Subgroups — #16807163

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 400 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	1	yes	maximal, characteristic, complemented
2	C₂×#11719162	C₂	1	1	yes	maximal, characteristic, complemented
3	#13131221	C₂	1	2	yes	maximal, complemented
4	#13131218	C₂	1	2	yes	maximal, complemented
5	#13131217	C₂	1	2	yes	maximal, complemented
6	#13131214	C₂	1	2	yes	maximal, complemented
7	C₂×#11718260	C₂	1	1	yes	maximal, characteristic, complemented
8	#13131068	C₂	1	1	yes	maximal, characteristic, complemented
9	#13131066	C₂	1	1	yes	maximal, characteristic, complemented
10	#13131042	C₂	1	1	yes	maximal, characteristic, complemented
11	#13123544	C₂	1	1	yes	maximal, characteristic
12	#11719537	C₄	1	1	yes	characteristic
13	C₂×#88202	C₂²	1	1	yes	characteristic
14	C₂×#88161	C₂²	1	1	yes	characteristic
15	#11719162	—	2	4	no	—
16	C₂×#88106	C₂²	1	2	yes	—
17	C₂×#88106	C₂²	1	1	yes	characteristic
18	C₂×#88106	C₂²	1	1	yes	characteristic
19	C₂²×#6092	C₂²	1	1	yes	characteristic
20	C₂²×#6092	C₂²	1	2	yes	—
21	C₂²×#6092	C₂²	1	1	yes	characteristic
22	#11719096	—	2	4	no	—
23	C₂×#88098	C₂²	1	2	yes	—
24	C₂×D₁₃⋊C₁₆	C₂²	1	2	yes	—
25	C₁₃×#408	C₄	1	1	yes	characteristic
26	#11718779	C₄	1	1	yes	characteristic
27	#11718771	C₄	1	1	yes	characteristic
28	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
29	#11718390	C₄	1	2	yes	—
30	#11718390	C₄	1	2	yes	—
31	#11718386	C₂²	1	2	yes	—
32	#11718386	C₂²	1	2	yes	—
33	C₂²⋊#6092	C₂²	1	2	yes	—
34	C₂²⋊#6092	C₂²	1	2	yes	—
35	#11718260	C₂²	1	1	yes	characteristic
36	#11718260	C₂²	1	1	yes	characteristic
37	#11718260	C₂²	1	1	yes	characteristic
38	#11718260	C₂²	1	1	yes	characteristic
39	#11718259	C₂²	1	2	yes	—
40	#11718259	C₂²	1	2	yes	—
41	#11718258	C₂²	1	2	yes	—
42	#11718257	C₂²	1	2	yes	—
43	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
44	C₂×#6127	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	#88202	—	2	2	no	—
48	#88202	—	2	2	no	—
49	#88201	C₈	1	1	yes	characteristic
50	#88198	C₈	1	1	yes	characteristic
51	#88194	C₈	1	1	yes	characteristic
52	#88193	C₈	1	2	yes	—
53	#88193	C₈	1	2	yes	—
54	#88190	C₈	1	1	yes	characteristic
55	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
56	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
57	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
58	C₂×#6099	C₂×C₄	1	1	yes	characteristic
59	C₂×#6099	C₂×C₄	1	2	yes	—
60	C₂²×#1055	C₂³	1	2	yes	—
61	C₂²×#1055	C₂³	1	1	yes	characteristic
62	#88161	—	2	2	no	—
63	#88161	—	2	2	no	—
64	C₂×#6094	C₂×C₄	1	2	yes	—
65	C₂×C₈×D₁₃	C₂³	1	2	yes	—
66	#88106	—	2	2	no	—
67	#88106	—	2	4	no	—
68	#88106	D₄	1	4	yes	complemented
69	C₂×#6092	—	2	2	no	—
70	C₂×#6092	—	2	2	no	—
71	C₂×#6092	—	2	4	no	—
72	C₂×#6092	C₂³	1	2	yes	—
73	C₂×#6092	C₂³	1	1	yes	characteristic
74	C₂×#6092	C₂³	1	1	yes	characteristic
75	C₂×#6092	C₂³	1	1	yes	characteristic
76	C₂×#6092	C₂³	1	1	yes	characteristic
77	C₂×#6092	C₂³	1	2	yes	—
78	#88098	—	2	4	no	—
79	D₁₃⋊C₁₆	—	2	4	no	—
80	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
81	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
82	#88060	C₂×C₄	1	2	yes	—
83	#88060	C₂×C₄	1	2	yes	—
84	#88054	C₂×C₄	1	2	yes	—
85	#88054	C₂×C₄	1	2	yes	—
86	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
87	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
88	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
89	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
90	#33438	—	13	1	no	maximal, Sylow
91	C₁₃×#41	—	2	2	no	—
92	C₁₃×#41	—	2	2	no	—
93	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
94	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
95	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
96	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
99	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
100	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
101	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
102	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
103	#6127	—	2	2	no	—
104	#6127	—	2	2	no	—
105	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
106	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
107	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
108	C₁₃×M₁₆	—	2	2	no	—
109	C₂×C₈×C₁₃	—	2	2	no	—
110	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
114	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
115	#6103	C₂×C₈	1	2	yes	—
116	#6103	C₂×C₈	1	2	yes	—
117	#6101	C₂×C₈	1	2	yes	—
118	#6101	C₂×C₈	1	2	yes	—
119	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
120	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
121	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
122	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
123	#6099	—	2	2	no	—
124	#6099	—	2	4	no	—
125	C₂×#1055	—	2	2	no	—
126	C₂×#1055	—	2	2	no	—
127	C₂×#1055	C₂²×C₄	1	2	yes	—
128	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
129	C₂×#1055	C₂×D₄	1	2	yes	—
130	C₂×#1055	#41	1	2	yes	—
131	C₂×#1055	C₂⁴	1	1	yes	characteristic
132	#6094	—	2	4	no	—
133	C₈×D₁₃	—	2	2	no	—
134	C₈×D₁₃	—	2	2	no	—
135	#6092	—	2	1	no	—
136	#6092	—	2	1	no	—
137	#6092	—	2	1	no	—
138	#6092	—	2	1	no	—
139	#6092	—	2	2	no	—
140	#6092	C₂×D₄	1	4	yes	—
141	C₂×#504	—	13	1	no	—
142	C₂×#504	—	13	1	no	—
143	#2931	—	13	1	no	—
144	#2169	—	13	2	no	—
145	#2168	—	13	2	no	—
146	#2168	—	13	2	no	—
147	D₄×C₁₆	—	13	2	no	—
148	C₂×C₄⋊C₁₆	—	13	1	no	—
149	#2113	—	13	1	no	—
150	#2113	—	13	1	no	—
151	C₄×M₃₂	—	13	1	no	—
152	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
153	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
154	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
155	Q₈×C₁₃	—	2	2	no	—
156	D₄×C₁₃	—	2	4	no	—
157	D₄×C₁₃	—	2	2	no	—
158	C₂×C₄×C₁₃	—	2	2	no	—
159	C₂×C₄×C₁₃	—	2	2	no	—
160	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
161	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
162	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
163	C₂×C₄×C₁₃	#131	1	2	yes	—
164	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
165	C₂²⋊D₁₃	—	2	4	no	—
166	C₂²⋊D₁₃	—	2	4	no	—
167	C₂²⋊D₁₃	—	4	2	no	—
168	C₂×Dic₁₃	—	2	2	no	—
169	C₂×Dic₁₃	—	4	1	no	—
170	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
171	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
172	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
173	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
174	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
175	D₅₂	—	2	2	no	—
176	C₄×D₁₃	—	2	2	no	—
177	C₄×D₁₃	—	2	2	no	—
178	Dic₂₆	—	2	2	no	—
179	C₈×C₁₃	—	2	2	no	—
180	C₈×C₁₃	—	2	1	no	—
181	C₈×C₁₃	—	2	1	no	—
182	#1055	—	2	2	no	—
183	#1055	C₄×D₄	1	2	yes	—
184	#1055	C₂²×D₄	1	1	yes	characteristic
185	#1055	C₂×#41	1	1	yes	characteristic
186	C₂×#131	—	13	1	no	—
187	C₂×#131	—	13	1	no	—
188	#532	—	13	1	no	—
189	#504	—	26	4	no	—
190	#504	—	26	4	no	—
191	C₂×M₃₂	—	13	1	no	—
192	C₂×M₃₂	—	13	1	no	—
193	C₂×M₃₂	—	13	2	no	—
194	C₂×M₃₂	—	13	2	no	—
195	C₂²×C₁₆	—	13	1	no	—
196	C₂²×C₁₆	—	13	1	no	—
197	C₂²×C₁₆	—	13	2	no	—
198	C₂²×C₁₆	—	13	2	no	—
199	#436	—	13	2	no	—
200	#436	—	13	2	no	—
201	C₈×D₄	—	13	2	no	—
202	C₈×D₄	—	13	2	no	—
203	#423	—	13	1	no	—
204	#408	—	13	1	no	—
205	#408	#261	1	1	yes	characteristic
206	C₂×C₄×C₈	—	13	1	no	—
207	C₄⋊C₁₆	—	13	1	no	—
208	C₄⋊C₁₆	—	13	1	no	—
209	C₄⋊C₁₆	—	13	1	no	—
210	C₄⋊C₁₆	—	13	1	no	—
211	C₂²⋊C₁₆	—	13	2	no	—
212	C₂²⋊C₁₆	—	13	2	no	—
213	C₂²⋊C₁₆	—	13	2	no	—
214	C₂²⋊C₁₆	—	13	2	no	—
215	#346	—	13	2	no	—
216	C₄×C₁₆	—	13	2	no	—
217	C₂²×C₁₃	—	2	1	no	—
218	C₂²×C₁₃	—	2	1	no	—
219	C₂²×C₁₃	—	2	2	no	—
220	C₂²×C₁₃	—	2	2	no	—
221	C₂²×C₁₃	—	2	2	no	—
222	C₂²×C₁₃	C₈×D₄	1	2	yes	—
223	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
224	C₂×D₁₃	—	2	2	no	—
225	C₂×D₁₃	—	2	2	no	—
226	C₂×D₁₃	—	2	2	no	—
227	C₄×C₁₃	—	2	2	no	—
228	C₄×C₁₃	C₈×D₄	1	2	yes	—
229	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
230	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
231	Dic₁₃	—	2	1	no	—
232	Dic₁₃	—	2	1	no	—
233	Dic₁₃	—	2	1	no	—
234	Dic₁₃	—	2	2	no	—
235	Dic₁₃	#504	1	2	yes	complemented
236	C₂×#41	—	13	1	no	—
237	C₂×#41	C₂×#261	1	1	yes	characteristic
238	#131	—	26	2	no	—
239	#131	—	26	2	no	—
240	#131	—	26	2	no	—
241	#131	—	26	2	no	—
242	C₂×M₁₆	—	13	1	no	—
243	C₂×M₁₆	—	13	2	no	—
244	C₂×M₁₆	—	13	2	no	—
245	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
246	C₂²×C₈	—	13	1	no	—
247	C₂²×C₈	—	13	2	no	—
248	C₂²×C₈	—	13	2	no	—
249	C₂²×C₈	C₂×#261	1	1	yes	characteristic
250	C₄⋊Q₈	—	13	1	no	—
251	#127	—	13	1	no	—
252	#124	—	13	1	no	—
253	#124	—	13	1	no	—
254	#121	—	13	2	no	—
255	#121	—	13	2	no	—
256	C₄×D₄	—	13	2	no	—
257	C₄×D₄	—	13	2	no	—
258	C₂×C₄²	—	13	1	no	—
259	M₃₂	—	13	4	no	—
260	M₃₂	—	26	4	no	—
261	M₃₂	—	26	4	no	—
262	M₃₂	—	26	2	no	—
263	C₂×C₁₆	—	13	2	no	—
264	C₂×C₁₆	—	13	2	no	—
265	C₂×C₁₆	—	13	1	no	—
266	C₂×C₁₆	—	13	1	no	—
267	C₂×C₁₆	—	13	1	no	—
268	C₂×C₁₆	—	13	1	no	—
269	C₂×C₁₆	—	26	4	no	—
270	C₂×C₁₆	—	26	4	no	—
271	C₂×C₁₆	—	26	2	no	—
272	C₂×C₁₆	—	26	2	no	—
273	C₄⋊C₈	—	13	2	no	—
274	C₄⋊C₈	—	13	2	no	—
275	C₂²⋊C₈	—	13	2	no	—
276	C₂²⋊C₈	—	13	2	no	—
277	C₂²⋊C₈	C₂×#261	1	2	yes	—
278	C₂²⋊C₈	C₂×#261	1	2	yes	—
279	C₄×C₈	—	13	1	no	—
280	C₄×C₈	—	13	1	no	—
281	C₄×C₈	—	13	1	no	—
282	C₄×C₈	—	13	1	no	—
283	C₂×C₁₃	—	2	2	no	—
284	C₂×C₁₃	—	4	2	no	—
285	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
286	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
287	C₂×C₁₃	#3407	1	1	yes	characteristic
288	D₁₃	—	4	2	no	—
289	#41	—	2	2	no	—
290	#41	—	2	2	no	—
291	#41	—	26	2	no	—
292	#41	—	26	2	no	—
293	C₂×Q₈	—	13	1	no	—
294	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
295	C₂×D₄	—	13	2	no	—
296	C₂×D₄	—	13	2	no	—
297	C₂×D₄	—	13	2	no	—
298	C₂×D₄	—	13	1	no	—
299	C₂×D₄	D₁₃⋊C₈	1	2	yes	—
300	C₂×D₄	D₁₃⋊C₈	1	1	yes	characteristic
301	C₂²×C₄	—	13	1	no	—
302	C₂²×C₄	—	13	1	no	—
303	C₂²×C₄	—	13	2	no	—
304	C₂²×C₄	C₂²×#261	1	2	yes	—
305	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
306	M₁₆	—	2	2	no	—
307	M₁₆	—	26	4	no	—
308	M₁₆	—	26	4	no	—
309	M₁₆	—	26	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	13	2	no	—
312	C₂×C₈	—	13	2	no	—
313	C₂×C₈	—	13	1	no	—
314	C₂×C₈	—	13	1	no	—
315	C₂×C₈	—	13	2	no	—
316	C₂×C₈	—	26	2	no	—
317	C₂×C₈	—	26	2	no	—
318	C₂×C₈	—	26	2	no	—
319	C₂×C₈	—	26	2	no	—
320	C₂×C₈	C₂²×#261	1	1	yes	characteristic
321	C₂×C₈	C₂²×#261	1	1	yes	characteristic
322	C₂×C₈	C₂²×#261	1	2	yes	—
323	C₄⋊C₄	—	13	2	no	—
324	C₄⋊C₄	—	13	2	no	—
325	C₂²⋊C₄	—	13	2	no	—
326	C₂²⋊C₄	—	13	2	no	—
327	C₂²⋊C₄	—	13	2	no	—
328	C₂²⋊C₄	—	13	2	no	—
329	C₄²	—	13	1	no	—
330	C₄²	—	13	1	no	—
331	C₄²	—	13	1	no	—
332	C₄²	—	13	1	no	—
333	C₁₆	—	13	4	no	—
334	C₁₆	—	26	1	no	—
335	C₁₆	—	26	1	no	—
336	C₁₆	—	26	1	no	—
337	C₁₆	—	26	1	no	—
338	C₁₆	—	26	2	no	—
339	C₁₃	#33438	1	1	yes	characteristic, complemented, Sylow
340	C₂³	—	13	2	no	—
341	C₂³	C₂×D₁₃⋊C₈	1	2	yes	—
342	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
343	Q₈	—	2	2	no	—
344	Q₈	—	26	2	no	—
345	D₄	—	2	4	no	—
346	D₄	—	2	2	no	—
347	D₄	—	26	4	no	—
348	D₄	—	26	4	no	—
349	D₄	—	26	2	no	—
350	D₄	—	52	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	13	1	no	—
354	C₂×C₄	—	13	1	no	—
355	C₂×C₄	—	13	1	no	—
356	C₂×C₄	—	13	1	no	—
357	C₂×C₄	—	13	2	no	—
358	C₂×C₄	—	26	2	no	—
359	C₂×C₄	—	26	2	no	—
360	C₂×C₄	—	26	2	no	—
361	C₂×C₄	—	52	1	no	—
362	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
363	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
364	C₂×C₄	#88238	1	2	yes	—
365	C₂×C₄	D₄×#261	1	2	yes	—
366	C₂×C₄	C₂³×#261	1	1	yes	characteristic
367	C₈	—	2	1	no	—
368	C₈	—	2	1	no	—
369	C₈	—	2	2	no	—
370	C₈	—	13	1	no	—
371	C₈	—	13	1	no	—
372	C₈	—	13	2	no	—
373	C₈	—	26	2	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	2	no	—
377	C₂²	—	2	2	no	—
378	C₂²	—	2	2	no	—
379	C₂²	—	26	2	no	—
380	C₂²	—	26	2	no	—
381	C₂²	—	26	2	no	—
382	C₂²	#11719063	1	2	yes	—
383	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
384	C₄	—	2	2	no	—
385	C₄	—	13	2	no	—
386	C₄	—	26	2	no	—
387	C₄	—	26	1	no	—
388	C₄	—	26	1	no	—
389	C₄	—	26	1	no	—
390	C₄	#11719063	1	2	yes	—
391	C₄	C₂×#88238	1	1	yes	characteristic
392	C₄	C₂×D₄×#261	1	1	yes	characteristic
393	C₂	—	2	2	no	—
394	C₂	—	4	2	no	—
395	C₂	—	52	2	no	—
396	C₂	C₂×#11719063	1	1	yes	characteristic
397	C₂	D₁₃⋊#504	1	1	yes	characteristic
398	C₂	#13131804	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#408	C₄
prime radical	#11719537	C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	#408	#261
hypocenter	C₁₃	#33438
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
1. C₂×#11719096	C₂	semidirect
2. C₂×#11719162	C₂	semidirect
3. #13131221	C₂	semidirect
4. #13131218	C₂	semidirect
5. #13131217	C₂	semidirect
6. #13131214	C₂	semidirect
7. C₂×#11718260	C₂	semidirect
8. #13131068	C₂	semidirect
9. #13131066	C₂	semidirect
10. #13131042	C₂	semidirect
68. #88106	D₄	semidirect
235. Dic₁₃	#504	semidirect
339. C₁₃	#33438	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
C₂×#11719162	2
#13131221	2
#13131218	2
#13131217	2
#13131214	2
C₂×#11718260	2
#13131068	2
#13131066	2
#13131042	2
#13123544	2
#33438	13