Subgroups — #16017745

Order 3328.

632 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	C₂×#11718776	C₂	1	1	yes	maximal, characteristic
2	#13117027	C₂	1	4	yes	maximal, complemented
3	#13117021	C₂	1	2	yes	maximal, complemented
4	#13117016	C₂	1	2	yes	maximal, complemented
5	#13116625	C₂	1	2	yes	maximal, complemented
6	#13116551	C₂	1	2	yes	maximal, complemented
7	#13116447	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#117	—	2	1	no	—
11	C₂×#88184	—	2	1	no	—
12	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
13	C₁₃×#427	—	2	1	no	—
14	C₁₃×#427	—	2	1	no	—
15	C₁₃×#427	C₂²	1	2	yes	—
16	C₁₃×#427	C₂²	1	2	yes	complemented
17	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₁₃×#106	—	2	1	no	—
20	#11718776	—	2	1	no	—
21	#11718776	—	2	1	no	—
22	#11718776	C₂²	1	2	yes	—
23	#11718776	C₂²	1	2	yes	—
24	C₂×#88061	—	2	1	no	—
25	C₂×#88058	C₂²	1	1	yes	characteristic
26	C₂×#88052	C₂²	1	2	yes	—
27	C₂²⋊#6090	C₂²	1	2	yes	complemented
28	C₂²⋊#6090	C₂²	1	2	yes	complemented
29	#11718083	C₂²	1	4	yes	complemented
30	#11718083	C₂²	1	4	yes	complemented
31	#11718082	C₂²	1	4	yes	complemented
32	#11718081	C₂²	1	4	yes	complemented
33	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₂²⋊D₁₃	—	4	1	no	—
35	C₂²×D₅₂	—	2	1	no	—
36	C₂²×C₈×C₁₃	—	2	4	no	—
37	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×#117	—	2	2	no	—
44	C₁₃×#117	—	2	2	no	—
45	C₁₃×#117	—	4	1	no	—
46	C₁₃×#117	—	4	1	no	—
47	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
48	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
49	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
50	C₂×C₄²×C₁₃	—	4	1	no	—
51	#88184	—	2	2	no	—
52	#88184	—	2	2	no	—
53	#88184	—	4	1	no	—
54	#88184	—	4	1	no	—
55	C₂×#6103	—	4	1	no	—
56	C₂×C₄⋊Dic₁₃	—	2	1	no	—
57	C₂×D₁₀₄	—	2	4	no	—
58	C₂×D₁₀₄	—	2	1	no	—
59	C₂×D₁₀₄	C₂³	1	2	yes	—
60	C₁₃×#107	—	2	1	no	—
61	C₁₃×#107	—	2	1	no	—
62	C₁₃×#107	C₂³	1	2	yes	—
63	C₁₃×#107	C₂³	1	2	yes	—
64	C₁₃×#106	—	2	2	no	—
65	C₁₃×#106	—	2	2	no	—
66	#88061	—	2	2	no	—
67	#88061	—	2	2	no	—
68	#88058	—	2	1	no	—
69	#88058	C₂³	1	2	yes	—
70	#88052	—	2	4	no	—
71	C₂×#6090	C₂³	1	4	yes	—
72	#33648	—	13	1	no	maximal, Sylow
73	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
74	C₂³×D₁₃	—	4	1	no	—
75	C₂²×C₄×C₁₃	—	2	2	no	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	—	4	1	no	—
78	C₂²×C₄×C₁₃	—	4	1	no	—
79	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₂²⋊D₁₃	—	4	2	no	—
86	C₂×C₂²⋊D₁₃	—	4	2	no	—
87	C₂×C₂²⋊D₁₃	—	8	4	no	—
88	C₂²×Dic₁₃	—	4	1	no	—
89	C₂×D₅₂	—	2	2	no	—
90	C₂×D₅₂	—	4	2	no	—
91	C₂×D₅₂	—	4	2	no	—
92	C₂×D₅₂	—	4	1	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	—	2	2	no	—
97	C₂×C₈×C₁₃	—	2	2	no	—
98	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	#41	1	4	yes	—
102	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	4	1	no	—
106	C₁₃×C₄⋊C₄	—	4	1	no	—
107	C₁₃×C₂²⋊C₄	—	4	2	no	—
108	C₁₃×C₂²⋊C₄	—	4	2	no	—
109	C₄²×C₁₃	—	4	2	no	—
110	C₄²×C₁₃	—	4	2	no	—
111	#6103	—	4	2	no	—
112	#6103	—	4	2	no	—
113	C₄⋊Dic₁₃	—	2	2	no	—
114	C₄⋊Dic₁₃	—	4	1	no	—
115	D₁₀₄	—	2	4	no	—
116	D₁₀₄	—	4	2	no	—
117	#6090	—	2	4	no	—
118	C₂×#466	—	13	1	no	—
119	C₂×#427	D₁₃	1	1	yes	characteristic, complemented
120	#2234	—	13	4	no	—
121	#2233	—	13	2	no	—
122	#2232	—	13	2	no	—
123	#2159	—	13	2	no	—
124	#2140	—	13	2	no	—
125	#2112	—	13	1	no	—
126	C₂³×C₁₃	—	2	2	no	—
127	C₂³×C₁₃	—	2	2	no	—
128	C₂³×C₁₃	—	4	1	no	—
129	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
130	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
131	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
132	C₂²×D₁₃	—	4	2	no	—
133	C₂²×D₁₃	—	8	2	no	—
134	C₂²×D₁₃	—	8	2	no	—
135	C₂²×D₁₃	—	8	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	1	no	—
138	C₂×C₄×C₁₃	—	2	1	no	—
139	C₂×C₄×C₁₃	—	2	1	no	—
140	C₂×C₄×C₁₃	—	2	1	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	—	8	2	no	—
146	C₂×C₄×C₁₃	—	8	2	no	—
147	C₂×C₄×C₁₃	#123	1	2	yes	—
148	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₁₃	#137	1	2	yes	—
152	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₂²⋊D₁₃	—	8	4	no	—
154	C₂²⋊D₁₃	—	8	4	no	—
155	C₂×Dic₁₃	—	4	2	no	—
156	C₂×Dic₁₃	—	8	2	no	—
157	D₅₂	—	4	2	no	—
158	D₅₂	—	4	2	no	—
159	D₅₂	—	8	2	no	—
160	C₈×C₁₃	—	2	1	no	—
161	C₈×C₁₃	—	2	1	no	—
162	C₈×C₁₃	#123	1	2	yes	—
163	C₈×C₁₃	C₂×#41	1	2	yes	—
164	C₂²×D₈	—	13	1	no	—
165	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
166	C₂×#121	—	26	1	no	—
167	C₂×#117	—	2	1	no	—
168	C₂×M₃₂	—	13	2	no	—
169	#466	—	13	2	no	—
170	#466	—	13	2	no	—
171	#466	—	26	1	no	—
172	#466	—	26	1	no	—
173	#427	—	2	1	no	—
174	#427	—	2	1	no	—
175	#427	C₂×D₁₃	1	2	yes	—
176	#427	C₂×D₁₃	1	2	yes	complemented
177	C₂×#107	—	13	1	no	—
178	C₂×#107	C₂×D₁₃	1	1	yes	characteristic, complemented
179	C₂×#107	C₂×D₁₃	1	1	yes	characteristic, complemented
180	C₂×#106	—	2	1	no	—
181	C₂×#102	—	26	1	no	—
182	#367	—	13	4	no	—
183	#366	—	13	4	no	—
184	#357	—	13	4	no	—
185	#357	—	13	4	no	—
186	C₂²⋊C₁₆	—	13	2	no	—
187	C₂²⋊C₁₆	—	13	2	no	—
188	C₂²×C₁₃	—	2	1	no	—
189	C₂²×C₁₃	—	2	1	no	—
190	C₂²×C₁₃	—	2	1	no	—
191	C₂²×C₁₃	—	2	1	no	—
192	C₂²×C₁₃	—	2	1	no	—
193	C₂²×C₁₃	—	2	1	no	—
194	C₂²×C₁₃	—	2	1	no	—
195	C₂²×C₁₃	—	4	4	no	—
196	C₂²×C₁₃	#480	1	2	yes	—
197	C₂²×C₁₃	#509	1	2	yes	—
198	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
199	C₂×D₁₃	—	8	2	no	—
200	C₂×D₁₃	—	8	2	no	—
201	C₂×D₁₃	—	8	2	no	—
202	C₂×D₁₃	—	16	4	no	—
203	C₄×C₁₃	—	2	1	no	—
204	C₄×C₁₃	—	2	1	no	—
205	C₄×C₁₃	—	8	2	no	—
206	C₄×C₁₃	—	8	2	no	—
207	C₄×C₁₃	#480	1	2	yes	—
208	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
209	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
210	Dic₁₃	—	8	2	no	—
211	C₂²×D₄	—	26	1	no	—
212	C₂²×D₄	—	52	1	no	—
213	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
214	C₂×D₈	—	13	2	no	—
215	C₂×D₈	—	26	4	no	—
216	C₂×D₈	—	26	1	no	—
217	C₂²×C₈	—	2	4	no	—
218	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
219	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
220	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
221	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
222	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
223	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
224	#121	—	26	2	no	—
225	#121	—	26	2	no	—
226	#121	—	52	1	no	—
227	#121	—	52	1	no	—
228	#117	—	2	2	no	—
229	#117	—	2	2	no	—
230	#117	—	4	1	no	—
231	#117	—	4	1	no	—
232	C₂×C₄⋊C₄	—	2	1	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	26	1	no	—
235	C₂×C₂²⋊C₄	—	4	1	no	—
236	C₂×C₂²⋊C₄	—	52	1	no	—
237	C₂×C₄²	—	4	1	no	—
238	M₃₂	—	26	4	no	—
239	C₂×C₁₆	—	13	4	no	—
240	#107	—	2	1	no	—
241	#107	—	2	1	no	—
242	#107	—	13	2	no	—
243	#107	—	26	1	no	—
244	#107	C₂²×D₁₃	1	2	yes	—
245	#107	C₂²×D₁₃	1	2	yes	—
246	#106	—	2	2	no	—
247	#106	—	2	2	no	—
248	#102	—	26	2	no	—
249	#102	—	26	2	no	—
250	C₂×C₁₃	—	2	2	no	—
251	C₂×C₁₃	—	2	2	no	—
252	C₂×C₁₃	—	4	1	no	—
253	C₂×C₁₃	C₂×#480	1	1	yes	characteristic
254	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
255	C₂×C₁₃	#3414	1	1	yes	characteristic
256	D₁₃	—	16	2	no	—
257	C₂⁴	—	52	1	no	—
258	C₂⁴	#6104	1	1	yes	characteristic
259	C₂×D₄	—	26	2	no	—
260	C₂×D₄	—	52	2	no	—
261	C₂×D₄	—	52	2	no	—
262	C₂×D₄	—	52	2	no	—
263	C₂×D₄	—	52	1	no	—
264	C₂×D₄	—	52	2	no	—
265	C₂×D₄	—	104	4	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	52	1	no	—
271	C₂²×C₄	#6104	1	1	yes	characteristic
272	C₂²×C₄	#6104	1	1	yes	characteristic
273	C₂²×C₄	#6104	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
277	D₈	—	26	4	no	—
278	D₈	—	52	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	#6127	1	4	yes	—
285	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
286	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
287	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
288	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	26	2	no	—
294	C₄⋊C₄	—	52	1	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	52	2	no	—
298	C₂²⋊C₄	—	52	2	no	—
299	C₄²	—	4	2	no	—
300	C₄²	—	4	2	no	—
301	C₁₆	—	26	4	no	—
302	C₁₃	#33648	1	1	yes	characteristic, complemented, Sylow
303	C₂³	—	2	2	no	—
304	C₂³	—	2	2	no	—
305	C₂³	—	4	1	no	—
306	C₂³	—	52	2	no	—
307	C₂³	—	104	2	no	—
308	C₂³	—	104	2	no	—
309	C₂³	—	104	2	no	—
310	C₂³	C₂×#6104	1	1	yes	characteristic
311	C₂³	C₂×#6104	1	1	yes	characteristic
312	C₂³	C₂×#6104	1	1	yes	characteristic
313	D₄	—	52	2	no	—
314	D₄	—	52	2	no	—
315	D₄	—	104	4	no	—
316	D₄	—	104	4	no	—
317	D₄	—	104	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	8	2	no	—
328	C₂×C₄	—	8	2	no	—
329	C₂×C₄	—	52	2	no	—
330	C₂×C₄	—	104	2	no	—
331	C₂×C₄	#88183	1	2	yes	—
332	C₂×C₄	C₂×#6104	1	1	yes	characteristic
333	C₂×C₄	C₂×#6104	1	1	yes	characteristic
334	C₂×C₄	C₂×#6104	1	1	yes	characteristic
335	C₂×C₄	#88196	1	2	yes	—
336	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	#88183	1	2	yes	—
340	C₈	C₂×#6127	1	2	yes	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	4	4	no	—
349	C₂²	—	104	2	no	—
350	C₂²	—	104	2	no	—
351	C₂²	—	104	2	no	—
352	C₂²	—	208	4	no	—
353	C₂²	#11718628	1	2	yes	—
354	C₂²	#11718854	1	2	yes	—
355	C₂²	C₂²×#6104	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	8	2	no	—
359	C₄	—	8	2	no	—
360	C₄	—	104	2	no	—
361	C₄	#11718628	1	2	yes	—
362	C₄	C₂×#88183	1	1	yes	characteristic
363	C₄	C₂×#88196	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	2	2	no	—
366	C₂	—	4	1	no	—
367	C₂	—	208	2	no	—
368	C₂	C₂×#11718628	1	1	yes	characteristic
369	C₂	C₂×#11718854	1	1	yes	characteristic
370	C₂	#13127970	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#427	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#11718776	C₂
hypercenter	C₂×#427	D₁₃
hypocenter	C₁₃	#33648
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. #13117027	C₂	semidirect
3. #13117021	C₂	semidirect
4. #13117016	C₂	semidirect
5. #13116625	C₂	semidirect
6. #13116551	C₂	semidirect
7. #13116447	C₂	semidirect
8. C₂×C₁₃×#427	C₂	semidirect
16. C₁₃×#427	C₂²	semidirect
17. C₂×C₁₃×#107	C₂²	semidirect
18. C₂×C₁₃×#107	C₂²	semidirect
27. C₂²⋊#6090	C₂²	semidirect
28. C₂²⋊#6090	C₂²	semidirect
29. #11718083	C₂²	semidirect
30. #11718083	C₂²	semidirect
31. #11718082	C₂²	semidirect
32. #11718081	C₂²	semidirect
119. C₂×#427	D₁₃	semidirect
176. #427	C₂×D₁₃	semidirect
178. C₂×#107	C₂×D₁₃	semidirect
179. C₂×#107	C₂×D₁₃	semidirect
302. C₁₃	#33648	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718776	2
#13117027	2
#13117021	2
#13117016	2
#13116625	2
#13116551	2
#13116447	2
C₂×C₁₃×#427	2
#33648	13