	1	1	yes	characteristic
324	C₄	#11719458	1	2	yes	—
325	C₂	—	2	2	no	—
326	C₂	M₁₆×D₅₂	1	1	yes	characteristic
327	C₂	#13123705	1	1	yes	characteristic
328	C₂	C₂×#11718361	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₈×M₁₆	D₁₃
hypocenter	C₁₃	#18058
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13113734	C₂	semidirect
4. #13113709	C₂	semidirect
5. C₈×Dic₅₂	C₂	semidirect
115. Dic₅₂	M₁₆	semidirect
279. C₁₃	#18058	semidirect

Maximal subgroups

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

subgroup	index
#13123923	2
C₂×C₄×Dic₅₂	2
#13113734	2
#13113709	2
C₈×Dic₅₂	2
#13113599	2
#13113284	2
C₈×C₁₃×M₁₆	2
#18058	13