Subgroups — #31816321

Order 17792.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581465	C₂	1	1	yes	maximal, characteristic
2	#117×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581297	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581296	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581290	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581285	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581280	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581279	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581273	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581268	C₂	1	1	yes	maximal, characteristic, complemented
11	#22580989	C₂	1	1	yes	maximal, characteristic, complemented
12	#22580950	C₂	1	1	yes	maximal, characteristic, complemented
13	#22580947	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×(C₂×D₁₃₉)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
15	#427×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555785	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
18	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
19	#17984826	—	2	1	no	—
20	#17984822	C₂²	1	1	yes	characteristic
21	#17984822	C₂²	1	1	yes	characteristic
22	#17984817	C₂²	1	1	yes	characteristic
23	#17984817	C₂²	1	1	yes	characteristic
24	#17984812	—	2	1	no	—
25	C₄×D₅₅₆	—	2	1	no	—
26	C₄×D₅₅₆	—	2	1	no	—
27	#17984801	C₂²	1	1	yes	characteristic
28	#17984800	C₂²	1	1	yes	characteristic
29	#17984800	C₂²	1	1	yes	characteristic
30	#17984799	—	2	1	no	—
31	#17984798	—	2	1	no	—
32	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
34	C₂²⋊C₄×D₁₃₉	—	2	1	no	—
35	C₄²×D₁₃₉	—	2	1	no	—
36	#17984775	C₂²	1	1	yes	characteristic, complemented
37	#17984775	C₂²	1	1	yes	characteristic, complemented
38	#17984773	C₂²	1	1	yes	characteristic, complemented
39	#17984773	C₂²	1	1	yes	characteristic, complemented
40	#17984767	C₂²	1	1	yes	characteristic, complemented
41	#17984767	C₂²	1	1	yes	characteristic, complemented
42	#17984765	C₂²	1	1	yes	characteristic, complemented
43	#17984765	C₂²	1	1	yes	characteristic, complemented
44	#17984764	C₂²	1	1	yes	characteristic, complemented
45	#17984764	C₂²	1	1	yes	characteristic, complemented
46	#17984762	C₂²	1	1	yes	characteristic, complemented
47	#17984762	C₂²	1	1	yes	characteristic, complemented
48	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
52	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
53	#117×C₁₃₉	C₂²	1	1	yes	characteristic
54	#117×C₁₃₉	C₂²	1	1	yes	characteristic
55	#107×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
56	#107×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
57	#106×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
58	#106×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₃₉	D₄	1	1	yes	characteristic
60	#14555785	—	2	2	no	—
61	#14555785	—	2	2	no	—
62	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
63	C₂×D₅₅₆	—	2	1	no	—
64	C₂×D₅₅₆	—	2	1	no	—
65	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
67	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
68	C₂×C₄×D₁₃₉	—	2	1	no	—
69	C₂×C₄×D₁₃₉	—	2	1	no	—
70	C₂×C₄×D₁₃₉	—	2	1	no	—
71	C₂×C₄×D₁₃₉	—	2	2	no	—
72	C₂×C₄×D₁₃₉	—	2	2	no	—
73	C₂×C₄×D₁₃₉	D₄	1	1	yes	characteristic
74	C₂×C₄×D₁₃₉	D₄	1	1	yes	characteristic
75	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
77	C₂×C₁₃₉⋊C₈	—	2	2	no	—
78	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₃₉	—	2	1	no	—
81	C₄⋊Dic₁₃₉	—	2	1	no	—
82	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
84	Dic₁₃₉⋊C₄	—	2	1	no	—
85	Dic₁₃₉⋊C₄	—	2	1	no	—
86	C₂²⋊Dic₁₃₉	—	2	1	no	—
87	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
88	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
90	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
91	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
92	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
93	C₄×Dic₁₃₉	—	2	1	no	—
94	C₄×Dic₁₃₉	—	2	1	no	—
95	C₄×Dic₁₃₉	—	2	1	no	—
96	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₃₉	—	2	2	no	—
98	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
105	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
106	C₄²×C₁₃₉	—	2	1	no	—
107	C₄²×C₁₃₉	—	2	1	no	—
108	C₂²×D₁₃₉	—	2	2	no	—
109	C₂²×D₁₃₉	—	2	2	no	—
110	C₂²×D₁₃₉	—	2	1	no	—
111	C₂²×D₁₃₉	—	4	1	no	—
112	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
114	Dic₂₇₈	—	2	1	no	—
115	Dic₂₇₈	—	2	1	no	—
116	D₅₅₆	—	2	1	no	—
117	D₅₅₆	—	2	1	no	—
118	D₅₅₆	—	4	2	no	—
119	D₅₅₆	—	4	2	no	—
120	C₂²⋊D₁₃₉	—	4	2	no	—
121	C₂×Dic₁₃₉	—	2	2	no	—
122	C₂×Dic₁₃₉	—	2	1	no	—
123	C₂×Dic₁₃₉	—	2	1	no	—
124	C₂×Dic₁₃₉	—	2	1	no	—
125	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
127	C₄×D₁₃₉	—	2	2	no	—
128	C₄×D₁₃₉	—	2	2	no	—
129	C₄×D₁₃₉	—	2	1	no	—
130	C₄×D₁₃₉	—	2	1	no	—
131	C₄×D₁₃₉	—	2	1	no	—
132	C₄×D₁₃₉	—	2	1	no	—
133	C₄×D₁₃₉	—	4	2	no	—
134	C₄×D₁₃₉	—	4	2	no	—
135	C₄×D₁₃₉	—	4	2	no	—
136	C₁₃₉⋊C₈	—	2	1	no	—
137	C₁₃₉⋊C₈	—	2	1	no	—
138	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
143	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃₉	#41	1	2	yes	—
145	C₂×C₄×C₁₃₉	#41	1	2	yes	—
146	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
147	C₈×C₁₃₉	—	2	1	no	—
148	C₈×C₁₃₉	—	2	1	no	—
149	C₂²×C₁₃₉	—	2	1	no	—
150	C₂²×C₁₃₉	—	2	1	no	—
151	C₂²×C₁₃₉	#123	1	2	yes	—
152	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₁₃₉	—	2	1	no	—
154	C₂×D₁₃₉	—	2	1	no	—
155	C₂×D₁₃₉	—	2	1	no	—
156	C₂×D₁₃₉	—	2	1	no	—
157	C₂×D₁₃₉	—	2	1	no	—
158	C₂×D₁₃₉	—	4	4	no	—
159	C₂×D₁₃₉	—	4	1	no	—
160	C₂×D₁₃₉	—	4	1	no	—
161	C₂×D₁₃₉	—	4	1	no	—
162	C₂×D₁₃₉	#135	1	2	yes	—
163	C₄×C₁₃₉	—	4	1	no	—
164	C₄×C₁₃₉	—	4	1	no	—
165	C₄×C₁₃₉	—	4	1	no	—
166	C₄×C₁₃₉	—	4	1	no	—
167	C₄×C₁₃₉	#123	1	2	yes	—
168	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
169	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
170	Dic₁₃₉	—	2	1	no	—
171	Dic₁₃₉	—	2	1	no	—
172	Dic₁₃₉	—	4	1	no	—
173	Dic₁₃₉	—	4	1	no	—
174	Dic₁₃₉	—	4	1	no	—
175	C₂×C₁₃₉	—	2	2	no	—
176	C₂×C₁₃₉	C₂×#123	1	1	yes	characteristic
177	C₂×C₁₃₉	C₂×#135	1	1	yes	characteristic
178	C₂×C₁₃₉	#575	1	1	yes	characteristic
179	D₁₃₉	—	2	2	no	—
180	D₁₃₉	—	4	1	no	—
181	D₁₃₉	—	8	1	no	—
182	C₁₃₉	#3091	1	1	yes	characteristic, complemented, Sylow
183	#3091	—	139	1	no	maximal, Sylow
184	#525	—	139	1	no	—
185	C₂×#117	—	139	1	no	—
186	#485	—	139	1	no	—
187	#485	—	139	1	no	—
188	#484	—	139	1	no	—
189	#483	—	139	1	no	—
190	#482	—	139	1	no	—
191	#482	—	139	1	no	—
192	#481	—	139	1	no	—
193	#480	—	139	1	no	—
194	#427	—	139	1	no	—
195	#427	D₁₃₉	1	1	yes	characteristic, complemented
196	#416	—	139	1	no	—
197	#416	—	139	1	no	—
198	C₂×C₂²⋊C₈	—	139	1	no	—
199	C₂×#41	—	139	1	no	—
200	C₂³×C₄	—	139	1	no	—
201	C₂²×C₈	—	139	1	no	—
202	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
203	#123	—	278	1	no	—
204	C₂²⋊Q₈	—	139	1	no	—
205	C₂²⋊Q₈	—	139	1	no	—
206	#121	—	139	1	no	—
207	#121	—	139	1	no	—
208	#120	—	278	1	no	—
209	C₄×D₄	—	278	1	no	—
210	C₄×D₄	—	278	1	no	—
211	#117	—	278	1	no	—
212	#117	—	278	1	no	—
213	#117	—	139	1	no	—
214	#117	—	139	1	no	—
215	#117	—	139	1	no	—
216	#117	C₂×D₁₃₉	1	1	yes	characteristic
217	#117	C₂×D₁₃₉	1	1	yes	characteristic
218	C₂×C₄⋊C₄	—	139	1	no	—
219	C₂×C₄⋊C₄	—	139	1	no	—
220	C₂×C₂²⋊C₄	—	278	1	no	—
221	C₂×C₄²	—	278	1	no	—
222	#107	—	139	1	no	—
223	#107	—	139	1	no	—
224	#107	C₂×D₁₃₉	1	1	yes	characteristic, complemented
225	#107	C₂×D₁₃₉	1	1	yes	characteristic, complemented
226	#106	—	139	1	no	—
227	#106	—	139	1	no	—
228	#106	C₂×D₁₃₉	1	1	yes	characteristic, complemented
229	#106	C₂×D₁₃₉	1	1	yes	characteristic, complemented
230	#103	—	139	1	no	—
231	#103	—	139	1	no	—
232	#103	—	139	1	no	—
233	#103	—	139	1	no	—
234	#102	—	139	1	no	—
235	#102	—	139	1	no	—
236	#102	—	139	1	no	—
237	#102	—	139	1	no	—
238	C₂²⋊C₈	—	139	1	no	—
239	C₂²⋊C₈	—	139	1	no	—
240	C₂²⋊C₈	—	139	1	no	—
241	C₂²⋊C₈	—	139	1	no	—
242	C₂⁴	—	139	1	no	—
243	#41	—	278	2	no	—
244	#41	—	278	2	no	—
245	C₂×Q₈	—	139	1	no	—
246	C₂×D₄	—	278	1	no	—
247	C₂×D₄	—	278	1	no	—
248	C₂×D₄	—	278	1	no	—
249	C₂×D₄	—	139	1	no	—
250	C₂²×C₄	—	278	1	no	—
251	C₂²×C₄	—	278	1	no	—
252	C₂²×C₄	—	278	2	no	—
253	C₂²×C₄	—	278	2	no	—
254	C₂²×C₄	—	278	1	no	—
255	C₂²×C₄	—	139	1	no	—
256	C₂²×C₄	—	139	1	no	—
257	C₂²×C₄	—	139	1	no	—
258	C₂²×C₄	—	139	1	no	—
259	C₂²×C₄	—	139	1	no	—
260	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	278	2	no	—
263	C₂×C₈	—	139	1	no	—
264	C₂×C₈	—	139	1	no	—
265	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
266	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
267	C₄⋊C₄	—	278	1	no	—
268	C₄⋊C₄	—	278	1	no	—
269	C₄⋊C₄	—	278	1	no	—
270	C₄⋊C₄	—	278	1	no	—
271	C₄⋊C₄	—	139	1	no	—
272	C₄⋊C₄	—	139	1	no	—
273	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
275	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	278	1	no	—
280	C₂²⋊C₄	—	278	1	no	—
281	C₂²⋊C₄	—	278	1	no	—
282	C₂²⋊C₄	—	278	1	no	—
283	C₂²⋊C₄	—	278	1	no	—
284	C₂²⋊C₄	—	278	1	no	—
285	C₂²⋊C₄	—	278	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	2	1	no	—
288	C₄²	—	278	1	no	—
289	C₄²	—	278	1	no	—
290	C₄²	—	278	1	no	—
291	C₂³	—	278	2	no	—
292	C₂³	—	278	2	no	—
293	C₂³	—	278	1	no	—
294	C₂³	—	556	1	no	—
295	C₂³	—	139	1	no	—
296	C₂³	—	139	1	no	—
297	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
298	Q₈	—	278	1	no	—
299	Q₈	—	278	1	no	—
300	D₄	—	278	1	no	—
301	D₄	—	278	1	no	—
302	D₄	—	556	2	no	—
303	D₄	—	556	2	no	—
304	D₄	—	556	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	278	2	no	—
310	C₂×C₄	—	278	2	no	—
311	C₂×C₄	—	278	2	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	2	no	—
320	C₂×C₄	—	556	2	no	—
321	C₂×C₄	—	556	2	no	—
322	C₂×C₄	—	139	1	no	—
323	C₂×C₄	—	139	1	no	—
324	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
325	C₂×C₄	#14555785	1	2	yes	—
326	C₂×C₄	#14555786	1	2	yes	—
327	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	278	1	no	—
331	C₈	—	278	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	278	1	no	—
335	C₂²	—	278	1	no	—
336	C₂²	—	278	1	no	—
337	C₂²	—	278	1	no	—
338	C₂²	—	278	1	no	—
339	C₂²	—	556	4	no	—
340	C₂²	—	556	1	no	—
341	C₂²	—	556	1	no	—
342	C₂²	—	556	1	no	—
343	C₂²	—	139	2	no	—
344	C₂²	#17984827	1	2	yes	—
345	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	1	no	—
350	C₄	—	278	1	no	—
351	C₄	—	278	1	no	—
352	C₄	—	556	1	no	—
353	C₄	—	556	1	no	—
354	C₄	—	556	1	no	—
355	C₄	#17984827	1	2	yes	—
356	C₄	C₂×#14555785	1	1	yes	characteristic
357	C₄	C₂×#14555786	1	1	yes	characteristic
358	C₂	—	2	2	no	—
359	C₂	—	278	2	no	—
360	C₂	—	556	1	no	—
361	C₂	—	1112	1	no	—
362	C₂	C₂×#17984827	1	1	yes	characteristic
363	C₂	#135×D₁₃₉	1	1	yes	characteristic
364	C₂	#22581750	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#427×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#22581465	C₂
hypercenter	#427	D₁₃₉
hypocenter	C₁₃₉	#3091
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

Complements

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

N	H ≅ G/N	type
2. #117×D₁₃₉	C₂	semidirect
3. #22581297	C₂	semidirect
4. #22581296	C₂	semidirect
5. #22581290	C₂	semidirect
6. #22581285	C₂	semidirect
7. #22581280	C₂	semidirect
8. #22581279	C₂	semidirect
9. #22581273	C₂	semidirect
10. #22581268	C₂	semidirect
11. #22580989	C₂	semidirect
12. #22580950	C₂	semidirect
13. #22580947	C₂	semidirect
14. C₂×(C₂×D₁₃₉)⋊C₈	C₂	semidirect
15. #427×C₁₃₉	C₂	semidirect
36. #17984775	C₂²	semidirect
37. #17984775	C₂²	semidirect
38. #17984773	C₂²	semidirect
39. #17984773	C₂²	semidirect
40. #17984767	C₂²	semidirect
41. #17984767	C₂²	semidirect
42. #17984765	C₂²	semidirect
43. #17984765	C₂²	semidirect
44. #17984764	C₂²	semidirect
45. #17984764	C₂²	semidirect
46. #17984762	C₂²	semidirect
47. #17984762	C₂²	semidirect
55. #107×C₁₃₉	C₂²	semidirect
56. #107×C₁₃₉	C₂²	semidirect
57. #106×C₁₃₉	C₂²	semidirect
58. #106×C₁₃₉	C₂²	semidirect
182. C₁₃₉	#3091	semidirect
195. #427	D₁₃₉	semidirect
224. #107	C₂×D₁₃₉	semidirect
225. #107	C₂×D₁₃₉	semidirect
228. #106	C₂×D₁₃₉	semidirect
229. #106	C₂×D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581465	2
#117×D₁₃₉	2
#22581297	2
#22581296	2
#22581290	2
#22581285	2
#22581280	2
#22581279	2
#22581273	2
#22581268	2
#22580989	2
#22580950	2
#22580947	2
C₂×(C₂×D₁₃₉)⋊C₈	2
#427×C₁₃₉	2
#3091	139