Subgroups — #31818319

Order 17792.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581671	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581605	C₂	1	1	yes	maximal, characteristic
3	#22581302	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581297	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581232	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581195	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581192	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581162	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581118	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581117	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581040	C₂	1	1	yes	maximal, characteristic, complemented
12	(C₂²⋊D₁₃₉)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	#22580948	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580947	C₂	1	1	yes	maximal, characteristic, complemented
15	#485×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555786	C₂²	1	1	yes	characteristic
17	C₂×#14555785	C₂²	1	1	yes	characteristic
18	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
19	C₂×#14555775	C₂²	1	1	yes	characteristic
20	C₂×#14555774	C₂²	1	1	yes	characteristic
21	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₃₉	C₂²	1	1	yes	characteristic
23	#17984841	C₂²	1	1	yes	characteristic
24	#17984840	C₂²	1	1	yes	characteristic
25	#17984836	—	2	1	no	—
26	#17984835	—	2	1	no	—
27	#17984828	—	2	1	no	—
28	#17984827	—	2	1	no	—
29	#17984824	—	2	1	no	—
30	#17984822	C₂²	1	1	yes	characteristic
31	#17984822	C₂²	1	1	yes	characteristic
32	#17984821	C₂²	1	1	yes	characteristic
33	#17984815	—	2	1	no	—
34	C₄×Dic₂₇₈	—	2	1	no	—
35	#17984807	C₂²	1	1	yes	characteristic
36	#17984804	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊D₁₃₉	C₂²	1	1	yes	characteristic
38	#17984800	C₂²	1	1	yes	characteristic
39	C₄⋊C₄×D₁₃₉	—	2	1	no	—
40	#17984775	C₂²	1	1	yes	characteristic
41	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
42	#17984767	C₂²	1	1	yes	characteristic
43	#17984766	C₂²	1	1	yes	characteristic
44	#17984765	C₂²	1	1	yes	characteristic
45	#17984765	C₂²	1	1	yes	characteristic
46	#17984765	C₂²	1	1	yes	characteristic
47	#17984764	C₂²	1	1	yes	characteristic
48	#17984762	C₂²	1	1	yes	characteristic
49	#17984762	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
53	#117×C₁₃₉	C₂²	1	1	yes	characteristic
54	#107×C₁₃₉	C₂²	1	1	yes	characteristic
55	#106×C₁₃₉	C₂²	1	1	yes	characteristic
56	#103×C₁₃₉	C₂²	1	1	yes	characteristic
57	#103×C₁₃₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
59	#14555786	—	2	2	no	—
60	#14555786	—	4	1	no	—
61	#14555785	—	2	2	no	—
62	#14555785	—	4	1	no	—
63	C₂×Dic₂₇₈	—	2	1	no	—
64	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
65	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
66	C₂×D₅₅₆	—	2	1	no	—
67	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
69	C₂×C₂²⋊D₁₃₉	D₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
74	Dic₅₅₆	—	4	1	no	—
75	#14555775	—	4	1	no	—
76	#14555774	—	2	2	no	—
77	C₂×C₁₃₉⋊C₈	—	4	1	no	—
78	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
80	C₈×D₁₃₉	—	4	1	no	—
81	C₄⋊Dic₁₃₉	—	2	1	no	—
82	C₄⋊Dic₁₃₉	—	2	1	no	—
83	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
84	Dic₁₃₉⋊C₄	—	2	1	no	—
85	Dic₁₃₉⋊C₄	—	2	1	no	—
86	Dic₁₃₉⋊C₄	—	2	1	no	—
87	Dic₁₃₉⋊C₄	—	2	1	no	—
88	Dic₁₃₉⋊C₄	—	2	1	no	—
89	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
90	C₂²⋊Dic₁₃₉	—	2	1	no	—
91	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
92	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
93	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
94	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
95	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
96	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
97	C₄×Dic₁₃₉	—	2	1	no	—
98	C₄×Dic₁₃₉	—	2	1	no	—
99	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
100	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₃₉	—	2	1	no	—
105	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
108	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
109	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
110	C₄²×C₁₃₉	—	2	1	no	—
111	C₂²×D₁₃₉	—	2	1	no	—
112	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
113	Dic₂₇₈	—	2	1	no	—
114	Dic₂₇₈	—	2	1	no	—
115	Dic₂₇₈	—	2	1	no	—
116	Dic₂₇₈	—	2	1	no	—
117	Dic₂₇₈	—	4	2	no	—
118	D₅₅₆	—	2	1	no	—
119	D₅₅₆	—	4	2	no	—
120	D₅₅₆	#41	1	2	yes	—
121	C₂²⋊D₁₃₉	—	2	2	no	—
122	C₂²⋊D₁₃₉	—	4	2	no	—
123	C₂×Dic₁₃₉	—	2	1	no	—
124	C₂×Dic₁₃₉	—	2	1	no	—
125	C₂×Dic₁₃₉	—	2	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	1	no	—
130	C₄×D₁₃₉	—	2	1	no	—
131	C₄×D₁₃₉	—	4	2	no	—
132	C₄×D₁₃₉	—	4	2	no	—
133	C₄×D₁₃₉	—	4	2	no	—
134	C₁₃₉⋊C₈	—	2	1	no	—
135	C₁₃₉⋊C₈	—	2	1	no	—
136	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
137	Q₈×C₁₃₉	—	2	1	no	—
138	Q₈×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	—	2	1	no	—
143	C₂×C₄×C₁₃₉	—	2	1	no	—
144	C₂×C₄×C₁₃₉	—	2	1	no	—
145	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
152	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₁₃₉	—	2	1	no	—
154	C₂×D₁₃₉	—	2	2	no	—
155	C₂×D₁₃₉	—	2	1	no	—
156	C₂×D₁₃₉	—	2	1	no	—
157	C₂×D₁₃₉	—	4	1	no	—
158	C₂×D₁₃₉	—	4	1	no	—
159	C₄×C₁₃₉	—	2	1	no	—
160	C₄×C₁₃₉	—	4	1	no	—
161	C₄×C₁₃₉	—	4	1	no	—
162	C₄×C₁₃₉	—	4	1	no	—
163	C₄×C₁₃₉	—	4	1	no	—
164	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
165	C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
166	Dic₁₃₉	—	2	1	no	—
167	Dic₁₃₉	—	2	1	no	—
168	Dic₁₃₉	—	4	1	no	—
169	Dic₁₃₉	—	4	1	no	—
170	Dic₁₃₉	—	4	1	no	—
171	Dic₁₃₉	#135	1	2	yes	—
172	C₂×C₁₃₉	—	4	1	no	—
173	C₂×C₁₃₉	#546	1	1	yes	characteristic
174	C₂×C₁₃₉	C₂×#135	1	1	yes	characteristic
175	C₂×C₁₃₉	#578	1	1	yes	characteristic
176	D₁₃₉	—	4	1	no	—
177	D₁₃₉	—	4	2	no	—
178	C₁₃₉	#3302	1	1	yes	characteristic, complemented, Sylow
179	#3302	—	139	1	no	maximal, Sylow
180	#556	—	139	1	no	—
181	#547	—	139	1	no	—
182	#487	—	139	1	no	—
183	#485	—	139	1	no	—
184	#485	D₁₃₉	1	1	yes	characteristic, complemented
185	#474	—	139	1	no	—
186	#467	—	139	1	no	—
187	#467	—	139	1	no	—
188	#462	—	139	1	no	—
189	#452	—	139	1	no	—
190	#452	—	139	1	no	—
191	C₄×SD₁₆	—	139	1	no	—
192	C₈×D₄	—	139	1	no	—
193	#416	—	139	1	no	—
194	#416	—	139	1	no	—
195	C₂×#41	—	139	1	no	—
196	C₂×#41	—	139	1	no	—
197	C₂×Q₁₆	—	139	1	no	—
198	C₂×Q₁₆	—	139	1	no	—
199	C₂×SD₁₆	—	139	1	no	—
200	C₂²×C₈	—	139	1	no	—
201	C₂²×C₈	—	139	1	no	—
202	C₄⋊Q₈	—	139	1	no	—
203	C₄⋊Q₈	—	139	1	no	—
204	#126	—	278	1	no	—
205	#126	—	278	1	no	—
206	#124	—	278	1	no	—
207	#123	—	278	1	no	—
208	#123	—	278	1	no	—
209	C₂²⋊Q₈	—	139	1	no	—
210	C₂²⋊Q₈	—	139	1	no	—
211	C₂²⋊Q₈	—	139	1	no	—
212	C₂²⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
213	#121	—	278	1	no	—
214	C₄×Q₈	—	278	1	no	—
215	C₄×Q₈	—	139	1	no	—
216	C₄×D₄	—	139	1	no	—
217	C₄×D₄	—	139	1	no	—
218	#117	—	139	1	no	—
219	#117	C₂×D₁₃₉	1	1	yes	characteristic
220	C₂×C₄⋊C₄	—	278	1	no	—
221	#107	—	139	1	no	—
222	#107	C₂×D₁₃₉	1	1	yes	characteristic
223	#106	C₂×D₁₃₉	1	1	yes	characteristic
224	C₄⋊C₈	—	139	1	no	—
225	#103	—	139	1	no	—
226	#103	—	139	1	no	—
227	#103	—	139	1	no	—
228	#103	—	139	1	no	—
229	#103	—	139	1	no	—
230	#103	C₂×D₁₃₉	1	1	yes	characteristic
231	#103	C₂×D₁₃₉	1	1	yes	characteristic
232	#102	—	139	1	no	—
233	#102	—	139	1	no	—
234	#102	—	139	1	no	—
235	C₂²⋊C₈	—	139	1	no	—
236	C₂²⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
237	C₄×C₈	—	139	1	no	—
238	#41	—	278	2	no	—
239	#41	—	278	2	no	—
240	#41	—	556	1	no	—
241	#41	—	556	1	no	—
242	C₂×Q₈	—	278	1	no	—
243	C₂×Q₈	—	139	1	no	—
244	C₂×Q₈	—	139	1	no	—
245	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
246	C₂×D₄	—	278	1	no	—
247	C₂×D₄	—	278	1	no	—
248	C₂×D₄	—	139	1	no	—
249	C₂×D₄	—	139	1	no	—
250	C₂²×C₄	—	278	1	no	—
251	C₂²×C₄	—	278	1	no	—
252	C₂²×C₄	—	278	1	no	—
253	C₂²×C₄	—	139	1	no	—
254	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
255	Q₁₆	—	556	1	no	—
256	Q₁₆	—	556	1	no	—
257	SD₁₆	—	278	2	no	—
258	C₂×C₈	—	556	1	no	—
259	C₂×C₈	—	556	1	no	—
260	C₂×C₈	—	139	1	no	—
261	C₂×C₈	—	139	1	no	—
262	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	278	1	no	—
266	C₄⋊C₄	—	278	1	no	—
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	—	278	1	no	—
272	C₄⋊C₄	—	139	1	no	—
273	C₄⋊C₄	—	139	1	no	—
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	—	139	1	no	—
285	C₂²⋊C₄	—	139	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	278	1	no	—
288	C₄²	—	278	1	no	—
289	C₄²	—	139	1	no	—
290	C₂³	—	278	1	no	—
291	C₂³	—	139	1	no	—
292	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	278	1	no	—
296	Q₈	—	278	1	no	—
297	Q₈	—	278	1	no	—
298	Q₈	—	278	1	no	—
299	Q₈	—	556	2	no	—
300	D₄	—	278	2	no	—
301	D₄	—	278	1	no	—
302	D₄	—	556	2	no	—
303	D₄	—	556	2	no	—
304	D₄	—	139	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	—	2	1	no	—
310	C₂×C₄	—	2	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	—	556	2	no	—
317	C₂×C₄	—	556	2	no	—
318	C₂×C₄	—	556	2	no	—
319	C₂×C₄	—	139	1	no	—
320	C₂×C₄	—	139	1	no	—
321	C₂×C₄	—	139	1	no	—
322	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	278	1	no	—
327	C₈	—	278	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	278	1	no	—
332	C₂²	—	278	1	no	—
333	C₂²	—	278	2	no	—
334	C₂²	—	278	1	no	—
335	C₂²	—	556	1	no	—
336	C₂²	—	556	1	no	—
337	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	278	1	no	—
344	C₄	—	278	1	no	—
345	C₄	—	556	1	no	—
346	C₄	—	556	1	no	—
347	C₄	—	556	1	no	—
348	C₄	—	139	2	no	—
349	C₄	#41×D₁₃₉	1	1	yes	characteristic
350	C₄	#17984888	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	556	1	no	—
353	C₂	—	556	2	no	—
354	C₂	#22581595	1	1	yes	characteristic
355	C₂	#135×D₁₃₉	1	1	yes	characteristic
356	C₂	#22581769	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#485×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#22581605	C₂
hypercenter	#485	D₁₃₉
hypocenter	C₁₃₉	#3302
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
1. #22581671	C₂	semidirect
3. #22581302	C₂	semidirect
4. #22581297	C₂	semidirect
5. #22581232	C₂	semidirect
6. #22581195	C₂	semidirect
7. #22581192	C₂	semidirect
8. #22581162	C₂	semidirect
9. #22581118	C₂	semidirect
10. #22581117	C₂	semidirect
11. #22581040	C₂	semidirect
12. (C₂²⋊D₁₃₉)⋊C₈	C₂	semidirect
13. #22580948	C₂	semidirect
14. #22580947	C₂	semidirect
15. #485×C₁₃₉	C₂	semidirect
178. C₁₃₉	#3302	semidirect
184. #485	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581671	2
#22581605	2
#22581302	2
#22581297	2
#22581232	2
#22581195	2
#22581192	2
#22581162	2
#22581118	2
#22581117	2
#22581040	2
(C₂²⋊D₁₃₉)⋊C₈	2
#22580948	2
#22580947	2
#485×C₁₃₉	2
#3302	139