Subgroups — #31814928

Order 17792.

778 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984888	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984850	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581434	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580959	C₂	1	1	yes	maximal, characteristic, complemented
5	#22580956	C₂	1	2	yes	maximal, complemented
6	#22580953	C₂	1	2	yes	maximal, complemented
7	#22580949	C₂	1	1	yes	maximal, characteristic, complemented
8	#22580948	C₂	1	2	yes	maximal, complemented
9	C₂×#17984762	C₂	1	1	yes	maximal, characteristic, complemented
10	#102×D₁₃₉	C₂	1	2	yes	maximal, complemented
11	#417×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#17984888	—	2	4	no	—
13	#17984888	C₄	1	8	yes	complemented
14	C₂×#14555786	C₂²	1	2	yes	—
15	C₂×#14555785	—	2	1	no	—
16	C₂×#14555785	C₂²	1	1	yes	characteristic
17	C₂²×D₅₅₆	—	2	1	no	—
18	C₂²×D₅₅₆	C₂²	1	1	yes	characteristic
19	C₂×D₄×D₁₃₉	—	2	2	no	—
20	C₂×D₄×D₁₃₉	C₂²	1	2	yes	—
21	#17984850	—	2	4	no	—
22	C₂×#14555768	C₂²	1	2	yes	complemented
23	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₈×D₁₃₉	C₂²	1	2	yes	complemented
25	#17984807	—	2	1	no	—
26	#17984804	—	2	1	no	—
27	C₄×C₂²⋊D₁₃₉	—	2	2	no	—
28	#17984801	C₂²	1	1	yes	characteristic, complemented
29	#17984800	C₂²	1	2	yes	complemented
30	C₄⋊C₄×D₁₃₉	C₂²	1	2	yes	complemented
31	#17984766	C₂²	1	1	yes	characteristic, complemented
32	#17984766	C₂²	1	1	yes	characteristic, complemented
33	#17984764	C₂²	1	2	yes	complemented
34	#17984764	C₂²	1	2	yes	complemented
35	#17984763	C₂²	1	1	yes	characteristic, complemented
36	#17984763	C₂²	1	1	yes	characteristic, complemented
37	#17984762	C₂²	1	2	yes	complemented
38	#17984762	C₂²	1	2	yes	complemented
39	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
41	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
42	#103×C₁₃₉	C₂²	1	2	yes	complemented
43	#102×C₁₃₉	C₂²	1	2	yes	complemented
44	C₂³×D₁₃₉	—	2	2	no	—
45	C₂³×D₁₃₉	—	2	1	no	—
46	#14555786	—	2	4	no	—
47	#14555786	C₂×C₄	1	8	yes	complemented
48	#14555785	—	2	8	no	—
49	#14555785	C₂×C₄	1	4	yes	—
50	#14555785	D₄	1	4	yes	—
51	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
52	C₂×D₅₅₆	—	2	2	no	—
53	C₂×D₅₅₆	—	2	8	no	—
54	C₂×D₅₅₆	—	2	1	no	—
55	C₂×D₅₅₆	—	2	2	no	—
56	C₂×D₅₅₆	C₂×C₄	1	4	yes	complemented
57	C₂×D₅₅₆	D₄	1	1	yes	characteristic
58	C₂×D₅₅₆	C₂³	1	2	yes	—
59	C₂×C₂²⋊D₁₃₉	—	2	2	no	—
60	C₂×C₂²⋊D₁₃₉	D₄	1	2	yes	—
61	D₄×D₁₃₉	—	2	16	no	—
62	D₄×D₁₃₉	—	2	4	no	—
63	D₄×D₁₃₉	C₂×C₄	1	8	yes	complemented
64	C₂×C₄×D₁₃₉	—	2	2	no	—
65	C₂×C₄×D₁₃₉	—	2	2	no	—
66	C₂×C₄×D₁₃₉	C₂³	1	2	yes	—
67	#14555768	—	2	4	no	—
68	C₂×C₁₃₉⋊C₈	—	2	2	no	—
69	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
71	C₈×D₁₃₉	—	2	4	no	—
72	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
74	Dic₁₃₉⋊C₄	—	2	2	no	—
75	C₂²⋊Dic₁₃₉	—	2	1	no	—
76	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
77	C₄×Dic₁₃₉	—	2	2	no	—
78	C₄×Dic₁₃₉	—	2	1	no	—
79	#41×C₁₃₉	—	2	2	no	—
80	#41×C₁₃₉	C₂×C₄	1	4	yes	complemented
81	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₁₃₉	—	2	1	no	—
83	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₁₃₉	—	2	1	no	—
85	C₂²×C₄×C₁₃₉	—	2	1	no	—
86	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
87	M₁₆×C₁₃₉	—	2	2	no	—
88	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
89	C₄⋊C₄×C₁₃₉	—	2	1	no	—
90	C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
91	C₂²×D₁₃₉	—	2	2	no	—
92	C₂²×D₁₃₉	—	2	4	no	—
93	C₂²×D₁₃₉	—	2	8	no	—
94	C₂²×D₁₃₉	—	8	2	no	—
95	C₂²×D₁₃₉	—	8	2	no	—
96	C₂²×D₁₃₉	—	8	1	no	—
97	C₂²×D₁₃₉	—	8	1	no	—
98	C₂²×D₁₃₉	C₂×D₄	1	2	yes	—
99	Dic₂₇₈	C₂²⋊C₄	1	4	yes	—
100	D₅₅₆	—	2	8	no	—
101	D₅₅₆	—	2	4	no	—
102	D₅₅₆	—	2	2	no	—
103	D₅₅₆	C₂²⋊C₄	1	4	yes	—
104	D₅₅₆	C₂²×C₄	1	4	yes	—
105	C₂²⋊D₁₃₉	—	2	8	no	—
106	C₂²⋊D₁₃₉	C₂²⋊C₄	1	8	yes	—
107	C₂×Dic₁₃₉	—	2	1	no	—
108	C₂×Dic₁₃₉	—	2	1	no	—
109	C₂×Dic₁₃₉	C₂×D₄	1	2	yes	—
110	C₄×D₁₃₉	—	2	8	no	—
111	C₄×D₁₃₉	—	8	2	no	—
112	C₄×D₁₃₉	C₂²×C₄	1	4	yes	—
113	C₄×D₁₃₉	C₂×D₄	1	4	yes	—
114	C₁₃₉⋊C₈	—	2	1	no	—
115	C₁₃₉⋊C₈	—	2	1	no	—
116	C₂³×C₁₃₉	—	2	1	no	—
117	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
118	Q₈×C₁₃₉	—	2	1	no	—
119	Q₈×C₁₃₉	C₂²×C₄	1	2	yes	—
120	D₄×C₁₃₉	—	2	4	no	—
121	D₄×C₁₃₉	—	2	1	no	—
122	D₄×C₁₃₉	C₂²×C₄	1	2	yes	—
123	C₂×C₄×C₁₃₉	—	2	1	no	—
124	C₂×C₄×C₁₃₉	—	2	4	no	—
125	C₂×C₄×C₁₃₉	—	2	2	no	—
126	C₂×C₄×C₁₃₉	—	8	1	no	—
127	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
128	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
130	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
131	C₈×C₁₃₉	—	2	2	no	—
132	C₂²×C₁₃₉	—	2	1	no	—
133	C₂²×C₁₃₉	—	2	1	no	—
134	C₂²×C₁₃₉	—	2	2	no	—
135	C₂²×C₁₃₉	—	4	1	no	—
136	C₂²×C₁₃₉	—	4	1	no	—
137	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	2	yes	—
138	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₁₃₉	—	2	2	no	—
140	C₂×D₁₃₉	—	2	2	no	—
141	C₂×D₁₃₉	—	2	4	no	—
142	C₂×D₁₃₉	—	4	2	no	—
143	C₂×D₁₃₉	—	4	2	no	—
144	C₂×D₁₃₉	—	8	8	no	—
145	C₂×D₁₃₉	—	8	4	no	—
146	C₂×D₁₃₉	C₂×C₂²⋊C₄	1	4	yes	—
147	C₄×C₁₃₉	—	2	2	no	—
148	C₄×C₁₃₉	—	4	2	no	—
149	C₄×C₁₃₉	C₂×C₂²⋊C₄	1	2	yes	—
150	C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
151	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
152	Dic₁₃₉	—	4	1	no	—
153	Dic₁₃₉	—	4	1	no	—
154	Dic₁₃₉	C₂×C₂²⋊C₄	1	4	yes	—
155	C₂×C₁₃₉	—	2	2	no	—
156	C₂×C₁₃₉	—	4	2	no	—
157	C₂×C₁₃₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₁₃₉	#576	1	1	yes	characteristic
159	C₂×C₁₃₉	#577	1	1	yes	characteristic
160	D₁₃₉	—	2	4	no	—
161	D₁₃₉	—	4	4	no	—
162	C₁₃₉	#2897	1	1	yes	characteristic, complemented, Sylow
163	#2897	—	139	1	no	maximal, Sylow
164	C₂×H(5,𝔽₂)	—	139	1	no	—
165	C₂×#131	—	139	1	no	—
166	#520	—	139	1	no	—
167	#418	—	139	1	no	—
168	#418	—	139	2	no	—
169	#417	—	139	2	no	—
170	#417	D₁₃₉	1	1	yes	characteristic, complemented
171	#416	—	139	1	no	—
172	#416	—	139	2	no	—
173	C₂×#102	—	139	1	no	—
174	C₂×#102	—	139	2	no	—
175	H(5,𝔽₂)	—	278	4	no	—
176	H(5,𝔽₂)	—	139	8	no	—
177	C₂×#41	—	278	1	no	—
178	C₂×#41	—	139	2	no	—
179	C₂×#41	—	139	1	no	—
180	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
181	C₂²×D₄	—	278	1	no	—
182	C₂²×D₄	—	278	2	no	—
183	C₂²×D₄	—	139	2	no	—
184	C₂²×D₄	—	139	1	no	—
185	#131	—	278	4	no	—
186	C₂×M₁₆	—	139	2	no	—
187	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic, complemented
188	C₂²×C₈	—	139	1	no	—
189	C₂²×C₈	—	139	2	no	—
190	C₄×Q₈	—	278	1	no	—
191	C₄×D₄	—	278	1	no	—
192	C₄×D₄	—	278	2	no	—
193	#117	—	139	1	no	—
194	#117	—	139	2	no	—
195	C₂×C₄⋊C₄	—	139	2	no	—
196	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic, complemented
197	#103	—	139	1	no	—
198	#103	—	139	1	no	—
199	#103	C₂×D₁₃₉	1	2	yes	complemented
200	#102	—	139	1	no	—
201	#102	—	139	1	no	—
202	#102	—	139	2	no	—
203	#102	—	139	2	no	—
204	#102	—	139	2	no	—
205	#102	—	139	2	no	—
206	#102	C₂×D₁₃₉	1	2	yes	complemented
207	C₂⁴	—	278	2	no	—
208	C₂⁴	—	278	1	no	—
209	#41	—	2	2	no	—
210	#41	—	278	8	no	—
211	#41	—	278	4	no	—
212	#41	—	139	8	no	—
213	#41	—	139	4	no	—
214	#41	—	139	4	no	—
215	#41	C₄×D₁₃₉	1	4	yes	complemented
216	C₂×Q₈	—	139	1	no	—
217	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	278	2	no	—
220	C₂×D₄	—	278	8	no	—
221	C₂×D₄	—	278	16	no	—
222	C₂×D₄	—	278	4	no	—
223	C₂×D₄	—	278	2	no	—
224	C₂×D₄	—	278	1	no	—
225	C₂×D₄	—	278	2	no	—
226	C₂×D₄	—	139	8	no	—
227	C₂×D₄	—	139	2	no	—
228	C₂×D₄	—	139	4	no	—
229	C₂×D₄	—	139	1	no	—
230	C₂×D₄	—	139	2	no	—
231	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	278	2	no	—
235	C₂²×C₄	—	278	2	no	—
236	C₂²×C₄	—	139	2	no	—
237	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
238	M₁₆	—	2	2	no	—
239	M₁₆	—	278	4	no	—
240	C₂×C₈	—	278	4	no	—
241	C₂×C₈	—	278	2	no	—
242	C₂×C₈	—	139	1	no	—
243	C₂×C₈	—	139	1	no	—
244	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	278	2	no	—
247	C₄⋊C₄	—	139	1	no	—
248	C₄⋊C₄	—	139	1	no	—
249	C₄⋊C₄	C₂²×D₁₃₉	1	2	yes	—
250	C₂²⋊C₄	—	278	2	no	—
251	C₂²⋊C₄	—	278	1	no	—
252	C₄²	—	278	2	no	—
253	C₄²	—	278	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	278	8	no	—
256	C₂³	—	278	2	no	—
257	C₂³	—	278	4	no	—
258	C₂³	—	1112	1	no	—
259	C₂³	—	1112	1	no	—
260	C₂³	—	1112	2	no	—
261	C₂³	—	1112	2	no	—
262	C₂³	—	139	2	no	—
263	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	—	139	4	no	—
266	Q₈	C₂×C₄×D₁₃₉	1	2	yes	—
267	D₄	—	2	4	no	—
268	D₄	—	2	1	no	—
269	D₄	—	278	8	no	—
270	D₄	—	278	8	no	—
271	D₄	—	278	4	no	—
272	D₄	—	278	2	no	—
273	D₄	—	139	8	no	—
274	D₄	—	139	4	no	—
275	D₄	—	139	4	no	—
276	D₄	C₂×C₄×D₁₃₉	1	2	yes	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	8	1	no	—
281	C₂×C₄	—	278	8	no	—
282	C₂×C₄	—	278	1	no	—
283	C₂×C₄	—	278	1	no	—
284	C₂×C₄	—	1112	2	no	—
285	C₂×C₄	—	139	2	no	—
286	C₂×C₄	—	139	4	no	—
287	C₂×C₄	—	139	4	no	—
288	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
289	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
290	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
291	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
292	C₈	—	2	2	no	—
293	C₈	—	278	1	no	—
294	C₈	—	278	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	278	2	no	—
301	C₂²	—	278	2	no	—
302	C₂²	—	278	4	no	—
303	C₂²	—	556	2	no	—
304	C₂²	—	556	2	no	—
305	C₂²	—	1112	4	no	—
306	C₂²	—	1112	8	no	—
307	C₂²	—	139	4	no	—
308	C₂²	C₂²⋊C₄×D₁₃₉	1	2	yes	—
309	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	556	1	no	—
313	C₄	—	556	1	no	—
314	C₄	—	139	4	no	—
315	C₄	C₂²⋊C₄×D₁₃₉	1	2	yes	—
316	C₄	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
317	C₄	C₂×D₄×D₁₃₉	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	2	no	—
320	C₂	—	278	4	no	—
321	C₂	—	556	4	no	—
322	C₂	C₂×C₂²⋊C₄×D₁₃₉	1	1	yes	characteristic
323	C₂	#22581757	1	1	yes	characteristic
324	C₂	#22581762	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator	C₄×C₁₃₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#417×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#17984888	C₂
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	#417	D₁₃₉
hypocenter	C₁₃₉	#2897
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
2. C₂×#17984850	C₂	semidirect
3. #22581434	C₂	semidirect
4. #22580959	C₂	semidirect
5. #22580956	C₂	semidirect
6. #22580953	C₂	semidirect
7. #22580949	C₂	semidirect
8. #22580948	C₂	semidirect
9. C₂×#17984762	C₂	semidirect
10. #102×D₁₃₉	C₂	semidirect
11. #417×C₁₃₉	C₂	semidirect
13. #17984888	C₄	semidirect
22. C₂×#14555768	C₂²	semidirect
23. C₂²×C₁₃₉⋊C₈	C₂²	semidirect
24. C₂×C₈×D₁₃₉	C₂²	semidirect
28. #17984801	C₂²	semidirect
29. #17984800	C₂²	semidirect
30. C₄⋊C₄×D₁₃₉	C₂²	semidirect
31. #17984766	C₂²	semidirect
32. #17984766	C₂²	semidirect
33. #17984764	C₂²	semidirect
34. #17984764	C₂²	semidirect
35. #17984763	C₂²	semidirect
36. #17984763	C₂²	semidirect
37. #17984762	C₂²	semidirect
38. #17984762	C₂²	semidirect
40. C₂×M₁₆×C₁₃₉	C₂²	semidirect
41. C₂×C₄⋊C₄×C₁₃₉	C₂²	semidirect
42. #103×C₁₃₉	C₂²	semidirect
43. #102×C₁₃₉	C₂²	semidirect
47. #14555786	C₂×C₄	semidirect
56. C₂×D₅₅₆	C₂×C₄	semidirect
63. D₄×D₁₃₉	C₂×C₄	semidirect
80. #41×C₁₃₉	C₂×C₄	semidirect
162. C₁₃₉	#2897	semidirect
170. #417	D₁₃₉	semidirect
187. C₂×M₁₆	C₂×D₁₃₉	semidirect
196. C₂×C₄⋊C₄	C₂×D₁₃₉	semidirect
199. #103	C₂×D₁₃₉	semidirect
206. #102	C₂×D₁₃₉	semidirect
215. #41	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×#17984888	2
C₂×#17984850	2
#22581434	2
#22580959	2
#22580956	2
#22580953	2
#22580949	2
#22580948	2
C₂×#17984762	2
#102×D₁₃₉	2
#417×C₁₃₉	2
#2897	139