Subgroups — #31815276

Order 17792.

484 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×#17984863	C₂	1	1	yes	maximal, characteristic
2	#22581435	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581432	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581055	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581054	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581052	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#14555775	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581045	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14555770	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581033	C₂	1	1	yes	maximal, characteristic, complemented
11	#107×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#22580953	C₂	1	1	yes	maximal, characteristic, complemented
13	#22580951	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580910	C₂	1	1	yes	maximal, characteristic, complemented
15	#440×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555786	C₂²	1	1	yes	characteristic
17	C₂×#14555784	C₂²	1	1	yes	characteristic
18	#17984863	C₄	1	8	yes	complemented
19	C₂×#14555775	C₂²	1	1	yes	characteristic
20	C₂×#14555774	C₂²	1	1	yes	characteristic
21	C₂×#14555770	C₂²	1	1	yes	characteristic
22	C₂×#14555768	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₁₃₉	C₂²	1	1	yes	characteristic
24	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
25	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
26	#17984807	C₂²	1	1	yes	characteristic
27	C₄×D₅₅₆	—	2	1	no	—
28	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
29	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
30	#17984804	—	2	1	no	—
31	#17984804	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₁₃₉	—	2	1	no	—
33	(C₂²⋊D₁₃₉)⋊C₄	—	2	1	no	—
34	#17984799	—	2	1	no	—
35	#17984799	C₂²	1	1	yes	characteristic
36	#17984798	—	2	1	no	—
37	C₂×Dic₁₃₉⋊C₄	—	2	1	no	—
38	C₄⋊C₄×D₁₃₉	—	2	1	no	—
39	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
40	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
41	#17984776	C₂²	1	1	yes	characteristic
42	#17984775	C₂²	1	1	yes	characteristic
43	#17984775	C₂²	1	1	yes	characteristic
44	#17984767	C₂²	1	1	yes	characteristic
45	#17984766	C₂²	1	1	yes	characteristic
46	#17984765	C₂²	1	1	yes	characteristic
47	#17984764	C₂²	1	1	yes	characteristic
48	#17984763	C₂²	1	1	yes	characteristic
49	#17984762	C₂²	1	1	yes	characteristic
50	#17984752	C₂²	1	1	yes	characteristic
51	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
52	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
56	#107×C₁₃₉	C₂²	1	1	yes	characteristic
57	#103×C₁₃₉	C₂²	1	1	yes	characteristic
58	#102×C₁₃₉	C₂²	1	1	yes	characteristic
59	#97×C₁₃₉	C₂²	1	1	yes	characteristic
60	#14555786	—	2	2	no	—
61	#14555786	C₂×C₄	1	4	yes	complemented
62	#14555784	—	2	2	no	—
63	#14555784	C₂×C₄	1	4	yes	complemented
64	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
65	C₂×D₅₅₆	—	2	1	no	—
66	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
68	C₂²×Dic₁₃₉	—	2	1	no	—
69	C₂²×Dic₁₃₉	—	2	1	no	—
70	C₂×C₄×D₁₃₉	—	2	1	no	—
71	C₂×C₄×D₁₃₉	—	2	1	no	—
72	C₂×C₄×D₁₃₉	—	2	1	no	—
73	C₂×C₄×D₁₃₉	—	2	1	no	—
74	C₂×C₄×D₁₃₉	—	2	1	no	—
75	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
76	#14555775	C₂×C₄	1	4	yes	complemented
77	#14555774	C₂×C₄	1	4	yes	complemented
78	#14555770	C₂×C₄	1	4	yes	complemented
79	#14555768	—	2	2	no	—
80	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
82	C₈×D₁₃₉	C₂×C₄	1	4	yes	complemented
83	C₄⋊Dic₁₃₉	—	2	1	no	—
84	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
86	Dic₁₃₉⋊C₄	—	2	1	no	—
87	Dic₁₃₉⋊C₄	—	2	1	no	—
88	Dic₁₃₉⋊C₄	—	2	1	no	—
89	Dic₁₃₉⋊C₄	—	2	1	no	—
90	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
91	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
92	C₂²⋊Dic₁₃₉	—	2	1	no	—
93	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
94	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
95	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
96	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
97	C₄×Dic₁₃₉	—	2	1	no	—
98	C₄×Dic₁₃₉	—	2	1	no	—
99	C₄×Dic₁₃₉	—	2	1	no	—
100	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₁₃₉	—	2	1	no	—
104	SD₁₆×C₁₃₉	C₂×C₄	1	4	yes	complemented
105	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₁₃₉	—	2	1	no	—
108	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
111	C₄²×C₁₃₉	—	2	1	no	—
112	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
113	C₂²×D₁₃₉	—	2	1	no	—
114	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
115	Dic₂₇₈	—	2	1	no	—
116	Dic₂₇₈	C₂²×C₄	1	2	yes	—
117	D₅₅₆	—	2	4	no	—
118	D₅₅₆	—	2	1	no	—
119	D₅₅₆	C₂²×C₄	1	2	yes	—
120	C₂²⋊D₁₃₉	—	2	4	no	—
121	C₂×Dic₁₃₉	—	2	1	no	—
122	C₂×Dic₁₃₉	—	2	4	no	—
123	C₂×Dic₁₃₉	—	2	1	no	—
124	C₂×Dic₁₃₉	—	2	1	no	—
125	C₂×Dic₁₃₉	—	8	1	no	—
126	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
129	C₄×D₁₃₉	—	2	4	no	—
130	C₄×D₁₃₉	—	8	1	no	—
131	C₄×D₁₃₉	—	8	1	no	—
132	C₄×D₁₃₉	—	8	1	no	—
133	C₄×D₁₃₉	—	8	1	no	—
134	C₄×D₁₃₉	C₂²×C₄	1	2	yes	—
135	C₄×D₁₃₉	#41	1	2	yes	—
136	C₁₃₉⋊C₈	—	2	1	no	—
137	C₁₃₉⋊C₈	C₂²×C₄	1	2	yes	—
138	C₂³×C₁₃₉	—	2	1	no	—
139	Q₈×C₁₃₉	—	2	1	no	—
140	Q₈×C₁₃₉	C₂²×C₄	1	2	yes	—
141	D₄×C₁₃₉	—	2	1	no	—
142	D₄×C₁₃₉	C₂²×C₄	1	2	yes	—
143	C₂×C₄×C₁₃₉	—	2	1	no	—
144	C₂×C₄×C₁₃₉	—	2	1	no	—
145	C₂×C₄×C₁₃₉	—	2	1	no	—
146	C₂×C₄×C₁₃₉	—	8	1	no	—
147	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
150	C₈×C₁₃₉	—	2	1	no	—
151	C₈×C₁₃₉	C₂²×C₄	1	2	yes	—
152	C₂²×C₁₃₉	—	2	2	no	—
153	C₂²×C₁₃₉	—	4	1	no	—
154	C₂²×C₁₃₉	—	4	1	no	—
155	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₁₃₉	—	2	1	no	—
157	C₂×D₁₃₉	—	2	1	no	—
158	C₂×D₁₃₉	—	2	2	no	—
159	C₂×D₁₃₉	—	4	1	no	—
160	C₂×D₁₃₉	—	4	1	no	—
161	C₂×D₁₃₉	C₄×D₄	1	2	yes	—
162	C₄×C₁₃₉	—	2	2	no	—
163	C₄×C₁₃₉	—	2	1	no	—
164	C₄×C₁₃₉	—	2	1	no	—
165	C₄×C₁₃₉	—	4	1	no	—
166	C₄×C₁₃₉	—	4	1	no	—
167	C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
168	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
169	Dic₁₃₉	—	2	2	no	—
170	Dic₁₃₉	—	2	1	no	—
171	Dic₁₃₉	—	2	1	no	—
172	Dic₁₃₉	—	4	1	no	—
173	Dic₁₃₉	—	4	1	no	—
174	Dic₁₃₉	C₄×D₄	1	2	yes	—
175	C₂×C₁₃₉	—	4	2	no	—
176	C₂×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂×C₁₃₉	#576	1	1	yes	characteristic
178	C₂×C₁₃₉	#578	1	1	yes	characteristic
179	D₁₃₉	—	2	2	no	—
180	D₁₃₉	—	4	2	no	—
181	C₁₃₉	#2951	1	1	yes	characteristic, complemented, Sylow
182	#2951	—	139	1	no	maximal, Sylow
183	C₂×#135	—	139	1	no	—
184	#520	—	139	1	no	—
185	#520	—	139	1	no	—
186	#440	—	139	1	no	—
187	#440	—	139	1	no	—
188	#440	—	139	1	no	—
189	#440	D₁₃₉	1	1	yes	characteristic, complemented
190	C₄×Q₁₆	—	139	1	no	—
191	C₄×Q₁₆	—	139	1	no	—
192	C₄×D₈	—	139	1	no	—
193	C₄×D₈	—	139	1	no	—
194	C₂×#107	—	139	1	no	—
195	#417	—	139	1	no	—
196	#417	—	139	1	no	—
197	#405	—	139	1	no	—
198	C₂×#41	—	139	1	no	—
199	C₂×#41	—	139	1	no	—
200	#135	—	139	8	no	—
201	C₂×Q₁₆	—	139	1	no	—
202	C₂×SD₁₆	—	139	1	no	—
203	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
204	C₂×D₈	—	139	1	no	—
205	C₂×M₁₆	—	139	1	no	—
206	C₂²×C₈	—	139	1	no	—
207	C₄×Q₈	—	139	1	no	—
208	C₄×Q₈	—	139	1	no	—
209	C₄×Q₈	—	139	1	no	—
210	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
211	C₄×D₄	—	278	1	no	—
212	C₄×D₄	—	278	1	no	—
213	C₄×D₄	—	278	1	no	—
214	C₄×D₄	—	278	1	no	—
215	C₄×D₄	—	139	1	no	—
216	C₄×D₄	—	139	1	no	—
217	C₄×D₄	—	139	1	no	—
218	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
219	#117	—	278	1	no	—
220	#117	—	278	1	no	—
221	#117	—	139	1	no	—
222	C₂×C₄⋊C₄	—	278	1	no	—
223	C₂×C₄⋊C₄	—	278	1	no	—
224	C₂×C₄⋊C₄	—	139	1	no	—
225	C₂×C₄⋊C₄	—	139	1	no	—
226	#107	—	139	1	no	—
227	#107	—	139	1	no	—
228	#107	—	139	1	no	—
229	#107	C₂×D₁₃₉	1	1	yes	characteristic
230	#103	—	139	1	no	—
231	#103	—	139	1	no	—
232	#103	—	139	1	no	—
233	#103	C₂×D₁₃₉	1	1	yes	characteristic
234	#102	—	139	1	no	—
235	#102	—	139	1	no	—
236	#102	—	139	1	no	—
237	#102	C₂×D₁₃₉	1	1	yes	characteristic
238	#97	—	139	1	no	—
239	#97	C₂×D₁₃₉	1	1	yes	characteristic
240	C₄×C₈	—	139	1	no	—
241	C₄×C₈	—	139	1	no	—
242	#41	—	278	2	no	—
243	#41	—	278	2	no	—
244	#41	—	139	4	no	—
245	#41	—	139	4	no	—
246	C₂×Q₈	—	139	1	no	—
247	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
248	C₂×D₄	—	278	1	no	—
249	C₂×D₄	—	278	1	no	—
250	C₂×D₄	—	139	1	no	—
251	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	278	1	no	—
254	C₂²×C₄	—	278	1	no	—
255	C₂²×C₄	—	278	1	no	—
256	C₂²×C₄	—	278	1	no	—
257	C₂²×C₄	—	278	1	no	—
258	C₂²×C₄	—	278	1	no	—
259	C₂²×C₄	—	278	1	no	—
260	C₂²×C₄	—	139	1	no	—
261	Q₁₆	—	139	4	no	—
262	SD₁₆	—	139	4	no	—
263	SD₁₆	C₄×D₁₃₉	1	4	yes	complemented
264	D₈	—	139	4	no	—
265	M₁₆	—	278	2	no	—
266	C₂×C₈	—	139	1	no	—
267	C₂×C₈	—	139	1	no	—
268	C₂×C₈	—	139	4	no	—
269	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	278	1	no	—
273	C₄⋊C₄	—	278	1	no	—
274	C₄⋊C₄	—	278	1	no	—
275	C₄⋊C₄	—	278	1	no	—
276	C₄⋊C₄	—	278	1	no	—
277	C₄⋊C₄	—	139	1	no	—
278	C₄⋊C₄	—	139	1	no	—
279	C₄⋊C₄	—	139	1	no	—
280	C₄⋊C₄	—	139	1	no	—
281	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
282	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	—	278	1	no	—
285	C₂²⋊C₄	—	278	1	no	—
286	C₂²⋊C₄	—	278	1	no	—
287	C₂²⋊C₄	—	139	1	no	—
288	C₂²⋊C₄	—	139	1	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	278	1	no	—
291	C₄²	—	278	1	no	—
292	C₄²	—	278	1	no	—
293	C₄²	—	139	1	no	—
294	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
295	C₂³	—	2	1	no	—
296	C₂³	—	278	1	no	—
297	C₂³	—	139	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	278	1	no	—
300	Q₈	—	139	2	no	—
301	Q₈	C₂×C₄×D₁₃₉	1	2	yes	—
302	D₄	—	2	1	no	—
303	D₄	—	278	4	no	—
304	D₄	—	278	4	no	—
305	D₄	—	278	1	no	—
306	D₄	—	139	2	no	—
307	D₄	C₂×C₄×D₁₃₉	1	2	yes	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	8	1	no	—
312	C₂×C₄	—	278	4	no	—
313	C₂×C₄	—	278	1	no	—
314	C₂×C₄	—	278	4	no	—
315	C₂×C₄	—	278	1	no	—
316	C₂×C₄	—	278	1	no	—
317	C₂×C₄	—	1112	1	no	—
318	C₂×C₄	—	1112	1	no	—
319	C₂×C₄	—	1112	1	no	—
320	C₂×C₄	—	1112	1	no	—
321	C₂×C₄	—	1112	1	no	—
322	C₂×C₄	—	139	1	no	—
323	C₂×C₄	—	139	1	no	—
324	C₂×C₄	—	139	1	no	—
325	C₂×C₄	—	139	2	no	—
326	C₂×C₄	—	139	2	no	—
327	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
328	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
329	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	278	1	no	—
332	C₈	—	139	2	no	—
333	C₈	C₂×C₄×D₁₃₉	1	2	yes	—
334	C₂²	—	2	2	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	278	1	no	—
338	C₂²	—	278	1	no	—
339	C₂²	—	278	2	no	—
340	C₂²	—	556	1	no	—
341	C₂²	—	556	1	no	—
342	C₂²	—	139	2	no	—
343	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	2	no	—
346	C₄	—	2	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	278	1	no	—
350	C₄	—	278	2	no	—
351	C₄	—	278	1	no	—
352	C₄	—	556	1	no	—
353	C₄	—	556	1	no	—
354	C₄	—	139	2	no	—
355	C₄	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
356	C₄	#41×D₁₃₉	1	1	yes	characteristic
357	C₂	—	4	2	no	—
358	C₂	—	278	2	no	—
359	C₂	—	556	2	no	—
360	C₂	C₄×D₄×D₁₃₉	1	1	yes	characteristic
361	C₂	#22581757	1	1	yes	characteristic
362	C₂	#22581769	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator	C₄×C₁₃₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#440×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#17984863	C₂
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	#440	D₁₃₉
hypocenter	C₁₃₉	#2951
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
2. #22581435	C₂	semidirect
3. #22581432	C₂	semidirect
4. #22581055	C₂	semidirect
5. #22581054	C₂	semidirect
6. #22581052	C₂	semidirect
7. C₄×#14555775	C₂	semidirect
8. #22581045	C₂	semidirect
9. C₄×#14555770	C₂	semidirect
10. #22581033	C₂	semidirect
11. #107×D₁₃₉	C₂	semidirect
12. #22580953	C₂	semidirect
13. #22580951	C₂	semidirect
14. #22580910	C₂	semidirect
15. #440×C₁₃₉	C₂	semidirect
18. #17984863	C₄	semidirect
61. #14555786	C₂×C₄	semidirect
63. #14555784	C₂×C₄	semidirect
76. #14555775	C₂×C₄	semidirect
77. #14555774	C₂×C₄	semidirect
78. #14555770	C₂×C₄	semidirect
82. C₈×D₁₃₉	C₂×C₄	semidirect
104. SD₁₆×C₁₃₉	C₂×C₄	semidirect
181. C₁₃₉	#2951	semidirect
189. #440	D₁₃₉	semidirect
263. SD₁₆	C₄×D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984863	2
#22581435	2
#22581432	2
#22581055	2
#22581054	2
#22581052	2
C₄×#14555775	2
#22581045	2
C₄×#14555770	2
#22581033	2
#107×D₁₃₉	2
#22580953	2
#22580951	2
#22580910	2
#440×C₁₃₉	2
#2951	139