Subgroups — #16727894

Order 3328.

1974 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123925	C₂	1	8	yes	maximal, complemented
2	#13123921	C₂	1	6	yes	maximal, complemented
3	#13123917	C₂	1	2	yes	maximal, complemented
4	Q₈×#6099	C₂	1	8	yes	maximal, complemented
5	#13123469	C₂	1	6	yes	maximal
6	C₂²×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₃	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₁₃	C₄	1	4	yes	—
9	C₂×C₄×Q₈×C₁₃	C₄	1	4	yes	—
10	C₂×C₄×Q₈×C₁₃	C₄	1	3	yes	—
11	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₁₃	C₂²	1	3	yes	—
14	C₂×C₄×Q₈×C₁₃	C₂²	1	4	yes	—
15	C₂×C₄×Q₈×C₁₃	C₂²	1	4	yes	—
16	C₂²×C₁₃×C₄⋊C₄	C₄	1	3	yes	—
17	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
18	#11718868	C₂²	1	4	yes	—
19	#11718685	C₂²	1	48	yes	complemented
20	Q₈×#1055	C₂²	1	16	yes	complemented
21	#11718587	C₂²	1	12	yes	—
22	#11718587	C₂²	1	12	yes	—
23	#11718586	C₂²	1	12	yes	—
24	#11718586	C₂²	1	12	yes	—
25	#11718585	C₂²	1	12	yes	—
26	C₄×#6099	C₂²	1	12	yes	—
27	C₂²×Q₈×C₁₃	—	2	8	no	—
28	C₂²×Q₈×C₁₃	C₂×C₄	1	4	yes	—
29	C₂²×Q₈×C₁₃	C₂×C₄	1	4	yes	—
30	C₂²×Q₈×C₁₃	C₂×C₄	1	3	yes	—
31	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
33	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₁₃	C₂×C₄	1	12	yes	—
36	C₄×Q₈×C₁₃	C₂×C₄	1	12	yes	—
37	C₄×Q₈×C₁₃	C₂×C₄	1	12	yes	—
38	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
39	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
40	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
41	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
42	C₄×Q₈×C₁₃	C₂³	1	12	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	12	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	6	yes	—
45	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	12	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
48	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
49	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
50	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
51	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
52	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
53	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
54	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
55	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
56	C₂×#6099	C₂³	1	8	yes	—
57	C₂²⋊#1055	C₂³	1	8	yes	—
58	C₂²⋊#1055	C₂³	1	8	yes	—
59	C₄⋊#1055	C₂³	1	24	yes	—
60	C₄⋊#1055	C₂³	1	24	yes	—
61	#88043	C₂³	1	24	yes	—
62	C₄×#1055	C₂³	1	24	yes	—
63	#60248	—	13	1	no	maximal, Sylow
64	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×Q₈×C₁₃	—	2	16	no	—
66	C₂×Q₈×C₁₃	—	2	16	no	—
67	C₂×Q₈×C₁₃	—	2	12	no	—
68	C₂×Q₈×C₁₃	M₁₆	1	8	yes	complemented
69	C₂×Q₈×C₁₃	C₂²×C₄	1	12	yes	—
70	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₁₃	—	2	12	no	—
72	C₂²×C₄×C₁₃	—	2	4	no	—
73	C₂²×C₄×C₁₃	—	4	6	no	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	6	yes	—
78	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
80	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
81	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
82	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
83	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
84	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
85	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₁₃×C₄⋊C₄	C₂²×C₄	1	24	yes	—
89	C₁₃×C₄⋊C₄	C₂²×C₄	1	12	yes	—
90	C₁₃×C₄⋊C₄	C₂²×C₄	1	12	yes	—
91	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
92	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
93	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
94	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
95	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
96	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
97	C₄²×C₁₃	C₂⁴	1	6	yes	—
98	C₄²×C₁₃	C₂⁴	1	6	yes	—
99	#6099	—	2	16	no	—
100	C₂×#1055	C₂⁴	1	16	yes	—
101	C₂²×C₄×Q₈	D₁₃	1	1	yes	characteristic
102	#2984	—	13	8	no	—
103	#2983	—	13	6	no	—
104	#2982	—	13	2	no	—
105	Q₈×M₁₆	—	13	8	no	—
106	#2920	—	13	6	no	—
107	C₂³×C₁₃	—	2	2	no	—
108	C₂³×C₁₃	—	2	2	no	—
109	C₂³×C₁₃	—	2	2	no	—
110	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
111	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
112	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	Q₈×C₁₃	—	2	24	no	—
114	Q₈×C₁₃	C₂×M₁₆	1	16	yes	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	6	no	—
117	C₂×C₄×C₁₃	—	2	12	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	12	no	—
121	C₂×C₄×C₁₃	—	2	12	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	4	6	no	—
125	C₂×C₄×C₁₃	—	4	3	no	—
126	C₂×C₄×C₁₃	—	4	3	no	—
127	C₂×C₄×C₁₃	C₂×M₁₆	1	12	yes	—
128	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
131	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
132	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
133	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
135	C₂×C₄×C₁₃	2₋¹⁺⁴	1	4	yes	—
136	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
137	#1055	—	2	16	no	—
138	C₂³×Q₈	Dic₁₃	1	1	yes	characteristic
139	C₂×C₄×Q₈	Dic₁₃	1	4	yes	—
140	C₂×C₄×Q₈	Dic₁₃	1	4	yes	—
141	C₂×C₄×Q₈	Dic₁₃	1	3	yes	—
142	C₂×C₄×Q₈	Dic₁₃	1	1	yes	characteristic
143	C₂×C₄×Q₈	C₂×D₁₃	1	4	yes	—
144	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
145	C₂×C₄×Q₈	C₂×D₁₃	1	3	yes	—
146	C₂×C₄×Q₈	C₂×D₁₃	1	4	yes	—
147	C₂²×C₄⋊C₄	Dic₁₃	1	3	yes	—
148	C₂²×C₄²	C₂×D₁₃	1	3	yes	—
149	#446	—	13	48	no	—
150	C₈×Q₈	—	13	16	no	—
151	#432	—	13	12	no	—
152	#432	—	13	12	no	—
153	#431	—	13	12	no	—
154	#431	—	13	12	no	—
155	#423	—	13	12	no	—
156	#407	—	13	4	no	—
157	C₄×M₁₆	—	13	12	no	—
158	C₂²×C₁₃	—	2	8	no	—
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	—	2	1	no	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	—	2	1	no	—
163	C₂²×C₁₃	—	2	1	no	—
164	C₂²×C₁₃	—	2	1	no	—
165	C₂²×C₁₃	#519	1	2	yes	—
166	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
167	C₂²×C₁₃	#568	1	2	yes	—
168	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
169	C₄×C₁₃	—	2	1	no	—
170	C₄×C₁₃	—	2	1	no	—
171	C₄×C₁₃	—	2	6	no	—
172	C₄×C₁₃	—	2	6	no	—
173	C₄×C₁₃	—	2	6	no	—
174	C₄×C₁₃	#519	1	2	yes	—
175	C₄×C₁₃	C₂²×M₁₆	1	12	yes	—
176	C₄×C₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
177	C₂²×Q₈	—	2	8	no	—
178	C₂²×Q₈	C₂×Dic₁₃	1	4	yes	—
179	C₂²×Q₈	C₂×Dic₁₃	1	4	yes	—
180	C₂²×Q₈	C₂×Dic₁₃	1	3	yes	—
181	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
182	C₂³×C₄	C₂×Dic₁₃	1	3	yes	—
183	C₂³×C₄	C₂×Dic₁₃	1	3	yes	—
184	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
185	C₂×M₁₆	—	13	8	no	—
186	C₄×Q₈	C₂×Dic₁₃	1	12	yes	—
187	C₄×Q₈	C₂×Dic₁₃	1	12	yes	—
188	C₄×Q₈	C₂×Dic₁₃	1	12	yes	—
189	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
190	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
191	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
192	C₄×Q₈	C₂²×D₁₃	1	4	yes	—
193	C₄×Q₈	C₂²×D₁₃	1	12	yes	—
194	C₂×C₄⋊C₄	C₂×Dic₁₃	1	12	yes	—
195	C₂×C₄⋊C₄	C₂×Dic₁₃	1	12	yes	—
196	C₂×C₄⋊C₄	C₂×Dic₁₃	1	6	yes	—
197	C₂×C₄⋊C₄	C₂×Dic₁₃	1	3	yes	—
198	C₂×C₄⋊C₄	C₂×Dic₁₃	1	3	yes	—
199	C₂×C₄²	C₂×Dic₁₃	1	6	yes	—
200	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
201	C₂×C₄²	C₂×Dic₁₃	1	6	yes	—
202	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
203	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
204	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
205	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
206	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
207	C₄⋊C₈	—	13	24	no	—
208	C₄⋊C₈	—	13	24	no	—
209	C₂²⋊C₈	—	13	8	no	—
210	C₂²⋊C₈	—	13	8	no	—
211	#97	—	13	24	no	—
212	C₄×C₈	—	13	24	no	—
213	C₂×C₁₃	—	2	2	no	—
214	C₂×C₁₃	—	2	2	no	—
215	C₂×C₁₃	—	2	2	no	—
216	C₂×C₁₃	C₂×#519	1	1	yes	characteristic
217	C₂×C₁₃	C₂³×M₁₆	1	1	yes	characteristic
218	C₂×C₁₃	C₂×#568	1	1	yes	characteristic
219	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
220	C₂×Q₈	—	2	16	no	—
221	C₂×Q₈	—	2	16	no	—
222	C₂×Q₈	—	2	12	no	—
223	C₂×Q₈	#6099	1	8	yes	complemented
224	C₂×Q₈	C₂²×Dic₁₃	1	12	yes	—
225	C₂×Q₈	C₂²×Dic₁₃	1	4	yes	—
226	C₂²×C₄	—	2	12	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	4	6	no	—
229	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
232	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
233	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
234	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
235	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
236	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
237	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
238	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
239	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
240	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
241	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
242	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
243	M₁₆	—	26	16	no	—
244	C₂×C₈	—	13	16	no	—
245	C₄⋊C₄	C₂²×Dic₁₃	1	24	yes	—
246	C₄⋊C₄	C₂²×Dic₁₃	1	12	yes	—
247	C₄⋊C₄	C₂²×Dic₁₃	1	12	yes	—
248	C₄²	C₂²×Dic₁₃	1	6	yes	—
249	C₄²	C₂²×Dic₁₃	1	6	yes	—
250	C₄²	C₂²×Dic₁₃	1	6	yes	—
251	C₄²	C₂²×Dic₁₃	1	6	yes	—
252	C₄²	C₂²×Dic₁₃	1	6	yes	—
253	C₄²	C₂²×Dic₁₃	1	6	yes	—
254	C₄²	C₂³×D₁₃	1	6	yes	—
255	C₄²	C₂³×D₁₃	1	6	yes	—
256	C₁₃	#60248	1	1	yes	characteristic, complemented, Sylow
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
261	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
262	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
263	Q₈	—	2	24	no	—
264	Q₈	C₂×#6099	1	16	yes	—
265	C₂×C₄	—	2	6	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	12	no	—
268	C₂×C₄	—	2	12	no	—
269	C₂×C₄	—	2	12	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	3	no	—
275	C₂×C₄	—	4	3	no	—
276	C₂×C₄	—	4	6	no	—
277	C₂×C₄	C₂×#6099	1	12	yes	—
278	C₂×C₄	#88254	1	4	yes	—
279	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
280	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
281	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
282	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
283	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
284	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
285	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
286	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
287	C₈	—	26	16	no	—
288	C₂²	—	2	8	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	C₂²×#6099	1	2	yes	—
296	C₂²	#11719517	1	2	yes	—
297	C₂²	#11719524	1	2	yes	—
298	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
299	C₄	—	2	6	no	—
300	C₄	—	2	6	no	—
301	C₄	—	2	6	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	C₂²×#6099	1	12	yes	—
305	C₄	#11719517	1	2	yes	—
306	C₄	C₂×#88254	1	2	yes	—
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	C₂×#11719517	1	1	yes	characteristic
311	C₂	C₂³×#6099	1	1	yes	characteristic
312	C₂	C₂×#11719524	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₁₃
commutator, socle	C₂²×C₁₃	C₂⁴×C₄
Fitting	C₂²×C₄×Q₈×C₁₃	C₂
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₁₃
hypocenter	C₁₃	#60248
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #13123925	C₂	semidirect
2. #13123921	C₂	semidirect
3. #13123917	C₂	semidirect
4. Q₈×#6099	C₂	semidirect
19. #11718685	C₂²	semidirect
20. Q₈×#1055	C₂²	semidirect
68. C₂×Q₈×C₁₃	M₁₆	semidirect
223. C₂×Q₈	#6099	semidirect
256. C₁₃	#60248	semidirect

Maximal subgroups

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

subgroup	index
#13123925	2
#13123921	2
#13123917	2
Q₈×#6099	2
#13123469	2
C₂²×C₄×Q₈×C₁₃	2
#60248	13