Subgroups — #16795082

Order 3328.

572 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131758	C₂	1	2	yes	maximal
2	#13130414	C₂	1	2	yes	maximal, complemented
3	#13130410	C₂	1	2	yes	maximal, complemented
4	#13130408	C₂	1	2	yes	maximal, complemented
5	#13130406	C₂	1	2	yes	maximal, complemented
6	#13130400	C₂	1	2	yes	maximal, complemented
7	#13130398	C₂	1	1	yes	maximal, characteristic, complemented
8	#13130393	C₂	1	1	yes	maximal, characteristic, complemented
9	#13123479	C₂	1	1	yes	maximal, characteristic
10	#11719227	C₂²	1	1	yes	characteristic
11	#11719130	C₂²	1	2	yes	—
12	#11719129	C₂²	1	2	yes	—
13	#11719078	—	2	2	no	—
14	#11719075	—	2	2	no	—
15	#11719066	—	2	2	no	—
16	#11719065	—	2	2	no	—
17	#11719054	C₂²	1	2	yes	—
18	#11719052	C₂²	1	2	yes	—
19	#11719051	C₂²	1	2	yes	—
20	D₁₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
21	C₁₃×#424	C₄	1	1	yes	characteristic
22	#11718790	C₄	1	1	yes	characteristic
23	M₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
24	#11718771	C₄	1	1	yes	characteristic
25	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
26	#11718660	C₄	1	1	yes	characteristic
27	#11718501	C₄	1	2	yes	—
28	#11718501	C₄	1	2	yes	—
29	#11718494	C₂²	1	2	yes	—
30	#11718494	C₂²	1	2	yes	—
31	#11718256	C₂²	1	2	yes	—
32	#11718256	C₂²	1	2	yes	—
33	#11718255	C₂²	1	2	yes	—
34	#11718255	C₂²	1	2	yes	—
35	#11718254	C₂²	1	1	yes	characteristic
36	#11718254	C₂²	1	1	yes	characteristic
37	#11718254	C₂²	1	1	yes	characteristic
38	#11718254	C₂²	1	1	yes	characteristic
39	#11718254	C₂²	1	2	yes	—
40	#11718253	C₂²	1	2	yes	—
41	C₂×#6127	C₂×C₄	1	1	yes	characteristic
42	C₂×#6122	—	2	2	no	—
43	C₂×#6118	—	2	2	no	—
44	C₂×D₁₃⋊C₈	—	2	2	no	—
45	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
49	C₂×#6099	C₂×C₄	1	1	yes	characteristic
50	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
51	#88147	D₄	1	1	yes	characteristic
52	#88147	Q₈	1	1	yes	characteristic
53	#88141	D₄	1	1	yes	characteristic
54	#88141	Q₈	1	1	yes	characteristic
55	#88135	D₄	1	2	yes	—
56	#88135	Q₈	1	2	yes	—
57	#88126	C₂×C₄	1	2	yes	—
58	C₄²×D₁₃	C₂³	1	2	yes	—
59	C₂²⋊#1057	—	2	2	no	—
60	#88112	—	2	2	no	—
61	#88112	—	2	2	no	—
62	#88111	—	2	2	no	—
63	#88111	—	2	2	no	—
64	#88110	—	2	2	no	—
65	#88110	—	2	2	no	—
66	C₄⋊#1057	—	2	2	no	—
67	C₄⋊#1057	C₂³	1	2	yes	—
68	C₄⋊#1057	C₂³	1	2	yes	—
69	C₄⋊#1057	C₂³	1	2	yes	—
70	C₄⋊#1057	C₂³	1	2	yes	—
71	C₄×#1057	—	2	2	no	—
72	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
73	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
74	#88060	C₂×C₄	1	2	yes	—
75	#88060	C₂×C₄	1	2	yes	—
76	#88060	C₂×C₄	1	2	yes	—
77	#88060	C₂×C₄	1	2	yes	—
78	#88055	C₂×C₄	1	2	yes	—
79	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
82	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
83	C₈×Dic₁₃	C₂³	1	2	yes	—
84	C₄⋊#1055	C₂×C₄	1	2	yes	—
85	C₄⋊#1055	C₂×C₄	1	2	yes	—
86	#17825	—	13	1	no	maximal, Sylow
87	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₂²⋊D₁₃	C₄⋊C₄	1	2	yes	—
89	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
91	#6127	—	4	1	no	—
92	#6127	—	4	1	no	—
93	C₂×D₅₂	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×D₁₃	C₂×Q₈	1	2	yes	—
97	C₂×Dic₂₆	C₄⋊C₄	1	1	yes	characteristic
98	#6122	—	8	2	no	—
99	C₂×#1057	—	2	2	no	—
100	C₂×#1057	—	2	2	no	—
101	C₂×#1057	—	2	2	no	—
102	C₂×#1057	—	2	2	no	—
103	#6118	—	8	2	no	—
104	D₁₃⋊C₈	—	8	2	no	—
105	C₁₃×M₁₆	—	4	1	no	—
106	C₂×C₈×C₁₃	—	4	1	no	—
107	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
110	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
111	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
112	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
113	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
114	#6103	C₄⋊C₄	1	2	yes	—
115	#6103	C₄⋊C₄	1	2	yes	—
116	#6101	C₄⋊C₄	1	2	yes	—
117	#6101	C₄⋊C₄	1	2	yes	—
118	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
120	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
122	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
123	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
124	#6099	—	4	1	no	—
125	C₂×#1055	—	4	1	no	—
126	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
127	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
128	C₂×#1055	C₂²×C₄	1	2	yes	—
129	#2989	—	13	2	no	—
130	#2922	—	13	1	no	—
131	#1574	—	13	2	no	—
132	#1573	—	13	2	no	—
133	#1572	—	13	2	no	—
134	#1571	—	13	2	no	—
135	#1569	—	13	2	no	—
136	#1568	—	13	1	no	—
137	#1567	—	13	1	no	—
138	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₁₃	—	2	1	no	—
141	C₂×C₄×C₁₃	—	2	1	no	—
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	C₂³×C₄	1	1	yes	characteristic
146	C₂²⋊D₁₃	—	4	2	no	—
147	C₂×Dic₁₃	—	2	1	no	—
148	C₂×Dic₁₃	—	2	1	no	—
149	C₂×Dic₁₃	—	4	1	no	—
150	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
153	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
155	D₅₂	—	4	1	no	—
156	C₄×D₁₃	—	2	2	no	—
157	C₄×D₁₃	—	2	2	no	—
158	C₄×D₁₃	—	2	2	no	—
159	C₄×D₁₃	—	2	2	no	—
160	C₄×D₁₃	—	4	2	no	—
161	Dic₂₆	—	4	1	no	—
162	#1057	—	4	2	no	—
163	#1057	—	4	2	no	—
164	#1057	—	4	2	no	—
165	#1057	—	4	2	no	—
166	C₈×C₁₃	—	2	1	no	—
167	C₈×C₁₃	—	2	1	no	—
168	C₈×C₁₃	—	2	2	no	—
169	#1055	—	2	1	no	—
170	#1055	—	2	1	no	—
171	#1055	—	2	2	no	—
172	C₄×#41	—	13	1	no	—
173	#446	—	26	2	no	—
174	#436	—	26	2	no	—
175	#435	—	26	2	no	—
176	#435	—	26	2	no	—
177	#433	—	13	2	no	—
178	#433	—	13	2	no	—
179	#432	—	13	2	no	—
180	#432	—	13	2	no	—
181	#431	—	13	2	no	—
182	#431	—	13	2	no	—
183	#431	—	13	2	no	—
184	#424	—	13	1	no	—
185	#424	#261	1	1	yes	characteristic
186	#423	—	13	2	no	—
187	#423	—	13	2	no	—
188	C₂×C₄⋊C₈	—	13	2	no	—
189	#408	—	13	1	no	—
190	#408	—	13	1	no	—
191	C₄×M₁₆	—	13	1	no	—
192	C₂×C₄×C₈	—	13	1	no	—
193	#335	—	13	2	no	—
194	#335	—	13	1	no	—
195	#335	—	13	1	no	—
196	#335	—	13	1	no	—
197	#335	—	13	1	no	—
198	#335	—	13	2	no	—
199	#335	—	13	2	no	—
200	#334	—	13	2	no	—
201	#334	—	13	2	no	—
202	#334	—	13	2	no	—
203	C₂²×C₁₃	—	2	1	no	—
204	C₂²×C₁₃	—	2	1	no	—
205	C₂²×C₁₃	—	2	1	no	—
206	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
207	C₂×D₁₃	—	2	2	no	—
208	C₂×D₁₃	—	2	2	no	—
209	C₂×D₁₃	—	2	2	no	—
210	C₄×C₁₃	—	2	1	no	—
211	C₄×C₁₃	—	2	2	no	—
212	C₄×C₁₃	—	2	2	no	—
213	C₄×C₁₃	#568	1	1	yes	characteristic
214	C₄×C₁₃	#568	1	1	yes	characteristic
215	Dic₁₃	—	2	2	no	—
216	Dic₁₃	—	2	1	no	—
217	Dic₁₃	—	2	1	no	—
218	Dic₁₃	—	2	1	no	—
219	Dic₁₃	#576	1	1	yes	characteristic
220	Dic₁₃	#578	1	1	yes	characteristic
221	C₂×#41	—	13	1	no	—
222	C₂×M₁₆	—	13	1	no	—
223	C₂×M₁₆	—	26	2	no	—
224	C₂×M₁₆	—	26	2	no	—
225	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
226	C₂²×C₈	—	13	1	no	—
227	C₂²×C₈	—	26	2	no	—
228	C₂²×C₈	C₂×#261	1	1	yes	characteristic
229	C₄×Q₈	—	13	1	no	—
230	C₄×Q₈	—	13	1	no	—
231	C₄×D₄	—	13	1	no	—
232	C₄×D₄	—	13	1	no	—
233	C₄×D₄	—	13	2	no	—
234	C₄×D₄	—	13	2	no	—
235	#117	—	13	2	no	—
236	#117	C₂×#261	1	1	yes	characteristic
237	C₂×C₄²	—	13	1	no	—
238	C₂×C₄²	—	13	2	no	—
239	C₄⋊C₈	—	13	2	no	—
240	C₄⋊C₈	—	13	2	no	—
241	C₄⋊C₈	—	13	2	no	—
242	C₄⋊C₈	—	13	2	no	—
243	C₄⋊C₈	—	13	2	no	—
244	C₄⋊C₈	—	13	2	no	—
245	C₄⋊C₈	—	26	2	no	—
246	C₄⋊C₈	—	26	2	no	—
247	C₄⋊C₈	—	26	2	no	—
248	C₄⋊C₈	C₂×#261	1	2	yes	—
249	C₄⋊C₈	C₂×#261	1	2	yes	—
250	C₂²⋊C₈	—	13	2	no	—
251	C₂²⋊C₈	—	13	2	no	—
252	C₂²⋊C₈	—	13	2	no	—
253	C₂²⋊C₈	—	13	2	no	—
254	C₂²⋊C₈	—	26	2	no	—
255	C₂²⋊C₈	—	26	2	no	—
256	C₂²⋊C₈	—	26	2	no	—
257	#97	—	13	2	no	—
258	#97	—	26	2	no	—
259	#97	—	26	2	no	—
260	C₄×C₈	—	13	1	no	—
261	C₄×C₈	—	13	2	no	—
262	C₄×C₈	—	13	1	no	—
263	C₄×C₈	—	13	1	no	—
264	C₄×C₈	—	13	1	no	—
265	C₄×C₈	—	26	2	no	—
266	C₂×C₁₃	—	4	1	no	—
267	C₂×C₁₃	#2905	1	1	yes	characteristic
268	C₂×C₁₃	#2913	1	1	yes	characteristic
269	C₂×C₁₃	#2916	1	1	yes	characteristic
270	D₁₃	—	4	2	no	—
271	#41	—	52	1	no	—
272	#41	—	52	1	no	—
273	C₂×Q₈	—	13	1	no	—
274	C₂×D₄	—	13	2	no	—
275	C₂×D₄	—	13	1	no	—
276	C₂²×C₄	—	13	1	no	—
277	C₂²×C₄	—	13	1	no	—
278	C₂²×C₄	—	13	2	no	—
279	C₂²×C₄	—	13	2	no	—
280	C₂²×C₄	—	13	2	no	—
281	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
282	M₁₆	—	4	1	no	—
283	M₁₆	—	52	1	no	—
284	M₁₆	—	104	2	no	—
285	M₁₆	—	104	2	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	13	1	no	—
288	C₂×C₈	—	13	1	no	—
289	C₂×C₈	—	13	2	no	—
290	C₂×C₈	—	26	2	no	—
291	C₂×C₈	—	26	2	no	—
292	C₂×C₈	—	26	2	no	—
293	C₂×C₈	—	26	2	no	—
294	C₂×C₈	—	52	1	no	—
295	C₂×C₈	—	104	2	no	—
296	C₂×C₈	C₂²×#261	1	1	yes	characteristic
297	C₂×C₈	C₂²×#261	1	1	yes	characteristic
298	C₂×C₈	C₂²×#261	1	2	yes	—
299	C₄⋊C₄	—	13	2	no	—
300	C₄⋊C₄	—	13	2	no	—
301	C₄⋊C₄	C₄⋊#261	1	1	yes	characteristic
302	C₄⋊C₄	C₄⋊#261	1	1	yes	characteristic
303	C₂²⋊C₄	—	13	2	no	—
304	C₂²⋊C₄	—	13	2	no	—
305	C₂²⋊C₄	C₄⋊#261	1	2	yes	—
306	C₄²	—	13	2	no	—
307	C₄²	—	13	1	no	—
308	C₄²	—	13	1	no	—
309	C₄²	—	13	2	no	—
310	C₄²	—	13	1	no	—
311	C₄²	—	13	1	no	—
312	C₄²	C₂²×#261	1	2	yes	—
313	C₁₃	#17825	1	1	yes	characteristic, complemented, Sylow
314	C₂³	—	13	2	no	—
315	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
316	Q₈	—	52	1	no	—
317	D₄	—	52	1	no	—
318	D₄	—	52	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	13	1	no	—
322	C₂×C₄	—	13	1	no	—
323	C₂×C₄	—	13	1	no	—
324	C₂×C₄	—	13	1	no	—
325	C₂×C₄	—	13	2	no	—
326	C₂×C₄	—	26	2	no	—
327	C₂×C₄	—	26	2	no	—
328	C₂×C₄	—	26	1	no	—
329	C₂×C₄	—	26	1	no	—
330	C₂×C₄	—	26	2	no	—
331	C₂×C₄	—	26	2	no	—
332	C₂×C₄	—	52	2	no	—
333	C₂×C₄	—	52	1	no	—
334	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
335	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
336	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
337	C₂×C₄	C₂³×#261	1	1	yes	characteristic
338	C₈	—	2	1	no	—
339	C₈	—	2	1	no	—
340	C₈	—	2	2	no	—
341	C₈	—	26	1	no	—
342	C₈	—	26	1	no	—
343	C₈	—	26	2	no	—
344	C₈	—	52	2	no	—
345	C₈	—	52	2	no	—
346	C₈	—	52	2	no	—
347	C₈	—	52	2	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	26	2	no	—
352	C₂²	—	26	2	no	—
353	C₂²	—	26	2	no	—
354	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
355	C₄	—	2	1	no	—
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	13	1	no	—
359	C₄	—	13	1	no	—
360	C₄	—	26	2	no	—
361	C₄	—	26	1	no	—
362	C₄	—	26	1	no	—
363	C₄	—	26	1	no	—
364	C₄	#11719621	1	1	yes	characteristic
365	C₄	#11719625	1	1	yes	characteristic
366	C₂	—	4	1	no	—
367	C₂	—	52	2	no	—
368	C₂	#13131480	1	1	yes	characteristic
369	C₂	#13131491	1	1	yes	characteristic
370	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×#424	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂×#6127	C₂×C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#424	#261
hypocenter	C₁₃	#17825
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

Complements

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

N	H ≅ G/N	type
2. #13130414	C₂	semidirect
3. #13130410	C₂	semidirect
4. #13130408	C₂	semidirect
5. #13130406	C₂	semidirect
6. #13130400	C₂	semidirect
7. #13130398	C₂	semidirect
8. #13130393	C₂	semidirect
313. C₁₃	#17825	semidirect

Maximal subgroups

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

subgroup	index
#13131758	2
#13130414	2
#13130410	2
#13130408	2
#13130406	2
#13130400	2
#13130398	2
#13130393	2
#13123479	2
#17825	13