Subgroups — #15889535

Order 3328.

600 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115777	C₂	1	1	yes	maximal, characteristic
2	#13115773	C₂	1	2	yes	maximal
3	#13115773	C₂	1	1	yes	maximal, characteristic
4	#13115402	C₂	1	2	yes	maximal
5	#13115395	C₂	1	2	yes	maximal
6	#13115395	C₂	1	2	yes	maximal
7	#13115395	C₂	1	2	yes	maximal
8	#13114981	C₂	1	2	yes	maximal
9	C₁₃×#2399	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#398	C₄	1	2	yes	—
13	C₁₃×#398	C₄	1	1	yes	characteristic
14	C₁₃×#398	C₄	1	1	yes	characteristic
15	C₁₃×#395	C₄	1	1	yes	characteristic
16	C₁₃×#395	C₄	1	1	yes	characteristic
17	C₁₃×#395	C₄	1	2	yes	—
18	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₁₃×#376	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊#1055	C₂²	1	2	yes	—
24	C₂×#88043	C₂²	1	1	yes	characteristic
25	C₂×C₄×#1055	C₂²	1	2	yes	—
26	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
27	#11718116	C₂²	1	2	yes	—
28	#11718116	C₂²	1	2	yes	—
29	#11718116	C₂²	1	2	yes	—
30	#11718116	C₂²	1	2	yes	—
31	#11718116	C₂²	1	2	yes	—
32	#11718116	C₂²	1	2	yes	—
33	#11718116	C₂²	1	2	yes	—
34	#11718116	C₂²	1	2	yes	—
35	#11718116	C₂²	1	2	yes	—
36	#11718116	C₂²	1	2	yes	—
37	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₁₃	—	4	1	no	—
40	C₄×Q₈×C₁₃	—	4	1	no	—
41	C₄×Q₈×C₁₃	—	4	1	no	—
42	C₄×Q₈×C₁₃	—	4	1	no	—
43	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
54	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₂²×#1055	C₂³	1	2	yes	—
58	C₂²×#1055	C₂³	1	2	yes	—
59	C₂²×#1055	C₂³	1	2	yes	—
60	C₂²×#1055	C₂³	1	2	yes	—
61	C₁₃×#95	C₂×C₄	1	2	yes	—
62	C₁₃×#95	C₂×C₄	1	2	yes	—
63	C₁₃×#95	C₂×C₄	1	2	yes	—
64	C₁₃×#95	C₂×C₄	1	2	yes	—
65	C₁₃×#95	C₂×C₄	1	1	yes	characteristic
66	C₁₃×#95	C₂×C₄	1	2	yes	—
67	C₁₃×#95	C₂×C₄	1	1	yes	characteristic
68	C₄⋊#1055	—	4	1	no	—
69	C₄⋊#1055	—	4	1	no	—
70	C₄⋊#1055	—	4	2	no	—
71	#88043	—	4	1	no	—
72	C₄×#1055	—	4	2	no	—
73	C₄×#1055	—	4	1	no	—
74	#23367	—	13	1	no	maximal, Sylow
75	C₂×Q₈×C₁₃	—	2	4	no	—
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	4	1	no	—
78	C₂×Q₈×C₁₃	—	4	1	no	—
79	C₂×Q₈×C₁₃	—	4	1	no	—
80	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₁₃×C₄⋊C₄	—	2	4	no	—
92	C₁₃×C₄⋊C₄	—	2	4	no	—
93	C₁₃×C₄⋊C₄	—	2	4	no	—
94	C₁₃×C₄⋊C₄	—	4	2	no	—
95	C₁₃×C₄⋊C₄	—	4	1	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₄⋊C₄	—	4	1	no	—
98	C₁₃×C₄⋊C₄	—	4	1	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₄²×C₁₃	—	2	1	no	—
101	C₄²×C₁₃	—	2	1	no	—
102	C₄²×C₁₃	—	2	1	no	—
103	C₄²×C₁₃	—	2	2	no	—
104	C₄²×C₁₃	—	2	1	no	—
105	C₄²×C₁₃	—	2	1	no	—
106	C₄²×C₁₃	—	2	2	no	—
107	C₄²×C₁₃	—	2	1	no	—
108	C₄²×C₁₃	—	2	2	no	—
109	C₄²×C₁₃	—	2	2	no	—
110	C₂×#1055	—	2	2	no	—
111	C₂×#1055	—	2	2	no	—
112	C₂×#1055	—	2	2	no	—
113	C₂×#1055	—	2	2	no	—
114	C₂×#1055	—	2	4	no	—
115	C₂×#1055	—	2	4	no	—
116	C₂×#1055	—	2	2	no	—
117	C₂×#1055	—	2	2	no	—
118	C₂×#1055	—	2	2	no	—
119	C₂×#1055	—	2	2	no	—
120	#2399	D₁₃	1	1	yes	characteristic
121	#1985	—	13	1	no	—
122	#1984	—	13	2	no	—
123	#1984	—	13	1	no	—
124	#1917	—	13	2	no	—
125	#1916	—	13	2	no	—
126	#1916	—	13	2	no	—
127	#1916	—	13	2	no	—
128	#1840	—	13	2	no	—
129	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
130	Q₈×C₁₃	—	4	4	no	—
131	Q₈×C₁₃	—	4	4	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	2	1	no	—
137	C₂×C₄×C₁₃	—	2	1	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	1	no	—
140	C₂×C₄×C₁₃	—	2	1	no	—
141	C₂×C₄×C₁₃	—	2	1	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	2	1	no	—
151	C₂×C₄×C₁₃	—	2	1	no	—
152	C₂×C₄×C₁₃	—	2	2	no	—
153	C₂×C₄×C₁₃	—	2	1	no	—
154	C₂×C₄×C₁₃	—	2	2	no	—
155	C₂×C₄×C₁₃	—	2	2	no	—
156	C₂×C₄×C₁₃	—	2	1	no	—
157	C₂×C₄×C₁₃	#131	1	2	yes	—
158	C₂×C₄×C₁₃	#131	1	4	yes	—
159	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
162	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
163	#1055	—	4	2	no	—
164	#1055	—	4	2	no	—
165	#1055	—	4	2	no	—
166	#1055	—	4	2	no	—
167	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
168	C₂×C₄×Q₈	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	C₂×#97	—	13	1	no	—
173	C₂×C₄×C₈	—	13	2	no	—
174	C₂×C₄×C₈	—	13	1	no	—
175	#398	Dic₁₃	1	2	yes	—
176	#398	Dic₁₃	1	1	yes	characteristic
177	#398	Dic₁₃	1	1	yes	characteristic
178	#395	Dic₁₃	1	1	yes	characteristic
179	#395	Dic₁₃	1	1	yes	characteristic
180	#395	Dic₁₃	1	2	yes	—
181	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
182	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
183	#376	C₂×D₁₃	1	1	yes	characteristic
184	#336	—	13	2	no	—
185	#336	—	13	2	no	—
186	#336	—	13	2	no	—
187	#336	—	13	2	no	—
188	#336	—	13	2	no	—
189	#336	—	13	2	no	—
190	#336	—	13	2	no	—
191	#336	—	13	2	no	—
192	#336	—	13	2	no	—
193	#336	—	13	2	no	—
194	C₂²×C₁₃	#518	1	1	yes	characteristic
195	C₂²×C₁₃	#519	1	2	yes	—
196	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
197	C₂²×C₁₃	C₂×#131	1	2	yes	—
198	C₂²×C₁₃	#568	1	1	yes	characteristic
199	C₄×C₁₃	—	2	2	no	—
200	C₄×C₁₃	—	2	4	no	—
201	C₄×C₁₃	—	4	2	no	—
202	C₄×C₁₃	—	4	1	no	—
203	C₄×C₁₃	—	4	1	no	—
204	C₄×C₁₃	—	4	1	no	—
205	C₄×C₁₃	—	4	1	no	—
206	C₄×C₁₃	—	4	1	no	—
207	C₄×C₁₃	—	4	2	no	—
208	C₄×C₁₃	—	4	2	no	—
209	C₄×C₁₃	#563	1	1	yes	characteristic
210	C₄×C₁₃	#563	1	1	yes	characteristic
211	C₄×C₁₃	#563	1	2	yes	—
212	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
213	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
214	C₂²×C₈	—	13	2	no	—
215	C₂²×C₈	—	13	2	no	—
216	C₂²×C₈	—	13	2	no	—
217	C₂²×C₈	—	13	2	no	—
218	C₄×Q₈	—	4	1	no	—
219	C₄×Q₈	—	4	1	no	—
220	C₄×Q₈	—	4	1	no	—
221	C₄×Q₈	—	4	1	no	—
222	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
223	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
224	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
225	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
226	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
227	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
228	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
229	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
230	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
231	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
233	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
234	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
236	C₄⋊C₈	—	52	1	no	—
237	C₄⋊C₈	—	52	1	no	—
238	C₄⋊C₈	—	52	2	no	—
239	#97	—	52	1	no	—
240	C₄×C₈	—	52	1	no	—
241	C₄×C₈	—	52	2	no	—
242	#95	C₂×Dic₁₃	1	1	yes	characteristic
243	#95	C₂×Dic₁₃	1	2	yes	—
244	#95	C₂×Dic₁₃	1	1	yes	characteristic
245	#95	C₂×Dic₁₃	1	2	yes	—
246	#95	C₂×Dic₁₃	1	2	yes	—
247	#95	C₂×Dic₁₃	1	2	yes	—
248	#95	C₂×Dic₁₃	1	2	yes	—
249	C₂×C₁₃	#2381	1	1	yes	characteristic
250	C₂×C₁₃	#2976	1	1	yes	characteristic
251	C₂×C₁₃	#2977	1	1	yes	characteristic
252	C₂×C₁₃	#2985	1	2	yes	—
253	C₂×C₁₃	#2986	1	2	yes	—
254	C₂×Q₈	—	2	4	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
260	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
261	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
262	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
263	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
264	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
265	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
266	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
267	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
268	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
269	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
270	C₂×C₈	—	26	2	no	—
271	C₂×C₈	—	26	2	no	—
272	C₂×C₈	—	26	2	no	—
273	C₂×C₈	—	26	2	no	—
274	C₂×C₈	—	26	4	no	—
275	C₂×C₈	—	26	2	no	—
276	C₂×C₈	—	26	2	no	—
277	C₂×C₈	—	26	4	no	—
278	C₂×C₈	—	26	2	no	—
279	C₂×C₈	—	26	2	no	—
280	C₄⋊C₄	—	2	4	no	—
281	C₄⋊C₄	—	2	4	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₁₃	#23367	1	1	yes	characteristic, complemented, Sylow
300	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
301	Q₈	—	4	4	no	—
302	Q₈	—	4	4	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	#88202	1	2	yes	—
329	C₂×C₄	#88202	1	4	yes	—
330	C₂×C₄	#88251	1	1	yes	characteristic
331	C₂×C₄	#88251	1	1	yes	characteristic
332	C₂×C₄	#88254	1	2	yes	—
333	C₂×C₄	#88254	1	2	yes	—
334	C₈	—	52	2	no	—
335	C₈	—	52	2	no	—
336	C₈	—	52	2	no	—
337	C₈	—	52	2	no	—
338	C₂²	#11719503	1	1	yes	characteristic
339	C₂²	#11719517	1	2	yes	—
340	C₂²	C₂×#88202	1	1	yes	characteristic
341	C₂²	C₂×#88202	1	2	yes	—
342	C₂²	#11719524	1	1	yes	characteristic
343	C₄	—	2	2	no	—
344	C₄	—	2	4	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	#11719440	1	1	yes	characteristic
354	C₄	#11719440	1	1	yes	characteristic
355	C₄	#11719440	1	2	yes	—
356	C₂	#13118017	1	1	yes	characteristic
357	C₂	#13123884	1	1	yes	characteristic
358	C₂	#13123892	1	1	yes	characteristic
359	C₂	#13123931	1	2	yes	—
360	C₂	#13123938	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle, prime radical	C₂³×C₁₃	C₂³×C₄
Fitting	C₁₃×#2399	C₂
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#2399	D₁₃
hypocenter	C₁₃	#23367
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2399

Complements

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

N	H ≅ G/N	type
299. C₁₃	#23367	semidirect

Maximal subgroups

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

subgroup	index
#13115777	2
#13115773	2
#13115402	2
#13115395	2
#13114981	2
C₁₃×#2399	2
#23367	13