Subgroups — #15831962

Order 3328.

666 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	#13123953	C₂	1	2	yes	maximal
2	#13113603	C₂	1	4	yes	maximal, complemented
3	#13113599	C₂	1	2	yes	maximal, complemented
4	#13113592	C₂	1	2	yes	maximal, complemented
5	#13113588	C₂	1	2	yes	maximal, complemented
6	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718049	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
10	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
11	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
12	C₁₃×#405	C₄	1	1	yes	characteristic
13	C₁₃×#405	C₄	1	2	yes	—
14	C₁₃×#405	C₄	1	2	yes	—
15	C₁₃×#405	C₄	1	1	yes	characteristic
16	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
17	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
20	#11718685	—	2	2	no	—
21	#11718673	—	2	2	no	—
22	#11718672	—	2	4	no	—
23	#11718587	C₂²	1	4	yes	—
24	#11718586	C₂²	1	2	yes	—
25	#11718585	C₂²	1	4	yes	—
26	C₂×C₄⋊#1055	C₂²	1	2	yes	—
27	#11718050	C₂²	1	2	yes	—
28	#11718050	C₂²	1	4	yes	—
29	#11718049	C₂²	1	2	yes	—
30	#11718049	C₂²	1	4	yes	—
31	#11718049	C₂²	1	2	yes	—
32	#11718049	C₂²	1	1	yes	characteristic
33	#11718049	C₂²	1	1	yes	characteristic
34	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
35	C₁₃×#131	—	2	4	no	—
36	C₁₃×#131	—	2	4	no	—
37	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
40	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
46	C₄×Q₈×C₁₃	Q₈	1	1	yes	characteristic
47	C₄×D₄×C₁₃	D₄	1	2	yes	—
48	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
49	C₄×D₄×C₁₃	Q₈	1	2	yes	—
50	C₄×D₄×C₁₃	Q₈	1	1	yes	characteristic
51	C₁₃×#117	C₂×C₄	1	2	yes	—
52	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
55	C₂×#6099	—	2	4	no	—
56	C₂²×#1055	—	2	2	no	—
57	C₁₃×#97	C₂×C₄	1	2	yes	—
58	C₁₃×#97	C₂×C₄	1	2	yes	—
59	C₁₃×#97	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₁₃	C₂³	1	2	yes	—
67	C₂²⋊#1055	—	2	2	no	—
68	C₂²⋊#1055	—	2	4	no	—
69	C₄⋊#1055	—	2	4	no	—
70	C₄⋊#1055	—	2	2	no	—
71	C₄⋊#1055	C₂³	1	4	yes	—
72	C₄⋊#1055	C₂³	1	2	yes	—
73	C₄⋊#1055	C₂³	1	2	yes	—
74	#88043	—	2	4	no	—
75	C₄×#1055	—	2	2	no	—
76	#17798	—	13	1	no	maximal, Sylow
77	C₁₃×#41	—	2	2	no	—
78	C₁₃×#41	—	2	2	no	—
79	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂×D₄×C₁₃	C₄⋊C₄	1	2	yes	—
81	C₂×D₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
87	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
88	C₁₃×M₁₆	—	2	2	no	—
89	C₁₃×M₁₆	—	2	4	no	—
90	C₁₃×M₁₆	—	2	4	no	—
91	C₁₃×M₁₆	M₁₆	1	4	yes	complemented
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	4	no	—
94	C₂×C₈×C₁₃	—	2	4	no	—
95	C₂×C₈×C₁₃	M₁₆	1	4	yes	complemented
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	2	yes	—
99	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
103	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
104	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
105	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
106	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
107	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
108	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
109	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₃	C₂×D₄	1	2	yes	—
113	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
114	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
115	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
116	#6099	—	8	4	no	—
117	C₂×#1055	—	2	2	no	—
118	C₂×#1055	—	2	4	no	—
119	C₂×#1055	—	2	2	no	—
120	C₂×#1055	—	8	2	no	—
121	#2989	—	13	2	no	—
122	C₄×#131	D₁₃	1	1	yes	characteristic
123	#1569	—	13	4	no	—
124	#1568	—	13	2	no	—
125	#1566	—	13	2	no	—
126	#1565	—	13	2	no	—
127	#1564	—	13	1	no	—
128	C₂×#334	—	13	1	no	—
129	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	Q₈×C₁₃	—	2	2	no	—
132	D₄×C₁₃	—	2	4	no	—
133	D₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	4	no	—
138	C₂×C₄×C₁₃	—	4	1	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
145	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
147	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
148	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
151	C₈×C₁₃	—	2	1	no	—
152	C₈×C₁₃	—	2	1	no	—
153	C₈×C₁₃	—	2	2	no	—
154	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
155	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
156	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
157	#1055	—	4	2	no	—
158	#1055	—	4	4	no	—
159	#1055	—	4	2	no	—
160	C₂×#131	Dic₁₃	1	1	yes	characteristic
161	C₂×#131	Dic₁₃	1	1	yes	characteristic
162	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
163	#446	—	26	2	no	—
164	#436	—	26	2	no	—
165	#435	—	26	4	no	—
166	#432	—	13	4	no	—
167	#431	—	13	2	no	—
168	#423	—	13	4	no	—
169	C₂×C₄⋊C₈	—	13	2	no	—
170	#405	Dic₁₃	1	1	yes	characteristic
171	#405	Dic₁₃	1	2	yes	—
172	#405	Dic₁₃	1	2	yes	—
173	#405	Dic₁₃	1	1	yes	characteristic
174	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
175	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
176	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
177	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
178	#335	—	13	2	no	—
179	#335	—	13	4	no	—
180	#334	—	13	2	no	—
181	#334	—	13	2	no	—
182	#334	—	13	1	no	—
183	#334	—	13	1	no	—
184	#334	—	13	4	no	—
185	C₂²×C₁₃	—	2	1	no	—
186	C₂²×C₁₃	—	2	1	no	—
187	C₂²×C₁₃	—	2	2	no	—
188	C₂²×C₁₃	—	2	2	no	—
189	C₂²×C₁₃	—	2	2	no	—
190	C₂²×C₁₃	#423	1	2	yes	—
191	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	2	1	no	—
194	C₄×C₁₃	—	2	2	no	—
195	C₄×C₁₃	—	2	1	no	—
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	2	no	—
198	C₄×C₁₃	#423	1	2	yes	—
199	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
200	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
201	C₄×C₁₃	#577	1	2	yes	—
202	C₂×#41	C₂×Dic₁₃	1	1	yes	characteristic
203	#131	—	2	4	no	—
204	#131	—	2	4	no	—
205	C₂×M₁₆	—	26	4	no	—
206	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
207	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
208	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
209	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
210	C₂²×C₈	—	26	2	no	—
211	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
212	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
213	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
214	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
215	C₄×Q₈	Dic₂₆	1	1	yes	characteristic
216	C₄×Q₈	D₅₂	1	1	yes	characteristic
217	C₄×D₄	Dic₂₆	1	2	yes	—
218	C₄×D₄	Dic₂₆	1	1	yes	characteristic
219	C₄×D₄	D₅₂	1	2	yes	—
220	C₄×D₄	D₅₂	1	1	yes	characteristic
221	#117	C₂×Dic₁₃	1	2	yes	—
222	#117	C₂×Dic₁₃	1	1	yes	characteristic
223	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
224	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
225	C₄⋊C₈	—	13	4	no	—
226	C₄⋊C₈	—	13	2	no	—
227	C₄⋊C₈	—	13	2	no	—
228	C₄⋊C₈	—	26	2	no	—
229	C₄⋊C₈	—	26	4	no	—
230	C₂²⋊C₈	—	26	2	no	—
231	C₂²⋊C₈	—	26	4	no	—
232	#97	—	26	4	no	—
233	#97	C₂×Dic₁₃	1	2	yes	—
234	#97	C₂×Dic₁₃	1	2	yes	—
235	#97	C₂×Dic₁₃	1	2	yes	—
236	C₄×C₈	—	26	2	no	—
237	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
238	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
239	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
240	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
241	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₁₃	1	2	yes	—
244	C₂×C₁₃	—	2	2	no	—
245	C₂×C₁₃	—	4	2	no	—
246	C₂×C₁₃	C₂×#423	1	1	yes	characteristic
247	C₂×C₁₃	#2912	1	1	yes	characteristic
248	C₂×C₁₃	#2916	1	1	yes	characteristic
249	#41	—	2	2	no	—
250	#41	—	2	2	no	—
251	C₂×Q₈	C₄⋊Dic₁₃	1	1	yes	characteristic
252	C₂×D₄	C₄⋊Dic₁₃	1	2	yes	—
253	C₂×D₄	C₄⋊Dic₁₃	1	1	yes	characteristic
254	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
255	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₅₂	1	2	yes	—
258	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
259	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
260	M₁₆	—	2	2	no	—
261	M₁₆	—	2	4	no	—
262	M₁₆	—	2	4	no	—
263	M₁₆	—	104	4	no	—
264	M₁₆	#6099	1	4	yes	complemented
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	—	26	2	no	—
269	C₂×C₈	—	26	2	no	—
270	C₂×C₈	—	26	4	no	—
271	C₂×C₈	—	104	2	no	—
272	C₂×C₈	#6099	1	4	yes	complemented
273	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
274	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
275	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
276	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
277	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
278	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
279	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
280	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
281	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
282	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
283	C₂²⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
284	C₂²⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
285	C₂²⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
286	C₄²	C₂×Dic₂₆	1	2	yes	—
287	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
288	C₄²	C₂×D₅₂	1	1	yes	characteristic
289	C₄²	C₂×D₅₂	1	2	yes	—
290	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
291	C₄²	C₂²×Dic₁₃	1	2	yes	—
292	C₄²	C₂³×D₁₃	1	1	yes	characteristic
293	C₁₃	#17798	1	1	yes	characteristic, complemented, Sylow
294	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
295	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
296	Q₈	—	2	2	no	—
297	D₄	—	2	4	no	—
298	D₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	C₂×#6099	1	2	yes	—
306	C₂×C₄	C₂×#6099	1	2	yes	—
307	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
308	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
309	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
310	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
311	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
312	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
313	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
315	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	2	no	—
319	C₈	—	52	2	no	—
320	C₈	—	52	4	no	—
321	C₈	—	52	2	no	—
322	C₈	C₂×#6099	1	2	yes	—
323	C₈	C₂×#6099	1	4	yes	—
324	C₈	C₂×#6099	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	#11718585	1	2	yes	—
331	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	2	no	—
338	C₄	#11718585	1	2	yes	—
339	C₄	#11719456	1	2	yes	—
340	C₄	C₂²×#6099	1	1	yes	characteristic
341	C₄	C₂²×#6099	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	4	2	no	—
344	C₂	C₂×#11718585	1	1	yes	characteristic
345	C₂	#13123431	1	1	yes	characteristic
346	C₂	#13123447	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₄×C₁₃×#131	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₄×#131	D₁₃
hypocenter	C₁₃	#17798
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #13113603	C₂	semidirect
3. #13113599	C₂	semidirect
4. #13113592	C₂	semidirect
5. #13113588	C₂	semidirect
6. #13113584	C₂	semidirect
7. C₂×#11718049	C₂	semidirect
91. C₁₃×M₁₆	M₁₆	semidirect
95. C₂×C₈×C₁₃	M₁₆	semidirect
264. M₁₆	#6099	semidirect
272. C₂×C₈	#6099	semidirect
293. C₁₃	#17798	semidirect

Maximal subgroups

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

subgroup	index
#13123953	2
#13113603	2
#13113599	2
#13113592	2
#13113588	2
#13113584	2
C₂×#11718049	2
C₄×C₁₃×#131	2
#17798	13