Subgroups — #16781759

Order 3328.

568 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130753	C₂	1	1	yes	maximal, characteristic
2	C₄×#88112	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#1057	C₂	1	2	yes	maximal
4	#13129672	C₂	1	2	yes	maximal
5	C₄×Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×Q₈×Dic₁₃	C₄	1	1	yes	characteristic
8	C₂×Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88112	—	2	1	no	—
10	C₂×C₄⋊#1057	C₂²	1	2	yes	—
11	C₂×C₄×#1057	C₂²	1	2	yes	—
12	C₄⋊C₄×Dic₁₃	—	2	1	no	—
13	C₄⋊C₄×Dic₁₃	—	2	1	no	—
14	C₄⋊C₄×Dic₁₃	C₄	1	1	yes	characteristic
15	C₄⋊C₄×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊Dic₁₃	—	2	1	no	—
17	C₄×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
18	C₄²×Dic₁₃	—	2	1	no	—
19	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
20	Q₈⋊#1057	—	2	4	no	—
21	#11718290	—	2	1	no	—
22	#11718290	—	2	1	no	—
23	#11718290	—	2	1	no	—
24	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1057	—	2	2	no	—
26	C₂²×#1057	—	2	2	no	—
27	C₄×Q₈×C₁₃	—	2	2	no	—
28	C₄×Q₈×C₁₃	—	2	2	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
30	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	—	2	1	no	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	Q₈×Dic₁₃	—	2	2	no	—
34	Q₈×Dic₁₃	—	2	1	no	—
35	Q₈×Dic₁₃	—	2	1	no	—
36	Q₈×Dic₁₃	C₈	1	4	yes	complemented
37	Q₈×Dic₁₃	D₄	1	2	yes	—
38	Q₈×Dic₁₃	Q₈	1	2	yes	—
39	C₂×C₄⋊Dic₁₃	—	2	1	no	—
40	C₂×C₄⋊Dic₁₃	—	2	1	no	—
41	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₁₃	—	2	1	no	—
44	C₂×C₄×Dic₁₃	—	2	1	no	—
45	C₂×C₄×Dic₁₃	—	2	1	no	—
46	C₂×C₄×Dic₁₃	—	2	1	no	—
47	C₂×C₄×Dic₁₃	—	2	1	no	—
48	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
50	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
51	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
52	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
53	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
54	#88112	—	2	4	no	—
55	C₄⋊#1057	—	2	2	no	—
56	C₄⋊#1057	—	2	2	no	—
57	C₄×#1057	—	2	2	no	—
58	C₄×#1057	—	2	2	no	—
59	#88071	—	2	1	no	—
60	#88071	—	2	1	no	—
61	#88071	—	2	1	no	—
62	#8566	—	13	1	no	maximal, Sylow
63	C₂×Q₈×C₁₃	—	2	1	no	—
64	C₂×Q₈×C₁₃	—	2	1	no	—
65	C₂×Q₈×C₁₃	—	2	1	no	—
66	C₂×Q₈×C₁₃	C₄⋊C₄	1	2	yes	—
67	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
68	C₂×Q₈×C₁₃	M₁₆	1	2	yes	—
69	C₂²×C₄×C₁₃	—	2	1	no	—
70	C₂²×C₄×C₁₃	—	2	1	no	—
71	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×Dic₁₃	—	2	1	no	—
75	C₂²×Dic₁₃	—	2	1	no	—
76	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×#1057	—	2	4	no	—
81	C₂×#1057	—	4	2	no	—
82	C₂×#1057	—	4	2	no	—
83	C₂×#1057	—	4	2	no	—
84	C₂×#1057	—	4	2	no	—
85	C₂×#1057	—	4	2	no	—
86	C₁₃×C₄⋊C₄	—	2	4	no	—
87	C₁₃×C₄⋊C₄	—	2	1	no	—
88	C₁₃×C₄⋊C₄	M₁₆	1	2	yes	—
89	C₄²×C₁₃	—	2	4	no	—
90	C₄²×C₁₃	—	2	1	no	—
91	C₄²×C₁₃	—	2	1	no	—
92	C₄⋊Dic₁₃	—	2	1	no	—
93	C₄⋊Dic₁₃	—	2	1	no	—
94	C₄⋊Dic₁₃	—	2	4	no	—
95	C₄⋊Dic₁₃	—	2	2	no	—
96	C₄⋊Dic₁₃	—	2	2	no	—
97	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
98	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
99	C₄×Dic₁₃	—	2	2	no	—
100	C₄×Dic₁₃	—	2	4	no	—
101	C₄×Dic₁₃	—	2	2	no	—
102	C₄×Dic₁₃	—	2	2	no	—
103	C₄×Dic₁₃	—	2	2	no	—
104	C₄×Dic₁₃	—	2	2	no	—
105	C₄×Dic₁₃	—	4	1	no	—
106	C₄×Dic₁₃	—	4	2	no	—
107	C₄×Dic₁₃	—	4	2	no	—
108	C₄×Dic₁₃	—	4	2	no	—
109	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
110	C₄×Dic₁₃	M₁₆	1	2	yes	—
111	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
113	C₄×Dic₁₃	#41	1	1	yes	characteristic
114	C₄×Dic₁₃	#41	1	1	yes	characteristic
115	C₄²×Q₈	—	13	1	no	—
116	#1831	—	13	1	no	—
117	C₄×C₄⋊C₈	—	13	1	no	—
118	C₂×Q₈⋊C₈	—	13	2	no	—
119	#1279	—	13	2	no	—
120	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	Q₈×C₁₃	—	2	2	no	—
122	Q₈×C₁₃	—	2	2	no	—
123	Q₈×C₁₃	—	2	2	no	—
124	Q₈×C₁₃	C₄⋊C₈	1	4	yes	complemented
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	1	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	—	4	1	no	—
135	C₂×C₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	#104	1	2	yes	—
137	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
143	C₂×Dic₁₃	—	2	1	no	—
144	C₂×Dic₁₃	—	2	1	no	—
145	C₂×Dic₁₃	—	2	1	no	—
146	C₂×Dic₁₃	—	2	1	no	—
147	C₂×Dic₁₃	—	2	1	no	—
148	C₂×Dic₁₃	—	2	1	no	—
149	C₂×Dic₁₃	—	2	2	no	—
150	C₂×Dic₁₃	—	2	2	no	—
151	C₂×Dic₁₃	—	4	1	no	—
152	C₂×Dic₁₃	—	4	1	no	—
153	C₂×Dic₁₃	—	4	1	no	—
154	C₂×Dic₁₃	—	4	1	no	—
155	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
156	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
157	C₂×Dic₁₃	#121	1	1	yes	characteristic
158	C₂×Dic₁₃	#121	1	1	yes	characteristic
159	C₂×Dic₁₃	C₂²⋊Q₈	1	1	yes	characteristic
160	C₂×Dic₁₃	C₂²⋊Q₈	1	1	yes	characteristic
161	C₂×Dic₁₃	#135	1	2	yes	—
162	C₂×Dic₁₃	#137	1	2	yes	—
163	#1057	—	4	4	no	—
164	#1057	—	8	2	no	—
165	C₂×C₄×Q₈	—	13	1	no	—
166	C₂×C₄×Q₈	—	13	1	no	—
167	C₂×C₄×Q₈	#261	1	1	yes	characteristic
168	C₂×C₄⋊C₈	—	13	2	no	—
169	C₂×C₄⋊C₈	—	26	1	no	—
170	C₂×C₄×C₈	—	13	2	no	—
171	C₄×C₄⋊C₄	—	13	1	no	—
172	C₄×C₄⋊C₄	—	13	1	no	—
173	C₄×C₄⋊C₄	—	13	1	no	—
174	C₄×C₄⋊C₄	—	26	1	no	—
175	C₄×C₄⋊C₄	—	26	1	no	—
176	C₄×C₄⋊C₄	—	26	1	no	—
177	C₄³	—	13	1	no	—
178	C₄³	—	26	1	no	—
179	#336	—	26	1	no	—
180	#336	—	26	1	no	—
181	#336	—	26	1	no	—
182	Q₈⋊C₈	—	26	4	no	—
183	C₂²×C₁₃	#380	1	1	yes	characteristic
184	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
185	C₂²×C₁₃	#407	1	1	yes	characteristic
186	C₂²×C₁₃	#416	1	1	yes	characteristic
187	C₂²×C₁₃	#419	1	1	yes	characteristic
188	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
189	C₂²×C₁₃	#421	1	1	yes	characteristic
190	C₄×C₁₃	—	2	4	no	—
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	4	1	no	—
193	C₄×C₁₃	—	4	2	no	—
194	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
195	C₄×C₁₃	#423	1	1	yes	characteristic
196	C₄×C₁₃	#431	1	1	yes	characteristic
197	C₄×C₁₃	#432	1	1	yes	characteristic
198	Dic₁₃	—	2	1	no	—
199	Dic₁₃	—	2	1	no	—
200	Dic₁₃	—	2	2	no	—
201	Dic₁₃	—	2	2	no	—
202	Dic₁₃	—	4	2	no	—
203	Dic₁₃	—	4	2	no	—
204	Dic₁₃	#464	1	2	yes	—
205	Dic₁₃	#479	1	2	yes	—
206	C₂²×Q₈	C₂×#261	1	1	yes	characteristic
207	C₂²×C₈	—	26	2	no	—
208	C₂²×C₈	—	26	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	13	4	no	—
212	C₄×Q₈	—	13	2	no	—
213	C₄×Q₈	—	13	2	no	—
214	C₄×Q₈	—	26	2	no	—
215	C₄×Q₈	—	26	1	no	—
216	C₄×Q₈	—	26	1	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	13	1	no	—
219	C₂×C₄⋊C₄	—	13	1	no	—
220	C₂×C₄⋊C₄	—	26	1	no	—
221	C₂×C₄⋊C₄	—	26	1	no	—
222	C₂×C₄⋊C₄	C₂×#261	1	1	yes	characteristic
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	—	13	1	no	—
225	C₂×C₄²	—	13	1	no	—
226	C₂×C₄²	—	13	1	no	—
227	C₂×C₄²	—	13	1	no	—
228	C₂×C₄²	—	13	1	no	—
229	C₂×C₄²	—	13	1	no	—
230	C₂×C₄²	—	26	1	no	—
231	C₂×C₄²	—	26	1	no	—
232	C₂×C₄²	—	26	1	no	—
233	C₂×C₄²	—	26	1	no	—
234	C₂×C₄²	—	26	1	no	—
235	C₂×C₄²	C₂×#261	1	1	yes	characteristic
236	C₄⋊C₈	—	26	2	no	—
237	C₄⋊C₈	—	26	2	no	—
238	C₄⋊C₈	—	26	4	no	—
239	C₄×C₈	—	26	2	no	—
240	C₄×C₈	—	26	2	no	—
241	#95	—	26	1	no	—
242	#95	—	26	1	no	—
243	#95	—	26	1	no	—
244	C₂×C₁₃	#1476	1	1	yes	characteristic
245	C₂×C₁₃	#1479	1	1	yes	characteristic
246	C₂×C₁₃	#1488	1	1	yes	characteristic
247	C₂×C₁₃	#1491	1	1	yes	characteristic
248	C₂×C₁₃	#1797	1	1	yes	characteristic
249	C₂×C₁₃	#1805	1	1	yes	characteristic
250	C₂×C₁₃	#1806	1	1	yes	characteristic
251	C₂×Q₈	—	2	1	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	D₁₃⋊C₈	1	2	yes	—
255	C₂×Q₈	C₄⋊#261	1	2	yes	—
256	C₂×Q₈	#6122	1	2	yes	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	13	1	no	—
260	C₂²×C₄	—	13	1	no	—
261	C₂²×C₄	—	13	1	no	—
262	C₂²×C₄	—	13	1	no	—
263	C₂²×C₄	—	26	1	no	—
264	C₂²×C₄	—	26	1	no	—
265	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
266	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
267	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
268	C₂×C₈	—	26	4	no	—
269	C₂×C₈	—	52	2	no	—
270	C₂×C₈	—	52	2	no	—
271	C₂×C₈	—	52	2	no	—
272	C₂×C₈	—	52	2	no	—
273	C₂×C₈	—	52	2	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	13	2	no	—
277	C₄⋊C₄	—	13	2	no	—
278	C₄⋊C₄	—	26	4	no	—
279	C₄⋊C₄	—	26	2	no	—
280	C₄⋊C₄	—	26	2	no	—
281	C₄⋊C₄	—	26	1	no	—
282	C₄⋊C₄	—	26	1	no	—
283	C₄⋊C₄	#6122	1	2	yes	—
284	C₄²	—	2	4	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	13	2	no	—
288	C₄²	—	13	2	no	—
289	C₄²	—	13	1	no	—
290	C₄²	—	13	1	no	—
291	C₄²	—	13	1	no	—
292	C₄²	—	13	1	no	—
293	C₄²	—	26	2	no	—
294	C₄²	—	26	2	no	—
295	C₄²	—	26	4	no	—
296	C₄²	—	26	2	no	—
297	C₄²	—	26	2	no	—
298	C₄²	—	26	2	no	—
299	C₄²	—	52	1	no	—
300	C₄²	—	52	2	no	—
301	C₄²	—	52	2	no	—
302	C₄²	—	52	2	no	—
303	C₁₃	#8566	1	1	yes	characteristic, complemented, Sylow
304	C₂³	C₂×C₂²⋊#261	1	1	yes	characteristic
305	Q₈	—	2	2	no	—
306	Q₈	—	2	2	no	—
307	Q₈	—	2	2	no	—
308	Q₈	#88112	1	4	yes	complemented
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	13	1	no	—
321	C₂×C₄	—	13	1	no	—
322	C₂×C₄	—	13	2	no	—
323	C₂×C₄	—	13	2	no	—
324	C₂×C₄	—	13	2	no	—
325	C₂×C₄	—	13	2	no	—
326	C₂×C₄	—	13	1	no	—
327	C₂×C₄	—	13	1	no	—
328	C₂×C₄	—	26	1	no	—
329	C₂×C₄	—	26	1	no	—
330	C₂×C₄	—	26	1	no	—
331	C₂×C₄	—	26	1	no	—
332	C₂×C₄	—	26	1	no	—
333	C₂×C₄	—	26	1	no	—
334	C₂×C₄	—	26	2	no	—
335	C₂×C₄	—	26	2	no	—
336	C₂×C₄	—	52	1	no	—
337	C₂×C₄	—	52	1	no	—
338	C₂×C₄	—	52	1	no	—
339	C₂×C₄	—	52	1	no	—
340	C₂×C₄	#88118	1	2	yes	—
341	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
342	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
343	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
344	C₂×C₄	#88237	1	1	yes	characteristic
345	C₂×C₄	C₂×#6122	1	1	yes	characteristic
346	C₂×C₄	C₂×#6122	1	1	yes	characteristic
347	C₈	—	52	4	no	—
348	C₈	—	104	2	no	—
349	C₂²	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
350	C₂²	#11719136	1	1	yes	characteristic
351	C₂²	#11719153	1	1	yes	characteristic
352	C₂²	C₂×#88118	1	1	yes	characteristic
353	C₂²	#11719165	1	1	yes	characteristic
354	C₂²	#11719166	1	1	yes	characteristic
355	C₂²	#11719169	1	1	yes	characteristic
356	C₄	—	2	2	no	—
357	C₄	—	2	4	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	2	no	—
360	C₄	—	13	2	no	—
361	C₄	—	13	2	no	—
362	C₄	—	26	2	no	—
363	C₄	—	26	2	no	—
364	C₄	—	26	1	no	—
365	C₄	—	26	1	no	—
366	C₄	—	52	2	no	—
367	C₄	—	52	2	no	—
368	C₄	C₂×#88112	1	1	yes	characteristic
369	C₄	#11719125	1	1	yes	characteristic
370	C₄	#11719128	1	1	yes	characteristic
371	C₄	#11719129	1	1	yes	characteristic
372	C₂	#13130130	1	1	yes	characteristic
373	C₂	#13130139	1	1	yes	characteristic
374	C₂	#13130167	1	1	yes	characteristic
375	C₂	#13130178	1	1	yes	characteristic
376	C₂	#13130692	1	1	yes	characteristic
377	C₂	#13130707	1	1	yes	characteristic
378	C₂	#13130709	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₄×Q₈×C₁₃	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×C₄×Q₈	#261
hypocenter	C₁₃	#8566
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
36. Q₈×Dic₁₃	C₈	semidirect
124. Q₈×C₁₃	C₄⋊C₈	semidirect
303. C₁₃	#8566	semidirect
308. Q₈	#88112	semidirect

Maximal subgroups

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

subgroup	index
#13130753	2
C₄×#88112	2
C₂×Q₈⋊#1057	2
#13129672	2
C₄×Q₈×Dic₁₃	2
#8566	13