Subgroups — #15834419

Order 3328.

630 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₂₆	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
3	#13113746	C₂	1	2	yes	maximal, complemented
4	#13113709	C₂	1	4	yes	maximal, complemented
5	#13113656	C₂	1	2	yes	maximal, complemented
6	#13113592	C₂	1	1	yes	maximal, characteristic
7	#13113266	C₂	1	2	yes	maximal
8	C₁₃×#1451	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	#11718779	—	2	2	no	—
15	M₁₆×Dic₁₃	—	2	2	no	—
16	C₂×#88058	C₄	1	1	yes	characteristic
17	C₂×#88056	C₄	1	2	yes	—
18	#11718585	C₂²	1	2	yes	—
19	#11718362	—	2	4	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	C₁₃×#322	C₂²	1	2	yes	—
30	#11718049	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	D₄	1	2	yes	—
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×#6099	—	2	2	no	—
43	C₂×Dic₅₂	—	2	4	no	—
44	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
45	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
46	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₂₆	—	2	2	no	—
48	C₄×Dic₂₆	—	2	2	no	—
49	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
50	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
51	C₄×Dic₂₆	C₂³	1	2	yes	—
52	C₄×Dic₂₆	C₂³	1	2	yes	—
53	C₁₃×C₄⋊C₈	—	2	1	no	—
54	C₁₃×C₂²⋊C₈	—	2	1	no	—
55	C₁₃×#97	D₄	1	2	yes	—
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	#88058	C₂×C₄	1	2	yes	—
63	#88058	C₂×C₄	1	1	yes	characteristic
64	#88058	C₂×C₄	1	1	yes	characteristic
65	#88056	C₂×C₄	1	2	yes	—
66	#88056	C₂×C₄	1	2	yes	—
67	#88056	C₂×C₄	1	2	yes	—
68	#88056	C₂×C₄	1	2	yes	—
69	#88055	—	2	4	no	—
70	#88054	—	2	4	no	—
71	#88054	—	2	4	no	—
72	C₈×Dic₁₃	—	2	4	no	—
73	C₄⋊#1055	—	2	2	no	—
74	C₄⋊#1055	C₂³	1	4	yes	—
75	#18042	—	13	1	no	maximal, Sylow
76	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
79	C₂²×Dic₁₃	—	2	2	no	—
80	C₂²×Dic₁₃	—	2	2	no	—
81	C₂×Dic₂₆	—	2	4	no	—
82	C₂×Dic₂₆	—	2	2	no	—
83	C₂×Dic₂₆	—	4	2	no	—
84	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
85	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
86	C₁₃×M₁₆	—	2	4	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	4	1	no	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	1	4	yes	—
95	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₃	#41	1	1	yes	characteristic
98	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₄⋊Dic₁₃	—	2	2	no	—
100	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
101	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
102	#6101	—	2	2	no	—
103	#6101	—	2	2	no	—
104	#6101	—	2	2	no	—
105	#6101	—	2	2	no	—
106	C₄×Dic₁₃	—	2	2	no	—
107	C₄×Dic₁₃	—	2	2	no	—
108	C₄×Dic₁₃	—	2	2	no	—
109	C₄×Dic₁₃	—	2	2	no	—
110	#6099	—	4	4	no	—
111	C₂×#1055	—	2	4	no	—
112	Dic₅₂	—	2	4	no	—
113	Dic₅₂	—	2	2	no	—
114	Dic₅₂	M₁₆	1	4	yes	complemented
115	Q₈×M₁₆	—	13	2	no	—
116	C₂×C₄×Q₁₆	—	13	1	no	—
117	#1593	—	13	2	no	—
118	#1587	—	13	4	no	—
119	#1579	—	13	2	no	—
120	#1566	—	13	1	no	—
121	#1487	—	13	2	no	—
122	#1451	D₁₃	1	1	yes	characteristic
123	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	4	1	no	—
127	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#137	1	4	yes	—
131	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
134	C₂×Dic₁₃	—	2	2	no	—
135	C₂×Dic₁₃	—	2	2	no	—
136	C₂×Dic₁₃	—	2	2	no	—
137	C₂×Dic₁₃	—	2	2	no	—
138	C₂×Dic₁₃	—	4	4	no	—
139	C₂×Dic₁₃	—	8	2	no	—
140	Dic₂₆	—	2	2	no	—
141	Dic₂₆	—	2	2	no	—
142	Dic₂₆	—	2	2	no	—
143	Dic₂₆	—	4	4	no	—
144	Dic₂₆	C₂×M₁₆	1	4	yes	—
145	C₈×C₁₃	—	2	1	no	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	—	2	4	no	—
149	C₈×C₁₃	—	4	2	no	—
150	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
151	#1055	—	4	4	no	—
152	C₂²×Q₁₆	—	13	1	no	—
153	C₂×C₄×Q₈	—	13	2	no	—
154	#446	—	13	4	no	—
155	#446	—	26	4	no	—
156	C₈×Q₈	—	13	4	no	—
157	C₄×Q₁₆	—	13	1	no	—
158	C₄×Q₁₆	—	13	2	no	—
159	C₄×Q₁₆	—	13	1	no	—
160	C₄×Q₁₆	—	13	1	no	—
161	C₄×Q₁₆	—	13	2	no	—
162	C₄×Q₁₆	—	13	1	no	—
163	#431	C₂×D₁₃	1	1	yes	characteristic
164	C₂×#107	—	13	1	no	—
165	#423	—	13	2	no	—
166	#423	—	26	2	no	—
167	C₂×#103	—	13	2	no	—
168	C₄×M₁₆	—	26	2	no	—
169	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
171	#334	—	13	2	no	—
172	#334	—	13	2	no	—
173	Q₈⋊C₈	—	13	4	no	—
174	Q₈⋊C₈	—	13	4	no	—
175	#322	C₂×D₁₃	1	2	yes	—
176	#322	C₂×D₁₃	1	2	yes	—
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	#441	1	2	yes	—
180	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	#441	1	2	yes	—
185	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
186	C₄×C₁₃	#568	1	1	yes	characteristic
187	C₄×C₁₃	C₂×#137	1	2	yes	—
188	Dic₁₃	—	2	4	no	—
189	Dic₁₃	—	4	2	no	—
190	Dic₁₃	—	4	2	no	—
191	Dic₁₃	—	4	2	no	—
192	C₂²×Q₈	—	13	2	no	—
193	C₂×Q₁₆	—	13	1	no	—
194	C₂×Q₁₆	—	13	2	no	—
195	C₂×Q₁₆	—	13	1	no	—
196	C₂×Q₁₆	—	26	4	no	—
197	C₂×M₁₆	—	26	2	no	—
198	C₂×M₁₆	D₅₂	1	2	yes	—
199	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
200	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
201	C₄×Q₈	—	13	2	no	—
202	C₄×Q₈	—	13	2	no	—
203	C₄×Q₈	—	13	2	no	—
204	C₄×Q₈	—	13	2	no	—
205	C₄×Q₈	—	26	2	no	—
206	C₄×Q₈	—	26	2	no	—
207	C₂×C₄⋊C₄	—	13	2	no	—
208	C₂×C₄⋊C₄	—	26	2	no	—
209	C₂×C₄²	—	26	2	no	—
210	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
211	#107	—	13	2	no	—
212	#107	—	13	1	no	—
213	#107	—	13	1	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	13	4	no	—
216	C₄⋊C₈	—	26	4	no	—
217	C₄⋊C₈	—	26	4	no	—
218	C₄⋊C₈	—	26	2	no	—
219	#103	—	13	2	no	—
220	#103	—	13	2	no	—
221	#103	—	13	2	no	—
222	#103	—	13	2	no	—
223	C₂²⋊C₈	—	2	1	no	—
224	#97	—	26	4	no	—
225	#97	D₅₂	1	2	yes	—
226	C₄×C₈	—	26	4	no	—
227	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
228	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₁₃	1	2	yes	—
230	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₁₃	1	2	yes	—
233	C₂×C₁₃	—	2	2	no	—
234	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
235	C₂×C₁₃	C₂×#441	1	1	yes	characteristic
236	C₂×C₁₃	#2957	1	1	yes	characteristic
237	C₂×Q₈	—	13	2	no	—
238	C₂×Q₈	—	13	2	no	—
239	C₂×Q₈	—	26	4	no	—
240	C₂×Q₈	—	26	2	no	—
241	C₂×Q₈	—	52	2	no	—
242	C₂²×C₄	—	26	2	no	—
243	C₂²×C₄	—	26	2	no	—
244	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
246	C₂²×C₄	#6127	1	1	yes	characteristic
247	Q₁₆	—	13	4	no	—
248	Q₁₆	—	26	4	no	—
249	Q₁₆	—	26	2	no	—
250	M₁₆	—	2	4	no	—
251	M₁₆	—	52	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	26	4	no	—
256	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
257	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
258	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂×D₅₂	1	4	yes	—
261	C₄⋊C₄	—	13	2	no	—
262	C₄⋊C₄	—	13	2	no	—
263	C₄⋊C₄	—	26	2	no	—
264	C₄⋊C₄	—	26	2	no	—
265	C₄⋊C₄	—	26	2	no	—
266	C₄⋊C₄	—	26	2	no	—
267	C₄⋊C₄	—	26	2	no	—
268	C₄²	—	26	2	no	—
269	C₄²	—	26	2	no	—
270	C₄²	—	26	2	no	—
271	C₄²	—	26	2	no	—
272	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
273	C₄²	C₂×D₅₂	1	1	yes	characteristic
274	C₄²	#6127	1	1	yes	characteristic
275	C₄²	C₂³×D₁₃	1	1	yes	characteristic
276	C₁₃	#18042	1	1	yes	characteristic, complemented, Sylow
277	C₂³	C₄×D₅₂	1	1	yes	characteristic
278	Q₈	—	13	4	no	—
279	Q₈	—	26	2	no	—
280	Q₈	—	26	2	no	—
281	Q₈	—	26	2	no	—
282	Q₈	—	52	4	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	26	2	no	—
287	C₂×C₄	—	26	2	no	—
288	C₂×C₄	—	26	2	no	—
289	C₂×C₄	—	26	2	no	—
290	C₂×C₄	—	52	4	no	—
291	C₂×C₄	—	104	2	no	—
292	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
293	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
294	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
295	C₂×C₄	#88163	1	4	yes	—
296	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
298	C₂×C₄	C₂×#6127	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	2	4	no	—
303	C₈	—	4	2	no	—
304	C₈	—	52	4	no	—
305	C₈	M₁₆×D₁₃	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#11718377	1	2	yes	—
309	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	26	4	no	—
314	C₄	—	52	2	no	—
315	C₄	—	52	2	no	—
316	C₄	—	52	2	no	—
317	C₄	#11718377	1	2	yes	—
318	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
319	C₄	#11719450	1	1	yes	characteristic
320	C₄	C₂×#88163	1	2	yes	—
321	C₂	—	2	2	no	—
322	C₂	M₁₆×D₅₂	1	1	yes	characteristic
323	C₂	C₂×#11718377	1	1	yes	characteristic
324	C₂	#13123754	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1451	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1451	D₁₃
hypocenter	C₁₃	#18042
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

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

N	H ≅ G/N	type
3. #13113746	C₂	semidirect
4. #13113709	C₂	semidirect
5. #13113656	C₂	semidirect
114. Dic₅₂	M₁₆	semidirect
276. C₁₃	#18042	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₂₆	2
C₂×C₄×Dic₅₂	2
#13113746	2
#13113709	2
#13113656	2
#13113592	2
#13113266	2
C₁₃×#1451	2
#18042	13