Subgroups — #31816659

Order 17792.

384 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	#22581570	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581436	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581289	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581264	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581246	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581231	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581173	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581166	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581115	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581054	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581050	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581045	C₂	1	1	yes	maximal, characteristic, complemented
13	D₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581011	C₂	1	1	yes	maximal, characteristic, complemented
15	#449×C₁₃₉	C₂	1	1	yes	maximal, characteristic
16	C₂×#14555775	C₂²	1	1	yes	characteristic
17	C₂×#14555773	C₂²	1	1	yes	characteristic
18	#17984841	C₂²	1	1	yes	characteristic
19	#17984835	—	2	1	no	—
20	#17984829	—	2	1	no	—
21	#17984829	C₂²	1	1	yes	characteristic
22	#17984823	—	2	1	no	—
23	#17984821	C₂²	1	1	yes	characteristic
24	#17984819	—	2	1	no	—
25	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
26	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
27	#17984807	C₂²	1	1	yes	characteristic
28	D₄×Dic₁₃₉	—	2	1	no	—
29	D₄×Dic₁₃₉	—	2	1	no	—
30	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
31	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
33	#17984801	—	2	1	no	—
34	#17984801	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₃₉	—	2	1	no	—
36	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
37	#17984776	C₂²	1	1	yes	characteristic
38	#17984776	C₂²	1	1	yes	characteristic
39	#17984775	C₂²	1	1	yes	characteristic
40	#17984775	C₂²	1	1	yes	characteristic
41	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
43	#17984767	C₂²	1	1	yes	characteristic
44	#17984767	C₂²	1	1	yes	characteristic
45	#17984766	C₂²	1	1	yes	characteristic
46	#17984763	C₂²	1	1	yes	characteristic
47	#17984763	C₂²	1	1	yes	characteristic
48	#17984763	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
50	#17984752	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₁₃₉	C₂²	1	1	yes	characteristic
55	#121×C₁₃₉	C₂²	1	1	yes	characteristic
56	#103×C₁₃₉	C₂²	1	1	yes	characteristic
57	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	—	2	1	no	—
60	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
61	C₂²×Dic₁₃₉	—	2	1	no	—
62	C₂²×Dic₁₃₉	—	2	1	no	—
63	C₂²×Dic₁₃₉	—	2	1	no	—
64	C₂²×Dic₁₃₉	—	2	1	no	—
65	#14555775	—	2	2	no	—
66	#14555773	—	2	2	no	—
67	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₁₃₉	—	2	1	no	—
70	C₄⋊Dic₁₃₉	—	2	1	no	—
71	C₄⋊Dic₁₃₉	—	2	1	no	—
72	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
73	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
74	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
77	Dic₁₃₉⋊C₄	—	2	1	no	—
78	Dic₁₃₉⋊C₄	—	2	1	no	—
79	Dic₁₃₉⋊C₄	—	2	1	no	—
80	C₂²⋊Dic₁₃₉	—	2	1	no	—
81	C₂²⋊Dic₁₃₉	—	2	1	no	—
82	C₂²⋊Dic₁₃₉	—	2	1	no	—
83	C₂²⋊Dic₁₃₉	—	2	1	no	—
84	C₂²⋊Dic₁₃₉	—	2	1	no	—
85	C₂²⋊Dic₁₃₉	—	2	1	no	—
86	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
88	C₄×Dic₁₃₉	—	2	1	no	—
89	C₄×Dic₁₃₉	—	2	1	no	—
90	C₄×Dic₁₃₉	—	2	1	no	—
91	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
93	#41×C₁₃₉	—	2	2	no	—
94	#41×C₁₃₉	—	2	2	no	—
95	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₁₃₉	—	2	1	no	—
97	C₂×D₄×C₁₃₉	—	2	1	no	—
98	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₁₃₉	—	2	1	no	—
101	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	SD₁₆×C₁₃₉	—	2	2	no	—
103	D₈×C₁₃₉	—	2	2	no	—
104	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
108	Dic₂₇₈	—	2	1	no	—
109	Dic₂₇₈	—	2	1	no	—
110	Dic₂₇₈	—	8	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	—	8	1	no	—
116	C₂×Dic₁₃₉	—	8	1	no	—
117	C₂×Dic₁₃₉	—	8	1	no	—
118	C₂×Dic₁₃₉	—	8	1	no	—
119	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
123	C₁₃₉⋊C₈	—	2	1	no	—
124	C₁₃₉⋊C₈	—	2	1	no	—
125	C₂³×C₁₃₉	—	2	1	no	—
126	C₂³×C₁₃₉	—	2	1	no	—
127	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
128	Q₈×C₁₃₉	—	2	1	no	—
129	Q₈×C₁₃₉	#41	1	2	yes	—
130	D₄×C₁₃₉	—	2	4	no	—
131	D₄×C₁₃₉	—	2	1	no	—
132	D₄×C₁₃₉	—	2	1	no	—
133	D₄×C₁₃₉	—	2	1	no	—
134	D₄×C₁₃₉	—	8	1	no	—
135	D₄×C₁₃₉	#41	1	2	yes	—
136	C₂×C₄×C₁₃₉	—	2	4	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	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
143	C₈×C₁₃₉	—	2	1	no	—
144	C₈×C₁₃₉	—	2	1	no	—
145	C₂²×C₁₃₉	—	2	1	no	—
146	C₂²×C₁₃₉	—	2	2	no	—
147	C₂²×C₁₃₉	—	2	1	no	—
148	C₂²×C₁₃₉	—	2	1	no	—
149	C₂²×C₁₃₉	—	4	1	no	—
150	C₂²×C₁₃₉	—	4	1	no	—
151	C₂²×C₁₃₉	—	4	1	no	—
152	C₂²×C₁₃₉	—	4	1	no	—
153	C₂²×C₁₃₉	—	4	1	no	—
154	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
155	C₄×C₁₃₉	—	2	1	no	—
156	C₄×C₁₃₉	—	2	2	no	—
157	C₄×C₁₃₉	—	4	1	no	—
158	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
159	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
160	Dic₁₃₉	—	2	1	no	—
161	Dic₁₃₉	—	2	1	no	—
162	Dic₁₃₉	—	2	1	no	—
163	Dic₁₃₉	—	2	1	no	—
164	Dic₁₃₉	—	4	1	no	—
165	Dic₁₃₉	—	4	1	no	—
166	Dic₁₃₉	—	4	1	no	—
167	Dic₁₃₉	—	4	1	no	—
168	C₂×C₁₃₉	—	4	1	no	—
169	C₂×C₁₃₉	—	4	2	no	—
170	C₂×C₁₃₉	—	8	1	no	—
171	C₂×C₁₃₉	#525	1	1	yes	characteristic
172	C₂×C₁₃₉	#576	1	1	yes	characteristic
173	C₂×C₁₃₉	#578	1	1	yes	characteristic
174	C₁₃₉	#3124	1	1	yes	characteristic, complemented, Sylow
175	#3124	—	139	1	no	maximal, Sylow
176	#542	—	139	1	no	—
177	#520	—	139	1	no	—
178	#484	—	139	1	no	—
179	#480	—	139	1	no	—
180	#477	—	139	1	no	—
181	#474	—	139	1	no	—
182	#464	—	139	1	no	—
183	#463	—	139	1	no	—
184	#452	—	139	1	no	—
185	#449	D₁₃₉	1	1	yes	characteristic
186	#440	—	139	1	no	—
187	#440	—	139	1	no	—
188	C₄×Q₁₆	—	139	1	no	—
189	C₄×D₈	—	139	1	no	—
190	#433	—	139	1	no	—
191	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
192	C₂×Q₁₆	—	139	1	no	—
193	C₂×SD₁₆	—	139	1	no	—
194	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
195	C₂×D₈	C₂×D₁₃₉	1	1	yes	characteristic
196	C₄⋊Q₈	—	139	1	no	—
197	#126	—	278	1	no	—
198	#124	—	278	1	no	—
199	#124	—	139	1	no	—
200	#123	—	278	1	no	—
201	C₂²⋊Q₈	—	278	1	no	—
202	C₂²⋊Q₈	—	139	1	no	—
203	#121	C₂×D₁₃₉	1	1	yes	characteristic
204	C₄×Q₈	—	139	1	no	—
205	C₄×Q₈	—	139	1	no	—
206	C₄×Q₈	—	139	1	no	—
207	C₄×D₄	—	278	1	no	—
208	C₄×D₄	—	278	1	no	—
209	C₄×D₄	—	139	1	no	—
210	C₄×D₄	—	139	1	no	—
211	C₄×D₄	—	139	1	no	—
212	#117	—	278	1	no	—
213	#117	—	139	1	no	—
214	C₂×C₄⋊C₄	—	278	1	no	—
215	C₂×C₄⋊C₄	—	139	1	no	—
216	#107	—	139	1	no	—
217	#107	—	139	1	no	—
218	#107	—	139	1	no	—
219	#107	—	139	1	no	—
220	C₄⋊C₈	—	139	1	no	—
221	C₄⋊C₈	—	139	1	no	—
222	#103	—	139	1	no	—
223	#103	—	139	1	no	—
224	#103	—	139	1	no	—
225	#103	C₂×D₁₃₉	1	1	yes	characteristic
226	#102	—	139	1	no	—
227	#102	—	139	1	no	—
228	#102	—	139	1	no	—
229	#102	C₂×D₁₃₉	1	1	yes	characteristic
230	C₂²⋊C₈	—	139	1	no	—
231	C₂²⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
232	#97	—	139	1	no	—
233	C₄×C₈	—	139	1	no	—
234	#41	—	2	2	no	—
235	#41	—	2	2	no	—
236	C₂×Q₈	—	278	1	no	—
237	C₂×Q₈	—	139	1	no	—
238	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
242	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	278	1	no	—
245	C₂²×C₄	—	278	1	no	—
246	C₂²×C₄	—	278	1	no	—
247	C₂²×C₄	—	278	1	no	—
248	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
249	Q₁₆	—	278	2	no	—
250	SD₁₆	—	2	2	no	—
251	SD₁₆	—	278	2	no	—
252	D₈	—	2	2	no	—
253	C₂×C₈	—	139	1	no	—
254	C₂×C₈	—	139	1	no	—
255	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
257	C₄⋊C₄	—	278	1	no	—
258	C₄⋊C₄	—	278	1	no	—
259	C₄⋊C₄	—	278	1	no	—
260	C₄⋊C₄	—	278	1	no	—
261	C₄⋊C₄	—	278	1	no	—
262	C₄⋊C₄	—	278	1	no	—
263	C₄⋊C₄	—	139	1	no	—
264	C₄⋊C₄	—	139	1	no	—
265	C₄⋊C₄	—	139	1	no	—
266	C₄⋊C₄	—	139	1	no	—
267	C₄⋊C₄	—	139	1	no	—
268	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	278	1	no	—
271	C₂²⋊C₄	—	278	1	no	—
272	C₂²⋊C₄	—	278	1	no	—
273	C₂²⋊C₄	—	278	1	no	—
274	C₂²⋊C₄	—	278	1	no	—
275	C₂²⋊C₄	—	278	1	no	—
276	C₂²⋊C₄	—	139	1	no	—
277	C₂²⋊C₄	—	139	1	no	—
278	C₄²	—	278	1	no	—
279	C₄²	—	278	1	no	—
280	C₄²	—	278	1	no	—
281	C₄²	—	139	1	no	—
282	C₄²	—	139	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
286	Q₈	—	2	1	no	—
287	Q₈	—	278	1	no	—
288	Q₈	—	278	1	no	—
289	Q₈	—	1112	1	no	—
290	Q₈	#14555784	1	2	yes	—
291	D₄	—	2	4	no	—
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	8	1	no	—
296	D₄	#14555784	1	2	yes	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	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	—	1112	1	no	—
307	C₂×C₄	—	1112	1	no	—
308	C₂×C₄	—	1112	1	no	—
309	C₂×C₄	—	1112	1	no	—
310	C₂×C₄	—	139	1	no	—
311	C₂×C₄	—	139	1	no	—
312	C₂×C₄	—	139	1	no	—
313	C₂×C₄	—	139	1	no	—
314	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	278	1	no	—
319	C₈	—	278	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	1	no	—
333	C₄	—	278	1	no	—
334	C₄	—	278	1	no	—
335	C₄	—	278	1	no	—
336	C₄	—	278	1	no	—
337	C₄	—	556	1	no	—
338	C₄	—	556	1	no	—
339	C₄	—	556	1	no	—
340	C₄	—	556	1	no	—
341	C₄	C₂×#14555784	1	1	yes	characteristic
342	C₄	C₂×#14555784	1	1	yes	characteristic
343	C₂	—	4	1	no	—
344	C₂	—	4	2	no	—
345	C₂	—	8	1	no	—
346	C₂	#22581459	1	1	yes	characteristic
347	C₂	#22581757	1	1	yes	characteristic
348	C₂	#22581767	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting, prime radical	#449×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	#449	D₁₃₉
hypocenter	C₁₃₉	#3124
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #449

Complements

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

N	H ≅ G/N	type
1. #22581570	C₂	semidirect
2. #22581436	C₂	semidirect
3. #22581289	C₂	semidirect
4. #22581264	C₂	semidirect
5. #22581246	C₂	semidirect
6. #22581231	C₂	semidirect
7. #22581173	C₂	semidirect
8. #22581166	C₂	semidirect
9. #22581115	C₂	semidirect
10. #22581054	C₂	semidirect
11. #22581050	C₂	semidirect
12. #22581045	C₂	semidirect
13. D₈×Dic₁₃₉	C₂	semidirect
14. #22581011	C₂	semidirect
174. C₁₃₉	#3124	semidirect

Maximal subgroups

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

subgroup	index
#22581570	2
#22581436	2
#22581289	2
#22581264	2
#22581246	2
#22581231	2
#22581173	2
#22581166	2
#22581115	2
#22581054	2
#22581050	2
#22581045	2
D₈×Dic₁₃₉	2
#22581011	2
#449×C₁₃₉	2
#3124	139