Subgroups — #15770977

Order 3328.

694 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88059	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718144	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718144	C₂	1	2	yes	maximal
4	#13114530	C₂	1	2	yes	maximal
5	C₂²×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	2	yes	—
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
12	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
13	C₂×C₁₃×C₄⋊C₈	C₄	1	4	yes	—
14	C₂×#88059	—	2	2	no	—
15	C₂×#88059	—	2	4	no	—
16	#11718144	—	2	2	no	—
17	#11718144	—	2	4	no	—
18	#11718116	—	2	4	no	—
19	C₂³×C₄×C₁₃	Q₈	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	Q₈	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	—	2	2	no	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	1	no	—
26	C₂²×C₈×C₁₃	—	2	1	no	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₃	D₄	1	2	yes	—
31	C₂²×C₈×C₁₃	D₄	1	2	yes	—
32	C₂×C₄²×C₁₃	—	2	1	no	—
33	C₂×C₄²×C₁₃	—	2	1	no	—
34	C₂×C₄²×C₁₃	—	2	1	no	—
35	C₂×C₄²×C₁₃	—	2	1	no	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₃	D₄	1	2	yes	—
38	C₂×#6099	—	2	2	no	—
39	C₂×#6099	—	2	2	no	—
40	C₂×#6099	—	2	2	no	—
41	C₂×#6099	—	2	2	no	—
42	C₂×#6099	—	2	2	no	—
43	C₂×#6099	—	2	2	no	—
44	C₂×#6099	—	2	2	no	—
45	C₂×#6099	—	2	2	no	—
46	C₂×#6099	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	D₄	1	4	yes	—
49	C₂×#6099	Q₈	1	4	yes	—
50	C₂²×#1055	—	2	2	no	—
51	C₂²×#1055	—	2	2	no	—
52	C₁₃×C₄⋊C₈	—	2	8	no	—
53	#88059	—	2	8	no	—
54	#11233	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₃	—	2	1	no	—
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	2	1	no	—
60	C₂²×C₄×C₁₃	—	2	1	no	—
61	C₂²×C₄×C₁₃	—	4	2	no	—
62	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	#41	1	2	yes	—
73	C₂²×C₄×C₁₃	#41	1	2	yes	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	—	4	1	no	—
81	C₂×C₈×C₁₃	—	4	2	no	—
82	C₂×C₈×C₁₃	—	4	1	no	—
83	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
84	C₄²×C₁₃	—	2	2	no	—
85	C₄²×C₁₃	—	2	2	no	—
86	C₄²×C₁₃	—	2	2	no	—
87	C₄²×C₁₃	—	2	2	no	—
88	#6099	—	2	4	no	—
89	#6099	—	2	4	no	—
90	#6099	—	2	4	no	—
91	#6099	—	2	4	no	—
92	#6099	—	4	2	no	—
93	#6099	—	4	2	no	—
94	#6099	—	4	2	no	—
95	#6099	—	4	2	no	—
96	C₂×#1055	—	2	4	no	—
97	C₂×#1055	—	2	4	no	—
98	C₂×#1055	—	4	2	no	—
99	C₂×#1055	—	4	2	no	—
100	C₂×#1055	—	4	2	no	—
101	C₂×#1055	—	4	2	no	—
102	C₂²×#108	—	13	1	no	—
103	C₂²×C₄⋊C₈	D₁₃	1	1	yes	characteristic
104	C₂×#341	—	13	2	no	—
105	C₂×#341	—	13	1	no	—
106	#1728	—	13	2	no	—
107	C₂³×C₁₃	—	2	2	no	—
108	C₂³×C₁₃	—	2	2	no	—
109	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂³×C₁₃	#125	1	2	yes	—
113	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	#108	1	4	yes	complemented
123	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#121	1	2	yes	—
129	C₂×C₄×C₁₃	#121	1	2	yes	—
130	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#123	1	2	yes	—
137	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#125	1	2	yes	—
139	C₂×C₄×C₁₃	#126	1	2	yes	—
140	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
141	C₈×C₁₃	—	2	4	no	—
142	C₈×C₁₃	—	4	2	no	—
143	#1055	—	4	4	no	—
144	#1055	—	4	4	no	—
145	C₂²×M₁₆	—	13	2	no	—
146	C₂²×M₁₆	—	13	2	no	—
147	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
148	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
149	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
150	C₂×#108	—	26	4	no	—
151	C₂×#108	—	26	2	no	—
152	C₂×C₄⋊C₈	—	2	2	no	—
153	C₂×C₄⋊C₈	—	2	2	no	—
154	C₂×C₄⋊C₈	Dic₁₃	1	4	yes	—
155	#341	—	26	4	no	—
156	#341	—	26	2	no	—
157	#336	—	26	4	no	—
158	C₂²×C₁₃	—	2	1	no	—
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	—	2	1	no	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	—	2	4	no	—
163	C₂²×C₁₃	—	2	4	no	—
164	C₂²×C₁₃	#380	1	1	yes	characteristic
165	C₂²×C₁₃	#389	1	2	yes	—
166	C₂²×C₁₃	#395	1	1	yes	characteristic
167	C₂²×C₁₃	#398	1	2	yes	—
168	C₂²×C₁₃	#400	1	1	yes	characteristic
169	C₂²×C₁₃	C₂×#108	1	2	yes	—
170	C₂²×C₁₃	#430	1	2	yes	—
171	C₄×C₁₃	—	2	4	no	—
172	C₄×C₁₃	—	4	2	no	—
173	C₄×C₁₃	#386	1	1	yes	characteristic
174	C₄×C₁₃	#387	1	2	yes	—
175	C₄×C₁₃	#391	1	1	yes	characteristic
176	C₄×C₁₃	#394	1	1	yes	characteristic
177	C₄×C₁₃	#396	1	2	yes	—
178	C₄×C₁₃	#398	1	1	yes	characteristic
179	C₂³×C₄	Dic₂₆	1	1	yes	characteristic
180	C₂³×C₄	Dic₂₆	1	1	yes	characteristic
181	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
182	C₂×M₁₆	—	13	4	no	—
183	C₂×M₁₆	—	13	4	no	—
184	C₂×M₁₆	—	26	2	no	—
185	C₂×M₁₆	—	26	2	no	—
186	C₂×M₁₆	—	26	2	no	—
187	C₂×M₁₆	—	26	2	no	—
188	C₂×M₁₆	—	26	2	no	—
189	C₂×M₁₆	—	26	2	no	—
190	C₂×M₁₆	—	26	2	no	—
191	C₂×M₁₆	—	26	2	no	—
192	C₂×M₁₆	—	26	2	no	—
193	C₂×M₁₆	—	26	2	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	2	1	no	—
200	C₂²×C₈	—	26	2	no	—
201	C₂²×C₈	—	26	2	no	—
202	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
203	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
204	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
205	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	D₅₂	1	2	yes	—
211	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
212	#108	—	26	8	no	—
213	C₄⋊C₈	—	2	8	no	—
214	C₂×C₁₃	—	2	1	no	—
215	C₂×C₁₃	—	2	2	no	—
216	C₂×C₁₃	—	2	1	no	—
217	C₂×C₁₃	#1441	1	1	yes	characteristic
218	C₂×C₁₃	#1810	1	1	yes	characteristic
219	C₂×C₁₃	#1811	1	1	yes	characteristic
220	C₂×C₁₃	#1950	1	4	yes	—
221	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
229	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
230	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
231	C₂²×C₄	#6127	1	1	yes	characteristic
232	C₂²×C₄	#6127	1	1	yes	characteristic
233	C₂²×C₄	D₄×D₁₃	1	2	yes	—
234	C₂²×C₄	#6129	1	2	yes	—
235	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
236	C₂²×C₄	#6131	1	2	yes	—
237	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
240	M₁₆	—	26	4	no	—
241	M₁₆	—	26	4	no	—
242	M₁₆	—	26	4	no	—
243	M₁₆	—	26	4	no	—
244	M₁₆	—	52	2	no	—
245	M₁₆	—	52	2	no	—
246	M₁₆	—	52	2	no	—
247	M₁₆	—	52	2	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	26	4	no	—
258	C₂×C₈	—	26	4	no	—
259	C₂×C₈	—	52	2	no	—
260	C₂×C₈	—	52	2	no	—
261	C₂×C₈	—	52	2	no	—
262	C₂×C₈	—	52	2	no	—
263	C₂×C₈	C₂²⋊Dic₁₃	1	4	yes	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₁₃	#11233	1	1	yes	characteristic, complemented, Sylow
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	#88142	1	2	yes	—
272	C₂³	#88144	1	2	yes	—
273	C₂³	#88178	1	1	yes	characteristic
274	C₂³	#88178	1	1	yes	characteristic
275	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	#88059	1	4	yes	complemented
285	C₂×C₄	#88143	1	2	yes	—
286	C₂×C₄	#88148	1	2	yes	—
287	C₂×C₄	#88149	1	2	yes	—
288	C₂×C₄	#88150	1	2	yes	—
289	C₂×C₄	#88151	1	2	yes	—
290	C₂×C₄	#88152	1	2	yes	—
291	C₂×C₄	#88180	1	1	yes	characteristic
292	C₂×C₄	#88184	1	1	yes	characteristic
293	C₂×C₄	#88184	1	1	yes	characteristic
294	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
295	C₂×C₄	#88189	1	1	yes	characteristic
296	C₂×C₄	#88191	1	1	yes	characteristic
297	C₂×C₄	#88193	1	2	yes	—
298	C₂×C₄	#88198	1	1	yes	characteristic
299	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
300	C₂×C₄	#88200	1	2	yes	—
301	C₂×C₄	#88201	1	1	yes	characteristic
302	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
303	C₈	—	2	4	no	—
304	C₈	—	4	2	no	—
305	C₈	—	52	4	no	—
306	C₈	—	52	4	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	#11718640	1	1	yes	characteristic
314	C₂²	#11718645	1	2	yes	—
315	C₂²	#11718646	1	2	yes	—
316	C₂²	#11718647	1	1	yes	characteristic
317	C₂²	C₂×#88059	1	2	yes	—
318	C₂²	#11718786	1	2	yes	—
319	C₂²	#11718873	1	1	yes	characteristic
320	C₄	—	2	4	no	—
321	C₄	—	4	2	no	—
322	C₄	#11718642	1	2	yes	—
323	C₄	#11718644	1	1	yes	characteristic
324	C₄	#11718652	1	1	yes	characteristic
325	C₄	#11718655	1	2	yes	—
326	C₄	#11718881	1	1	yes	characteristic
327	C₄	#11718891	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	2	no	—
330	C₂	—	2	1	no	—
331	C₂	#13113089	1	1	yes	characteristic
332	C₂	#13114874	1	1	yes	characteristic
333	C₂	#13114878	1	1	yes	characteristic
334	C₂	#13115612	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂²×C₁₃×C₄⋊C₈	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂²×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#11233
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
122. C₂×C₄×C₁₃	#108	semidirect
268. C₁₃	#11233	semidirect
284. C₂×C₄	#88059	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88059	2
C₂×#11718144	2
#13114530	2
C₂²×C₁₃×C₄⋊C₈	2
#11233	13