Subgroups — #16795104

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	#13131532	C₂	1	1	yes	maximal, characteristic
2	#13131481	C₂	1	1	yes	maximal, characteristic
3	#13130412	C₂	1	2	yes	maximal, complemented
4	#13130411	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	#13130400	C₂	1	2	yes	maximal, complemented
8	#13130394	C₂	1	1	yes	maximal, characteristic, complemented
9	#13130387	C₂	1	2	yes	maximal, complemented
10	D₁₃⋊#334	C₂	1	1	yes	maximal, characteristic, complemented
11	#13123799	C₂	1	1	yes	maximal, characteristic
12	C₂×#88238	—	2	1	no	—
13	C₄×#6129	C₂²	1	1	yes	characteristic
14	#11719147	—	2	1	no	—
15	#11719130	C₂²	1	2	yes	—
16	#11719129	C₂²	1	2	yes	—
17	#11719119	C₂²	1	2	yes	—
18	C₂×C₄⋊#1057	C₂²	1	2	yes	—
19	#11719079	—	2	1	no	—
20	#11719072	—	2	1	no	—
21	#11719068	—	2	2	no	—
22	#11719067	—	2	2	no	—
23	#11719052	C₂²	1	1	yes	characteristic
24	#11719052	C₂²	1	1	yes	characteristic
25	D₁₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
26	D₁₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
27	C₁₃×#436	C₄	1	1	yes	characteristic
28	#11718788	C₄	1	2	yes	—
29	M₁₆×Dic₁₃	C₂²	1	2	yes	—
30	#11718673	C₄	1	1	yes	characteristic
31	#11718494	C₂²	1	1	yes	characteristic
32	#11718492	C₄	1	1	yes	characteristic
33	#11718389	C₄	1	2	yes	—
34	#11718379	C₂²	1	2	yes	—
35	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
36	#11718340	C₄	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×#6129	D₄	1	1	yes	characteristic
48	C₂×#6122	—	2	2	no	—
49	C₂²×#1057	—	2	2	no	—
50	#88238	—	4	4	no	—
51	C₂×#6118	—	2	1	no	—
52	C₂×#6118	—	2	1	no	—
53	C₂×D₁₃⋊C₈	—	2	1	no	—
54	C₂×D₁₃⋊C₈	—	2	1	no	—
55	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
56	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
57	D₄×Dic₁₃	Q₈	1	1	yes	characteristic
58	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
59	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
60	C₂×#6099	C₂×C₄	1	2	yes	—
61	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
63	#88146	D₄	1	1	yes	characteristic
64	#88141	Q₈	1	1	yes	characteristic
65	#88135	Q₈	1	2	yes	—
66	#88131	D₄	1	2	yes	—
67	C₄²×D₁₃	C₂³	1	1	yes	characteristic
68	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
69	C₂²⋊#1057	—	2	2	no	—
70	C₂²⋊#1057	—	2	2	no	—
71	#88112	—	2	1	no	—
72	#88112	—	2	1	no	—
73	#88112	—	2	2	no	—
74	#88110	—	2	2	no	—
75	C₄⋊#1057	—	2	1	no	—
76	C₄⋊#1057	—	2	1	no	—
77	C₄⋊#1057	—	2	2	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	1	yes	characteristic
86	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
87	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
88	C₂²⋊#1055	C₂×C₄	1	2	yes	—
89	#88060	C₂×C₄	1	2	yes	—
90	#88055	C₂×C₄	1	2	yes	—
91	#88055	C₂×C₄	1	2	yes	—
92	#88054	C₂×C₄	1	2	yes	—
93	#88054	C₂×C₄	1	2	yes	—
94	C₈×Dic₁₃	C₂³	1	2	yes	—
95	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
96	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
97	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
98	C₄×#1055	C₂×C₄	1	1	yes	characteristic
99	#17830	—	13	1	no	maximal, Sylow
100	C₂×D₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
102	C₂×C₂²⋊D₁₃	C₄⋊C₄	1	2	yes	—
103	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
104	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
105	#6129	—	2	2	no	—
106	#6129	—	2	2	no	—
107	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
110	C₂×Dic₂₆	C₄⋊C₄	1	1	yes	characteristic
111	#6122	—	4	4	no	—
112	C₂×#1057	—	2	1	no	—
113	C₂×#1057	—	2	2	no	—
114	C₂×#1057	—	2	1	no	—
115	C₂×#1057	—	2	2	no	—
116	C₂×#1057	—	2	1	no	—
117	C₂×#1057	—	2	1	no	—
118	C₂×#1057	—	4	4	no	—
119	#6118	—	4	2	no	—
120	#6118	—	4	2	no	—
121	D₁₃⋊C₈	—	4	2	no	—
122	D₁₃⋊C₈	—	4	2	no	—
123	C₁₃×M₁₆	—	2	4	no	—
124	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
127	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
128	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
129	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
130	C₂²⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
131	#6103	C₄⋊C₄	1	2	yes	—
132	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
133	#6101	C₄⋊C₄	1	2	yes	—
134	#6101	C₄⋊C₄	1	2	yes	—
135	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
136	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
137	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
138	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
139	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
140	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
141	#6099	—	4	2	no	—
142	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
143	C₂×#1055	C₂²×C₄	1	2	yes	—
144	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
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	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
159	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
160	D₄×C₁₃	—	2	2	no	—
161	C₂×C₄×C₁₃	—	2	2	no	—
162	C₂×C₄×C₁₃	—	2	2	no	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
167	C₂²⋊D₁₃	—	2	4	no	—
168	C₂×Dic₁₃	—	2	2	no	—
169	C₂×Dic₁₃	—	2	2	no	—
170	C₂×Dic₁₃	—	4	2	no	—
171	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
172	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
173	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
175	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
176	C₄×D₁₃	—	2	1	no	—
177	C₄×D₁₃	—	2	1	no	—
178	C₄×D₁₃	—	4	1	no	—
179	C₄×D₁₃	AGL(1,ℤ/8)	1	2	yes	—
180	C₄×D₁₃	#137	1	2	yes	—
181	Dic₂₆	—	2	2	no	—
182	#1057	—	4	1	no	—
183	#1057	—	4	1	no	—
184	#1057	—	4	2	no	—
185	#1057	—	4	2	no	—
186	#1057	—	4	1	no	—
187	#1057	—	4	1	no	—
188	C₈×C₁₃	—	2	2	no	—
189	C₈×C₁₃	—	2	1	no	—
190	C₈×C₁₃	#131	1	2	yes	complemented
191	#1055	—	2	1	no	—
192	#1055	—	2	1	no	—
193	#1055	—	2	2	no	—
194	C₂×#131	—	26	1	no	—
195	C₄×#41	—	13	1	no	—
196	#446	—	13	1	no	—
197	#446	—	13	1	no	—
198	#436	—	13	1	no	—
199	#436	#261	1	1	yes	characteristic
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	2	no	—
208	#424	—	26	1	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	—	2	2	no	—
230	C₂²×C₁₃	—	2	2	no	—
231	C₂²×C₁₃	—	2	2	no	—
232	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
233	C₂×D₁₃	—	2	1	no	—
234	C₂×D₁₃	—	2	1	no	—
235	C₂×D₁₃	#428	1	2	yes	—
236	C₄×C₁₃	—	2	2	no	—
237	C₄×C₁₃	—	2	1	no	—
238	C₄×C₁₃	#428	1	2	yes	—
239	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
240	C₄×C₁₃	#568	1	1	yes	characteristic
241	Dic₁₃	—	2	1	no	—
242	Dic₁₃	—	2	2	no	—
243	Dic₁₃	—	2	1	no	—
244	Dic₁₃	—	2	2	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	—	26	1	no	—
251	C₂×M₁₆	—	26	2	no	—
252	C₂×M₁₆	—	26	1	no	—
253	C₂×M₁₆	C₂×#261	1	2	yes	—
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	—	13	1	no	—
261	C₄×D₄	—	13	1	no	—
262	C₄×D₄	—	13	2	no	—
263	C₄×D₄	—	13	2	no	—
264	C₄×D₄	C₂×#261	1	1	yes	characteristic
265	#117	—	13	1	no	—
266	#117	—	13	2	no	—
267	C₂×C₄²	—	13	2	no	—
268	C₂×C₄²	—	13	1	no	—
269	C₄⋊C₈	—	13	2	no	—
270	C₄⋊C₈	—	13	2	no	—
271	C₄⋊C₈	—	13	1	no	—
272	C₄⋊C₈	—	13	1	no	—
273	C₄⋊C₈	—	13	1	no	—
274	C₄⋊C₈	—	13	2	no	—
275	C₄⋊C₈	—	13	1	no	—
276	C₄⋊C₈	—	13	2	no	—
277	C₄⋊C₈	—	13	1	no	—
278	C₄⋊C₈	—	26	2	no	—
279	C₄⋊C₈	—	26	1	no	—
280	C₄⋊C₈	—	26	1	no	—
281	C₄⋊C₈	—	26	1	no	—
282	C₄⋊C₈	—	26	1	no	—
283	C₄⋊C₈	—	26	2	no	—
284	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
285	C₂²⋊C₈	—	13	2	no	—
286	C₂²⋊C₈	—	13	2	no	—
287	C₂²⋊C₈	—	26	2	no	—
288	C₂²⋊C₈	—	26	2	no	—
289	C₂²⋊C₈	C₂×#261	1	2	yes	—
290	#97	—	13	2	no	—
291	#97	—	13	2	no	—
292	#97	—	26	2	no	—
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	—	4	2	no	—
300	C₂×C₁₃	#2906	1	1	yes	characteristic
301	C₂×C₁₃	C₂×#428	1	1	yes	characteristic
302	C₂×C₁₃	#2916	1	1	yes	characteristic
303	D₁₃	—	2	2	no	—
304	#41	—	26	2	no	—
305	#41	—	26	2	no	—
306	C₂×Q₈	—	13	1	no	—
307	C₂×D₄	—	13	2	no	—
308	C₂×D₄	C₄⋊#261	1	1	yes	characteristic
309	C₂²×C₄	—	13	1	no	—
310	C₂²×C₄	—	13	2	no	—
311	C₂²×C₄	—	13	2	no	—
312	C₂²×C₄	—	13	1	no	—
313	C₂²×C₄	—	13	1	no	—
314	C₂²×C₄	C₂²×#261	1	2	yes	—
315	M₁₆	—	2	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	2	no	—
329	C₂×C₈	—	26	1	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	2	no	—
339	C₄⋊C₄	—	13	1	no	—
340	C₄⋊C₄	C₄⋊#261	1	1	yes	characteristic
341	C₂²⋊C₄	—	13	2	no	—
342	C₂²⋊C₄	—	13	2	no	—
343	C₂²⋊C₄	C₄⋊#261	1	2	yes	—
344	C₄²	—	13	2	no	—
345	C₄²	—	13	1	no	—
346	C₄²	—	13	1	no	—
347	C₄²	—	13	2	no	—
348	C₄²	—	13	1	no	—
349	C₄²	—	13	2	no	—
350	C₄²	C₂²×#261	1	1	yes	characteristic
351	C₁₃	#17830	1	1	yes	characteristic, complemented, Sylow
352	C₂³	—	13	1	no	—
353	C₂³	C₂×C₄⋊#261	1	2	yes	—
354	Q₈	—	26	2	no	—
355	D₄	—	2	2	no	—
356	D₄	—	26	4	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	13	1	no	—
360	C₂×C₄	—	13	2	no	—
361	C₂×C₄	—	13	2	no	—
362	C₂×C₄	—	13	1	no	—
363	C₂×C₄	—	13	1	no	—
364	C₂×C₄	—	13	2	no	—
365	C₂×C₄	—	13	2	no	—
366	C₂×C₄	—	26	2	no	—
367	C₂×C₄	—	26	2	no	—
368	C₂×C₄	—	26	1	no	—
369	C₂×C₄	—	26	1	no	—
370	C₂×C₄	—	52	1	no	—
371	C₂×C₄	—	52	2	no	—
372	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
373	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
374	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
375	C₂×C₄	C₂³×#261	1	1	yes	characteristic
376	C₈	—	2	2	no	—
377	C₈	—	2	1	no	—
378	C₈	—	26	1	no	—
379	C₈	—	26	1	no	—
380	C₈	—	26	2	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	#88238	1	2	yes	complemented
388	C₂²	—	2	2	no	—
389	C₂²	—	2	2	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	13	2	no	—
392	C₂²	—	26	1	no	—
393	C₂²	—	26	1	no	—
394	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
395	C₄	—	2	2	no	—
396	C₄	—	2	1	no	—
397	C₄	—	13	1	no	—
398	C₄	—	13	1	no	—
399	C₄	—	26	1	no	—
400	C₄	—	26	2	no	—
401	C₄	—	26	2	no	—
402	C₄	—	26	1	no	—
403	C₄	#11719098	1	2	yes	—
404	C₄	C₂×#88238	1	1	yes	characteristic
405	C₄	#11719629	1	1	yes	characteristic
406	C₂	—	4	2	no	—
407	C₂	—	26	2	no	—
408	C₂	#13131481	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×#436	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂×#6129	D₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#436	#261
hypocenter	C₁₃	#17830
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #436

Complements

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

N	H ≅ G/N	type
3. #13130412	C₂	semidirect
4. #13130411	C₂	semidirect
5. #13130406	C₂	semidirect
6. #13130404	C₂	semidirect
7. #13130400	C₂	semidirect
8. #13130394	C₂	semidirect
9. #13130387	C₂	semidirect
10. D₁₃⋊#334	C₂	semidirect
190. C₈×C₁₃	#131	semidirect
351. C₁₃	#17830	semidirect
387. C₈	#88238	semidirect

Maximal subgroups

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

subgroup	index
#13131532	2
#13131481	2
#13130412	2
#13130411	2
#13130406	2
#13130404	2
#13130400	2
#13130394	2
#13130387	2
D₁₃⋊#334	2
#13123799	2
#17830	13