Subgroups — #31809407

Order 17792.

620 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984821	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₇₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#22580859	C₂	1	1	yes	maximal, characteristic, complemented
5	#22580851	C₂	1	1	yes	maximal, characteristic, complemented
6	#22580850	C₂	1	2	yes	maximal, complemented
7	#22580840	C₂	1	2	yes	maximal, complemented
8	#22580825	C₂	1	2	yes	maximal
9	C₄×C₄⋊Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊C₄×Dic₁₃₉	C₂	1	2	yes	maximal
11	C₄×C₂²⋊C₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₂₇₈	C₂²	1	1	yes	characteristic
13	C₂³×Dic₁₃₉	C₂²	1	2	yes	—
14	#17984821	C₄	1	8	yes	complemented
15	C₄×Dic₂₇₈	—	2	4	no	—
16	C₂×C₄⋊Dic₁₃₉	—	2	4	no	—
17	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₃₉⋊C₄	C₂²	1	2	yes	—
22	C₂×Dic₁₃₉⋊C₄	C₂²	1	2	yes	—
23	C₂×Dic₁₃₉⋊C₄	C₂²	1	2	yes	—
24	C₂×C₂²⋊Dic₁₃₉	C₂²	1	2	yes	—
25	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
26	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
27	C₁₃₉⋊#95	C₂²	1	2	yes	—
28	C₁₃₉⋊#95	C₂²	1	2	yes	—
29	C₁₃₉⋊#95	C₂²	1	2	yes	—
30	C₁₃₉⋊#95	C₂²	1	2	yes	—
31	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
36	#95×C₁₃₉	C₂²	1	1	yes	characteristic
37	#95×C₁₃₉	C₂²	1	1	yes	characteristic
38	C₂×Dic₂₇₈	—	2	2	no	—
39	C₂×Dic₂₇₈	—	2	2	no	—
40	C₂×Dic₂₇₈	C₂×C₄	1	4	yes	complemented
41	C₂²×Dic₁₃₉	—	2	4	no	—
42	C₂²×Dic₁₃₉	—	2	4	no	—
43	C₂²×Dic₁₃₉	—	4	2	no	—
44	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
45	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
46	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
47	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
48	C₄⋊Dic₁₃₉	—	2	4	no	—
49	C₄⋊Dic₁₃₉	—	2	2	no	—
50	C₄⋊Dic₁₃₉	C₂×C₄	1	4	yes	complemented
51	C₄⋊Dic₁₃₉	D₄	1	4	yes	—
52	Dic₁₃₉⋊C₄	—	2	4	no	—
53	Dic₁₃₉⋊C₄	—	4	2	no	—
54	Dic₁₃₉⋊C₄	C₂×C₄	1	8	yes	complemented
55	C₂²⋊Dic₁₃₉	C₂×C₄	1	8	yes	—
56	C₄×Dic₁₃₉	—	2	4	no	—
57	C₄×Dic₁₃₉	—	4	2	no	—
58	C₂⁴×C₁₃₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₉	—	2	2	no	—
60	C₂²×C₄×C₁₃₉	—	2	2	no	—
61	C₂²×C₄×C₁₃₉	C₂×C₄	1	4	yes	—
62	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
68	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
69	C₂²⋊C₄×C₁₃₉	Q₈	1	4	yes	complemented
70	C₄²×C₁₃₉	—	2	2	no	—
71	C₄²×C₁₃₉	—	2	2	no	—
72	Dic₂₇₈	—	2	8	no	—
73	C₂×Dic₁₃₉	—	2	2	no	—
74	C₂×Dic₁₃₉	—	2	2	no	—
75	C₂×Dic₁₃₉	—	2	2	no	—
76	C₂×Dic₁₃₉	—	2	2	no	—
77	C₂×Dic₁₃₉	—	2	2	no	—
78	C₂×Dic₁₃₉	—	2	2	no	—
79	C₂×Dic₁₃₉	—	2	2	no	—
80	C₂×Dic₁₃₉	—	2	2	no	—
81	C₂×Dic₁₃₉	—	2	2	no	—
82	C₂×Dic₁₃₉	—	4	8	no	—
83	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
84	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
85	C₂×Dic₁₃₉	C₂×D₄	1	4	yes	—
86	C₂³×C₁₃₉	—	2	1	no	—
87	C₂³×C₁₃₉	—	2	1	no	—
88	C₂³×C₁₃₉	—	2	1	no	—
89	C₂³×C₁₃₉	—	2	1	no	—
90	C₂³×C₁₃₉	C₂²×C₄	1	2	yes	—
91	C₂³×C₁₃₉	C₂×Q₈	1	2	yes	—
92	C₂³×C₁₃₉	#41	1	2	yes	—
93	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
94	C₂×C₄×C₁₃₉	—	2	1	no	—
95	C₂×C₄×C₁₃₉	—	2	1	no	—
96	C₂×C₄×C₁₃₉	—	2	1	no	—
97	C₂×C₄×C₁₃₉	—	2	1	no	—
98	C₂×C₄×C₁₃₉	—	2	1	no	—
99	C₂×C₄×C₁₃₉	—	2	1	no	—
100	C₂×C₄×C₁₃₉	—	2	8	no	—
101	C₂×C₄×C₁₃₉	—	2	1	no	—
102	C₂×C₄×C₁₃₉	—	2	1	no	—
103	C₂×C₄×C₁₃₉	—	2	1	no	—
104	C₂×C₄×C₁₃₉	—	2	1	no	—
105	C₂×C₄×C₁₃₉	—	2	1	no	—
106	C₂×C₄×C₁₃₉	—	2	1	no	—
107	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
111	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
112	C₂×C₄×C₁₃₉	#41	1	2	yes	—
113	C₂²×C₁₃₉	—	2	2	no	—
114	C₂²×C₁₃₉	—	2	2	no	—
115	C₂²×C₁₃₉	—	2	2	no	—
116	C₂²×C₁₃₉	—	2	2	no	—
117	C₂²×C₁₃₉	—	2	2	no	—
118	C₂²×C₁₃₉	—	2	2	no	—
119	C₂²×C₁₃₉	C₄×Q₈	1	4	yes	complemented
120	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
121	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
123	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
124	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
125	C₂²×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂²×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
127	C₄×C₁₃₉	—	2	2	no	—
128	C₄×C₁₃₉	—	2	2	no	—
129	C₄×C₁₃₉	—	2	2	no	—
130	C₄×C₁₃₉	—	4	1	no	—
131	C₄×C₁₃₉	—	4	1	no	—
132	C₄×C₁₃₉	C₄×D₄	1	4	yes	—
133	Dic₁₃₉	—	2	4	no	—
134	Dic₁₃₉	—	2	4	no	—
135	Dic₁₃₉	—	2	4	no	—
136	Dic₁₃₉	—	4	2	no	—
137	C₂×C₁₃₉	—	2	4	no	—
138	C₂×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂×C₁₃₉	C₂×C₄×Q₈	1	1	yes	characteristic
140	C₂×C₁₃₉	#520	1	1	yes	characteristic
141	C₂×C₁₃₉	#546	1	1	yes	characteristic
142	C₂×C₁₃₉	D₄×Q₈	1	1	yes	characteristic
143	C₂×C₁₃₉	#552	1	1	yes	characteristic
144	C₂×C₁₃₉	#554	1	1	yes	characteristic
145	C₁₃₉	#2360	1	1	yes	characteristic, complemented, Sylow
146	#2360	—	139	1	no	maximal, Sylow
147	C₂×C₂²⋊Q₈	—	139	1	no	—
148	C₂×C₄×Q₈	—	139	1	no	—
149	C₂²×C₄⋊C₄	—	139	1	no	—
150	#391	—	139	1	no	—
151	#389	—	139	1	no	—
152	#389	—	139	2	no	—
153	#387	—	139	2	no	—
154	#385	—	139	2	no	—
155	C₄×C₄⋊C₄	—	139	1	no	—
156	C₄×C₂²⋊C₄	—	139	2	no	—
157	C₄×C₂²⋊C₄	D₁₃₉	1	1	yes	characteristic
158	C₂²×Q₈	—	139	1	no	—
159	C₂³×C₄	—	139	2	no	—
160	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
161	C₂²⋊Q₈	—	139	8	no	—
162	C₄×Q₈	—	278	4	no	—
163	C₂×C₄⋊C₄	—	278	4	no	—
164	C₂×C₄⋊C₄	—	139	2	no	—
165	C₂×C₄⋊C₄	—	139	1	no	—
166	C₂×C₄⋊C₄	—	139	2	no	—
167	C₂×C₄⋊C₄	—	139	2	no	—
168	C₂×C₄⋊C₄	—	139	1	no	—
169	C₂×C₄⋊C₄	—	139	2	no	—
170	C₂×C₄⋊C₄	—	139	1	no	—
171	C₂×C₂²⋊C₄	—	139	2	no	—
172	C₂×C₂²⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
174	C₂×C₄²	—	139	2	no	—
175	C₂×C₄²	—	139	2	no	—
176	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
177	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
178	#95	—	139	2	no	—
179	#95	—	139	2	no	—
180	#95	—	139	2	no	—
181	#95	—	139	2	no	—
182	#95	C₂×D₁₃₉	1	1	yes	characteristic
183	#95	C₂×D₁₃₉	1	1	yes	characteristic
184	C₂⁴	C₂²×D₁₃₉	1	1	yes	characteristic
185	C₂×Q₈	—	278	2	no	—
186	C₂×Q₈	—	278	2	no	—
187	C₂×Q₈	—	139	4	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	278	4	no	—
191	C₂²×C₄	—	278	4	no	—
192	C₂²×C₄	—	556	2	no	—
193	C₂²×C₄	—	139	2	no	—
194	C₂²×C₄	—	139	2	no	—
195	C₂²×C₄	—	139	2	no	—
196	C₂²×C₄	—	139	2	no	—
197	C₂²×C₄	C₄×D₁₃₉	1	4	yes	—
198	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
199	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
204	C₄⋊C₄	—	278	4	no	—
205	C₄⋊C₄	—	278	4	no	—
206	C₄⋊C₄	—	278	2	no	—
207	C₄⋊C₄	—	556	2	no	—
208	C₄⋊C₄	—	139	8	no	—
209	C₄⋊C₄	—	139	4	no	—
210	C₄⋊C₄	—	139	4	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	139	8	no	—
213	C₂²⋊C₄	Dic₂₇₈	1	4	yes	complemented
214	C₄²	—	2	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	278	4	no	—
217	C₄²	—	556	2	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	1	no	—
222	C₂³	C₂×C₄×D₁₃₉	1	2	yes	—
223	C₂³	C₂×Dic₂₇₈	1	2	yes	—
224	C₂³	#14555785	1	2	yes	—
225	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
226	Q₈	—	278	8	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	278	2	no	—
241	C₂×C₄	—	278	2	no	—
242	C₂×C₄	—	278	2	no	—
243	C₂×C₄	—	278	2	no	—
244	C₂×C₄	—	278	2	no	—
245	C₂×C₄	—	278	2	no	—
246	C₂×C₄	—	278	2	no	—
247	C₂×C₄	—	278	2	no	—
248	C₂×C₄	—	278	2	no	—
249	C₂×C₄	—	556	8	no	—
250	C₂×C₄	—	139	4	no	—
251	C₂×C₄	—	139	4	no	—
252	C₂×C₄	—	139	4	no	—
253	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
254	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
255	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
256	C₂×C₄	C₂×Dic₂₇₈	1	2	yes	—
257	C₂×C₄	C₂×Dic₂₇₈	1	2	yes	—
258	C₂×C₄	#14555784	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	C₄×Dic₂₇₈	1	4	yes	complemented
266	C₂²	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
267	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
268	C₂²	C₂²×Dic₂₇₈	1	1	yes	characteristic
269	C₂²	C₂×#14555784	1	1	yes	characteristic
270	C₂²	C₂×#14555785	1	1	yes	characteristic
271	C₂²	#17984888	1	1	yes	characteristic
272	C₂²	#17984889	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	278	4	no	—
279	C₄	—	278	4	no	—
280	C₄	—	278	4	no	—
281	C₄	—	556	2	no	—
282	C₄	(C₂²⋊D₁₃₉)⋊C₄	1	4	yes	—
283	C₂	—	2	4	no	—
284	C₂	C₂×(C₂²⋊D₁₃₉)⋊C₄	1	1	yes	characteristic
285	C₂	C₂×C₄×Dic₂₇₈	1	1	yes	characteristic
286	C₂	#22581433	1	1	yes	characteristic
287	C₂	#22581599	1	1	yes	characteristic
288	C₂	D₄×Dic₂₇₈	1	1	yes	characteristic
289	C₂	#22581643	1	1	yes	characteristic
290	C₂	#22581660	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃₉
commutator	C₂²×C₁₃₉	C₂³×C₄
Fitting	C₄×C₂²⋊C₄×C₁₃₉	C₂
socle, cosocle	C₂³×C₁₃₉	C₂⁴
prime radical	C₂⁴×C₁₃₉	C₂³
hypercenter	C₄×C₂²⋊C₄	D₁₃₉
hypocenter	C₁₃₉	#2360
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₂₇₈	C₂	semidirect
4. #22580859	C₂	semidirect
5. #22580851	C₂	semidirect
6. #22580850	C₂	semidirect
7. #22580840	C₂	semidirect
9. C₄×C₄⋊Dic₁₃₉	C₂	semidirect
14. #17984821	C₄	semidirect
40. C₂×Dic₂₇₈	C₂×C₄	semidirect
50. C₄⋊Dic₁₃₉	C₂×C₄	semidirect
54. Dic₁₃₉⋊C₄	C₂×C₄	semidirect
69. C₂²⋊C₄×C₁₃₉	Q₈	semidirect
119. C₂²×C₁₃₉	C₄×Q₈	semidirect
145. C₁₃₉	#2360	semidirect
213. C₂²⋊C₄	Dic₂₇₈	semidirect
265. C₂²	C₄×Dic₂₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984821	2
C₂×C₄×Dic₂₇₈	2
C₂²×C₄⋊Dic₁₃₉	2
#22580859	2
#22580851	2
#22580850	2
#22580840	2
#22580825	2
C₄×C₄⋊Dic₁₃₉	2
C₂²⋊C₄×Dic₁₃₉	2
C₄×C₂²⋊C₄×C₁₃₉	2
#2360	139