Subgroups — #31809701

Order 17792.

432 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃₉⋊#401	C₂	1	1	yes	maximal, characteristic, complemented
2	#22580895	C₂	1	1	yes	maximal, characteristic, complemented
3	#22580894	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580891	C₂	1	1	yes	maximal, characteristic, complemented
5	#22580881	C₂	1	1	yes	maximal, characteristic
6	#22580881	C₂	1	1	yes	maximal, characteristic
7	#22580876	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₄×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#22580804	C₂	1	1	yes	maximal, characteristic, complemented
10	#22580804	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊C₄×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
12	C₂²⋊C₄×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
13	C₂²⋊C₄×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
14	#22580798	C₂	1	1	yes	maximal, characteristic, complemented
15	#397×C₁₃₉	C₂	1	1	yes	maximal, characteristic
16	C₂³×Dic₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
31	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
32	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
33	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
34	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
35	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
36	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
37	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
38	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
39	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
40	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
41	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
42	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
43	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
45	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
48	#95×C₁₃₉	C₂²	1	1	yes	characteristic
49	#95×C₁₃₉	C₂²	1	1	yes	characteristic
50	#95×C₁₃₉	C₂²	1	1	yes	characteristic
51	C₂²×Dic₁₃₉	—	2	2	no	—
52	C₂²×Dic₁₃₉	—	2	2	no	—
53	C₂²×Dic₁₃₉	—	2	2	no	—
54	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
55	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
59	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
60	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
61	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
62	C₄⋊Dic₁₃₉	—	2	2	no	—
63	Dic₁₃₉⋊C₄	—	2	2	no	—
64	Dic₁₃₉⋊C₄	—	2	2	no	—
65	Dic₁₃₉⋊C₄	—	2	2	no	—
66	Dic₁₃₉⋊C₄	—	2	2	no	—
67	C₂²⋊Dic₁₃₉	—	2	2	no	—
68	C₂²⋊Dic₁₃₉	—	2	2	no	—
69	C₂²⋊Dic₁₃₉	—	2	2	no	—
70	C₄×Dic₁₃₉	—	2	2	no	—
71	C₄×Dic₁₃₉	—	2	2	no	—
72	C₄×Dic₁₃₉	—	2	2	no	—
73	C₄×Dic₁₃₉	—	2	2	no	—
74	C₄×Dic₁₃₉	—	2	2	no	—
75	C₄×Dic₁₃₉	—	2	2	no	—
76	C₂⁴×C₁₃₉	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₃₉	—	2	2	no	—
84	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
85	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
86	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
87	C₂×Dic₁₃₉	—	2	1	no	—
88	C₂×Dic₁₃₉	—	2	1	no	—
89	C₂×Dic₁₃₉	—	2	1	no	—
90	C₂×Dic₁₃₉	—	2	1	no	—
91	C₂×Dic₁₃₉	—	2	1	no	—
92	C₂×Dic₁₃₉	—	2	1	no	—
93	C₂×Dic₁₃₉	—	2	1	no	—
94	C₂×Dic₁₃₉	—	2	1	no	—
95	C₂×Dic₁₃₉	—	2	1	no	—
96	C₂×Dic₁₃₉	—	2	1	no	—
97	C₂×Dic₁₃₉	—	2	1	no	—
98	C₂×Dic₁₃₉	—	2	1	no	—
99	C₂×Dic₁₃₉	—	2	1	no	—
100	C₂×Dic₁₃₉	—	2	1	no	—
101	C₂×Dic₁₃₉	—	2	1	no	—
102	C₂×Dic₁₃₉	—	2	1	no	—
103	C₂×Dic₁₃₉	—	2	1	no	—
104	C₂×Dic₁₃₉	—	2	1	no	—
105	C₂×Dic₁₃₉	—	2	1	no	—
106	C₂×Dic₁₃₉	—	4	4	no	—
107	C₂×Dic₁₃₉	#41	1	2	yes	—
108	C₂×Dic₁₃₉	#41	1	2	yes	—
109	C₂×Dic₁₃₉	#41	1	2	yes	—
110	C₂×Dic₁₃₉	#41	1	2	yes	—
111	C₂×Dic₁₃₉	#41	1	2	yes	—
112	C₂³×C₁₃₉	—	2	1	no	—
113	C₂³×C₁₃₉	—	2	1	no	—
114	C₂³×C₁₃₉	—	2	1	no	—
115	C₂³×C₁₃₉	—	2	1	no	—
116	C₂³×C₁₃₉	—	2	1	no	—
117	C₂³×C₁₃₉	—	2	1	no	—
118	C₂³×C₁₃₉	#41	1	2	yes	—
119	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
120	C₂×C₄×C₁₃₉	—	2	1	no	—
121	C₂×C₄×C₁₃₉	—	2	1	no	—
122	C₂×C₄×C₁₃₉	—	2	1	no	—
123	C₂×C₄×C₁₃₉	—	2	1	no	—
124	C₂×C₄×C₁₃₉	—	2	1	no	—
125	C₂×C₄×C₁₃₉	—	2	1	no	—
126	C₂×C₄×C₁₃₉	—	2	1	no	—
127	C₂×C₄×C₁₃₉	—	2	1	no	—
128	C₂×C₄×C₁₃₉	—	2	1	no	—
129	C₂×C₄×C₁₃₉	—	2	1	no	—
130	C₂×C₄×C₁₃₉	—	2	1	no	—
131	C₂×C₄×C₁₃₉	—	2	1	no	—
132	C₂×C₄×C₁₃₉	—	2	1	no	—
133	C₂×C₄×C₁₃₉	—	2	1	no	—
134	C₂×C₄×C₁₃₉	—	2	1	no	—
135	C₂×C₄×C₁₃₉	—	2	1	no	—
136	C₂×C₄×C₁₃₉	—	2	1	no	—
137	C₂×C₄×C₁₃₉	#41	1	2	yes	—
138	C₂²×C₁₃₉	—	2	2	no	—
139	C₂²×C₁₃₉	—	2	2	no	—
140	C₂²×C₁₃₉	—	2	2	no	—
141	C₂²×C₁₃₉	—	4	1	no	—
142	C₂²×C₁₃₉	—	4	1	no	—
143	C₂²×C₁₃₉	—	4	1	no	—
144	C₂²×C₁₃₉	—	4	1	no	—
145	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
146	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
147	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
148	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
149	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
150	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
151	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
152	C₄×C₁₃₉	—	2	2	no	—
153	C₄×C₁₃₉	—	4	1	no	—
154	C₄×C₁₃₉	—	4	1	no	—
155	C₄×C₁₃₉	—	4	1	no	—
156	C₄×C₁₃₉	—	4	1	no	—
157	C₄×C₁₃₉	—	4	1	no	—
158	Dic₁₃₉	—	2	2	no	—
159	Dic₁₃₉	—	2	2	no	—
160	Dic₁₃₉	—	4	1	no	—
161	Dic₁₃₉	—	4	1	no	—
162	Dic₁₃₉	—	4	1	no	—
163	Dic₁₃₉	—	4	1	no	—
164	Dic₁₃₉	—	4	1	no	—
165	Dic₁₃₉	#126	1	4	yes	—
166	C₂×C₁₃₉	—	4	2	no	—
167	C₂×C₁₃₉	C₂×#126	1	1	yes	characteristic
168	C₂×C₁₃₉	#529	1	1	yes	characteristic
169	C₂×C₁₃₉	#529	1	1	yes	characteristic
170	C₂×C₁₃₉	#529	1	1	yes	characteristic
171	C₂×C₁₃₉	#529	1	1	yes	characteristic
172	C₂×C₁₃₉	#529	1	1	yes	characteristic
173	C₂×C₁₃₉	#529	1	1	yes	characteristic
174	C₁₃₉	#2401	1	1	yes	characteristic, complemented, Sylow
175	#2401	—	139	1	no	maximal, Sylow
176	#401	—	139	1	no	—
177	#400	—	139	1	no	—
178	#400	—	139	1	no	—
179	#400	—	139	1	no	—
180	#397	—	139	1	no	—
181	#397	—	139	1	no	—
182	#397	D₁₃₉	1	1	yes	characteristic
183	#396	—	139	1	no	—
184	C₄×C₄⋊C₄	—	139	1	no	—
185	C₄×C₄⋊C₄	—	139	1	no	—
186	C₄×C₄⋊C₄	—	139	1	no	—
187	C₄×C₂²⋊C₄	—	139	1	no	—
188	C₄×C₂²⋊C₄	—	139	1	no	—
189	C₄×C₂²⋊C₄	—	139	1	no	—
190	#376	—	139	1	no	—
191	C₂³×C₄	—	139	1	no	—
192	C₂×C₄⋊C₄	—	139	1	no	—
193	C₂×C₄⋊C₄	—	139	1	no	—
194	C₂×C₄⋊C₄	—	139	1	no	—
195	C₂×C₄⋊C₄	—	139	1	no	—
196	C₂×C₄⋊C₄	—	139	1	no	—
197	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
198	C₂×C₂²⋊C₄	—	139	1	no	—
199	C₂×C₂²⋊C₄	—	139	1	no	—
200	C₂×C₂²⋊C₄	—	139	1	no	—
201	C₂×C₂²⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
202	C₂×C₂²⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
204	C₂×C₄²	—	139	1	no	—
205	C₂×C₄²	—	139	1	no	—
206	C₂×C₄²	—	139	1	no	—
207	C₂×C₄²	—	139	1	no	—
208	C₂×C₄²	—	139	1	no	—
209	C₂×C₄²	—	139	1	no	—
210	#95	—	139	1	no	—
211	#95	—	139	1	no	—
212	#95	—	139	1	no	—
213	#95	—	139	1	no	—
214	#95	—	139	1	no	—
215	#95	—	139	1	no	—
216	#95	—	139	1	no	—
217	#95	—	139	1	no	—
218	#95	—	139	1	no	—
219	#95	—	139	1	no	—
220	#95	—	139	1	no	—
221	#95	—	139	1	no	—
222	#95	—	139	1	no	—
223	#95	C₂×D₁₃₉	1	1	yes	characteristic
224	#95	C₂×D₁₃₉	1	1	yes	characteristic
225	#95	C₂×D₁₃₉	1	1	yes	characteristic
226	C₂⁴	C₂²×D₁₃₉	1	1	yes	characteristic
227	C₂²×C₄	—	278	2	no	—
228	C₂²×C₄	—	278	2	no	—
229	C₂²×C₄	—	278	2	no	—
230	C₂²×C₄	—	139	1	no	—
231	C₂²×C₄	—	139	1	no	—
232	C₂²×C₄	—	139	1	no	—
233	C₂²×C₄	—	139	1	no	—
234	C₂²×C₄	—	139	1	no	—
235	C₂²×C₄	—	139	1	no	—
236	C₂²×C₄	—	139	1	no	—
237	C₂²×C₄	—	139	1	no	—
238	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
239	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
240	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
241	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
243	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	278	2	no	—
246	C₄⋊C₄	—	278	2	no	—
247	C₄⋊C₄	—	278	2	no	—
248	C₄⋊C₄	—	278	2	no	—
249	C₄⋊C₄	—	278	2	no	—
250	C₂²⋊C₄	—	2	2	no	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	—	278	2	no	—
254	C₂²⋊C₄	—	278	2	no	—
255	C₂²⋊C₄	—	278	2	no	—
256	C₄²	—	278	2	no	—
257	C₄²	—	278	2	no	—
258	C₄²	—	278	2	no	—
259	C₄²	—	278	2	no	—
260	C₄²	—	278	2	no	—
261	C₄²	—	278	2	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	#14555784	1	2	yes	—
269	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	278	1	no	—
288	C₂×C₄	—	278	1	no	—
289	C₂×C₄	—	278	1	no	—
290	C₂×C₄	—	278	1	no	—
291	C₂×C₄	—	278	1	no	—
292	C₂×C₄	—	278	1	no	—
293	C₂×C₄	—	278	1	no	—
294	C₂×C₄	—	278	1	no	—
295	C₂×C₄	—	278	1	no	—
296	C₂×C₄	—	278	1	no	—
297	C₂×C₄	—	278	1	no	—
298	C₂×C₄	—	278	1	no	—
299	C₂×C₄	—	278	1	no	—
300	C₂×C₄	—	278	1	no	—
301	C₂×C₄	—	278	1	no	—
302	C₂×C₄	—	278	1	no	—
303	C₂×C₄	—	278	1	no	—
304	C₂×C₄	—	278	1	no	—
305	C₂×C₄	—	278	1	no	—
306	C₂×C₄	—	556	4	no	—
307	C₂×C₄	—	139	2	no	—
308	C₂×C₄	—	139	2	no	—
309	C₂×C₄	—	139	2	no	—
310	C₂×C₄	—	139	2	no	—
311	C₂×C₄	—	139	2	no	—
312	C₂×C₄	#14555784	1	2	yes	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	#41×D₁₃₉	1	1	yes	characteristic
321	C₂²	#41×D₁₃₉	1	1	yes	characteristic
322	C₂²	#41×D₁₃₉	1	1	yes	characteristic
323	C₂²	#41×D₁₃₉	1	1	yes	characteristic
324	C₂²	#41×D₁₃₉	1	1	yes	characteristic
325	C₂²	C₂×#14555784	1	1	yes	characteristic
326	C₂²	C₂×#14555784	1	1	yes	characteristic
327	C₄	—	2	2	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	278	2	no	—
334	C₄	—	278	2	no	—
335	C₄	—	556	1	no	—
336	C₄	—	556	1	no	—
337	C₄	—	556	1	no	—
338	C₄	—	556	1	no	—
339	C₄	—	556	1	no	—
340	C₄	—	139	4	no	—
341	C₂	—	4	2	no	—
342	C₂	#126×D₁₃₉	1	1	yes	characteristic
343	C₂	#22581480	1	1	yes	characteristic
344	C₂	#22581480	1	1	yes	characteristic
345	C₂	#22581480	1	1	yes	characteristic
346	C₂	#22581482	1	1	yes	characteristic
347	C₂	#22581484	1	1	yes	characteristic
348	C₂	#22581486	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃₉
commutator, socle, cosocle	C₂³×C₁₃₉	C₂⁴
Fitting	#397×C₁₃₉	C₂
prime radical	C₂⁴×C₁₃₉	C₂³
hypercenter	#397	D₁₃₉
hypocenter	C₁₃₉	#2401
LCS	G ▷ C₂³×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ #397

Complements

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

N	H ≅ G/N	type
1. C₁₃₉⋊#401	C₂	semidirect
2. #22580895	C₂	semidirect
3. #22580894	C₂	semidirect
4. #22580891	C₂	semidirect
8. C₄⋊C₄×Dic₁₃₉	C₂	semidirect
9. #22580804	C₂	semidirect
10. #22580804	C₂	semidirect
14. #22580798	C₂	semidirect
174. C₁₃₉	#2401	semidirect

Maximal subgroups

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

subgroup	index
C₁₃₉⋊#401	2
#22580895	2
#22580894	2
#22580891	2
#22580881	2
#22580876	2
C₄⋊C₄×Dic₁₃₉	2
#22580804	2
C₂²⋊C₄×Dic₁₃₉	2
#22580798	2
#397×C₁₃₉	2
#2401	139