Subgroups — #31816188

Order 17792.

388 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581506	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#22581262	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581258	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581255	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581251	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581246	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581241	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581236	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581231	C₂	1	1	yes	maximal, characteristic, complemented
11	#22580991	C₂	1	1	yes	maximal, characteristic, complemented
12	#22580989	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×Dic₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580949	C₂	1	1	yes	maximal, characteristic
15	#416×C₁₃₉	C₂	1	1	yes	maximal, characteristic
16	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
17	#17984841	C₂²	1	1	yes	characteristic
18	#17984841	C₂²	1	1	yes	characteristic
19	#17984833	C₂²	1	1	yes	characteristic
20	#17984833	C₂²	1	1	yes	characteristic
21	#17984819	—	2	1	no	—
22	#17984819	—	2	1	no	—
23	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
24	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
25	C₄×Dic₂₇₈	—	2	1	no	—
26	C₄×Dic₂₇₈	—	2	1	no	—
27	D₄×Dic₁₃₉	—	2	1	no	—
28	D₄×Dic₁₃₉	—	2	1	no	—
29	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
30	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
31	#17984801	—	2	1	no	—
32	#17984801	C₂²	1	1	yes	characteristic
33	#17984801	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₁₃₉	—	2	1	no	—
35	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
36	#17984776	C₂²	1	1	yes	characteristic
37	#17984776	C₂²	1	1	yes	characteristic
38	#17984775	C₂²	1	1	yes	characteristic
39	#17984775	C₂²	1	1	yes	characteristic
40	#17984774	C₂²	1	1	yes	characteristic
41	#17984774	C₂²	1	1	yes	characteristic
42	#17984773	C₂²	1	1	yes	characteristic
43	#17984773	C₂²	1	1	yes	characteristic
44	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
46	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
47	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
48	#17984766	C₂²	1	1	yes	characteristic
49	#17984766	C₂²	1	1	yes	characteristic
50	#17984763	C₂²	1	1	yes	characteristic
51	#17984763	C₂²	1	1	yes	characteristic
52	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	#117×C₁₃₉	C₂²	1	1	yes	characteristic
55	#103×C₁₃₉	C₂²	1	1	yes	characteristic
56	#103×C₁₃₉	C₂²	1	1	yes	characteristic
57	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	#102×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	—	2	1	no	—
60	C₂×Dic₂₇₈	—	2	1	no	—
61	C₂²×Dic₁₃₉	—	2	1	no	—
62	C₂²×Dic₁₃₉	—	2	1	no	—
63	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
64	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
65	C₂×C₁₃₉⋊C₈	—	2	2	no	—
66	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₁₃₉	—	2	1	no	—
69	C₄⋊Dic₁₃₉	—	2	1	no	—
70	C₄⋊Dic₁₃₉	—	2	1	no	—
71	C₄⋊Dic₁₃₉	—	2	1	no	—
72	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
76	Dic₁₃₉⋊C₄	—	2	1	no	—
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	C₄×Dic₁₃₉	—	2	1	no	—
85	C₄×Dic₁₃₉	—	2	1	no	—
86	C₄×Dic₁₃₉	—	2	1	no	—
87	C₄×Dic₁₃₉	—	2	1	no	—
88	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
89	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
90	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
91	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
92	#41×C₁₃₉	—	2	2	no	—
93	#41×C₁₃₉	Q₈	1	4	yes	complemented
94	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₃₉	—	2	1	no	—
96	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₁₃₉	—	2	1	no	—
98	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₃₉	—	2	2	no	—
100	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×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	C₄²×C₁₃₉	—	2	1	no	—
106	Dic₂₇₈	—	4	2	no	—
107	Dic₂₇₈	—	4	2	no	—
108	C₂×Dic₁₃₉	—	2	2	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	—	4	1	no	—
114	C₂×Dic₁₃₉	—	4	2	no	—
115	C₂×Dic₁₃₉	—	4	2	no	—
116	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
120	C₁₃₉⋊C₈	—	2	1	no	—
121	C₁₃₉⋊C₈	—	2	1	no	—
122	C₂³×C₁₃₉	—	2	1	no	—
123	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
124	Q₈×C₁₃₉	—	2	1	no	—
125	Q₈×C₁₃₉	C₂×Q₈	1	2	yes	—
126	D₄×C₁₃₉	—	2	4	no	—
127	D₄×C₁₃₉	—	2	1	no	—
128	D₄×C₁₃₉	C₂×Q₈	1	2	yes	—
129	C₂×C₄×C₁₃₉	—	2	1	no	—
130	C₂×C₄×C₁₃₉	—	2	4	no	—
131	C₂×C₄×C₁₃₉	—	2	1	no	—
132	C₂×C₄×C₁₃₉	—	2	1	no	—
133	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
135	C₂×C₄×C₁₃₉	#41	1	2	yes	—
136	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
137	C₈×C₁₃₉	—	2	1	no	—
138	C₈×C₁₃₉	—	2	1	no	—
139	C₂²×C₁₃₉	—	2	1	no	—
140	C₂²×C₁₃₉	—	2	1	no	—
141	C₂²×C₁₃₉	—	2	2	no	—
142	C₂²×C₁₃₉	—	4	1	no	—
143	C₂²×C₁₃₉	—	4	1	no	—
144	C₂²×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
145	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
146	C₄×C₁₃₉	—	2	2	no	—
147	C₄×C₁₃₉	—	4	1	no	—
148	C₄×C₁₃₉	—	4	1	no	—
149	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
150	C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
151	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
152	Dic₁₃₉	—	2	1	no	—
153	Dic₁₃₉	—	2	1	no	—
154	Dic₁₃₉	—	2	1	no	—
155	Dic₁₃₉	—	4	1	no	—
156	Dic₁₃₉	—	4	1	no	—
157	Dic₁₃₉	—	4	1	no	—
158	Dic₁₃₉	—	4	1	no	—
159	Dic₁₃₉	#135	1	2	yes	—
160	C₂×C₁₃₉	—	2	2	no	—
161	C₂×C₁₃₉	—	4	2	no	—
162	C₂×C₁₃₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
163	C₂×C₁₃₉	C₂×#135	1	1	yes	characteristic
164	C₂×C₁₃₉	#575	1	1	yes	characteristic
165	C₁₃₉	#3076	1	1	yes	characteristic, complemented, Sylow
166	#3076	—	139	1	no	maximal, Sylow
167	#533	—	139	1	no	—
168	C₄×#41	—	139	1	no	—
169	#479	—	139	1	no	—
170	#479	—	139	1	no	—
171	#478	—	139	1	no	—
172	#478	—	139	1	no	—
173	#477	—	139	1	no	—
174	#476	—	139	1	no	—
175	#475	—	139	1	no	—
176	#474	—	139	1	no	—
177	#427	—	139	1	no	—
178	#427	—	139	1	no	—
179	C₂×C₄⋊C₈	—	139	1	no	—
180	#416	—	139	1	no	—
181	#416	D₁₃₉	1	1	yes	characteristic
182	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
183	C₂²×C₈	—	139	1	no	—
184	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
185	C₄⋊Q₈	—	139	1	no	—
186	C₄⋊Q₈	—	139	1	no	—
187	#125	—	139	1	no	—
188	#125	—	139	1	no	—
189	C₂²⋊Q₈	—	278	1	no	—
190	C₂²⋊Q₈	—	278	1	no	—
191	C₄×Q₈	—	278	1	no	—
192	C₄×Q₈	—	278	1	no	—
193	C₄×Q₈	—	139	1	no	—
194	C₄×Q₈	—	139	1	no	—
195	C₄×D₄	—	278	1	no	—
196	C₄×D₄	—	278	1	no	—
197	C₄×D₄	—	139	1	no	—
198	C₄×D₄	—	139	1	no	—
199	#117	—	278	1	no	—
200	#117	—	139	1	no	—
201	#117	—	139	1	no	—
202	#117	C₂×D₁₃₉	1	1	yes	characteristic
203	C₂×C₄²	—	278	1	no	—
204	C₂×C₄²	—	139	1	no	—
205	#107	—	139	1	no	—
206	#107	—	139	1	no	—
207	#107	—	139	1	no	—
208	#107	—	139	1	no	—
209	#106	—	139	1	no	—
210	#106	—	139	1	no	—
211	#106	—	139	1	no	—
212	#106	—	139	1	no	—
213	C₄⋊C₈	—	139	1	no	—
214	C₄⋊C₈	—	139	1	no	—
215	C₄⋊C₈	—	139	1	no	—
216	C₄⋊C₈	—	139	1	no	—
217	#103	—	139	1	no	—
218	#103	—	139	1	no	—
219	#103	C₂×D₁₃₉	1	1	yes	characteristic
220	#103	C₂×D₁₃₉	1	1	yes	characteristic
221	#102	—	139	1	no	—
222	#102	—	139	1	no	—
223	#102	C₂×D₁₃₉	1	1	yes	characteristic
224	#102	C₂×D₁₃₉	1	1	yes	characteristic
225	#41	—	2	2	no	—
226	#41	Dic₂₇₈	1	4	yes	complemented
227	C₂×Q₈	—	278	1	no	—
228	C₂×Q₈	—	278	1	no	—
229	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	278	1	no	—
234	C₂²×C₄	—	278	1	no	—
235	C₂²×C₄	—	139	1	no	—
236	C₂²×C₄	—	139	1	no	—
237	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	278	2	no	—
240	C₂×C₈	—	139	1	no	—
241	C₂×C₈	—	139	1	no	—
242	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
244	C₄⋊C₄	—	278	1	no	—
245	C₄⋊C₄	—	278	1	no	—
246	C₄⋊C₄	—	278	1	no	—
247	C₄⋊C₄	—	278	1	no	—
248	C₄⋊C₄	—	278	1	no	—
249	C₄⋊C₄	—	278	1	no	—
250	C₄⋊C₄	—	278	1	no	—
251	C₄⋊C₄	—	278	1	no	—
252	C₄⋊C₄	—	139	1	no	—
253	C₄⋊C₄	—	139	1	no	—
254	C₄⋊C₄	—	139	1	no	—
255	C₄⋊C₄	—	139	1	no	—
256	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	278	1	no	—
260	C₂²⋊C₄	—	278	1	no	—
261	C₂²⋊C₄	—	278	1	no	—
262	C₂²⋊C₄	—	278	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	278	1	no	—
265	C₄²	—	278	1	no	—
266	C₄²	—	278	1	no	—
267	C₄²	—	278	1	no	—
268	C₄²	—	139	1	no	—
269	C₄²	—	139	1	no	—
270	C₄²	—	139	1	no	—
271	C₄²	—	139	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
274	Q₈	—	2	1	no	—
275	Q₈	—	556	2	no	—
276	Q₈	—	556	2	no	—
277	Q₈	C₂×Dic₂₇₈	1	2	yes	—
278	D₄	—	2	4	no	—
279	D₄	—	2	1	no	—
280	D₄	C₂×Dic₂₇₈	1	2	yes	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	278	2	no	—
286	C₂×C₄	—	278	1	no	—
287	C₂×C₄	—	278	1	no	—
288	C₂×C₄	—	278	1	no	—
289	C₂×C₄	—	278	1	no	—
290	C₂×C₄	—	556	1	no	—
291	C₂×C₄	—	556	2	no	—
292	C₂×C₄	—	556	2	no	—
293	C₂×C₄	—	139	1	no	—
294	C₂×C₄	—	139	1	no	—
295	C₂×C₄	—	139	1	no	—
296	C₂×C₄	—	139	1	no	—
297	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
298	C₂×C₄	C₂×Dic₂₇₈	1	2	yes	—
299	C₂×C₄	#14555784	1	2	yes	—
300	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	278	1	no	—
304	C₈	—	278	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	#17984819	1	2	yes	—
311	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	278	1	no	—
316	C₄	—	278	1	no	—
317	C₄	—	278	1	no	—
318	C₄	—	556	1	no	—
319	C₄	—	556	1	no	—
320	C₄	—	556	1	no	—
321	C₄	—	556	1	no	—
322	C₄	—	139	2	no	—
323	C₄	#17984819	1	2	yes	—
324	C₄	C₂²×Dic₂₇₈	1	1	yes	characteristic
325	C₄	C₂×#14555784	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	4	2	no	—
328	C₂	C₂×#17984819	1	1	yes	characteristic
329	C₂	#135×D₁₃₉	1	1	yes	characteristic
330	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	#416×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#41×C₁₃₉	C₂²
hypercenter	#416	D₁₃₉
hypocenter	C₁₃₉	#3076
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
1. #22581506	C₂	semidirect
3. #22581262	C₂	semidirect
4. #22581258	C₂	semidirect
5. #22581255	C₂	semidirect
6. #22581251	C₂	semidirect
7. #22581246	C₂	semidirect
8. #22581241	C₂	semidirect
9. #22581236	C₂	semidirect
10. #22581231	C₂	semidirect
11. #22580991	C₂	semidirect
12. #22580989	C₂	semidirect
13. C₂×Dic₁₃₉⋊C₈	C₂	semidirect
93. #41×C₁₃₉	Q₈	semidirect
165. C₁₃₉	#3076	semidirect
226. #41	Dic₂₇₈	semidirect

Maximal subgroups

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

subgroup	index
#22581506	2
#41×Dic₁₃₉	2
#22581262	2
#22581258	2
#22581255	2
#22581251	2
#22581246	2
#22581241	2
#22581236	2
#22581231	2
#22580991	2
#22580989	2
C₂×Dic₁₃₉⋊C₈	2
#22580949	2
#416×C₁₃₉	2
#3076	139