Subgroups — #16799413

Order 3328.

814 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131005	C₂	1	2	yes	maximal
2	#13130933	C₂	1	2	yes	maximal
3	#13130889	C₂	1	4	yes	maximal
4	C₄×C₄⋊#1057	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊#1057	C₂	1	2	yes	maximal
6	#13130550	C₂	1	2	yes	maximal
7	C₄²×#1057	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
10	C₂×Q₈×Dic₁₃	C₄	1	1	yes	characteristic
11	C₂×Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#1057	C₂²	1	4	yes	—
13	C₂×C₄⋊#1057	C₂²	1	2	yes	—
14	C₂×C₄×#1057	C₂²	1	2	yes	—
15	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
16	C₂×C₄×#1057	C₂²	1	2	yes	—
17	C₂×C₄×#1057	C₂²	1	2	yes	—
18	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×Dic₁₃	C₄	1	2	yes	—
20	C₄⋊C₄×Dic₁₃	C₄	1	1	yes	characteristic
21	C₄⋊C₄×Dic₁₃	C₂²	1	2	yes	—
22	C₄⋊C₄×Dic₁₃	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
24	C₄×C₄⋊Dic₁₃	C₄	1	2	yes	—
25	C₄²×Dic₁₃	C₂²	1	2	yes	—
26	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
27	#11718290	C₂²	1	4	yes	—
28	#11718290	C₂²	1	4	yes	—
29	#11718290	C₂²	1	2	yes	—
30	#11718290	C₂²	1	4	yes	—
31	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1057	C₂³	1	4	yes	—
33	C₂²×#1057	C₂³	1	2	yes	—
34	C₂²×#1057	C₂³	1	2	yes	—
35	C₄×Q₈×C₁₃	C₈	1	8	yes	complemented
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	Q₈×Dic₁₃	—	2	4	no	—
41	Q₈×Dic₁₃	—	2	4	no	—
42	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
50	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
53	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
55	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
56	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
57	C₄⋊#1057	—	2	8	no	—
58	C₄⋊#1057	—	2	4	no	—
59	C₄×#1057	—	2	4	no	—
60	C₄×#1057	—	2	4	no	—
61	C₄×#1057	—	2	2	no	—
62	C₄×#1057	—	2	4	no	—
63	C₄×#1057	Q₈	1	4	yes	complemented
64	#88071	C₂×C₄	1	2	yes	—
65	#88071	C₂×C₄	1	2	yes	—
66	#88071	C₂×C₄	1	2	yes	—
67	#21339	—	13	1	no	maximal, Sylow
68	C₂×Q₈×C₁₃	—	2	2	no	—
69	C₂×Q₈×C₁₃	—	2	2	no	—
70	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
77	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
78	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×Dic₁₃	C₂⁴	1	1	yes	characteristic
82	C₂×#1057	—	2	2	no	—
83	C₂×#1057	—	2	2	no	—
84	C₂×#1057	—	2	2	no	—
85	C₂×#1057	—	2	2	no	—
86	C₂×#1057	—	2	4	no	—
87	C₂×#1057	—	2	4	no	—
88	C₂×#1057	—	2	4	no	—
89	C₂×#1057	C₂×Q₈	1	4	yes	—
90	C₂×#1057	#41	1	4	yes	—
91	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
92	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
93	C₄²×C₁₃	C₂×C₈	1	8	yes	—
94	C₄²×C₁₃	C₂×C₈	1	4	yes	—
95	C₄⋊Dic₁₃	—	2	2	no	—
96	C₄⋊Dic₁₃	—	2	2	no	—
97	C₄⋊Dic₁₃	—	2	4	no	—
98	C₄⋊Dic₁₃	—	2	4	no	—
99	C₄×Dic₁₃	—	2	2	no	—
100	C₄×Dic₁₃	—	2	4	no	—
101	C₄×Dic₁₃	—	2	4	no	—
102	C₄×Dic₁₃	—	2	2	no	—
103	C₄×Dic₁₃	—	2	4	no	—
104	C₄×Dic₁₃	—	2	2	no	—
105	C₄×Dic₁₃	—	2	4	no	—
106	C₄×Dic₁₃	—	2	4	no	—
107	C₄×Dic₁₃	M₁₆	1	4	yes	—
108	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
109	C₄×Dic₁₃	#41	1	4	yes	—
110	C₄×Dic₁₃	#41	1	2	yes	—
111	C₄²×Q₈	—	13	1	no	—
112	#1984	—	13	2	no	—
113	#1939	—	13	2	no	—
114	#1916	—	13	4	no	—
115	C₄×C₄⋊C₈	—	13	1	no	—
116	C₄×C₄⋊C₈	—	13	2	no	—
117	#1744	—	13	2	no	—
118	C₄²×C₈	—	13	1	no	—
119	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
120	Q₈×C₁₃	—	2	8	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	#117	1	2	yes	—
128	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
130	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
132	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
133	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
135	C₂×C₄×C₁₃	#131	1	2	yes	—
136	C₂×Dic₁₃	—	2	1	no	—
137	C₂×Dic₁₃	—	2	1	no	—
138	C₂×Dic₁₃	—	2	1	no	—
139	C₂×Dic₁₃	—	2	1	no	—
140	C₂×Dic₁₃	—	2	2	no	—
141	C₂×Dic₁₃	—	2	2	no	—
142	C₂×Dic₁₃	—	2	2	no	—
143	C₂×Dic₁₃	—	2	2	no	—
144	C₂×Dic₁₃	—	2	2	no	—
145	C₂×Dic₁₃	—	2	2	no	—
146	C₂×Dic₁₃	#117	1	2	yes	—
147	C₂×Dic₁₃	C₄×Q₈	1	2	yes	—
148	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
149	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
150	C₂×Dic₁₃	#131	1	2	yes	—
151	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
152	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
153	C₂×Dic₁₃	C₂×#41	1	2	yes	—
154	C₂×Dic₁₃	C₂×#41	1	2	yes	—
155	#1057	—	2	4	no	—
156	#1057	—	2	4	no	—
157	#1057	—	4	4	no	—
158	C₂×C₄×Q₈	—	13	1	no	—
159	C₂×C₄×Q₈	—	13	1	no	—
160	C₂×C₄×Q₈	#261	1	1	yes	characteristic
161	C₂×C₄⋊C₈	—	13	4	no	—
162	C₂×C₄⋊C₈	—	13	2	no	—
163	C₂×C₄×C₈	—	13	1	no	—
164	C₂×C₄×C₈	—	13	2	no	—
165	C₂×C₄×C₈	—	13	2	no	—
166	C₂×C₄×C₈	—	13	1	no	—
167	C₂×C₄×C₈	—	13	2	no	—
168	C₄×C₄⋊C₄	—	13	2	no	—
169	C₄×C₄⋊C₄	—	13	1	no	—
170	C₄×C₄⋊C₄	—	13	1	no	—
171	C₄×C₄⋊C₄	—	13	2	no	—
172	C₄×C₄⋊C₄	—	13	2	no	—
173	C₄×C₄⋊C₄	—	13	1	no	—
174	C₄³	—	13	1	no	—
175	C₄³	—	13	2	no	—
176	#336	—	13	2	no	—
177	#336	—	13	4	no	—
178	#336	—	13	4	no	—
179	#336	—	13	4	no	—
180	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
181	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
182	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
183	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
184	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
185	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
186	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
187	C₄×C₁₃	—	2	4	no	—
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	2	4	no	—
190	C₄×C₁₃	#431	1	4	yes	—
191	C₄×C₁₃	C₈×Q₈	1	4	yes	—
192	Dic₁₃	—	2	2	no	—
193	Dic₁₃	—	2	2	no	—
194	Dic₁₃	—	4	2	no	—
195	Dic₁₃	—	4	2	no	—
196	Dic₁₃	#446	1	4	yes	—
197	Dic₁₃	#529	1	2	yes	—
198	Dic₁₃	#533	1	2	yes	—
199	C₂²×Q₈	C₂×#261	1	1	yes	characteristic
200	C₂²×C₈	—	13	2	no	—
201	C₂²×C₈	—	13	2	no	—
202	C₂²×C₈	—	13	4	no	—
203	C₄×Q₈	—	26	4	no	—
204	C₄×Q₈	—	26	4	no	—
205	C₄×Q₈	#1057	1	8	yes	complemented
206	C₂×C₄⋊C₄	—	13	2	no	—
207	C₂×C₄⋊C₄	—	13	2	no	—
208	C₂×C₄⋊C₄	—	13	1	no	—
209	C₂×C₄⋊C₄	—	13	1	no	—
210	C₂×C₄⋊C₄	C₂×#261	1	2	yes	—
211	C₂×C₄⋊C₄	C₂×#261	1	1	yes	characteristic
212	C₂×C₄²	—	13	1	no	—
213	C₂×C₄²	—	13	1	no	—
214	C₂×C₄²	—	13	1	no	—
215	C₂×C₄²	—	13	2	no	—
216	C₂×C₄²	—	13	2	no	—
217	C₂×C₄²	—	13	1	no	—
218	C₂×C₄²	—	13	2	no	—
219	C₂×C₄²	—	13	1	no	—
220	C₂×C₄²	—	13	2	no	—
221	C₂×C₄²	—	13	1	no	—
222	C₂×C₄²	—	13	2	no	—
223	C₂×C₄²	C₂×#261	1	2	yes	—
224	C₂×C₄²	C₂×#261	1	1	yes	characteristic
225	C₄⋊C₈	—	26	4	no	—
226	C₄⋊C₈	—	26	8	no	—
227	C₄×C₈	—	13	4	no	—
228	C₄×C₈	—	26	4	no	—
229	C₄×C₈	—	26	2	no	—
230	C₄×C₈	—	26	4	no	—
231	C₄×C₈	—	26	4	no	—
232	#95	—	13	2	no	—
233	#95	—	13	2	no	—
234	#95	—	13	2	no	—
235	C₂×C₁₃	#2295	1	1	yes	characteristic
236	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
237	C₂×C₁₃	#2922	1	1	yes	characteristic
238	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
239	C₂×C₁₃	C₂×#446	1	1	yes	characteristic
240	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
241	C₂×C₁₃	#2965	1	1	yes	characteristic
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	C₂×#1057	1	4	yes	—
245	C₂²×C₄	—	13	1	no	—
246	C₂²×C₄	—	13	1	no	—
247	C₂²×C₄	—	13	1	no	—
248	C₂²×C₄	—	13	1	no	—
249	C₂²×C₄	—	13	2	no	—
250	C₂²×C₄	—	13	2	no	—
251	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
252	C₂²×C₄	C₂²×#261	1	2	yes	—
253	C₂²×C₄	C₂²×#261	1	2	yes	—
254	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
255	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
256	C₂×C₈	—	13	4	no	—
257	C₂×C₈	—	13	4	no	—
258	C₂×C₈	—	26	2	no	—
259	C₂×C₈	—	26	2	no	—
260	C₂×C₈	—	26	2	no	—
261	C₂×C₈	—	26	2	no	—
262	C₂×C₈	—	26	4	no	—
263	C₂×C₈	—	26	4	no	—
264	C₂×C₈	—	26	4	no	—
265	C₄⋊C₄	—	26	4	no	—
266	C₄⋊C₄	—	26	4	no	—
267	C₄⋊C₄	—	26	2	no	—
268	C₄⋊C₄	—	26	2	no	—
269	C₄⋊C₄	C₂×#1057	1	8	yes	—
270	C₄⋊C₄	C₂×#1057	1	4	yes	—
271	C₄²	—	13	4	no	—
272	C₄²	—	13	4	no	—
273	C₄²	—	13	2	no	—
274	C₄²	—	13	2	no	—
275	C₄²	—	26	2	no	—
276	C₄²	—	26	4	no	—
277	C₄²	—	26	4	no	—
278	C₄²	—	26	4	no	—
279	C₄²	—	26	2	no	—
280	C₄²	—	26	4	no	—
281	C₄²	—	26	2	no	—
282	C₄²	—	26	4	no	—
283	C₄²	C₂×#1057	1	8	yes	—
284	C₄²	C₂×#1057	1	4	yes	—
285	C₁₃	#21339	1	1	yes	characteristic, complemented, Sylow
286	C₂³	C₂³×#261	1	1	yes	characteristic
287	Q₈	—	2	8	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	13	2	no	—
295	C₂×C₄	—	13	2	no	—
296	C₂×C₄	—	13	1	no	—
297	C₂×C₄	—	13	1	no	—
298	C₂×C₄	—	13	2	no	—
299	C₂×C₄	—	13	2	no	—
300	C₂×C₄	—	13	2	no	—
301	C₂×C₄	—	13	2	no	—
302	C₂×C₄	—	13	2	no	—
303	C₂×C₄	—	26	1	no	—
304	C₂×C₄	—	26	1	no	—
305	C₂×C₄	—	26	1	no	—
306	C₂×C₄	—	26	1	no	—
307	C₂×C₄	—	26	2	no	—
308	C₂×C₄	—	26	2	no	—
309	C₂×C₄	—	26	2	no	—
310	C₂×C₄	—	26	2	no	—
311	C₂×C₄	—	26	2	no	—
312	C₂×C₄	—	26	2	no	—
313	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
314	C₂×C₄	#88237	1	2	yes	—
315	C₂×C₄	#88240	1	2	yes	—
316	C₂×C₄	Q₈×#261	1	2	yes	—
317	C₂×C₄	C₂²×#1057	1	4	yes	—
318	C₂×C₄	C₂²×#1057	1	2	yes	—
319	C₂×C₄	C₂²×#1057	1	4	yes	—
320	C₂×C₄	C₂²×#1057	1	2	yes	—
321	C₂×C₄	C₂²×#1057	1	2	yes	—
322	C₈	—	26	4	no	—
323	C₈	—	26	4	no	—
324	C₈	—	52	4	no	—
325	C₂²	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
326	C₂²	C₂×#88237	1	1	yes	characteristic
327	C₂²	C₂×#88240	1	1	yes	characteristic
328	C₂²	C₂×Q₈×#261	1	1	yes	characteristic
329	C₂²	D₁₃⋊#131	1	1	yes	characteristic
330	C₂²	#41×#261	1	1	yes	characteristic
331	C₂²	C₂³×#1057	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	4	no	—
334	C₄	—	2	4	no	—
335	C₄	—	13	2	no	—
336	C₄	—	13	2	no	—
337	C₄	—	13	4	no	—
338	C₄	—	26	2	no	—
339	C₄	—	26	2	no	—
340	C₄	—	52	2	no	—
341	C₄	—	52	2	no	—
342	C₄	#11719119	1	4	yes	—
343	C₄	Q₈×#1057	1	4	yes	—
344	C₂	#13131295	1	1	yes	characteristic
345	C₂	C₂×#11719119	1	1	yes	characteristic
346	C₂	#13131521	1	1	yes	characteristic
347	C₂	C₂×Q₈×#1057	1	1	yes	characteristic
348	C₂	D₁₃⋊#446	1	1	yes	characteristic
349	C₂	#41×#1057	1	1	yes	characteristic
350	C₂	#13131644	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂³×C₈
Frattini	C₂³	C₂³×#261
Fitting	C₂×C₄×Q₈×C₁₃	C₄
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×Dic₁₃	C₂⁴
hypercenter	C₂×C₄×Q₈	#261
hypocenter	C₁₃	#21339
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
35. C₄×Q₈×C₁₃	C₈	semidirect
63. C₄×#1057	Q₈	semidirect
205. C₄×Q₈	#1057	semidirect
285. C₁₃	#21339	semidirect

Maximal subgroups

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

subgroup	index
#13131005	2
#13130933	2
#13130889	2
C₄×C₄⋊#1057	2
#13130550	2
C₄²×#1057	2
C₄×Q₈×Dic₁₃	2
#21339	13