Subgroups — #31815131

Order 17792.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984849	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14555769	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581031	C₂	1	2	yes	maximal, complemented
4	#22581025	C₂	1	2	yes	maximal, complemented
5	#22581025	C₂	1	2	yes	maximal, complemented
6	D₄×C₁₃₉⋊C₈	C₂	1	2	yes	maximal, complemented
7	#22580980	C₂	1	1	yes	maximal, characteristic, complemented
8	#22580920	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂²⋊(C₁₃₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	#22580909	C₂	1	1	yes	maximal, characteristic, complemented
11	#525×C₁₃₉	C₂	1	1	yes	maximal, characteristic
12	#17984849	—	2	4	no	—
13	C₂×#14555769	—	2	4	no	—
14	C₂×#14555769	C₂²	1	2	yes	—
15	C₂×#14555769	C₂²	1	1	yes	characteristic
16	C₂×#14555769	C₂²	1	1	yes	characteristic
17	C₂×#14555769	C₂²	1	2	yes	complemented
18	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
19	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
21	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	complemented
22	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	complemented
23	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	complemented
24	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
25	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
26	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
27	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
28	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	complemented
29	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	complemented
30	#17984753	C₂²	1	2	yes	complemented
31	C₄×C₁₃₉⋊C₈	C₂²	1	2	yes	complemented
32	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
33	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
34	#123×C₁₃₉	C₄	1	2	yes	—
35	C₂²⋊Q₈×C₁₃₉	C₄	1	1	yes	characteristic
36	C₂²⋊Q₈×C₁₃₉	C₄	1	1	yes	characteristic
37	#121×C₁₃₉	C₄	1	1	yes	characteristic
38	#121×C₁₃₉	C₄	1	1	yes	characteristic
39	#120×C₁₃₉	C₄	1	2	yes	—
40	C₄×D₄×C₁₃₉	C₂²	1	2	yes	—
41	C₄×D₄×C₁₃₉	C₂²	1	2	yes	—
42	#117×C₁₃₉	C₂²	1	1	yes	characteristic
43	#14555769	—	2	4	no	—
44	#14555769	—	2	4	no	—
45	#14555769	—	2	2	no	—
46	#14555769	D₄	1	4	yes	complemented
47	C₂×C₁₃₉⋊C₈	—	2	4	no	—
48	C₂×C₁₃₉⋊C₈	—	2	4	no	—
49	C₂×C₁₃₉⋊C₈	—	2	2	no	—
50	C₂×C₁₃₉⋊C₈	—	2	2	no	—
51	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
52	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
53	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
54	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
55	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
56	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
57	C₂⁴×C₁₃₉	C₂×C₄	1	1	yes	characteristic
58	#41×C₁₃₉	—	2	2	no	—
59	#41×C₁₃₉	—	2	2	no	—
60	C₂×Q₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
61	C₂×D₄×C₁₃₉	C₂×C₄	1	2	yes	—
62	C₂×D₄×C₁₃₉	C₂×C₄	1	2	yes	—
63	C₂×D₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
64	C₂×D₄×C₁₃₉	C₂×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃₉	—	2	2	no	—
66	C₂²×C₄×C₁₃₉	—	2	2	no	—
67	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
72	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
75	C₄⋊C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
76	C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
77	C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
78	C₂²⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
79	C₂²⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
80	C₂²⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
81	C₂²⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
82	C₂²⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
83	C₄²×C₁₃₉	C₂³	1	2	yes	—
84	C₁₃₉⋊C₈	—	2	1	no	—
85	C₁₃₉⋊C₈	—	2	2	no	—
86	C₁₃₉⋊C₈	—	2	1	no	—
87	C₁₃₉⋊C₈	—	2	1	no	—
88	C₁₃₉⋊C₈	—	2	1	no	—
89	C₁₃₉⋊C₈	C₂×D₄	1	4	yes	—
90	C₂³×C₁₃₉	—	2	2	no	—
91	C₂³×C₁₃₉	—	2	2	no	—
92	C₂³×C₁₃₉	—	4	1	no	—
93	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
94	C₂³×C₁₃₉	C₂²×C₄	1	2	yes	—
95	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
96	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
97	Q₈×C₁₃₉	—	2	2	no	—
98	D₄×C₁₃₉	—	2	4	no	—
99	D₄×C₁₃₉	—	2	4	no	—
100	D₄×C₁₃₉	—	2	2	no	—
101	D₄×C₁₃₉	—	4	2	no	—
102	C₂×C₄×C₁₃₉	—	2	2	no	—
103	C₂×C₄×C₁₃₉	—	2	2	no	—
104	C₂×C₄×C₁₃₉	—	2	2	no	—
105	C₂×C₄×C₁₃₉	—	2	1	no	—
106	C₂×C₄×C₁₃₉	—	2	1	no	—
107	C₂×C₄×C₁₃₉	—	2	2	no	—
108	C₂×C₄×C₁₃₉	—	2	1	no	—
109	C₂×C₄×C₁₃₉	—	2	1	no	—
110	C₂×C₄×C₁₃₉	—	2	2	no	—
111	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
112	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
113	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
116	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₁₃₉	#41	1	2	yes	—
119	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
120	C₂²×C₁₃₉	—	2	1	no	—
121	C₂²×C₁₃₉	—	2	1	no	—
122	C₂²×C₁₃₉	—	2	1	no	—
123	C₂²×C₁₃₉	—	2	1	no	—
124	C₂²×C₁₃₉	—	2	1	no	—
125	C₂²×C₁₃₉	—	2	2	no	—
126	C₂²×C₁₃₉	—	2	2	no	—
127	C₂²×C₁₃₉	—	2	1	no	—
128	C₂²×C₁₃₉	—	2	1	no	—
129	C₂²×C₁₃₉	—	2	2	no	—
130	C₂²×C₁₃₉	—	4	4	no	—
131	C₂²×C₁₃₉	C₄×D₄	1	2	yes	—
132	C₂²×C₁₃₉	#131	1	2	yes	complemented
133	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
134	C₄×C₁₃₉	—	2	1	no	—
135	C₄×C₁₃₉	—	2	1	no	—
136	C₄×C₁₃₉	—	2	2	no	—
137	C₄×C₁₃₉	—	2	2	no	—
138	C₄×C₁₃₉	—	2	2	no	—
139	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
140	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
142	C₂×C₁₃₉	—	2	2	no	—
143	C₂×C₁₃₉	—	2	2	no	—
144	C₂×C₁₃₉	—	4	2	no	—
145	C₂×C₁₃₉	—	4	1	no	—
146	C₂×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂×C₁₃₉	C₂×#131	1	1	yes	characteristic
148	C₂×C₁₃₉	#568	1	1	yes	characteristic
149	C₁₃₉	#2933	1	1	yes	characteristic, complemented, Sylow
150	#2933	—	139	1	no	maximal, Sylow
151	C₂×#131	—	139	1	no	—
152	C₂²×M₁₆	—	139	1	no	—
153	#525	D₁₃₉	1	1	yes	characteristic
154	#436	—	139	2	no	—
155	#435	—	139	2	no	—
156	#435	—	139	2	no	—
157	C₈×D₄	—	139	2	no	—
158	#424	—	139	1	no	—
159	#408	—	139	1	no	—
160	C₂×C₂²⋊C₈	—	139	1	no	—
161	#405	—	139	1	no	—
162	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
163	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
164	#131	—	278	4	no	—
165	C₂×M₁₆	—	278	4	no	—
166	C₂×M₁₆	—	139	1	no	—
167	C₂×M₁₆	—	139	2	no	—
168	C₂×M₁₆	—	139	2	no	—
169	C₂×M₁₆	—	139	1	no	—
170	C₂²×C₈	—	139	2	no	—
171	C₂²×C₈	—	139	1	no	—
172	C₂²×C₈	—	139	2	no	—
173	C₂²×C₈	—	139	1	no	—
174	#123	Dic₁₃₉	1	2	yes	—
175	C₂²⋊Q₈	Dic₁₃₉	1	1	yes	characteristic
176	C₂²⋊Q₈	Dic₁₃₉	1	1	yes	characteristic
177	#121	Dic₁₃₉	1	1	yes	characteristic
178	#121	Dic₁₃₉	1	1	yes	characteristic
179	#120	Dic₁₃₉	1	2	yes	—
180	C₄×D₄	C₂×D₁₃₉	1	2	yes	—
181	C₄×D₄	C₂×D₁₃₉	1	2	yes	—
182	#117	C₂×D₁₃₉	1	1	yes	characteristic
183	C₄⋊C₈	—	139	2	no	—
184	C₄⋊C₈	—	139	2	no	—
185	C₂²⋊C₈	—	139	1	no	—
186	C₂²⋊C₈	—	139	1	no	—
187	C₂²⋊C₈	—	139	1	no	—
188	C₂²⋊C₈	—	139	1	no	—
189	C₂²⋊C₈	—	139	2	no	—
190	C₂²⋊C₈	—	139	2	no	—
191	#97	—	139	2	no	—
192	C₄×C₈	—	139	2	no	—
193	C₂⁴	C₂×Dic₁₃₉	1	1	yes	characteristic
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	C₂×Q₈	C₂×Dic₁₃₉	1	1	yes	characteristic
197	C₂×D₄	C₂×Dic₁₃₉	1	2	yes	—
198	C₂×D₄	C₂×Dic₁₃₉	1	2	yes	—
199	C₂×D₄	C₂×Dic₁₃₉	1	1	yes	characteristic
200	C₂×D₄	C₂×Dic₁₃₉	1	2	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₂×Dic₁₃₉	1	1	yes	characteristic
204	C₂²×C₄	C₂×Dic₁₃₉	1	1	yes	characteristic
205	C₂²×C₄	C₂×Dic₁₃₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
208	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
210	M₁₆	—	278	2	no	—
211	M₁₆	—	278	4	no	—
212	M₁₆	—	278	4	no	—
213	M₁₆	—	139	4	no	—
214	C₂×C₈	—	278	2	no	—
215	C₂×C₈	—	278	2	no	—
216	C₂×C₈	—	278	4	no	—
217	C₂×C₈	—	278	4	no	—
218	C₂×C₈	—	139	1	no	—
219	C₂×C₈	—	139	1	no	—
220	C₂×C₈	—	139	1	no	—
221	C₂×C₈	—	139	1	no	—
222	C₂×C₈	—	139	2	no	—
223	C₂×C₈	—	139	2	no	—
224	C₄⋊C₄	C₂×Dic₁₃₉	1	1	yes	characteristic
225	C₄⋊C₄	C₂×Dic₁₃₉	1	1	yes	characteristic
226	C₄⋊C₄	C₂×Dic₁₃₉	1	2	yes	—
227	C₄⋊C₄	C₂×Dic₁₃₉	1	2	yes	—
228	C₂²⋊C₄	C₂×Dic₁₃₉	1	2	yes	—
229	C₂²⋊C₄	C₂×Dic₁₃₉	1	2	yes	—
230	C₂²⋊C₄	C₂×Dic₁₃₉	1	2	yes	—
231	C₂²⋊C₄	C₂×Dic₁₃₉	1	2	yes	—
232	C₂²⋊C₄	C₂×Dic₁₃₉	1	2	yes	—
233	C₄²	C₂²×D₁₃₉	1	2	yes	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	1	no	—
237	C₂³	C₂²×Dic₁₃₉	1	1	yes	characteristic
238	C₂³	C₂²×Dic₁₃₉	1	2	yes	—
239	C₂³	C₂²×Dic₁₃₉	1	1	yes	characteristic
240	C₂³	C₂²×Dic₁₃₉	1	1	yes	characteristic
241	Q₈	—	2	2	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	4	no	—
244	D₄	—	2	2	no	—
245	D₄	—	4	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	C₂²×Dic₁₃₉	1	2	yes	—
256	C₂×C₄	C₂²×Dic₁₃₉	1	2	yes	—
257	C₂×C₄	C₂²×Dic₁₃₉	1	1	yes	characteristic
258	C₂×C₄	C₂²×Dic₁₃₉	1	1	yes	characteristic
259	C₂×C₄	C₂²×Dic₁₃₉	1	2	yes	—
260	C₂×C₄	C₂²×Dic₁₃₉	1	1	yes	characteristic
261	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
262	C₂×C₄	#14555784	1	2	yes	—
263	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
264	C₈	—	278	1	no	—
265	C₈	—	278	1	no	—
266	C₈	—	278	2	no	—
267	C₈	—	278	1	no	—
268	C₈	—	278	1	no	—
269	C₈	—	139	4	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	4	no	—
281	C₂²	D₄×Dic₁₃₉	1	2	yes	—
282	C₂²	#17984849	1	2	yes	complemented
283	C₂²	C₂³×Dic₁₃₉	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	D₄×Dic₁₃₉	1	2	yes	—
290	C₄	C₂×D₄×D₁₃₉	1	1	yes	characteristic
291	C₄	C₂×#14555784	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	2	no	—
296	C₂	C₂×D₄×Dic₁₃₉	1	1	yes	characteristic
297	C₂	C₂×#17984849	1	1	yes	characteristic
298	C₂	#22581720	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₉
commutator, socle	C₂²×C₁₃₉	C₂³×C₄
Fitting, prime radical	#525×C₁₃₉	C₂
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	#525	D₁₃₉
hypocenter	C₁₃₉	#2933
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#17984849	C₂	semidirect
2. C₂²×#14555769	C₂	semidirect
3. #22581031	C₂	semidirect
4. #22581025	C₂	semidirect
5. #22581025	C₂	semidirect
6. D₄×C₁₃₉⋊C₈	C₂	semidirect
7. #22580980	C₂	semidirect
8. #22580920	C₂	semidirect
9. C₂×C₂²⋊(C₁₃₉⋊C₈)	C₂	semidirect
10. #22580909	C₂	semidirect
17. C₂×#14555769	C₂²	semidirect
21. C₂²×C₁₃₉⋊C₈	C₂²	semidirect
22. C₄⋊(C₁₃₉⋊C₈)	C₂²	semidirect
23. C₄⋊(C₁₃₉⋊C₈)	C₂²	semidirect
28. C₂²⋊(C₁₃₉⋊C₈)	C₂²	semidirect
29. C₂²⋊(C₁₃₉⋊C₈)	C₂²	semidirect
30. #17984753	C₂²	semidirect
31. C₄×C₁₃₉⋊C₈	C₂²	semidirect
46. #14555769	D₄	semidirect
132. C₂²×C₁₃₉	#131	semidirect
149. C₁₃₉	#2933	semidirect
282. C₂²	#17984849	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984849	2
C₂²×#14555769	2
#22581031	2
#22581025	2
D₄×C₁₃₉⋊C₈	2
#22580980	2
#22580920	2
C₂×C₂²⋊(C₁₃₉⋊C₈)	2
#22580909	2
#525×C₁₃₉	2
#2933	139