Subgroups — #15834418

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	Q₈×#6099	C₂	1	2	yes	maximal
2	#13113745	C₂	1	2	yes	maximal, complemented
3	#13113708	C₂	1	4	yes	maximal, complemented
4	#13113655	C₂	1	2	yes	maximal, complemented
5	#13113592	C₂	1	1	yes	maximal, characteristic
6	#13113265	C₂	1	2	yes	maximal
7	#13113125	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₄×C₁₃×Q₁₆	C₄	1	2	yes	—
12	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
13	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
14	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
16	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
18	C₂×C₁₃×#103	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	#11718685	—	2	4	no	—
21	#11718685	C₂²	1	4	yes	—
22	Q₈×#1055	C₂²	1	4	yes	—
23	#11718586	C₂²	1	1	yes	characteristic
24	#11718585	—	2	2	no	—
25	#11718585	C₂²	1	2	yes	—
26	C₄×#6099	—	2	2	no	—
27	C₄×#6099	C₂²	1	1	yes	characteristic
28	Q₈⋊#1055	C₂²	1	4	yes	—
29	Q₈⋊#1055	C₂²	1	4	yes	—
30	#11718049	C₂²	1	2	yes	—
31	#11718049	C₂²	1	2	yes	—
32	#11718047	C₂²	1	2	yes	—
33	#11718046	C₂²	1	2	yes	—
34	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×Q₁₆	—	2	4	no	—
36	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₄×Q₈×C₁₃	—	2	2	no	—
42	C₄×Q₈×C₁₃	—	2	2	no	—
43	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
44	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
45	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
46	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₁₃	—	2	2	no	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×#6099	—	2	2	no	—
52	C₂×#6099	D₄	1	2	yes	—
53	C₁₃×#107	C₂×C₄	1	2	yes	—
54	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
55	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
56	C₁₃×#103	C₂×C₄	1	2	yes	—
57	C₁₃×#103	C₂×C₄	1	2	yes	—
58	C₁₃×#103	C₂×C₄	1	2	yes	—
59	C₁₃×#103	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₂²⋊#1055	—	2	1	no	—
65	C₄⋊#1055	—	2	2	no	—
66	C₄⋊#1055	—	2	1	no	—
67	C₄⋊#1055	—	2	4	no	—
68	C₄⋊#1055	—	2	4	no	—
69	C₄⋊#1055	C₂³	1	4	yes	—
70	#88043	—	2	4	no	—
71	#88043	D₄	1	2	yes	—
72	C₄×#1055	—	2	4	no	—
73	C₄×#1055	C₂³	1	2	yes	—
74	C₄×#1055	C₂³	1	2	yes	—
75	#18042	—	13	1	no	maximal, Sylow
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	2	4	no	—
78	C₂×Q₈×C₁₃	—	4	2	no	—
79	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
86	C₁₃×Q₁₆	—	2	4	no	—
87	C₁₃×Q₁₆	—	2	2	no	—
88	C₁₃×Q₁₆	M₁₆	1	4	yes	complemented
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	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
101	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
102	C₄²×C₁₃	—	2	2	no	—
103	C₄²×C₁₃	—	2	2	no	—
104	C₄²×C₁₃	—	2	2	no	—
105	C₄²×C₁₃	—	2	2	no	—
106	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	#41	1	1	yes	characteristic
109	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
110	#6099	—	2	4	no	—
111	#6099	—	4	4	no	—
112	C₂×#1055	—	2	4	no	—
113	C₂×#1055	—	2	2	no	—
114	C₂×#1055	C₂×D₄	1	4	yes	—
115	Q₈×M₁₆	—	13	2	no	—
116	C₂×C₄×Q₁₆	D₁₃	1	1	yes	characteristic
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	—	13	1	no	—
123	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
124	Q₈×C₁₃	—	2	2	no	—
125	Q₈×C₁₃	—	2	2	no	—
126	Q₈×C₁₃	—	2	2	no	—
127	Q₈×C₁₃	—	4	4	no	—
128	Q₈×C₁₃	C₂×M₁₆	1	4	yes	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	—	4	4	no	—
137	C₂×C₄×C₁₃	—	8	2	no	—
138	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#137	1	4	yes	—
142	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
145	C₈×C₁₃	—	2	1	no	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
149	#1055	—	2	4	no	—
150	#1055	—	4	2	no	—
151	#1055	—	4	4	no	—
152	C₂²×Q₁₆	Dic₁₃	1	1	yes	characteristic
153	C₂×C₄×Q₈	C₂×D₁₃	1	2	yes	—
154	#446	—	13	4	no	—
155	#446	—	26	4	no	—
156	C₈×Q₈	—	13	4	no	—
157	C₄×Q₁₆	Dic₁₃	1	2	yes	—
158	C₄×Q₁₆	Dic₁₃	1	1	yes	characteristic
159	C₄×Q₁₆	Dic₁₃	1	1	yes	characteristic
160	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
161	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
162	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
163	#431	—	13	1	no	—
164	C₂×#107	Dic₁₃	1	1	yes	characteristic
165	#423	—	13	2	no	—
166	#423	—	26	2	no	—
167	C₂×#103	Dic₁₃	1	2	yes	—
168	C₄×M₁₆	—	13	1	no	—
169	C₄×M₁₆	—	26	2	no	—
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	—	13	2	no	—
176	#322	—	13	2	no	—
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	4	no	—
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	4	2	no	—
186	C₄×C₁₃	—	4	2	no	—
187	C₄×C₁₃	—	4	2	no	—
188	C₄×C₁₃	#441	1	2	yes	—
189	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
190	C₄×C₁₃	#568	1	1	yes	characteristic
191	C₄×C₁₃	C₂×#137	1	2	yes	—
192	C₂²×Q₈	C₂×Dic₁₃	1	2	yes	—
193	C₂×Q₁₆	—	2	4	no	—
194	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
195	C₂×Q₁₆	C₂×Dic₁₃	1	1	yes	characteristic
196	C₂×Q₁₆	C₂×Dic₁₃	1	1	yes	characteristic
197	C₂×M₁₆	—	13	2	no	—
198	C₂×M₁₆	—	26	2	no	—
199	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
200	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
201	C₄×Q₈	—	2	2	no	—
202	C₄×Q₈	—	2	2	no	—
203	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
204	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
205	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
206	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
211	#107	C₂×Dic₁₃	1	2	yes	—
212	#107	C₂×Dic₁₃	1	1	yes	characteristic
213	#107	C₂×Dic₁₃	1	1	yes	characteristic
214	C₄⋊C₈	—	13	4	no	—
215	C₄⋊C₈	—	26	4	no	—
216	C₄⋊C₈	—	26	4	no	—
217	C₄⋊C₈	—	26	2	no	—
218	C₄⋊C₈	—	26	1	no	—
219	#103	C₂×Dic₁₃	1	2	yes	—
220	#103	C₂×Dic₁₃	1	2	yes	—
221	#103	C₂×Dic₁₃	1	2	yes	—
222	#103	C₂×Dic₁₃	1	2	yes	—
223	C₂²⋊C₈	—	26	1	no	—
224	#97	—	13	2	no	—
225	#97	—	26	4	no	—
226	C₄×C₈	—	13	2	no	—
227	C₄×C₈	—	13	2	no	—
228	C₄×C₈	—	26	4	no	—
229	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
230	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
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	—	2	2	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
241	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
245	C₂²×C₄	#6129	1	1	yes	characteristic
246	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
247	Q₁₆	—	2	4	no	—
248	Q₁₆	—	2	2	no	—
249	Q₁₆	#6099	1	4	yes	complemented
250	M₁₆	—	26	4	no	—
251	M₁₆	—	52	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	13	4	no	—
255	C₂×C₈	—	26	4	no	—
256	C₂×C₈	—	26	2	no	—
257	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
258	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
267	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	D₄×D₁₃	1	1	yes	characteristic
273	C₄²	#6129	1	1	yes	characteristic
274	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
275	C₄²	C₂³×D₁₃	1	1	yes	characteristic
276	C₁₃	#18042	1	1	yes	characteristic, complemented, Sylow
277	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
278	Q₈	—	2	2	no	—
279	Q₈	—	2	2	no	—
280	Q₈	—	2	2	no	—
281	Q₈	—	4	4	no	—
282	Q₈	C₂×#6099	1	4	yes	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	#88172	1	4	yes	—
293	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
294	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
295	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
297	C₂×C₄	C₂×#6129	1	1	yes	characteristic
298	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	26	4	no	—
303	C₈	—	52	2	no	—
304	C₈	—	52	4	no	—
305	C₈	C₂×#6099	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#11718845	1	2	yes	—
309	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	4	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	#11718845	1	2	yes	—
318	C₄	C₂×#88172	1	2	yes	—
319	C₄	C₂²×#6099	1	1	yes	characteristic
320	C₄	#11719524	1	1	yes	characteristic
321	C₂	—	2	2	no	—
322	C₂	D₄×#6099	1	1	yes	characteristic
323	C₂	C₂×#11718845	1	1	yes	characteristic
324	C₂	#13123753	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₁₃
hypocenter	C₁₃	#18042
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #13113745	C₂	semidirect
3. #13113708	C₂	semidirect
4. #13113655	C₂	semidirect
88. C₁₃×Q₁₆	M₁₆	semidirect
249. Q₁₆	#6099	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
Q₈×#6099	2
#13113745	2
#13113708	2
#13113655	2
#13113592	2
#13113265	2
#13113125	2
C₂×C₄×C₁₃×Q₁₆	2
#18042	13