Subgroups — #15866473

Order 3328.

874 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123426	C₂	1	2	yes	maximal
2	C₂×#11718767	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718762	C₂	1	1	yes	maximal, characteristic
4	#13115622	C₂	1	4	yes	maximal, complemented
5	#13115617	C₂	1	4	yes	maximal, complemented
6	#13114574	C₂	1	2	yes	maximal
7	C₂²×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88180	C₂²	1	1	yes	characteristic
11	C₂×#88180	C₂²	1	2	yes	—
12	C₂²×#6099	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#108	C₄	1	8	yes	complemented
14	C₂×C₁₃×#108	C₂²	1	2	yes	—
15	C₂×C₁₃×#108	C₂²	1	2	yes	—
16	#11718782	—	2	4	no	—
17	#11718782	C₂²	1	8	yes	—
18	#11718767	—	2	1	no	—
19	#11718767	—	2	1	no	—
20	#11718767	—	2	1	no	—
21	#11718767	—	2	1	no	—
22	#11718767	—	2	4	no	—
23	C₂×#88058	C₂²	1	2	yes	—
24	C₂×#88057	C₂²	1	2	yes	—
25	#11718762	—	2	1	no	—
26	#11718762	—	2	1	no	—
27	#11718762	C₂²	1	2	yes	—
28	#11718762	C₂²	1	2	yes	—
29	C₂×#88055	—	2	1	no	—
30	C₂×C₈×Dic₁₃	—	2	1	no	—
31	#11718146	C₂²	1	8	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₁₃×M₁₆	—	2	2	no	—
34	C₂×C₁₃×M₁₆	—	2	4	no	—
35	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
36	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
37	C₂²×C₈×C₁₃	—	2	2	no	—
38	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
44	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊Dic₁₃	—	2	1	no	—
47	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
48	#88180	—	2	1	no	—
49	#88180	—	2	1	no	—
50	#88180	—	2	2	no	—
51	#88180	—	2	2	no	—
52	#88180	—	2	2	no	—
53	#88180	—	2	2	no	—
54	#88180	D₄	1	2	yes	—
55	#88180	C₂³	1	2	yes	—
56	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
57	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
58	C₂×C₄×Dic₁₃	—	2	1	no	—
59	C₂×C₄×Dic₁₃	—	2	2	no	—
60	C₂×#6099	—	2	1	no	—
61	C₂×#6099	—	2	2	no	—
62	C₂×#6099	Q₈	1	4	yes	—
63	C₂×#6099	C₂³	1	2	yes	—
64	C₂²×#1055	—	2	1	no	—
65	C₁₃×#108	C₂×C₄	1	8	yes	complemented
66	C₁₃×#108	C₂×C₄	1	8	yes	complemented
67	#88058	—	2	2	no	—
68	#88058	—	2	2	no	—
69	#88058	—	2	8	no	—
70	#88057	—	2	2	no	—
71	#88057	—	2	2	no	—
72	#88057	—	2	8	no	—
73	#88055	—	2	2	no	—
74	#88055	—	2	2	no	—
75	C₈×Dic₁₃	—	2	2	no	—
76	C₈×Dic₁₃	—	2	2	no	—
77	#21181	—	13	1	no	maximal, Sylow
78	C₂⁴×C₁₃	#41	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₁₃	#41	1	2	yes	—
87	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₂²×Dic₁₃	—	2	2	no	—
90	C₂²×Dic₁₃	—	2	4	no	—
91	C₁₃×M₁₆	—	2	8	no	—
92	C₁₃×M₁₆	—	2	4	no	—
93	C₁₃×M₁₆	C₂²×C₄	1	8	yes	—
94	C₂×C₈×C₁₃	—	2	8	no	—
95	C₂×C₈×C₁₃	—	2	4	no	—
96	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
97	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
98	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
99	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
100	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
107	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
108	C₂²⋊Dic₁₃	—	2	2	no	—
109	C₂²⋊Dic₁₃	—	2	2	no	—
110	C₂²⋊Dic₁₃	—	4	2	no	—
111	C₂²⋊Dic₁₃	—	4	2	no	—
112	C₄⋊Dic₁₃	—	2	2	no	—
113	C₄⋊Dic₁₃	—	2	4	no	—
114	C₄⋊Dic₁₃	—	2	4	no	—
115	C₄⋊Dic₁₃	C₂×D₄	1	4	yes	—
116	C₄×Dic₁₃	—	2	2	no	—
117	C₄×Dic₁₃	—	2	2	no	—
118	C₄×Dic₁₃	—	4	2	no	—
119	C₄×Dic₁₃	—	4	2	no	—
120	#6099	—	2	4	no	—
121	#6099	—	2	2	no	—
122	#6099	C₂×Q₈	1	4	yes	—
123	C₂×#1055	—	2	4	no	—
124	C₂×#1055	—	2	2	no	—
125	C₂²×#108	D₁₃	1	1	yes	characteristic
126	#2911	—	13	2	no	—
127	C₂×#427	—	13	1	no	—
128	C₂×#405	—	13	1	no	—
129	#1952	—	13	4	no	—
130	#1951	—	13	4	no	—
131	#1742	—	13	2	no	—
132	C₂³×C₁₃	—	4	2	no	—
133	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
134	C₂³×C₁₃	#121	1	1	yes	characteristic
135	C₂³×C₁₃	#121	1	1	yes	characteristic
136	C₂³×C₁₃	#125	1	2	yes	—
137	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
143	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
146	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
147	C₂×C₄×C₁₃	#125	1	2	yes	—
148	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
149	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
150	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
156	C₂×Dic₁₃	—	2	4	no	—
157	C₂×Dic₁₃	—	4	4	no	—
158	C₂×Dic₁₃	—	4	4	no	—
159	C₂×Dic₁₃	—	8	4	no	—
160	C₈×C₁₃	—	2	8	no	—
161	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
162	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
163	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
164	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
165	#1055	—	2	4	no	—
166	C₂²×M₁₆	—	13	1	no	—
167	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
168	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
169	C₂×#117	—	13	1	no	—
170	C₂×#117	—	13	2	no	—
171	C₂×#108	Dic₁₃	1	8	yes	complemented
172	C₂×#108	C₂×D₁₃	1	2	yes	—
173	C₂×#108	C₂×D₁₃	1	2	yes	—
174	#428	—	13	8	no	—
175	#428	—	26	4	no	—
176	#427	—	26	4	no	—
177	#427	—	26	1	no	—
178	#427	—	26	1	no	—
179	#427	—	26	1	no	—
180	#427	—	26	1	no	—
181	C₂×#107	—	13	2	no	—
182	C₂×#106	—	13	2	no	—
183	#405	—	13	2	no	—
184	#405	—	13	2	no	—
185	#405	—	26	1	no	—
186	#405	—	26	1	no	—
187	C₂×#97	—	26	1	no	—
188	C₂×C₄×C₈	—	26	1	no	—
189	#343	—	13	8	no	—
190	C₂²×C₁₃	—	2	2	no	—
191	C₂²×C₁₃	—	2	2	no	—
192	C₂²×C₁₃	—	2	2	no	—
193	C₂²×C₁₃	—	4	2	no	—
194	C₂²×C₁₃	—	4	2	no	—
195	C₂²×C₁₃	#389	1	2	yes	—
196	C₂²×C₁₃	#389	1	2	yes	—
197	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
198	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
199	C₂²×C₁₃	C₂×#125	1	1	yes	characteristic
200	C₄×C₁₃	#389	1	2	yes	—
201	C₄×C₁₃	#389	1	2	yes	—
202	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
203	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
204	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
205	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
206	Dic₁₃	—	4	4	no	—
207	Dic₁₃	—	8	4	no	—
208	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
209	C₂×M₁₆	—	2	2	no	—
210	C₂×M₁₆	—	2	4	no	—
211	C₂×M₁₆	—	13	4	no	—
212	C₂×M₁₆	—	13	2	no	—
213	C₂×M₁₆	—	26	1	no	—
214	C₂×M₁₆	—	26	2	no	—
215	C₂×M₁₆	C₂×Dic₁₃	1	8	yes	—
216	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
217	C₂²×C₈	—	2	2	no	—
218	C₂²×C₈	—	26	1	no	—
219	C₂²×C₈	Dic₂₆	1	4	yes	—
220	C₂²×C₈	D₅₂	1	1	yes	characteristic
221	C₂²×C₈	D₅₂	1	1	yes	characteristic
222	C₂²×C₈	D₅₂	1	1	yes	characteristic
223	C₂²×C₈	D₅₂	1	1	yes	characteristic
224	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
225	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
226	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
227	#117	—	13	2	no	—
228	#117	—	13	2	no	—
229	#117	—	26	1	no	—
230	#117	—	26	1	no	—
231	#117	—	26	2	no	—
232	#117	—	26	2	no	—
233	#117	—	26	2	no	—
234	#117	—	26	2	no	—
235	C₂×C₄⋊C₄	—	13	2	no	—
236	C₂×C₄⋊C₄	—	13	4	no	—
237	C₂×C₂²⋊C₄	—	26	1	no	—
238	C₂×C₂²⋊C₄	—	26	2	no	—
239	C₂×C₄²	—	26	1	no	—
240	C₂×C₄²	—	26	2	no	—
241	#108	C₂×Dic₁₃	1	8	yes	complemented
242	#108	C₂×Dic₁₃	1	8	yes	complemented
243	#107	—	26	8	no	—
244	#107	—	26	2	no	—
245	#107	—	26	2	no	—
246	#106	—	26	8	no	—
247	#106	—	26	2	no	—
248	#106	—	26	2	no	—
249	#97	—	26	2	no	—
250	#97	—	26	2	no	—
251	C₄×C₈	—	26	2	no	—
252	C₄×C₈	—	26	2	no	—
253	C₂×C₁₃	—	2	1	no	—
254	C₂×C₁₃	—	2	1	no	—
255	C₂×C₁₃	—	2	2	no	—
256	C₂×C₁₃	—	2	2	no	—
257	C₂×C₁₃	C₂×#389	1	1	yes	characteristic
258	C₂×C₁₃	#3067	1	2	yes	—
259	C₂⁴	#6131	1	1	yes	characteristic
260	C₂²×C₄	—	26	2	no	—
261	C₂²×C₄	—	26	4	no	—
262	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
263	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
265	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
266	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
267	C₂²×C₄	#6129	1	2	yes	—
268	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
269	C₂²×C₄	#6131	1	1	yes	characteristic
270	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
271	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
272	M₁₆	—	2	8	no	—
273	M₁₆	—	2	4	no	—
274	M₁₆	—	13	4	no	—
275	M₁₆	—	26	4	no	—
276	M₁₆	—	26	2	no	—
277	M₁₆	C₂²×Dic₁₃	1	8	yes	—
278	C₂×C₈	—	2	8	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	26	4	no	—
281	C₂×C₈	—	26	2	no	—
282	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
283	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
284	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
285	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
286	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
287	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
288	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
289	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
290	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
291	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
292	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
293	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
294	C₄⋊C₄	—	13	4	no	—
295	C₄⋊C₄	—	26	2	no	—
296	C₄⋊C₄	—	26	4	no	—
297	C₄⋊C₄	—	26	4	no	—
298	C₂²⋊C₄	—	26	2	no	—
299	C₂²⋊C₄	—	26	2	no	—
300	C₂²⋊C₄	—	52	2	no	—
301	C₂²⋊C₄	—	52	2	no	—
302	C₄²	—	26	2	no	—
303	C₄²	—	26	2	no	—
304	C₄²	—	52	2	no	—
305	C₄²	—	52	2	no	—
306	C₁₃	#21181	1	1	yes	characteristic, complemented, Sylow
307	C₂³	—	4	2	no	—
308	C₂³	#88144	1	2	yes	—
309	C₂³	#88149	1	1	yes	characteristic
310	C₂³	#88149	1	1	yes	characteristic
311	C₂³	Q₈×Dic₁₃	1	2	yes	—
312	C₂³	C₂×#6131	1	1	yes	characteristic
313	C₂×C₄	—	26	4	no	—
314	C₂×C₄	—	52	4	no	—
315	C₂×C₄	—	52	4	no	—
316	C₂×C₄	—	104	4	no	—
317	C₂×C₄	#88142	1	2	yes	—
318	C₂×C₄	#88142	1	2	yes	—
319	C₂×C₄	#88144	1	2	yes	—
320	C₂×C₄	#88149	1	1	yes	characteristic
321	C₂×C₄	#88149	1	1	yes	characteristic
322	C₂×C₄	#88151	1	2	yes	—
323	C₂×C₄	#88151	1	2	yes	—
324	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
325	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
326	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
327	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
328	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
329	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
330	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
331	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
332	C₂×C₄	C₂×#6129	1	1	yes	characteristic
333	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
334	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
335	C₈	—	2	8	no	—
336	C₈	—	26	4	no	—
337	C₈	C₂×C₄⋊Dic₁₃	1	2	yes	—
338	C₈	C₂×C₄⋊Dic₁₃	1	2	yes	—
339	C₈	C₂×C₄⋊Dic₁₃	1	2	yes	—
340	C₈	C₂×C₄⋊Dic₁₃	1	2	yes	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	#11718646	1	2	yes	—
347	C₂²	#11718646	1	2	yes	—
348	C₂²	C₂×#88144	1	1	yes	characteristic
349	C₂²	C₂×#88149	1	1	yes	characteristic
350	C₂²	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
351	C₄	—	52	4	no	—
352	C₄	—	104	4	no	—
353	C₄	#11718646	1	2	yes	—
354	C₄	#11718646	1	2	yes	—
355	C₄	C₂×#88142	1	1	yes	characteristic
356	C₄	C₂×#88151	1	1	yes	characteristic
357	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
358	C₄	C₂×D₄×Dic₁₃	1	1	yes	characteristic
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	2	2	no	—
362	C₂	—	2	2	no	—
363	C₂	C₂×#11718646	1	1	yes	characteristic
364	C₂	#13124557	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#88149
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₁₃×#108	C₂
socle	C₂²×C₁₃	C₂×#121
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×#108	D₁₃
hypocenter	C₁₃	#21181
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
4. #13115622	C₂	semidirect
5. #13115617	C₂	semidirect
13. C₂×C₁₃×#108	C₄	semidirect
65. C₁₃×#108	C₂×C₄	semidirect
66. C₁₃×#108	C₂×C₄	semidirect
171. C₂×#108	Dic₁₃	semidirect
241. #108	C₂×Dic₁₃	semidirect
242. #108	C₂×Dic₁₃	semidirect
306. C₁₃	#21181	semidirect

Maximal subgroups

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

subgroup	index
#13123426	2
C₂×#11718767	2
C₂×#11718762	2
#13115622	2
#13115617	2
#13114574	2
C₂²×C₁₃×#108	2
#21181	13