Subgroups — #15860955

Order 3328.

1010 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718767	C₂	1	1	yes	maximal, characteristic
2	#13123254	C₂	1	2	yes	maximal
3	C₂×#11718762	C₂	1	1	yes	maximal, characteristic
4	#13114723	C₂	1	4	yes	maximal, complemented
5	#13114718	C₂	1	4	yes	maximal, complemented
6	#13114547	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	2	yes	—
11	C₂×#88180	C₂²	1	1	yes	characteristic
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	#11718783	C₂²	1	8	yes	—
17	M₁₆×Dic₁₃	—	2	4	no	—
18	#11718767	C₄	1	2	yes	complemented
19	#11718767	C₄	1	2	yes	complemented
20	#11718767	C₂²	1	2	yes	—
21	#11718767	C₂²	1	2	yes	—
22	C₂×#88058	C₄	1	2	yes	complemented
23	C₂×#88057	C₄	1	2	yes	complemented
24	#11718762	—	2	4	no	—
25	#11718762	C₄	1	8	yes	complemented
26	C₂×#88055	C₂²	1	2	yes	—
27	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
28	#11718147	C₂²	1	8	yes	—
29	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₁₃×M₁₆	—	2	2	no	—
31	C₂×C₁₃×M₁₆	—	2	4	no	—
32	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
33	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
34	C₂²×C₈×C₁₃	—	2	2	no	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₂²⋊Dic₁₃	—	2	1	no	—
44	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
45	#88180	—	2	2	no	—
46	#88180	—	2	2	no	—
47	#88180	—	2	2	no	—
48	#88180	C₂×C₄	1	4	yes	—
49	#88180	C₂×C₄	1	4	yes	—
50	#88180	C₂³	1	4	yes	—
51	C₂×C₄⋊Dic₁₃	—	2	1	no	—
52	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
53	C₂×C₄×Dic₁₃	—	2	2	no	—
54	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
55	C₂×#6099	C₂×C₄	1	4	yes	—
56	C₂×#6099	D₄	1	2	yes	—
57	C₂×#6099	D₄	1	2	yes	—
58	C₂×#6099	Q₈	1	4	yes	—
59	C₂²×#1055	C₂³	1	2	yes	—
60	C₁₃×#108	C₂×C₄	1	8	yes	complemented
61	C₁₃×#108	C₂×C₄	1	8	yes	complemented
62	#88058	C₂×C₄	1	4	yes	—
63	#88058	C₂×C₄	1	4	yes	—
64	#88057	C₂×C₄	1	4	yes	—
65	#88057	C₂×C₄	1	4	yes	—
66	#88055	—	2	8	no	—
67	#88055	C₂×C₄	1	8	yes	complemented
68	C₈×Dic₁₃	—	2	8	no	—
69	C₈×Dic₁₃	C₂×C₄	1	8	yes	complemented
70	#20446	—	13	1	no	maximal, Sylow
71	C₂⁴×C₁₃	#41	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₁₃	#41	1	2	yes	—
80	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₁₃	—	2	2	no	—
83	C₂²×Dic₁₃	—	2	4	no	—
84	C₁₃×M₁₆	—	2	8	no	—
85	C₁₃×M₁₆	—	2	4	no	—
86	C₁₃×M₁₆	C₂²×C₄	1	8	yes	—
87	C₂×C₈×C₁₃	—	2	8	no	—
88	C₂×C₈×C₁₃	—	2	4	no	—
89	C₂×C₈×C₁₃	C₄²	1	4	yes	—
90	C₂×C₈×C₁₃	C₄²	1	4	yes	—
91	C₂×C₈×C₁₃	C₄²	1	4	yes	—
92	C₂×C₈×C₁₃	C₄²	1	4	yes	—
93	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
101	C₂²⋊Dic₁₃	—	2	4	no	—
102	C₂²⋊Dic₁₃	—	2	4	no	—
103	C₂²⋊Dic₁₃	—	2	4	no	—
104	C₄⋊Dic₁₃	—	2	4	no	—
105	C₄⋊Dic₁₃	—	2	4	no	—
106	C₄⋊Dic₁₃	C₂²×C₄	1	8	yes	—
107	C₄×Dic₁₃	—	2	4	no	—
108	C₄×Dic₁₃	—	2	4	no	—
109	C₄×Dic₁₃	—	2	2	no	—
110	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
111	#6099	C₄⋊C₄	1	8	yes	—
112	#6099	C₄⋊C₄	1	8	yes	—
113	C₂×#1055	C₂²×C₄	1	4	yes	—
114	C₂×#1055	C₂×D₄	1	4	yes	—
115	C₂×#1055	C₂×Q₈	1	4	yes	—
116	C₂²×#108	D₁₃	1	1	yes	characteristic
117	C₂×#427	—	13	1	no	—
118	#2873	—	13	2	no	—
119	C₂×#405	—	13	1	no	—
120	#1773	—	13	4	no	—
121	#1772	—	13	4	no	—
122	#1733	—	13	2	no	—
123	C₂³×C₁₃	—	4	2	no	—
124	C₂³×C₁₃	#117	1	2	yes	—
125	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
126	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
127	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
128	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
130	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₁₃	#117	1	2	yes	—
134	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
139	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
140	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
141	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
147	C₂×Dic₁₃	—	2	4	no	—
148	C₂×Dic₁₃	—	2	8	no	—
149	C₂×Dic₁₃	—	4	4	no	—
150	C₂×Dic₁₃	—	8	4	no	—
151	C₈×C₁₃	—	2	8	no	—
152	C₈×C₁₃	C₂×C₄²	1	2	yes	—
153	C₈×C₁₃	C₂×C₄²	1	2	yes	—
154	C₈×C₁₃	C₂×C₄²	1	2	yes	—
155	C₈×C₁₃	C₂×C₄²	1	2	yes	—
156	#1055	C₂×C₄⋊C₄	1	8	yes	—
157	C₂²×M₁₆	—	13	1	no	—
158	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
159	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
160	C₂×#117	—	13	1	no	—
161	C₂×#117	—	13	2	no	—
162	C₂×#108	Dic₁₃	1	8	yes	complemented
163	C₂×#108	C₂×D₁₃	1	2	yes	—
164	C₂×#108	C₂×D₁₃	1	2	yes	—
165	#427	—	13	2	no	—
166	#427	—	13	2	no	—
167	#427	—	13	2	no	—
168	#427	—	13	2	no	—
169	C₂×#107	—	13	2	no	—
170	C₂×#106	—	13	2	no	—
171	#405	—	13	8	no	—
172	#405	—	13	8	no	—
173	#405	—	26	4	no	—
174	C₄×M₁₆	—	26	4	no	—
175	C₂×#97	—	13	2	no	—
176	C₂×C₄×C₈	—	13	2	no	—
177	#339	—	13	8	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	—	2	2	no	—
180	C₂²×C₁₃	—	2	2	no	—
181	C₂²×C₁₃	—	4	2	no	—
182	C₂²×C₁₃	—	4	2	no	—
183	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
184	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
185	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
186	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
187	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
188	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
189	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
190	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
191	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
192	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
193	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
194	Dic₁₃	—	4	4	no	—
195	Dic₁₃	—	4	8	no	—
196	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	4	no	—
199	C₂×M₁₆	—	13	4	no	—
200	C₂×M₁₆	—	13	2	no	—
201	C₂×M₁₆	—	13	2	no	—
202	C₂×M₁₆	—	13	4	no	—
203	C₂×M₁₆	C₂×Dic₁₃	1	8	yes	—
204	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	13	2	no	—
207	C₂²×C₈	C₄×D₁₃	1	4	yes	—
208	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
209	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
210	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
211	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
212	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
213	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
215	#117	—	13	4	no	—
216	#117	—	13	4	no	—
217	#117	—	13	4	no	—
218	#117	—	26	2	no	—
219	#117	—	26	2	no	—
220	#117	—	26	2	no	—
221	C₂×C₄⋊C₄	—	13	4	no	—
222	C₂×C₄⋊C₄	—	26	1	no	—
223	C₂×C₂²⋊C₄	—	26	1	no	—
224	C₂×C₂²⋊C₄	—	26	2	no	—
225	C₂×C₄²	—	13	2	no	—
226	C₂×C₄²	—	26	2	no	—
227	#108	C₂×Dic₁₃	1	8	yes	complemented
228	#108	C₂×Dic₁₃	1	8	yes	complemented
229	#107	—	13	4	no	—
230	#107	—	13	4	no	—
231	#106	—	13	4	no	—
232	#106	—	13	4	no	—
233	#97	—	13	8	no	—
234	#97	—	26	8	no	—
235	C₄×C₈	—	13	8	no	—
236	C₄×C₈	—	26	8	no	—
237	C₂×C₁₃	—	2	1	no	—
238	C₂×C₁₃	—	2	1	no	—
239	C₂×C₁₃	—	2	2	no	—
240	C₂×C₁₃	—	2	2	no	—
241	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
242	C₂×C₁₃	#2957	1	2	yes	—
243	C₂⁴	#6131	1	1	yes	characteristic
244	C₂²×C₄	—	26	2	no	—
245	C₂²×C₄	—	26	4	no	—
246	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
249	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
250	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
251	C₂²×C₄	#6129	1	2	yes	—
252	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
253	C₂²×C₄	#6131	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
255	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
256	M₁₆	—	2	8	no	—
257	M₁₆	—	2	4	no	—
258	M₁₆	—	13	8	no	—
259	M₁₆	—	13	8	no	—
260	M₁₆	C₂²×Dic₁₃	1	8	yes	—
261	C₂×C₈	—	2	8	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	13	4	no	—
264	C₂×C₈	—	13	4	no	—
265	C₂×C₈	—	13	4	no	—
266	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
267	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
268	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
269	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
270	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
271	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
272	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
273	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
274	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
275	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
276	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
277	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
278	C₄⋊C₄	—	13	8	no	—
279	C₄⋊C₄	—	26	4	no	—
280	C₄⋊C₄	—	26	4	no	—
281	C₂²⋊C₄	—	26	4	no	—
282	C₂²⋊C₄	—	26	4	no	—
283	C₂²⋊C₄	—	26	4	no	—
284	C₄²	—	13	4	no	—
285	C₄²	—	26	4	no	—
286	C₄²	—	26	4	no	—
287	C₄²	—	26	2	no	—
288	C₁₃	#20446	1	1	yes	characteristic, complemented, Sylow
289	C₂³	—	4	2	no	—
290	C₂³	#88146	1	2	yes	—
291	C₂³	#88147	1	1	yes	characteristic
292	C₂³	#88147	1	1	yes	characteristic
293	C₂³	Q₈×Dic₁₃	1	2	yes	—
294	C₂³	C₂×#6131	1	1	yes	characteristic
295	C₂×C₄	—	26	4	no	—
296	C₂×C₄	—	26	8	no	—
297	C₂×C₄	—	52	4	no	—
298	C₂×C₄	—	104	4	no	—
299	C₂×C₄	#88141	1	2	yes	—
300	C₂×C₄	#88141	1	2	yes	—
301	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
302	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
303	C₂×C₄	#88146	1	2	yes	—
304	C₂×C₄	#88147	1	1	yes	characteristic
305	C₂×C₄	#88147	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
307	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
308	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
309	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
310	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
311	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
312	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
313	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
314	C₂×C₄	C₂×#6129	1	1	yes	characteristic
315	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
316	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
317	C₈	—	2	8	no	—
318	C₈	—	13	8	no	—
319	C₈	C₂×C₄×Dic₁₃	1	2	yes	—
320	C₈	C₂×C₄×Dic₁₃	1	2	yes	—
321	C₈	C₂×C₄×Dic₁₃	1	2	yes	—
322	C₈	C₂×C₄×Dic₁₃	1	2	yes	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	2	no	—
328	C₂²	C₄⋊C₄×Dic₁₃	1	2	yes	—
329	C₂²	C₄⋊C₄×Dic₁₃	1	2	yes	—
330	C₂²	C₂×#88146	1	1	yes	characteristic
331	C₂²	C₂×#88147	1	1	yes	characteristic
332	C₂²	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
333	C₄	—	52	4	no	—
334	C₄	—	52	8	no	—
335	C₄	C₄⋊C₄×Dic₁₃	1	2	yes	—
336	C₄	C₄⋊C₄×Dic₁₃	1	2	yes	—
337	C₄	C₂×#88141	1	1	yes	characteristic
338	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
339	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
340	C₄	C₂×D₄×Dic₁₃	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	2	2	no	—
344	C₂	—	2	2	no	—
345	C₂	C₂×C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
346	C₂	#13123752	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₁₃×#108	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×#108	D₁₃
hypocenter	C₁₃	#20446
LCS	G ▷ 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. #13114723	C₂	semidirect
5. #13114718	C₂	semidirect
13. C₂×C₁₃×#108	C₄	semidirect
18. #11718767	C₄	semidirect
19. #11718767	C₄	semidirect
22. C₂×#88058	C₄	semidirect
23. C₂×#88057	C₄	semidirect
25. #11718762	C₄	semidirect
60. C₁₃×#108	C₂×C₄	semidirect
61. C₁₃×#108	C₂×C₄	semidirect
67. #88055	C₂×C₄	semidirect
69. C₈×Dic₁₃	C₂×C₄	semidirect
162. C₂×#108	Dic₁₃	semidirect
227. #108	C₂×Dic₁₃	semidirect
228. #108	C₂×Dic₁₃	semidirect
288. C₁₃	#20446	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718767	2
#13123254	2
C₂×#11718762	2
#13114723	2
#13114718	2
#13114547	2
C₂²×C₁₃×#108	2
#20446	13