Subgroups — #15889516

Order 3328.

636 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115777	C₂	1	2	yes	maximal
2	#13115773	C₂	1	2	yes	maximal
3	#13115402	C₂	1	2	yes	maximal
4	#13115402	C₂	1	2	yes	maximal
5	#13115395	C₂	1	2	yes	maximal
6	#13115395	C₂	1	2	yes	maximal
7	#13114979	C₂	1	1	yes	maximal, characteristic
8	#13114979	C₂	1	1	yes	maximal, characteristic
9	C₄×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
12	C₁₃×#391	C₄	1	2	yes	—
13	C₁₃×#389	C₄	1	2	yes	—
14	C₁₃×#389	C₄	1	1	yes	characteristic
15	C₁₃×#389	C₄	1	1	yes	characteristic
16	C₁₃×#384	C₄	1	1	yes	characteristic
17	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₄³×C₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#1055	C₂²	1	2	yes	—
22	C₂×C₄⋊#1055	C₂²	1	2	yes	—
23	C₂×#88043	C₂²	1	2	yes	—
24	C₂×C₄×#1055	C₂²	1	2	yes	—
25	#11718116	C₂²	1	1	yes	characteristic
26	#11718116	C₂²	1	2	yes	—
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	1	yes	characteristic
33	#11718116	C₂²	1	2	yes	—
34	#11718116	C₂²	1	1	yes	characteristic
35	#11718116	C₂²	1	2	yes	—
36	#11718116	C₂²	1	1	yes	characteristic
37	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
38	C₁₃×C₄⋊Q₈	—	2	4	no	—
39	C₄×Q₈×C₁₃	—	4	2	no	—
40	C₄×Q₈×C₁₃	—	4	2	no	—
41	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	C₂²×#1055	C₂³	1	2	yes	—
60	C₂²×#1055	C₂³	1	2	yes	—
61	C₂²×#1055	C₂³	1	2	yes	—
62	C₂²×#1055	C₂³	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	2	yes	—
66	C₁₃×#95	C₂×C₄	1	2	yes	—
67	C₄⋊#1055	—	4	2	no	—
68	C₄⋊#1055	—	4	2	no	—
69	#88043	—	4	2	no	—
70	C₄×#1055	—	4	2	no	—
71	#23364	—	13	1	no	maximal, Sylow
72	C₂×Q₈×C₁₃	—	2	4	no	—
73	C₂×Q₈×C₁₃	—	2	4	no	—
74	C₂×Q₈×C₁₃	—	4	2	no	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₁₃×C₄⋊C₄	—	2	4	no	—
87	C₁₃×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₄⋊C₄	—	2	4	no	—
89	C₁₃×C₄⋊C₄	—	4	1	no	—
90	C₁₃×C₄⋊C₄	—	4	1	no	—
91	C₁₃×C₄⋊C₄	—	4	2	no	—
92	C₁₃×C₄⋊C₄	—	4	2	no	—
93	C₁₃×C₄⋊C₄	—	4	2	no	—
94	C₁₃×C₄⋊C₄	M₁₆	1	4	yes	—
95	C₄²×C₁₃	—	2	2	no	—
96	C₄²×C₁₃	—	2	2	no	—
97	C₄²×C₁₃	—	2	2	no	—
98	C₄²×C₁₃	—	2	2	no	—
99	C₄²×C₁₃	—	2	2	no	—
100	C₄²×C₁₃	—	2	2	no	—
101	C₄²×C₁₃	—	2	2	no	—
102	C₄²×C₁₃	—	2	2	no	—
103	C₄²×C₁₃	—	2	1	no	—
104	C₄²×C₁₃	—	2	1	no	—
105	C₄²×C₁₃	—	2	2	no	—
106	C₄²×C₁₃	—	4	1	no	—
107	C₂×#1055	—	2	2	no	—
108	C₂×#1055	—	2	2	no	—
109	C₂×#1055	—	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	2	no	—
115	C₂×#1055	—	2	2	no	—
116	C₂×#1055	—	2	2	no	—
117	C₂×#1055	—	2	2	no	—
118	C₂×#1055	—	2	2	no	—
119	C₄×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
120	#1985	—	13	2	no	—
121	#1984	—	13	2	no	—
122	#1917	—	13	2	no	—
123	#1917	—	13	2	no	—
124	#1916	—	13	2	no	—
125	#1916	—	13	2	no	—
126	#1839	—	13	1	no	—
127	#1839	—	13	1	no	—
128	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
129	Q₈×C₁₃	—	4	8	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	1	no	—
138	C₂×C₄×C₁₃	—	2	1	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	2	1	no	—
143	C₂×C₄×C₁₃	—	2	1	no	—
144	C₂×C₄×C₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	—	2	1	no	—
146	C₂×C₄×C₁₃	—	2	1	no	—
147	C₂×C₄×C₁₃	—	2	1	no	—
148	C₂×C₄×C₁₃	—	2	1	no	—
149	C₂×C₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	—	2	1	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	C₂×M₁₆	1	2	yes	—
157	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
158	C₂×C₄×C₁₃	#131	1	2	yes	—
159	C₂×C₄×C₁₃	#131	1	2	yes	—
160	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
161	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
162	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
163	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
164	#1055	—	4	2	no	—
165	#1055	—	4	2	no	—
166	#1055	—	4	2	no	—
167	#1055	—	4	2	no	—
168	C₂×C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
169	C₂×C₄×Q₈	C₂×D₁₃	1	2	yes	—
170	C₂×C₄⋊C₈	—	13	2	no	—
171	C₂×C₄⋊C₈	—	13	2	no	—
172	C₂×#97	—	13	2	no	—
173	C₂×C₄×C₈	—	13	2	no	—
174	#391	Dic₁₃	1	2	yes	—
175	#389	Dic₁₃	1	2	yes	—
176	#389	Dic₁₃	1	1	yes	characteristic
177	#389	Dic₁₃	1	1	yes	characteristic
178	#384	Dic₁₃	1	1	yes	characteristic
179	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
180	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
181	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
182	C₄³	C₂×D₁₃	1	1	yes	characteristic
183	#336	—	13	1	no	—
184	#336	—	13	2	no	—
185	#336	—	13	2	no	—
186	#336	—	13	2	no	—
187	#336	—	13	1	no	—
188	#336	—	13	1	no	—
189	#336	—	13	2	no	—
190	#336	—	13	1	no	—
191	#336	—	13	2	no	—
192	#336	—	13	2	no	—
193	#336	—	13	2	no	—
194	#336	—	13	2	no	—
195	C₂²×C₁₃	#518	1	1	yes	characteristic
196	C₂²×C₁₃	#519	1	1	yes	characteristic
197	C₂²×C₁₃	#519	1	1	yes	characteristic
198	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
199	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
200	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
201	C₂²×C₁₃	#568	1	1	yes	characteristic
202	C₄×C₁₃	—	2	4	no	—
203	C₄×C₁₃	—	2	2	no	—
204	C₄×C₁₃	—	2	2	no	—
205	C₄×C₁₃	—	2	2	no	—
206	C₄×C₁₃	—	4	2	no	—
207	C₄×C₁₃	—	4	1	no	—
208	C₄×C₁₃	—	4	2	no	—
209	C₄×C₁₃	—	4	2	no	—
210	C₄×C₁₃	—	4	1	no	—
211	C₄×C₁₃	—	4	1	no	—
212	C₄×C₁₃	#563	1	2	yes	—
213	C₄×C₁₃	#563	1	2	yes	—
214	C₂²×Q₈	C₂×Dic₁₃	1	2	yes	—
215	C₂²×C₈	—	13	2	no	—
216	C₂²×C₈	—	13	2	no	—
217	C₂²×C₈	—	13	2	no	—
218	C₂²×C₈	—	13	2	no	—
219	C₄⋊Q₈	—	2	4	no	—
220	C₄×Q₈	—	4	2	no	—
221	C₄×Q₈	—	4	2	no	—
222	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
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	2	yes	—
226	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
227	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
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	1	yes	characteristic
231	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
232	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
233	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
234	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
236	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
237	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
238	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
239	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
240	C₄⋊C₈	—	52	2	no	—
241	C₄⋊C₈	—	52	2	no	—
242	#97	—	52	2	no	—
243	C₄×C₈	—	52	2	no	—
244	#95	C₂×Dic₁₃	1	2	yes	—
245	#95	C₂×Dic₁₃	1	2	yes	—
246	#95	C₂×Dic₁₃	1	2	yes	—
247	#95	C₂×Dic₁₃	1	2	yes	—
248	C₂×C₁₃	#2381	1	1	yes	characteristic
249	C₂×C₁₃	#2974	1	1	yes	characteristic
250	C₂×C₁₃	#2981	1	1	yes	characteristic
251	C₂×C₁₃	#2983	1	1	yes	characteristic
252	C₂×C₁₃	#2984	1	1	yes	characteristic
253	C₂×C₁₃	#2985	1	1	yes	characteristic
254	C₂×C₁₃	#2986	1	1	yes	characteristic
255	C₂×Q₈	—	2	4	no	—
256	C₂×Q₈	—	2	4	no	—
257	C₂×Q₈	—	4	2	no	—
258	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
259	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
260	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
261	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
262	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
263	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
264	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
265	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
266	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
267	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
268	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
269	C₂×C₈	—	26	2	no	—
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	2	no	—
275	C₂×C₈	—	26	2	no	—
276	C₂×C₈	—	26	2	no	—
277	C₂×C₈	—	26	2	no	—
278	C₂×C₈	—	26	2	no	—
279	C₂×C₈	—	26	2	no	—
280	C₂×C₈	—	26	2	no	—
281	C₄⋊C₄	—	2	4	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	#6099	1	4	yes	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	4	1	no	—
302	C₁₃	#23364	1	1	yes	characteristic, complemented, Sylow
303	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
304	Q₈	—	4	8	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	C₂×#6099	1	2	yes	—
332	C₂×C₄	C₂×#6099	1	4	yes	—
333	C₂×C₄	#88202	1	2	yes	—
334	C₂×C₄	#88202	1	2	yes	—
335	C₂×C₄	#88251	1	2	yes	—
336	C₂×C₄	#88254	1	1	yes	characteristic
337	C₂×C₄	#88254	1	1	yes	characteristic
338	C₂×C₄	#88254	1	2	yes	—
339	C₈	—	52	2	no	—
340	C₈	—	52	2	no	—
341	C₈	—	52	2	no	—
342	C₈	—	52	2	no	—
343	C₂²	C₂²×#6099	1	1	yes	characteristic
344	C₂²	#11719503	1	1	yes	characteristic
345	C₂²	#11719517	1	1	yes	characteristic
346	C₂²	#11719517	1	1	yes	characteristic
347	C₂²	C₂×#88202	1	1	yes	characteristic
348	C₂²	C₂×#88202	1	1	yes	characteristic
349	C₂²	#11719524	1	1	yes	characteristic
350	C₄	—	2	4	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	2	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	1	no	—
360	C₄	#11719440	1	2	yes	—
361	C₄	#11719440	1	2	yes	—
362	C₂	#13118017	1	1	yes	characteristic
363	C₂	#13123871	1	1	yes	characteristic
364	C₂	#13123912	1	1	yes	characteristic
365	C₂	#13123921	1	1	yes	characteristic
366	C₂	#13123925	1	1	yes	characteristic
367	C₂	#13123931	1	1	yes	characteristic
368	C₂	#13123938	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle, prime radical	C₂³×C₁₃	C₂³×C₄
Fitting	C₄×C₁₃×C₄⋊Q₈	C₂
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#23364
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
302. C₁₃	#23364	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
#13114979	2
C₄×C₁₃×C₄⋊Q₈	2
#23364	13