Subgroups — #15867639

Order 3328.

666 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13117310	C₂	1	1	yes	maximal, characteristic
2	#13115776	C₂	1	1	yes	maximal, characteristic
3	#13115775	C₂	1	1	yes	maximal, characteristic
4	#13115560	C₂	1	1	yes	maximal, characteristic
5	#13115396	C₂	1	2	yes	maximal
6	#13115390	C₂	1	2	yes	maximal
7	#13114695	C₂	1	2	yes	maximal
8	C₄⋊C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
9	#13114586	C₂	1	2	yes	maximal
10	#97×Dic₁₃	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#1940	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×#88055	C₂²	1	1	yes	characteristic
18	C₂×#88055	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
21	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
22	#11718635	C₄	1	2	yes	—
23	#11718634	C₄	1	2	yes	—
24	#11718593	C₂²	1	2	yes	—
25	C₄×#6101	C₂²	1	2	yes	—
26	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊#1055	C₂²	1	2	yes	—
28	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
29	C₂×#88043	C₂²	1	1	yes	characteristic
30	#11718313	C₄	1	4	yes	—
31	C₁₃×#336	C₂²	1	2	yes	—
32	#11718142	C₂²	1	2	yes	—
33	#11718142	C₂²	1	2	yes	—
34	#11718142	C₂²	1	2	yes	—
35	#11718142	C₂²	1	2	yes	—
36	#11718142	C₂²	1	2	yes	—
37	#11718116	C₂²	1	2	yes	—
38	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂×#6101	C₂×C₄	1	2	yes	—
47	C₂×#6101	C₂×C₄	1	4	yes	—
48	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
50	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
51	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
53	C₂²×#1055	C₂³	1	2	yes	—
54	C₂²×#1055	C₂³	1	1	yes	characteristic
55	C₂²×#1055	C₂³	1	1	yes	characteristic
56	C₄×Dic₂₆	—	2	2	no	—
57	C₄×Dic₂₆	—	2	2	no	—
58	C₁₃×C₄⋊C₈	—	2	4	no	—
59	C₁₃×C₄⋊C₈	—	2	2	no	—
60	C₁₃×#97	—	2	2	no	—
61	C₁₃×#95	C₂×C₄	1	2	yes	—
62	#88071	C₂×C₄	1	4	yes	—
63	#88071	C₂×C₄	1	4	yes	—
64	#88071	C₂×C₄	1	4	yes	—
65	#88055	—	2	2	no	—
66	#88055	Q₈	1	4	yes	complemented
67	C₈×Dic₁₃	—	2	2	no	—
68	C₈×Dic₁₃	—	2	4	no	—
69	C₈×Dic₁₃	—	2	2	no	—
70	C₄⋊#1055	—	2	2	no	—
71	C₄⋊#1055	—	2	4	no	—
72	#88043	—	2	2	no	—
73	#21344	—	13	1	no	maximal, Sylow
74	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
79	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
80	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
81	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
82	C₂×Dic₂₆	—	2	2	no	—
83	C₂×Dic₂₆	—	4	2	no	—
84	C₂×C₈×C₁₃	—	2	1	no	—
85	C₂×C₈×C₁₃	—	2	1	no	—
86	C₂×C₈×C₁₃	—	2	1	no	—
87	C₂×C₈×C₁₃	—	2	1	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₁₃	#41	1	2	yes	—
93	C₁₃×C₄⋊C₄	—	4	2	no	—
94	C₄²×C₁₃	—	2	2	no	—
95	C₄²×C₁₃	—	2	2	no	—
96	C₄²×C₁₃	#41	1	2	yes	—
97	C₄²×C₁₃	#41	1	2	yes	—
98	C₄⋊Dic₁₃	—	2	2	no	—
99	#6101	—	2	4	no	—
100	#6101	—	2	8	no	—
101	C₄×Dic₁₃	—	2	2	no	—
102	C₄×Dic₁₃	—	2	2	no	—
103	C₄×Dic₁₃	—	2	1	no	—
104	C₄×Dic₁₃	—	2	1	no	—
105	C₄×Dic₁₃	—	2	4	no	—
106	C₄×Dic₁₃	—	4	2	no	—
107	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
108	C₄×Dic₁₃	#41	1	2	yes	—
109	C₂×#1055	—	2	1	no	—
110	C₂×#1055	—	2	1	no	—
111	C₂×#1055	—	2	1	no	—
112	C₂×#1055	—	2	1	no	—
113	C₂×#1055	—	2	2	no	—
114	C₂×#1055	—	2	2	no	—
115	C₂×#1055	—	2	2	no	—
116	C₂×#1055	C₂×Q₈	1	2	yes	—
117	C₂×#1055	#41	1	2	yes	—
118	#2295	—	13	1	no	—
119	#1985	—	13	1	no	—
120	#1985	—	13	1	no	—
121	#1940	—	13	1	no	—
122	#1940	D₁₃	1	1	yes	characteristic
123	#1916	—	13	2	no	—
124	#1916	—	13	2	no	—
125	#1768	—	13	2	no	—
126	C₄×C₄⋊C₈	—	13	1	no	—
127	#1746	—	13	2	no	—
128	C₄×#97	—	13	1	no	—
129	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	4	no	—
133	C₂×C₄×C₁₃	—	2	4	no	—
134	C₂×C₄×C₁₃	—	2	4	no	—
135	C₂×C₄×C₁₃	#117	1	2	yes	—
136	C₂×C₄×C₁₃	#117	1	2	yes	—
137	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
143	C₂×Dic₁₃	—	2	2	no	—
144	C₂×Dic₁₃	—	2	2	no	—
145	C₂×Dic₁₃	—	2	2	no	—
146	C₂×Dic₁₃	—	2	2	no	—
147	C₂×Dic₁₃	—	2	2	no	—
148	C₂×Dic₁₃	—	2	2	no	—
149	C₂×Dic₁₃	—	2	4	no	—
150	C₂×Dic₁₃	—	2	4	no	—
151	C₂×Dic₁₃	C₄×Q₈	1	4	yes	—
152	C₂×Dic₁₃	#131	1	4	yes	—
153	Dic₂₆	—	4	4	no	—
154	C₈×C₁₃	—	2	2	no	—
155	C₈×C₁₃	—	2	2	no	—
156	C₈×C₁₃	—	4	2	no	—
157	#1055	—	2	2	no	—
158	#1055	—	2	2	no	—
159	#1055	—	4	2	no	—
160	C₂×C₄×Q₈	—	13	1	no	—
161	C₂×C₄⋊C₈	—	13	2	no	—
162	C₂×C₄⋊C₈	—	13	1	no	—
163	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
164	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
165	C₂×#97	—	13	1	no	—
166	C₂×#97	—	13	1	no	—
167	C₂×#97	—	13	1	no	—
168	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
169	C₂×C₄×C₈	—	13	1	no	—
170	C₂×C₄×C₈	—	13	1	no	—
171	C₂×C₄×C₈	—	13	2	no	—
172	#391	—	13	2	no	—
173	#389	—	13	2	no	—
174	#387	—	13	4	no	—
175	C₄×C₄⋊C₄	—	13	2	no	—
176	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
177	#376	—	13	2	no	—
178	C₄³	—	13	1	no	—
179	#336	—	13	2	no	—
180	#336	—	13	2	no	—
181	#336	—	13	2	no	—
182	#336	—	13	2	no	—
183	#336	—	13	2	no	—
184	#336	—	13	2	no	—
185	#336	C₂×D₁₃	1	2	yes	—
186	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
187	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
188	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
189	C₂²×C₁₃	C₂×#131	1	2	yes	—
190	C₂²×C₁₃	#568	1	2	yes	—
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	4	4	no	—
194	C₄×C₁₃	#529	1	1	yes	characteristic
195	C₄×C₁₃	#529	1	1	yes	characteristic
196	C₄×C₁₃	#533	1	1	yes	characteristic
197	C₄×C₁₃	#533	1	1	yes	characteristic
198	Dic₁₃	—	2	4	no	—
199	Dic₁₃	—	2	4	no	—
200	Dic₁₃	—	4	2	no	—
201	Dic₁₃	—	4	2	no	—
202	C₂²×Q₈	—	13	1	no	—
203	C₂²×C₈	—	13	1	no	—
204	C₂²×C₈	—	13	1	no	—
205	C₂²×C₈	—	13	2	no	—
206	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
207	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
208	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
209	C₄×Q₈	—	26	2	no	—
210	C₄×Q₈	—	26	2	no	—
211	C₂×C₄⋊C₄	—	13	2	no	—
212	C₂×C₄⋊C₄	—	13	1	no	—
213	C₂×C₄⋊C₄	—	13	4	no	—
214	C₂×C₄⋊C₄	C₄×D₁₃	1	2	yes	—
215	C₂×C₄²	—	13	1	no	—
216	C₂×C₄²	—	13	2	no	—
217	C₂×C₄²	—	13	2	no	—
218	C₂×C₄²	—	13	1	no	—
219	C₂×C₄²	—	13	2	no	—
220	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
221	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
222	C₄⋊C₈	—	2	4	no	—
223	C₄⋊C₈	—	2	2	no	—
224	C₄⋊C₈	—	26	4	no	—
225	C₄⋊C₈	—	26	2	no	—
226	#97	—	2	2	no	—
227	#97	—	13	4	no	—
228	#97	—	26	2	no	—
229	#97	—	26	2	no	—
230	C₄×C₈	—	26	4	no	—
231	C₄×C₈	—	26	2	no	—
232	C₄×C₈	—	26	2	no	—
233	#95	—	13	4	no	—
234	#95	—	13	4	no	—
235	#95	—	13	4	no	—
236	#95	C₄×D₁₃	1	2	yes	—
237	C₂×C₁₃	#2295	1	1	yes	characteristic
238	C₂×C₁₃	#2921	1	1	yes	characteristic
239	C₂×C₁₃	#2922	1	1	yes	characteristic
240	C₂×C₁₃	#2962	1	2	yes	—
241	C₂×C₁₃	#2968	1	2	yes	—
242	C₂×Q₈	—	26	2	no	—
243	C₂×Q₈	—	52	2	no	—
244	C₂²×C₄	—	13	2	no	—
245	C₂²×C₄	—	13	2	no	—
246	C₂²×C₄	—	13	2	no	—
247	C₂²×C₄	—	13	2	no	—
248	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
249	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
251	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	13	2	no	—
260	C₂×C₈	—	13	2	no	—
261	C₂×C₈	—	26	1	no	—
262	C₂×C₈	—	26	1	no	—
263	C₂×C₈	—	26	1	no	—
264	C₂×C₈	—	26	1	no	—
265	C₂×C₈	—	26	2	no	—
266	C₂×C₈	—	26	2	no	—
267	C₂×C₈	—	26	2	no	—
268	C₂×C₈	#6129	1	2	yes	—
269	C₂×C₈	Q₈×D₁₃	1	2	yes	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	26	8	no	—
272	C₄⋊C₄	—	26	4	no	—
273	C₄⋊C₄	—	26	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	13	2	no	—
277	C₄²	—	13	2	no	—
278	C₄²	—	26	4	no	—
279	C₄²	—	26	1	no	—
280	C₄²	—	26	2	no	—
281	C₄²	—	26	2	no	—
282	C₄²	—	26	1	no	—
283	C₄²	—	52	2	no	—
284	C₄²	#6129	1	2	yes	—
285	C₄²	#6131	1	2	yes	—
286	C₁₃	#21344	1	1	yes	characteristic, complemented, Sylow
287	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
288	Q₈	—	52	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	13	4	no	—
295	C₂×C₄	—	13	4	no	—
296	C₂×C₄	—	26	2	no	—
297	C₂×C₄	—	26	2	no	—
298	C₂×C₄	—	26	2	no	—
299	C₂×C₄	—	26	2	no	—
300	C₂×C₄	—	26	4	no	—
301	C₂×C₄	—	26	4	no	—
302	C₂×C₄	—	26	2	no	—
303	C₂×C₄	—	26	2	no	—
304	C₂×C₄	#88146	1	2	yes	—
305	C₂×C₄	#88146	1	2	yes	—
306	C₂×C₄	C₂×#6129	1	1	yes	characteristic
307	C₂×C₄	C₂×#6129	1	1	yes	characteristic
308	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
309	C₂×C₄	C₂×#6131	1	1	yes	characteristic
310	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
311	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	2	2	no	—
314	C₈	—	4	2	no	—
315	C₈	—	26	2	no	—
316	C₈	—	26	2	no	—
317	C₈	—	52	2	no	—
318	C₂²	C₂×#88146	1	1	yes	characteristic
319	C₂²	C₄×#6129	1	1	yes	characteristic
320	C₂²	C₄×Q₈×D₁₃	1	1	yes	characteristic
321	C₂²	#11719450	1	2	yes	—
322	C₂²	D₁₃×#131	1	2	yes	—
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	4	4	no	—
326	C₄	—	26	4	no	—
327	C₄	—	26	4	no	—
328	C₄	—	52	2	no	—
329	C₄	—	52	2	no	—
330	C₄	#11719305	1	1	yes	characteristic
331	C₄	#11719327	1	1	yes	characteristic
332	C₄	#11719396	1	1	yes	characteristic
333	C₄	#11719433	1	1	yes	characteristic
334	C₂	#13117307	1	1	yes	characteristic
335	C₂	#13123473	1	1	yes	characteristic
336	C₂	#13123479	1	1	yes	characteristic
337	C₂	#13123781	1	2	yes	—
338	C₂	#13123827	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle, prime radical	C₂³×C₁₃	C₂³×C₄
Fitting	C₁₃×#1940	C₂
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1940	D₁₃
hypocenter	C₁₃	#21344
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1940

Complements

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

N	H ≅ G/N	type
66. #88055	Q₈	semidirect
286. C₁₃	#21344	semidirect

Maximal subgroups

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

subgroup	index
#13117310	2
#13115776	2
#13115775	2
#13115560	2
#13115396	2
#13115390	2
#13114695	2
C₄⋊C₈×Dic₁₃	2
#13114586	2
#97×Dic₁₃	2
C₁₃×#1940	2
#21344	13