Subgroups — #31809003

Order 17792.

512 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×D₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#22580840	C₂	1	1	yes	maximal, characteristic, complemented
3	#22580839	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580838	C₂	1	1	yes	maximal, characteristic, complemented
5	#22580837	C₂	1	1	yes	maximal, characteristic, complemented
6	#22580836	C₂	1	1	yes	maximal, characteristic, complemented
7	#22580835	C₂	1	1	yes	maximal, characteristic, complemented
8	#22580834	C₂	1	1	yes	maximal, characteristic, complemented
9	#22580815	C₂	1	1	yes	maximal, characteristic
10	#22580815	C₂	1	1	yes	maximal, characteristic
11	#22580812	C₂	1	1	yes	maximal, characteristic
12	#22580810	C₂	1	1	yes	maximal, characteristic
13	C₄×(C₂×D₁₃₉)⋊C₄	C₂	1	1	yes	maximal, characteristic
14	#22580800	C₂	1	1	yes	maximal, characteristic
15	#387×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×D₁₃₉	C₄	1	8	yes	complemented
26	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₁₃₉	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	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
47	C₂×C₄²×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	#95×C₁₃₉	C₂²	1	1	yes	characteristic
52	C₂³×D₁₃₉	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
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	C₂×C₄×D₁₃₉	—	2	2	no	—
61	C₂×C₄×D₁₃₉	—	2	2	no	—
62	C₂×C₄×D₁₃₉	—	4	1	no	—
63	C₂×C₄×D₁₃₉	—	4	1	no	—
64	C₂×C₄×D₁₃₉	—	4	1	no	—
65	C₂×C₄×D₁₃₉	—	4	1	no	—
66	C₂×C₄×D₁₃₉	C₂×C₄	1	4	yes	—
67	C₂×C₄×D₁₃₉	C₂×C₄	1	4	yes	—
68	C₂×C₄×D₁₃₉	C₂×C₄	1	4	yes	—
69	C₄⋊Dic₁₃₉	—	4	1	no	—
70	C₄⋊Dic₁₃₉	C₂×C₄	1	4	yes	—
71	Dic₁₃₉⋊C₄	—	4	1	no	—
72	Dic₁₃₉⋊C₄	—	4	1	no	—
73	Dic₁₃₉⋊C₄	C₂×C₄	1	4	yes	—
74	Dic₁₃₉⋊C₄	C₂×C₄	1	4	yes	—
75	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
76	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
77	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
78	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
79	C₄×Dic₁₃₉	—	4	1	no	—
80	C₄×Dic₁₃₉	—	4	1	no	—
81	C₄×Dic₁₃₉	—	4	1	no	—
82	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
87	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
89	C₄⋊C₄×C₁₃₉	—	4	1	no	—
90	C₄⋊C₄×C₁₃₉	C₂×C₄	1	4	yes	—
91	C₄²×C₁₃₉	—	4	1	no	—
92	C₂²×D₁₃₉	—	2	1	no	—
93	C₂²×D₁₃₉	—	2	1	no	—
94	C₂²×D₁₃₉	—	2	1	no	—
95	C₂²×D₁₃₉	—	2	1	no	—
96	C₂²×D₁₃₉	C₂²×C₄	1	2	yes	—
97	C₂²×D₁₃₉	#41	1	2	yes	—
98	C₂²×D₁₃₉	#41	1	2	yes	—
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	—	2	1	no	—
107	C₂×Dic₁₃₉	—	2	1	no	—
108	C₂×Dic₁₃₉	—	2	1	no	—
109	C₂×Dic₁₃₉	—	2	1	no	—
110	C₂×Dic₁₃₉	—	2	1	no	—
111	C₂×Dic₁₃₉	—	2	1	no	—
112	C₂×Dic₁₃₉	—	2	1	no	—
113	C₂×Dic₁₃₉	—	2	1	no	—
114	C₂×Dic₁₃₉	—	2	1	no	—
115	C₂×Dic₁₃₉	—	2	1	no	—
116	C₂×Dic₁₃₉	—	2	1	no	—
117	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
118	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
119	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
120	C₄×D₁₃₉	—	2	8	no	—
121	C₄×D₁₃₉	—	4	4	no	—
122	C₄×D₁₃₉	—	4	4	no	—
123	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
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	—	2	1	no	—
138	C₂×C₄×C₁₃₉	—	2	1	no	—
139	C₂×C₄×C₁₃₉	—	2	1	no	—
140	C₂×C₄×C₁₃₉	—	2	1	no	—
141	C₂×C₄×C₁₃₉	—	2	1	no	—
142	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
143	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
144	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
145	C₂²×C₁₃₉	C₂³×C₄	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	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂²×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂²×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
151	C₂²×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
152	C₂×D₁₃₉	—	2	2	no	—
153	C₂×D₁₃₉	—	2	2	no	—
154	C₂×D₁₃₉	—	2	2	no	—
155	C₂×D₁₃₉	—	2	2	no	—
156	C₂×D₁₃₉	—	2	2	no	—
157	C₂×D₁₃₉	—	2	2	no	—
158	C₂×D₁₃₉	#117	1	4	yes	—
159	C₄×C₁₃₉	—	2	2	no	—
160	C₄×C₁₃₉	—	2	2	no	—
161	C₄×C₁₃₉	—	2	2	no	—
162	C₄×C₁₃₉	—	4	1	no	—
163	C₄×C₁₃₉	—	4	1	no	—
164	C₄×C₁₃₉	—	4	1	no	—
165	C₄×C₁₃₉	—	4	1	no	—
166	Dic₁₃₉	—	2	2	no	—
167	Dic₁₃₉	—	2	2	no	—
168	Dic₁₃₉	—	2	2	no	—
169	Dic₁₃₉	—	4	1	no	—
170	Dic₁₃₉	—	4	1	no	—
171	Dic₁₃₉	—	4	1	no	—
172	Dic₁₃₉	—	4	1	no	—
173	C₂×C₁₃₉	C₂×#117	1	1	yes	characteristic
174	C₂×C₁₃₉	#518	1	1	yes	characteristic
175	C₂×C₁₃₉	#519	1	1	yes	characteristic
176	C₂×C₁₃₉	#539	1	1	yes	characteristic
177	C₂×C₁₃₉	#539	1	1	yes	characteristic
178	C₂×C₁₃₉	#539	1	1	yes	characteristic
179	C₂×C₁₃₉	#539	1	1	yes	characteristic
180	D₁₃₉	—	2	4	no	—
181	C₁₃₉	#2313	1	1	yes	characteristic, complemented, Sylow
182	#2313	—	139	1	no	maximal, Sylow
183	C₂²×C₄⋊C₄	—	139	1	no	—
184	#387	—	139	1	no	—
185	#387	—	139	1	no	—
186	#387	—	139	1	no	—
187	#387	—	139	1	no	—
188	#387	—	139	1	no	—
189	#387	—	139	1	no	—
190	#387	—	139	1	no	—
191	#387	D₁₃₉	1	1	yes	characteristic, complemented
192	#381	—	139	1	no	—
193	#381	—	139	1	no	—
194	#380	—	139	1	no	—
195	#380	—	139	1	no	—
196	C₄×C₂²⋊C₄	—	139	1	no	—
197	C₄×C₂²⋊C₄	—	139	1	no	—
198	C₂³×C₄	—	139	1	no	—
199	C₂³×C₄	—	139	1	no	—
200	C₂³×C₄	—	139	1	no	—
201	C₂×C₄⋊C₄	—	139	1	no	—
202	C₂×C₄⋊C₄	—	139	1	no	—
203	C₂×C₄⋊C₄	—	139	1	no	—
204	C₂×C₄⋊C₄	—	139	1	no	—
205	C₂×C₄⋊C₄	—	139	1	no	—
206	C₂×C₄⋊C₄	—	139	1	no	—
207	C₂×C₄⋊C₄	—	139	8	no	—
208	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
209	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	—	139	1	no	—
211	C₂×C₂²⋊C₄	—	139	1	no	—
212	C₂×C₂²⋊C₄	—	139	1	no	—
213	C₂×C₂²⋊C₄	—	139	1	no	—
214	C₂×C₄²	—	139	1	no	—
215	C₂×C₄²	—	139	1	no	—
216	C₂×C₄²	—	139	1	no	—
217	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
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	—	139	1	no	—
224	#95	—	139	1	no	—
225	#95	—	139	1	no	—
226	#95	—	139	1	no	—
227	#95	—	139	1	no	—
228	#95	—	139	1	no	—
229	#95	—	139	1	no	—
230	#95	C₂×D₁₃₉	1	1	yes	characteristic
231	#95	C₂×D₁₃₉	1	1	yes	characteristic
232	#95	C₂×D₁₃₉	1	1	yes	characteristic
233	#95	C₂×D₁₃₉	1	1	yes	characteristic
234	C₂⁴	—	139	1	no	—
235	C₂²×C₄	—	278	2	no	—
236	C₂²×C₄	—	278	2	no	—
237	C₂²×C₄	—	556	1	no	—
238	C₂²×C₄	—	556	1	no	—
239	C₂²×C₄	—	556	1	no	—
240	C₂²×C₄	—	556	1	no	—
241	C₂²×C₄	—	139	1	no	—
242	C₂²×C₄	—	139	1	no	—
243	C₂²×C₄	—	139	1	no	—
244	C₂²×C₄	—	139	1	no	—
245	C₂²×C₄	—	139	1	no	—
246	C₂²×C₄	—	139	1	no	—
247	C₂²×C₄	—	139	1	no	—
248	C₂²×C₄	—	139	4	no	—
249	C₂²×C₄	—	139	4	no	—
250	C₂²×C₄	—	139	4	no	—
251	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
252	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
254	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
255	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
256	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
257	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	556	1	no	—
260	C₄⋊C₄	—	556	1	no	—
261	C₄⋊C₄	—	556	1	no	—
262	C₄⋊C₄	—	139	4	no	—
263	C₄⋊C₄	—	139	4	no	—
264	C₄⋊C₄	—	139	4	no	—
265	C₄⋊C₄	C₄×D₁₃₉	1	4	yes	—
266	C₂²⋊C₄	—	278	2	no	—
267	C₂²⋊C₄	—	278	2	no	—
268	C₂²⋊C₄	—	278	2	no	—
269	C₂²⋊C₄	—	278	2	no	—
270	C₄²	—	4	1	no	—
271	C₄²	—	556	1	no	—
272	C₄²	—	556	1	no	—
273	C₄²	—	556	1	no	—
274	C₂³	—	278	1	no	—
275	C₂³	—	278	1	no	—
276	C₂³	—	278	1	no	—
277	C₂³	—	278	1	no	—
278	C₂³	—	139	2	no	—
279	C₂³	—	139	2	no	—
280	C₂³	—	139	2	no	—
281	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
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	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	278	8	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	—	278	1	no	—
307	C₂×C₄	—	278	1	no	—
308	C₂×C₄	—	278	1	no	—
309	C₂×C₄	—	278	1	no	—
310	C₂×C₄	—	278	1	no	—
311	C₂×C₄	—	278	1	no	—
312	C₂×C₄	—	278	1	no	—
313	C₂×C₄	—	278	1	no	—
314	C₂×C₄	—	278	1	no	—
315	C₂×C₄	—	278	1	no	—
316	C₂×C₄	—	278	1	no	—
317	C₂×C₄	—	278	1	no	—
318	C₂×C₄	—	278	1	no	—
319	C₂×C₄	—	556	4	no	—
320	C₂×C₄	—	556	4	no	—
321	C₂×C₄	—	139	2	no	—
322	C₂×C₄	—	139	2	no	—
323	C₂×C₄	—	139	2	no	—
324	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
325	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
326	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
327	C₂²	—	278	2	no	—
328	C₂²	—	278	2	no	—
329	C₂²	—	278	2	no	—
330	C₂²	—	278	2	no	—
331	C₂²	—	278	2	no	—
332	C₂²	—	278	2	no	—
333	C₂²	—	139	4	no	—
334	C₂²	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
335	C₂²	#41×D₁₃₉	1	1	yes	characteristic
336	C₂²	#41×D₁₃₉	1	1	yes	characteristic
337	C₂²	#17984887	1	1	yes	characteristic
338	C₂²	#17984888	1	1	yes	characteristic
339	C₂²	#17984889	1	1	yes	characteristic
340	C₂²	#17984890	1	1	yes	characteristic
341	C₄	—	2	2	no	—
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	278	2	no	—
349	C₄	—	278	2	no	—
350	C₄	—	278	2	no	—
351	C₄	—	556	1	no	—
352	C₄	—	556	1	no	—
353	C₄	—	556	1	no	—
354	C₄	—	556	1	no	—
355	C₂	—	278	4	no	—
356	C₂	#117×D₁₃₉	1	1	yes	characteristic
357	C₂	#22581425	1	1	yes	characteristic
358	C₂	#22581429	1	1	yes	characteristic
359	C₂	#22581544	1	1	yes	characteristic
360	C₂	#22581546	1	1	yes	characteristic
361	C₂	#22581549	1	1	yes	characteristic
362	C₂	#22581551	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃₉
commutator	C₂²×C₁₃₉	C₂³×C₄
Fitting	#387×C₁₃₉	C₂
socle, cosocle	C₂³×C₁₃₉	C₂⁴
prime radical	C₂³×D₁₃₉	C₂³
hypercenter	#387	D₁₃₉
hypocenter	C₁₃₉	#2313
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ #387

Complements

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

N	H ≅ G/N	type
2. #22580840	C₂	semidirect
3. #22580839	C₂	semidirect
4. #22580838	C₂	semidirect
5. #22580837	C₂	semidirect
6. #22580836	C₂	semidirect
7. #22580835	C₂	semidirect
8. #22580834	C₂	semidirect
15. #387×C₁₃₉	C₂	semidirect
25. C₄⋊C₄×D₁₃₉	C₄	semidirect
181. C₁₃₉	#2313	semidirect
191. #387	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×D₁₃₉	2
#22580840	2
#22580839	2
#22580838	2
#22580837	2
#22580836	2
#22580835	2
#22580834	2
#22580815	2
#22580812	2
#22580810	2
C₄×(C₂×D₁₃₉)⋊C₄	2
#22580800	2
#387×C₁₃₉	2
#2313	139