Subgroups — #16807162

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719096	C₂	1	2	yes	maximal, complemented
2	#13131222	C₂	1	1	yes	maximal, characteristic, complemented
3	#13131220	C₂	1	1	yes	maximal, characteristic, complemented
4	#13131218	C₂	1	2	yes	maximal, complemented
5	#13131217	C₂	1	2	yes	maximal, complemented
6	#13131215	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×#6092	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₃⋊(C₄⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
9	#13131067	C₂	1	2	yes	maximal, complemented
10	C₄×#88098	C₂	1	1	yes	maximal, characteristic, complemented
11	#13123545	C₂	1	1	yes	maximal, characteristic
12	C₂×#88161	C₂²	1	2	yes	—
13	#11719253	C₄	1	1	yes	characteristic
14	C₂×#88106	C₂²	1	2	yes	—
15	C₂²×#6092	C₂²	1	2	yes	—
16	#11719096	—	2	8	no	—
17	C₂×#88098	C₂²	1	2	yes	—
18	C₂×#88098	C₂²	1	1	yes	characteristic
19	C₂×#88098	C₂²	1	1	yes	characteristic
20	C₂×D₁₃⋊C₁₆	C₂²	1	2	yes	—
21	C₂×D₁₃⋊C₁₆	C₂²	1	2	yes	—
22	C₁₃×#436	C₄	1	1	yes	characteristic
23	#11718789	C₄	1	2	yes	—
24	D₄×#1055	C₂²	1	1	yes	characteristic
25	#11718498	C₄	1	1	yes	characteristic
26	#11718495	C₂²	1	1	yes	characteristic
27	#11718388	C₄	1	2	yes	—
28	#11718386	C₂²	1	2	yes	—
29	#11718349	C₄	1	1	yes	characteristic
30	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
31	C₂²⋊#6092	C₂²	1	2	yes	—
32	C₂²⋊#6092	C₂²	1	2	yes	—
33	#11718260	C₂²	1	2	yes	—
34	#11718259	C₂²	1	2	yes	—
35	#11718259	C₂²	1	2	yes	—
36	#11718258	C₂²	1	1	yes	characteristic
37	#11718258	C₂²	1	1	yes	characteristic
38	C₄⋊#6092	C₂²	1	1	yes	characteristic
39	C₄⋊#6092	C₂²	1	1	yes	characteristic
40	C₄×#6092	C₂²	1	1	yes	characteristic
41	C₄×#6092	C₂²	1	1	yes	characteristic
42	C₂×#6127	C₂×C₄	1	2	yes	—
43	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
44	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
45	#88194	C₈	1	1	yes	characteristic
46	#88192	C₈	1	1	yes	characteristic
47	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
48	C₂²×#1055	C₂³	1	2	yes	—
49	#88161	—	2	4	no	—
50	#88161	—	2	4	no	—
51	C₂×#6094	C₂×C₄	1	2	yes	—
52	C₂×#6094	C₂×C₄	1	2	yes	—
53	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₁₃	C₂³	1	2	yes	—
55	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
56	#88149	C₈	1	1	yes	characteristic
57	#88142	C₈	1	1	yes	characteristic
58	#88139	C₈	1	2	yes	—
59	#88138	C₈	1	2	yes	—
60	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
61	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
62	#88106	—	2	4	no	—
63	C₂×#6092	—	2	4	no	—
64	C₂×#6092	C₂³	1	1	yes	characteristic
65	C₂×#6092	C₂³	1	1	yes	characteristic
66	C₂×#6092	C₂³	1	2	yes	—
67	C₂×#6092	C₂³	1	2	yes	—
68	C₂×#6092	C₂³	1	1	yes	characteristic
69	C₂×#6092	C₂³	1	1	yes	characteristic
70	#88098	—	2	2	no	—
71	#88098	—	2	4	no	—
72	#88098	D₄	1	4	yes	complemented
73	D₁₃⋊C₁₆	—	2	4	no	—
74	D₁₃⋊C₁₆	—	2	4	no	—
75	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
76	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
77	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
78	C₂²⋊#1055	C₂×C₄	1	2	yes	—
79	#88060	C₂×C₄	1	2	yes	—
80	#88060	C₂×C₄	1	2	yes	—
81	#88054	C₂×C₄	1	2	yes	—
82	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
84	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
85	C₄×#1055	C₂³	1	1	yes	characteristic
86	#33438	—	13	1	no	maximal, Sylow
87	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
90	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
91	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
92	#6127	—	2	4	no	—
93	#6127	—	2	4	no	—
94	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
95	C₂×D₅₂	C₂×C₈	1	2	yes	—
96	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
97	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
98	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
99	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
101	C₁₃×M₁₆	—	2	4	no	—
102	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
106	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
107	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
109	#6103	C₂×C₈	1	2	yes	—
110	#6103	C₂×C₈	1	2	yes	—
111	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
112	#6101	C₂×C₈	1	2	yes	—
113	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
114	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
115	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×#1055	—	2	2	no	—
117	C₂×#1055	—	2	2	no	—
118	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
119	C₂×#1055	C₂²×C₄	1	2	yes	—
120	C₂×#1055	C₂⁴	1	1	yes	characteristic
121	#6094	—	2	4	no	—
122	#6094	—	2	4	no	—
123	C₈×D₁₃	—	2	2	no	—
124	C₈×D₁₃	—	2	2	no	—
125	C₈×D₁₃	—	2	2	no	—
126	C₈×D₁₃	C₂×D₄	1	2	yes	—
127	C₈×D₁₃	#41	1	2	yes	—
128	#6092	—	2	2	no	—
129	#6092	—	2	2	no	—
130	#6092	—	2	1	no	—
131	#6092	—	2	1	no	—
132	#6092	C₂×D₄	1	2	yes	—
133	#6092	C₂×D₄	1	2	yes	—
134	C₂×#504	—	13	2	no	—
135	#2931	—	13	1	no	—
136	#2169	—	13	1	no	—
137	#2169	—	13	1	no	—
138	#2168	—	13	2	no	—
139	#2168	—	13	2	no	—
140	D₄×C₁₆	—	13	1	no	—
141	D₄×C₁₆	—	13	1	no	—
142	C₂×C₄⋊C₁₆	—	13	1	no	—
143	#2113	—	13	2	no	—
144	C₄×M₃₂	—	13	1	no	—
145	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
146	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
147	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
148	D₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
153	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
155	C₂²⋊D₁₃	—	2	4	no	—
156	C₂²⋊D₁₃	—	2	4	no	—
157	C₂²⋊D₁₃	—	4	2	no	—
158	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
159	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
160	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
161	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
162	D₅₂	—	2	4	no	—
163	D₅₂	—	2	2	no	—
164	C₄×D₁₃	—	2	2	no	—
165	C₄×D₁₃	—	2	2	no	—
166	C₄×D₁₃	—	2	2	no	—
167	C₄×D₁₃	—	4	1	no	—
168	C₄×D₁₃	C₄×D₄	1	2	yes	—
169	C₄×D₁₃	#131	1	2	yes	—
170	Dic₂₆	—	2	4	no	—
171	C₈×C₁₃	—	2	2	no	—
172	C₈×C₁₃	—	2	1	no	—
173	C₈×C₁₃	C₄×D₄	1	2	yes	—
174	#1055	—	2	2	no	—
175	#1055	—	2	1	no	—
176	#1055	C₂²×D₄	1	1	yes	characteristic
177	#1055	C₂×#41	1	1	yes	characteristic
178	C₂×#131	—	13	2	no	—
179	#532	—	13	1	no	—
180	#504	—	26	8	no	—
181	C₂×M₃₂	—	13	1	no	—
182	C₂×M₃₂	—	13	1	no	—
183	C₂×M₃₂	—	13	2	no	—
184	C₂×M₃₂	—	13	2	no	—
185	C₂²×C₁₆	—	13	2	no	—
186	C₂²×C₁₆	—	13	2	no	—
187	C₂²×C₁₆	—	13	2	no	—
188	#436	—	13	2	no	—
189	#436	—	13	1	no	—
190	#436	#261	1	1	yes	characteristic
191	C₈×D₄	—	13	1	no	—
192	C₈×D₄	—	13	1	no	—
193	C₈×D₄	—	13	2	no	—
194	#423	—	13	1	no	—
195	#408	—	13	2	no	—
196	C₂×C₄×C₈	—	13	1	no	—
197	C₄⋊C₁₆	—	13	1	no	—
198	C₄⋊C₁₆	—	13	1	no	—
199	C₄⋊C₁₆	—	13	2	no	—
200	C₂²⋊C₁₆	—	13	2	no	—
201	C₂²⋊C₁₆	—	13	2	no	—
202	C₂²⋊C₁₆	—	13	2	no	—
203	C₂²⋊C₁₆	—	13	2	no	—
204	#346	—	13	1	no	—
205	#346	—	13	1	no	—
206	C₄×C₁₆	—	13	1	no	—
207	C₄×C₁₆	—	13	1	no	—
208	C₂²×C₁₃	—	2	2	no	—
209	C₂²×C₁₃	—	2	2	no	—
210	C₂²×C₁₃	—	2	2	no	—
211	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
212	C₂×D₁₃	—	2	1	no	—
213	C₂×D₁₃	—	2	1	no	—
214	C₂×D₁₃	—	2	2	no	—
215	C₂×D₁₃	—	2	2	no	—
216	C₂×D₁₃	—	2	2	no	—
217	C₂×D₁₃	C₈×D₄	1	2	yes	—
218	C₄×C₁₃	—	2	1	no	—
219	C₄×C₁₃	—	2	2	no	—
220	C₄×C₁₃	#504	1	2	yes	complemented
221	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
222	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
223	Dic₁₃	—	2	1	no	—
224	Dic₁₃	—	2	1	no	—
225	Dic₁₃	—	2	2	no	—
226	Dic₁₃	C₈×D₄	1	2	yes	—
227	C₂×#41	—	13	2	no	—
228	#131	—	26	4	no	—
229	#131	—	26	4	no	—
230	C₂×M₁₆	—	13	2	no	—
231	C₂×M₁₆	—	13	2	no	—
232	C₂×M₁₆	C₂×#261	1	2	yes	—
233	C₂²×C₈	—	13	1	no	—
234	C₂²×C₈	—	13	2	no	—
235	C₂²×C₈	—	13	2	no	—
236	C₂²×C₈	—	13	1	no	—
237	C₄⋊Q₈	—	13	1	no	—
238	#127	—	13	1	no	—
239	#124	—	13	2	no	—
240	#121	—	13	1	no	—
241	#121	—	13	1	no	—
242	#121	—	13	2	no	—
243	C₄×D₄	—	13	2	no	—
244	C₄×D₄	—	13	1	no	—
245	C₄×D₄	C₂×#261	1	1	yes	characteristic
246	C₂×C₄²	—	13	1	no	—
247	M₃₂	—	13	4	no	—
248	M₃₂	—	26	4	no	—
249	M₃₂	—	26	4	no	—
250	M₃₂	—	26	2	no	—
251	C₂×C₁₆	—	13	1	no	—
252	C₂×C₁₆	—	13	1	no	—
253	C₂×C₁₆	—	13	1	no	—
254	C₂×C₁₆	—	13	1	no	—
255	C₂×C₁₆	—	13	2	no	—
256	C₂×C₁₆	—	13	2	no	—
257	C₂×C₁₆	—	26	4	no	—
258	C₂×C₁₆	—	26	4	no	—
259	C₂×C₁₆	—	26	4	no	—
260	C₄⋊C₈	—	13	2	no	—
261	C₄⋊C₈	—	13	1	no	—
262	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
263	C₂²⋊C₈	—	13	2	no	—
264	C₂²⋊C₈	—	13	2	no	—
265	C₂²⋊C₈	—	13	2	no	—
266	C₂²⋊C₈	C₂×#261	1	2	yes	—
267	C₄×C₈	—	13	1	no	—
268	C₄×C₈	—	13	1	no	—
269	C₄×C₈	—	13	1	no	—
270	C₄×C₈	C₂×#261	1	1	yes	characteristic
271	C₂×C₁₃	—	4	2	no	—
272	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
273	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
274	C₂×C₁₃	#3407	1	1	yes	characteristic
275	D₁₃	—	2	2	no	—
276	D₁₃	—	4	2	no	—
277	#41	—	26	4	no	—
278	#41	—	26	4	no	—
279	C₂×Q₈	—	13	2	no	—
280	C₂×D₄	—	13	2	no	—
281	C₂×D₄	—	13	2	no	—
282	C₂×D₄	—	13	2	no	—
283	C₂×D₄	—	13	1	no	—
284	C₂×D₄	—	13	2	no	—
285	C₂×D₄	D₁₃⋊C₈	1	1	yes	characteristic
286	C₂²×C₄	—	13	1	no	—
287	C₂²×C₄	—	13	1	no	—
288	C₂²×C₄	—	13	2	no	—
289	C₂²×C₄	—	13	1	no	—
290	C₂²×C₄	C₂²×#261	1	2	yes	—
291	M₁₆	—	2	4	no	—
292	M₁₆	—	26	4	no	—
293	M₁₆	—	26	4	no	—
294	C₂×C₈	—	13	1	no	—
295	C₂×C₈	—	13	2	no	—
296	C₂×C₈	—	13	2	no	—
297	C₂×C₈	—	13	1	no	—
298	C₂×C₈	—	13	2	no	—
299	C₂×C₈	—	26	2	no	—
300	C₂×C₈	—	26	2	no	—
301	C₂×C₈	—	26	2	no	—
302	C₂×C₈	—	26	2	no	—
303	C₂×C₈	—	26	2	no	—
304	C₂×C₈	C₂²×#261	1	1	yes	characteristic
305	C₂×C₈	C₂²×#261	1	2	yes	—
306	C₂×C₈	C₂²×#261	1	1	yes	characteristic
307	C₄⋊C₄	—	13	2	no	—
308	C₄⋊C₄	—	13	1	no	—
309	C₄⋊C₄	D₁₃⋊C₈	1	1	yes	characteristic
310	C₂²⋊C₄	—	13	2	no	—
311	C₂²⋊C₄	—	13	2	no	—
312	C₂²⋊C₄	—	13	2	no	—
313	C₂²⋊C₄	D₁₃⋊C₈	1	2	yes	—
314	C₄²	—	13	1	no	—
315	C₄²	—	13	1	no	—
316	C₄²	—	13	1	no	—
317	C₄²	C₂²×#261	1	1	yes	characteristic
318	C₁₆	—	13	2	no	—
319	C₁₆	—	13	2	no	—
320	C₁₆	—	26	2	no	—
321	C₁₆	—	26	2	no	—
322	C₁₆	—	26	1	no	—
323	C₁₆	—	26	1	no	—
324	C₁₃	#33438	1	1	yes	characteristic, complemented, Sylow
325	C₂³	—	13	1	no	—
326	C₂³	—	13	2	no	—
327	C₂³	C₂×D₁₃⋊C₈	1	2	yes	—
328	Q₈	—	26	4	no	—
329	D₄	—	2	2	no	—
330	D₄	—	26	4	no	—
331	D₄	—	26	4	no	—
332	D₄	—	26	4	no	—
333	D₄	—	26	2	no	—
334	D₄	—	52	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	13	1	no	—
338	C₂×C₄	—	13	2	no	—
339	C₂×C₄	—	13	2	no	—
340	C₂×C₄	—	13	1	no	—
341	C₂×C₄	—	13	1	no	—
342	C₂×C₄	—	13	2	no	—
343	C₂×C₄	—	26	2	no	—
344	C₂×C₄	—	26	2	no	—
345	C₂×C₄	—	26	2	no	—
346	C₂×C₄	—	52	1	no	—
347	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
348	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
349	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
350	C₂×C₄	C₂³×#261	1	1	yes	characteristic
351	C₈	—	2	2	no	—
352	C₈	—	2	1	no	—
353	C₈	—	13	1	no	—
354	C₈	—	13	1	no	—
355	C₈	—	26	2	no	—
356	C₈	—	26	1	no	—
357	C₈	D₄×#261	1	2	yes	—
358	C₂²	—	2	2	no	—
359	C₂²	—	2	2	no	—
360	C₂²	—	2	2	no	—
361	C₂²	—	13	2	no	—
362	C₂²	—	26	1	no	—
363	C₂²	—	26	1	no	—
364	C₂²	—	26	2	no	—
365	C₂²	—	26	2	no	—
366	C₂²	—	26	2	no	—
367	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
368	C₄	—	2	2	no	—
369	C₄	—	2	1	no	—
370	C₄	—	13	2	no	—
371	C₄	—	26	2	no	—
372	C₄	—	26	1	no	—
373	C₄	—	26	1	no	—
374	C₄	#11719096	1	2	yes	complemented
375	C₄	C₂×D₄×#261	1	1	yes	characteristic
376	C₄	D₁₃⋊#131	1	1	yes	characteristic
377	C₂	—	4	2	no	—
378	C₂	—	26	2	no	—
379	C₂	—	52	2	no	—
380	C₂	D₄×D₁₃⋊C₈	1	1	yes	characteristic
381	C₂	C₂×#11719096	1	1	yes	characteristic
382	C₂	#13131804	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#436	C₄
prime radical	#11719253	C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	#436	#261
hypocenter	C₁₃	#33438
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₂×#11719096	C₂	semidirect
2. #13131222	C₂	semidirect
3. #13131220	C₂	semidirect
4. #13131218	C₂	semidirect
5. #13131217	C₂	semidirect
6. #13131215	C₂	semidirect
7. D₄×#6092	C₂	semidirect
8. D₁₃⋊(C₄⋊C₁₆)	C₂	semidirect
9. #13131067	C₂	semidirect
10. C₄×#88098	C₂	semidirect
72. #88098	D₄	semidirect
220. C₄×C₁₃	#504	semidirect
324. C₁₃	#33438	semidirect
374. C₄	#11719096	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719096	2
#13131222	2
#13131220	2
#13131218	2
#13131217	2
#13131215	2
D₄×#6092	2
D₁₃⋊(C₄⋊C₁₆)	2
#13131067	2
C₄×#88098	2
#13123545	2
#33438	13