Subgroups — #16795105

Order 3328.

634 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131536	C₂	1	1	yes	maximal, characteristic
2	#13131482	C₂	1	1	yes	maximal, characteristic
3	#13130414	C₂	1	2	yes	maximal, complemented
4	#13130410	C₂	1	2	yes	maximal, complemented
5	#13130406	C₂	1	1	yes	maximal, characteristic, complemented
6	#13130404	C₂	1	1	yes	maximal, characteristic, complemented
7	#13130401	C₂	1	2	yes	maximal, complemented
8	#13130394	C₂	1	1	yes	maximal, characteristic, complemented
9	#13130388	C₂	1	2	yes	maximal, complemented
10	D₁₃⋊#334	C₂	1	1	yes	maximal, characteristic, complemented
11	#13123801	C₂	1	1	yes	maximal, characteristic
12	C₂×#88240	—	2	1	no	—
13	C₄×#6131	C₂²	1	1	yes	characteristic
14	#11719159	—	2	1	no	—
15	#11719079	—	2	1	no	—
16	#11719079	—	2	2	no	—
17	#11719077	—	2	2	no	—
18	#11719072	—	2	1	no	—
19	#11719054	C₂²	1	2	yes	—
20	#11719052	C₂²	1	1	yes	characteristic
21	#11719052	C₂²	1	1	yes	characteristic
22	#11719052	C₂²	1	2	yes	—
23	#11719051	C₂²	1	2	yes	—
24	D₁₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
25	D₁₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
26	D₁₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
27	C₁₃×#446	C₄	1	1	yes	characteristic
28	#11718685	C₄	1	1	yes	characteristic
29	#11718499	C₄	1	1	yes	characteristic
30	#11718497	C₄	1	2	yes	—
31	#11718494	C₂²	1	1	yes	characteristic
32	#11718494	C₂²	1	2	yes	—
33	#11718368	C₂²	1	2	yes	—
34	#11718367	C₄	1	2	yes	—
35	#11718349	C₄	1	1	yes	characteristic
36	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
37	#11718256	C₂²	1	2	yes	—
38	#11718255	C₂²	1	1	yes	characteristic
39	#11718255	C₂²	1	1	yes	characteristic
40	#11718255	C₂²	1	2	yes	—
41	#11718254	C₂²	1	2	yes	—
42	#11718254	C₂²	1	2	yes	—
43	#11718254	C₂²	1	2	yes	—
44	#11718253	C₂²	1	1	yes	characteristic
45	#11718253	C₂²	1	1	yes	characteristic
46	#11718253	C₂²	1	2	yes	—
47	C₂×#6131	D₄	1	1	yes	characteristic
48	#88240	—	4	4	no	—
49	C₂×#6118	—	2	1	no	—
50	C₂×#6118	—	2	2	no	—
51	C₂×#6118	—	2	1	no	—
52	C₂×D₁₃⋊C₈	—	2	1	no	—
53	C₂×D₁₃⋊C₈	—	2	2	no	—
54	C₂×D₁₃⋊C₈	—	2	1	no	—
55	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
56	Q₈×Dic₁₃	Q₈	1	1	yes	characteristic
57	C₂×#6094	C₂×C₄	1	2	yes	—
58	C₂×#6094	C₂×C₄	1	2	yes	—
59	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
61	#88147	Q₈	1	2	yes	—
62	#88147	Q₈	1	1	yes	characteristic
63	#88146	D₄	1	1	yes	characteristic
64	#88146	D₄	1	2	yes	—
65	C₄×D₅₂	C₂×C₄	1	2	yes	—
66	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
67	C₄²×D₁₃	C₂³	1	1	yes	characteristic
68	C₄²×D₁₃	C₂³	1	2	yes	—
69	#88112	—	2	1	no	—
70	#88112	—	2	2	no	—
71	#88112	—	2	1	no	—
72	#88111	—	2	2	no	—
73	#88111	—	2	2	no	—
74	#88110	—	2	2	no	—
75	C₄⋊#1057	—	2	1	no	—
76	C₄⋊#1057	—	2	2	no	—
77	C₄⋊#1057	—	2	1	no	—
78	C₄⋊#1057	C₂³	1	1	yes	characteristic
79	C₄⋊#1057	C₂³	1	1	yes	characteristic
80	C₄⋊#1057	C₂³	1	2	yes	—
81	C₄⋊#1057	C₂³	1	1	yes	characteristic
82	C₄⋊#1057	C₂³	1	1	yes	characteristic
83	C₄⋊#1057	C₂³	1	2	yes	—
84	C₄×#1057	—	2	2	no	—
85	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
86	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
87	C₁₃×#97	C₂×C₄	1	2	yes	—
88	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
89	#88060	C₂×C₄	1	2	yes	—
90	#88060	C₂×C₄	1	2	yes	—
91	#88060	C₂×C₄	1	2	yes	—
92	C₈×Dic₁₃	C₂³	1	2	yes	—
93	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
94	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
95	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
96	C₄⋊#1055	C₂×C₄	1	2	yes	—
97	#88043	C₂×C₄	1	2	yes	—
98	C₄×#1055	C₂×C₄	1	1	yes	characteristic
99	#17830	—	13	1	no	maximal, Sylow
100	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
101	#6131	—	2	2	no	—
102	#6131	—	2	2	no	—
103	C₂×D₅₂	C₄⋊C₄	1	2	yes	—
104	C₂×D₅₂	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
107	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
108	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×D₁₃	C₂×Q₈	1	2	yes	—
110	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
111	C₂×#1057	—	2	1	no	—
112	C₂×#1057	—	2	2	no	—
113	C₂×#1057	—	2	2	no	—
114	C₂×#1057	—	2	1	no	—
115	C₂×#1057	—	2	1	no	—
116	C₂×#1057	—	2	1	no	—
117	#6118	—	4	4	no	—
118	#6118	—	4	2	no	—
119	#6118	—	4	2	no	—
120	D₁₃⋊C₈	—	4	4	no	—
121	D₁₃⋊C₈	—	4	2	no	—
122	D₁₃⋊C₈	—	4	2	no	—
123	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
125	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
126	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
127	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
128	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
130	#6103	C₄⋊C₄	1	2	yes	—
131	#6103	C₄⋊C₄	1	2	yes	—
132	#6103	C₄⋊C₄	1	2	yes	—
133	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
134	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
135	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
136	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
137	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
138	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
139	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
140	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
141	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
142	C₂×#1055	C₂²×C₄	1	2	yes	—
143	#6094	—	2	4	no	—
144	#6094	—	4	2	no	—
145	C₈×D₁₃	—	2	2	no	—
146	C₈×D₁₃	—	4	1	no	—
147	#2965	—	13	1	no	—
148	#2925	—	13	1	no	—
149	#2906	—	13	1	no	—
150	#1574	—	13	2	no	—
151	#1573	—	13	2	no	—
152	#1571	—	13	1	no	—
153	#1570	—	13	1	no	—
154	#1569	—	13	2	no	—
155	#1567	—	13	1	no	—
156	#1565	—	13	2	no	—
157	C₂×#334	—	13	1	no	—
158	C₂²×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
159	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
160	Q₈×C₁₃	—	2	2	no	—
161	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
162	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
166	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
169	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
170	D₅₂	—	2	4	no	—
171	D₅₂	—	2	2	no	—
172	C₄×D₁₃	—	2	1	no	—
173	C₄×D₁₃	—	2	2	no	—
174	C₄×D₁₃	—	2	1	no	—
175	C₄×D₁₃	—	2	2	no	—
176	C₄×D₁₃	—	2	2	no	—
177	C₄×D₁₃	—	2	2	no	—
178	C₄×D₁₃	—	4	1	no	—
179	C₄×D₁₃	—	4	2	no	—
180	C₄×D₁₃	AGL(1,ℤ/8)	1	2	yes	—
181	C₄×D₁₃	#137	1	2	yes	—
182	#1057	—	4	1	no	—
183	#1057	—	4	2	no	—
184	#1057	—	4	1	no	—
185	#1057	—	4	2	no	—
186	#1057	—	4	1	no	—
187	#1057	—	4	1	no	—
188	C₈×C₁₃	—	2	1	no	—
189	C₈×C₁₃	—	2	2	no	—
190	C₈×C₁₃	#131	1	2	yes	complemented
191	#1055	—	2	1	no	—
192	#1055	—	2	2	no	—
193	#1055	—	2	1	no	—
194	C₂×#131	—	26	1	no	—
195	C₄×#41	—	13	1	no	—
196	#446	—	13	1	no	—
197	#446	#261	1	1	yes	characteristic
198	#436	—	13	1	no	—
199	#436	—	13	1	no	—
200	#435	—	13	2	no	—
201	#435	—	13	2	no	—
202	#433	—	26	2	no	—
203	#432	—	13	2	no	—
204	#431	—	13	1	no	—
205	#431	—	13	2	no	—
206	#431	—	13	2	no	—
207	#424	—	26	1	no	—
208	#424	—	26	2	no	—
209	#423	—	13	1	no	—
210	#423	—	13	1	no	—
211	#423	—	13	2	no	—
212	C₂×C₄⋊C₈	—	13	1	no	—
213	C₂×C₄⋊C₈	—	13	1	no	—
214	C₂×C₄⋊C₈	—	13	2	no	—
215	#408	—	26	1	no	—
216	#405	—	26	1	no	—
217	C₄×M₁₆	—	13	2	no	—
218	C₂×C₄×C₈	—	13	1	no	—
219	#335	—	13	2	no	—
220	#335	—	13	2	no	—
221	#335	—	13	2	no	—
222	#335	—	13	2	no	—
223	#334	—	13	1	no	—
224	#334	—	13	1	no	—
225	#334	—	13	1	no	—
226	#334	—	13	1	no	—
227	#334	—	13	2	no	—
228	#334	—	13	2	no	—
229	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
230	C₂×D₁₃	—	2	1	no	—
231	C₂×D₁₃	—	2	1	no	—
232	C₂×D₁₃	—	2	2	no	—
233	C₂×D₁₃	—	2	2	no	—
234	C₂×D₁₃	—	2	2	no	—
235	C₂×D₁₃	#428	1	2	yes	—
236	C₄×C₁₃	—	2	2	no	—
237	C₄×C₁₃	—	2	1	no	—
238	C₄×C₁₃	—	2	2	no	—
239	C₄×C₁₃	#428	1	2	yes	—
240	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
241	C₄×C₁₃	#568	1	1	yes	characteristic
242	Dic₁₃	—	2	1	no	—
243	Dic₁₃	—	2	2	no	—
244	Dic₁₃	—	2	1	no	—
245	Dic₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
246	Dic₁₃	C₂×#137	1	1	yes	characteristic
247	C₂×#41	—	13	1	no	—
248	#131	—	52	4	no	—
249	C₂×M₁₆	—	13	2	no	—
250	C₂×M₁₆	—	13	2	no	—
251	C₂×M₁₆	—	26	1	no	—
252	C₂×M₁₆	—	26	2	no	—
253	C₂×M₁₆	—	26	1	no	—
254	C₂²×C₈	—	13	1	no	—
255	C₂²×C₈	—	13	1	no	—
256	C₂²×C₈	—	26	1	no	—
257	C₂²×C₈	—	26	2	no	—
258	C₂²×C₈	—	26	1	no	—
259	C₄×Q₈	—	13	1	no	—
260	C₄×Q₈	C₂×#261	1	1	yes	characteristic
261	C₄×D₄	—	13	2	no	—
262	C₄×D₄	—	13	1	no	—
263	C₄×D₄	—	13	2	no	—
264	C₄×D₄	—	13	1	no	—
265	#117	—	13	1	no	—
266	#117	—	13	2	no	—
267	C₂×C₄²	—	13	1	no	—
268	C₂×C₄²	—	13	2	no	—
269	C₄⋊C₈	—	13	1	no	—
270	C₄⋊C₈	—	13	1	no	—
271	C₄⋊C₈	—	13	1	no	—
272	C₄⋊C₈	—	13	1	no	—
273	C₄⋊C₈	—	13	2	no	—
274	C₄⋊C₈	—	13	2	no	—
275	C₄⋊C₈	—	13	2	no	—
276	C₄⋊C₈	—	13	1	no	—
277	C₄⋊C₈	—	26	2	no	—
278	C₄⋊C₈	—	26	1	no	—
279	C₄⋊C₈	—	26	1	no	—
280	C₄⋊C₈	—	26	2	no	—
281	C₄⋊C₈	—	26	1	no	—
282	C₄⋊C₈	—	26	1	no	—
283	C₄⋊C₈	C₂×#261	1	2	yes	—
284	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
285	C₂²⋊C₈	—	13	2	no	—
286	C₂²⋊C₈	—	13	2	no	—
287	C₂²⋊C₈	—	13	2	no	—
288	C₂²⋊C₈	—	26	2	no	—
289	C₂²⋊C₈	—	26	2	no	—
290	#97	—	13	2	no	—
291	#97	—	26	2	no	—
292	#97	C₂×#261	1	2	yes	—
293	C₄×C₈	—	13	1	no	—
294	C₄×C₈	—	13	1	no	—
295	C₄×C₈	—	13	2	no	—
296	C₄×C₈	—	13	1	no	—
297	C₄×C₈	—	26	2	no	—
298	C₄×C₈	C₂×#261	1	1	yes	characteristic
299	C₂×C₁₃	#2906	1	1	yes	characteristic
300	C₂×C₁₃	C₂×#428	1	1	yes	characteristic
301	C₂×C₁₃	#2916	1	1	yes	characteristic
302	D₁₃	—	2	2	no	—
303	D₁₃	—	4	2	no	—
304	#41	—	26	2	no	—
305	#41	—	26	2	no	—
306	C₂×Q₈	C₄⋊#261	1	1	yes	characteristic
307	C₂×D₄	—	13	2	no	—
308	C₂×D₄	—	13	1	no	—
309	C₂²×C₄	—	13	1	no	—
310	C₂²×C₄	—	13	2	no	—
311	C₂²×C₄	—	13	1	no	—
312	C₂²×C₄	—	13	2	no	—
313	C₂²×C₄	—	13	1	no	—
314	C₂²×C₄	—	13	2	no	—
315	M₁₆	—	26	4	no	—
316	M₁₆	—	52	2	no	—
317	M₁₆	—	52	2	no	—
318	M₁₆	—	52	2	no	—
319	M₁₆	—	52	4	no	—
320	C₂×C₈	—	13	1	no	—
321	C₂×C₈	—	13	1	no	—
322	C₂×C₈	—	13	2	no	—
323	C₂×C₈	—	26	2	no	—
324	C₂×C₈	—	26	1	no	—
325	C₂×C₈	—	26	2	no	—
326	C₂×C₈	—	26	1	no	—
327	C₂×C₈	—	26	1	no	—
328	C₂×C₈	—	26	1	no	—
329	C₂×C₈	—	26	2	no	—
330	C₂×C₈	—	52	1	no	—
331	C₂×C₈	—	52	2	no	—
332	C₂×C₈	—	52	2	no	—
333	C₂×C₈	—	52	4	no	—
334	C₂×C₈	C₂²×#261	1	1	yes	characteristic
335	C₂×C₈	C₂²×#261	1	2	yes	—
336	C₂×C₈	C₂²×#261	1	1	yes	characteristic
337	C₄⋊C₄	—	13	2	no	—
338	C₄⋊C₄	—	13	1	no	—
339	C₄⋊C₄	C₄⋊#261	1	2	yes	—
340	C₄⋊C₄	C₄⋊#261	1	1	yes	characteristic
341	C₂²⋊C₄	—	13	2	no	—
342	C₂²⋊C₄	—	13	2	no	—
343	C₂²⋊C₄	—	13	2	no	—
344	C₄²	—	13	2	no	—
345	C₄²	—	13	1	no	—
346	C₄²	—	13	2	no	—
347	C₄²	—	13	1	no	—
348	C₄²	—	13	1	no	—
349	C₄²	C₂²×#261	1	1	yes	characteristic
350	C₄²	C₂²×#261	1	2	yes	—
351	C₁₃	#17830	1	1	yes	characteristic, complemented, Sylow
352	C₂³	—	13	1	no	—
353	C₂³	—	13	2	no	—
354	Q₈	—	2	2	no	—
355	D₄	—	26	4	no	—
356	D₄	—	26	2	no	—
357	C₂×C₄	—	13	1	no	—
358	C₂×C₄	—	13	2	no	—
359	C₂×C₄	—	13	2	no	—
360	C₂×C₄	—	13	1	no	—
361	C₂×C₄	—	13	2	no	—
362	C₂×C₄	—	13	1	no	—
363	C₂×C₄	—	26	2	no	—
364	C₂×C₄	—	26	2	no	—
365	C₂×C₄	—	26	1	no	—
366	C₂×C₄	—	26	2	no	—
367	C₂×C₄	—	26	1	no	—
368	C₂×C₄	—	26	2	no	—
369	C₂×C₄	—	52	1	no	—
370	C₂×C₄	—	52	2	no	—
371	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
372	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
373	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
374	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
375	C₂×C₄	C₂³×#261	1	1	yes	characteristic
376	C₈	—	2	1	no	—
377	C₈	—	2	2	no	—
378	C₈	—	26	1	no	—
379	C₈	—	26	2	no	—
380	C₈	—	26	1	no	—
381	C₈	—	52	1	no	—
382	C₈	—	52	1	no	—
383	C₈	—	52	1	no	—
384	C₈	—	52	1	no	—
385	C₈	—	52	2	no	—
386	C₈	—	52	2	no	—
387	C₈	#88240	1	2	yes	complemented
388	C₂²	—	13	2	no	—
389	C₂²	—	26	1	no	—
390	C₂²	—	26	1	no	—
391	C₂²	—	26	2	no	—
392	C₂²	—	26	2	no	—
393	C₂²	—	26	2	no	—
394	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
395	C₄	—	2	2	no	—
396	C₄	—	2	1	no	—
397	C₄	—	2	2	no	—
398	C₄	—	13	1	no	—
399	C₄	—	13	1	no	—
400	C₄	—	26	1	no	—
401	C₄	—	26	2	no	—
402	C₄	—	26	1	no	—
403	C₄	#11719098	1	2	yes	—
404	C₄	C₂×#88240	1	1	yes	characteristic
405	C₄	#11719629	1	1	yes	characteristic
406	C₂	—	26	2	no	—
407	C₂	—	52	2	no	—
408	C₂	#13131482	1	1	yes	characteristic
409	C₂	C₂×#11719098	1	1	yes	characteristic
410	C₂	#13131498	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#446	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂×#6131	D₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#446	#261
hypocenter	C₁₃	#17830
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
3. #13130414	C₂	semidirect
4. #13130410	C₂	semidirect
5. #13130406	C₂	semidirect
6. #13130404	C₂	semidirect
7. #13130401	C₂	semidirect
8. #13130394	C₂	semidirect
9. #13130388	C₂	semidirect
10. D₁₃⋊#334	C₂	semidirect
190. C₈×C₁₃	#131	semidirect
351. C₁₃	#17830	semidirect
387. C₈	#88240	semidirect

Maximal subgroups

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

subgroup	index
#13131536	2
#13131482	2
#13130414	2
#13130410	2
#13130406	2
#13130404	2
#13130401	2
#13130394	2
#13130388	2
D₁₃⋊#334	2
#13123801	2
#17830	13