Subgroups — C₄×#17984889 #31819614

Order 17792.

1492 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984889	C₂	1	1	yes	maximal, characteristic
2	#22581436	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581433	C₂	1	3	yes	maximal, complemented
4	#22581432	C₂	1	6	yes	maximal, complemented
5	#22581430	C₂	1	3	yes	maximal, complemented
6	#22581429	C₂	1	2	yes	maximal, complemented
7	C₄×#14555785	C₂	1	3	yes	maximal, complemented
8	C₄×#14555784	C₂	1	6	yes	maximal, complemented
9	C₂×C₄×Dic₂₇₈	C₂	1	3	yes	maximal, complemented
10	C₄×Q₈×D₁₃₉	C₂	1	2	yes	maximal, complemented
11	C₄×#41×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#17984889	C₄	1	16	yes	direct product factor
13	C₂×#14555785	C₂²	1	3	yes	—
14	C₂×#14555784	C₂²	1	6	yes	—
15	C₂²×Dic₂₇₈	C₂²	1	3	yes	—
16	C₂×Q₈×D₁₃₉	C₂²	1	2	yes	—
17	Q₈×Dic₁₃₉	C₂²	1	2	yes	—
18	C₄×Dic₂₇₈	C₂²	1	6	yes	—
19	C₄×Dic₂₇₈	C₂²	1	3	yes	—
20	C₄×Dic₂₇₈	C₂²	1	6	yes	—
21	C₄×Dic₂₇₈	C₂²	1	6	yes	—
22	C₄×Dic₂₇₈	C₂²	1	3	yes	—
23	#17984807	C₂²	1	6	yes	—
24	#17984807	C₂²	1	6	yes	—
25	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
26	C₄×D₅₅₆	C₂²	1	3	yes	—
27	D₄×Dic₁₃₉	C₂²	1	6	yes	—
28	C₄×C₂²⋊D₁₃₉	C₂²	1	6	yes	—
29	C₄×C₂²⋊D₁₃₉	C₂²	1	6	yes	—
30	(C₂²⋊D₁₃₉)⋊C₄	C₂²	1	12	yes	—
31	#17984801	C₂²	1	3	yes	—
32	#17984800	C₂²	1	6	yes	—
33	#17984799	C₂²	1	6	yes	—
34	#17984798	C₂²	1	12	yes	—
35	C₂×C₄⋊Dic₁₃₉	C₂²	1	3	yes	—
36	C₂×Dic₁₃₉⋊C₄	C₂²	1	6	yes	—
37	C₄⋊C₄×D₁₃₉	C₂²	1	6	yes	—
38	C₂×C₄×Dic₁₃₉	C₂²	1	6	yes	—
39	C₄²×D₁₃₉	C₂²	1	6	yes	—
40	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
41	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
42	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
43	C₄×D₄×C₁₃₉	C₂²	1	3	yes	—
44	C₄×D₄×C₁₃₉	C₂²	1	3	yes	—
45	#117×C₁₃₉	C₂²	1	3	yes	—
46	C₂×C₄²×C₁₃₉	C₂²	1	3	yes	—
47	#14555785	C₂×C₄	1	24	yes	complemented
48	#14555784	C₂×C₄	1	48	yes	complemented
49	C₂×Dic₂₇₈	C₂×C₄	1	24	yes	complemented
50	C₂×Dic₂₇₈	C₂³	1	6	yes	—
51	C₂×Dic₂₇₈	C₂³	1	3	yes	—
52	Q₈×D₁₃₉	C₂×C₄	1	16	yes	complemented
53	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₃₉	C₂³	1	6	yes	—
55	C₂²×Dic₁₃₉	C₂³	1	6	yes	—
56	C₂²×Dic₁₃₉	C₂³	1	6	yes	—
57	C₂×C₄×D₁₃₉	C₂³	1	2	yes	—
58	C₂×C₄×D₁₃₉	C₂³	1	6	yes	—
59	C₂×C₄×D₁₃₉	C₂³	1	6	yes	—
60	C₄⋊Dic₁₃₉	C₂³	1	6	yes	—
61	C₄⋊Dic₁₃₉	C₂³	1	3	yes	—
62	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
63	Dic₁₃₉⋊C₄	C₂³	1	12	yes	—
64	Dic₁₃₉⋊C₄	C₂³	1	6	yes	—
65	Dic₁₃₉⋊C₄	C₂³	1	6	yes	—
66	C₂²⋊Dic₁₃₉	C₂³	1	6	yes	—
67	(C₂×D₁₃₉)⋊C₄	C₂³	1	2	yes	—
68	(C₂×D₁₃₉)⋊C₄	C₂³	1	6	yes	—
69	C₄×Dic₁₃₉	C₂³	1	6	yes	—
70	C₄×Dic₁₃₉	C₂³	1	6	yes	—
71	C₄×Dic₁₃₉	C₂³	1	6	yes	—
72	C₄×Dic₁₃₉	C₂³	1	12	yes	—
73	#41×C₁₃₉	C₂×C₄	1	8	yes	complemented
74	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
75	C₂×D₄×C₁₃₉	C₂³	1	3	yes	—
76	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
77	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
78	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
79	C₄⋊C₄×C₁₃₉	C₂³	1	3	yes	—
80	C₄⋊C₄×C₁₃₉	C₂³	1	3	yes	—
81	C₂²⋊C₄×C₁₃₉	C₂³	1	6	yes	—
82	C₄²×C₁₃₉	C₂³	1	3	yes	—
83	C₄²×C₁₃₉	C₂³	1	3	yes	—
84	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
85	C₄²×C₁₃₉	C₂³	1	3	yes	—
86	C₂²×D₁₃₉	C₂⁴	1	2	yes	—
87	Dic₂₇₈	C₂²×C₄	1	24	yes	—
88	Dic₂₇₈	C₂²×C₄	1	12	yes	—
89	D₅₅₆	C₂²×C₄	1	4	yes	—
90	C₂²⋊D₁₃₉	C₂²×C₄	1	24	yes	—
91	C₂×Dic₁₃₉	—	4	6	no	—
92	C₂×Dic₁₃₉	C₂²×C₄	1	24	yes	—
93	C₂×Dic₁₃₉	C₂⁴	1	2	yes	—
94	C₂×Dic₁₃₉	C₂⁴	1	6	yes	—
95	C₂×Dic₁₃₉	C₂⁴	1	6	yes	—
96	C₄×D₁₃₉	—	2	4	no	—
97	C₄×D₁₃₉	—	4	6	no	—
98	C₄×D₁₃₉	C₂²×C₄	1	24	yes	—
99	C₂³×C₁₃₉	C₂⁴	1	3	yes	—
100	Q₈×C₁₃₉	C₂²×C₄	1	4	yes	—
101	D₄×C₁₃₉	C₂²×C₄	1	12	yes	—
102	C₂×C₄×C₁₃₉	—	2	6	no	—
103	C₂×C₄×C₁₃₉	—	4	3	no	—
104	C₂×C₄×C₁₃₉	C₂²×C₄	1	12	yes	—
105	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
106	C₂×C₄×C₁₃₉	C₂⁴	1	3	yes	—
107	C₂×C₄×C₁₃₉	C₂⁴	1	3	yes	—
108	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
109	C₂×C₄×C₁₃₉	C₂⁴	1	3	yes	—
110	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
111	C₂²×C₁₃₉	—	2	3	no	—
112	C₂²×C₁₃₉	—	2	3	no	—
113	C₂²×C₁₃₉	C₂³×C₄	1	6	yes	—
114	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
115	C₂×D₁₃₉	—	2	2	no	—
116	C₂×D₁₃₉	—	2	2	no	—
117	C₂×D₁₃₉	C₂³×C₄	1	4	yes	—
118	C₄×C₁₃₉	—	2	3	no	—
119	C₄×C₁₃₉	—	2	1	no	—
120	C₄×C₁₃₉	—	2	3	no	—
121	C₄×C₁₃₉	C₂³×C₄	1	2	yes	—
122	C₄×C₁₃₉	C₂³×C₄	1	6	yes	—
123	C₄×C₁₃₉	2₋¹⁺⁴	1	2	yes	complemented
124	Dic₁₃₉	—	2	2	no	—
125	Dic₁₃₉	—	2	6	no	—
126	Dic₁₃₉	C₂³×C₄	1	12	yes	—
127	C₂×C₁₃₉	—	2	6	no	—
128	C₂×C₁₃₉	C₂⁴×C₄	1	1	yes	characteristic
129	C₂×C₁₃₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
130	C₂×C₁₃₉	#585	1	1	yes	characteristic
131	D₁₃₉	—	2	4	no	—
132	C₁₃₉	C₄×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
133	C₄×2₋¹⁺⁴	—	139	1	no	maximal, Sylow
134	C₂×2₋¹⁺⁴	—	139	1	no	—
135	#520	—	139	1	no	—
136	#520	—	139	6	no	—
137	#520	—	139	3	no	—
138	#519	—	139	2	no	—
139	#519	—	139	3	no	—
140	C₄×#41	—	139	6	no	—
141	C₄×#41	—	139	3	no	—
142	C₄×#41	D₁₃₉	1	1	yes	characteristic, complemented
143	C₂×C₄×Q₈	—	139	2	no	—
144	C₂×C₄×Q₈	—	139	3	no	—
145	2₋¹⁺⁴	—	139	16	no	—
146	C₂×#41	—	139	6	no	—
147	C₂×#41	—	139	3	no	—
148	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
149	C₂²×Q₈	—	139	2	no	—
150	C₂²×Q₈	—	139	3	no	—
151	C₄×Q₈	—	139	2	no	—
152	C₄×Q₈	—	139	6	no	—
153	C₄×Q₈	—	139	6	no	—
154	C₄×Q₈	—	139	6	no	—
155	C₄×Q₈	—	139	3	no	—
156	C₄×Q₈	—	139	6	no	—
157	C₄×Q₈	—	139	6	no	—
158	C₄×Q₈	—	139	3	no	—
159	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
160	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
161	C₄×D₄	—	139	6	no	—
162	C₄×D₄	—	139	6	no	—
163	C₄×D₄	—	139	6	no	—
164	C₄×D₄	—	139	12	no	—
165	C₄×D₄	—	139	1	no	—
166	C₄×D₄	—	139	3	no	—
167	C₄×D₄	C₂×D₁₃₉	1	3	yes	—
168	C₄×D₄	C₂×D₁₃₉	1	3	yes	—
169	#117	—	139	3	no	—
170	#117	—	139	6	no	—
171	#117	—	139	12	no	—
172	#117	—	139	6	no	—
173	#117	C₂×D₁₃₉	1	3	yes	—
174	C₂×C₄⋊C₄	—	139	6	no	—
175	C₂×C₄⋊C₄	—	139	3	no	—
176	C₂×C₄⋊C₄	—	139	6	no	—
177	C₂×C₄²	—	139	6	no	—
178	C₂×C₄²	—	139	6	no	—
179	C₂×C₄²	C₂×D₁₃₉	1	3	yes	—
180	#41	—	139	48	no	—
181	#41	—	139	24	no	—
182	#41	C₄×D₁₃₉	1	8	yes	complemented
183	C₂×Q₈	—	139	16	no	—
184	C₂×Q₈	—	139	24	no	—
185	C₂×Q₈	—	139	6	no	—
186	C₂×Q₈	—	139	3	no	—
187	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
188	C₂×D₄	—	139	6	no	—
189	C₂×D₄	—	139	1	no	—
190	C₂×D₄	C₂²×D₁₃₉	1	3	yes	—
191	C₂²×C₄	—	139	2	no	—
192	C₂²×C₄	—	139	6	no	—
193	C₂²×C₄	—	139	6	no	—
194	C₂²×C₄	—	139	6	no	—
195	C₂²×C₄	—	139	6	no	—
196	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
197	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
198	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
199	C₄⋊C₄	—	139	12	no	—
200	C₄⋊C₄	—	139	6	no	—
201	C₄⋊C₄	—	139	6	no	—
202	C₄⋊C₄	—	139	3	no	—
203	C₄⋊C₄	—	139	6	no	—
204	C₄⋊C₄	—	139	1	no	—
205	C₄⋊C₄	C₂²×D₁₃₉	1	3	yes	—
206	C₄⋊C₄	C₂²×D₁₃₉	1	3	yes	—
207	C₂²⋊C₄	—	139	6	no	—
208	C₂²⋊C₄	—	139	2	no	—
209	C₂²⋊C₄	—	139	6	no	—
210	C₂²⋊C₄	C₂²×D₁₃₉	1	6	yes	—
211	C₄²	—	139	6	no	—
212	C₄²	—	139	6	no	—
213	C₄²	—	139	12	no	—
214	C₄²	—	139	6	no	—
215	C₄²	C₂²×D₁₃₉	1	3	yes	—
216	C₄²	C₂²×D₁₃₉	1	3	yes	—
217	C₄²	C₂²×D₁₃₉	1	3	yes	—
218	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
219	C₂³	—	139	2	no	—
220	C₂³	C₂³×D₁₃₉	1	3	yes	—
221	Q₈	—	139	24	no	—
222	Q₈	—	139	12	no	—
223	Q₈	C₂×C₄×D₁₃₉	1	4	yes	—
224	D₄	—	139	24	no	—
225	D₄	—	139	4	no	—
226	D₄	C₂×C₄×D₁₃₉	1	12	yes	—
227	C₂×C₄	—	2	6	no	—
228	C₂×C₄	—	4	3	no	—
229	C₂×C₄	—	278	4	no	—
230	C₂×C₄	—	556	6	no	—
231	C₂×C₄	—	556	6	no	—
232	C₂×C₄	—	139	2	no	—
233	C₂×C₄	—	139	6	no	—
234	C₂×C₄	—	139	24	no	—
235	C₂×C₄	—	139	6	no	—
236	C₂×C₄	—	139	24	no	—
237	C₂×C₄	C₂×C₄×D₁₃₉	1	12	yes	—
238	C₂×C₄	C₂³×D₁₃₉	1	3	yes	—
239	C₂×C₄	C₂³×D₁₃₉	1	3	yes	—
240	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
241	C₂×C₄	C₂³×D₁₃₉	1	3	yes	—
242	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
244	C₂²	—	2	3	no	—
245	C₂²	—	2	3	no	—
246	C₂²	—	278	2	no	—
247	C₂²	—	278	2	no	—
248	C₂²	—	139	4	no	—
249	C₂²	C₂²×C₄×D₁₃₉	1	6	yes	—
250	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
251	C₄	—	2	3	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	3	no	—
254	C₄	—	278	2	no	—
255	C₄	—	278	6	no	—
256	C₄	—	139	12	no	—
257	C₄	C₂²×C₄×D₁₃₉	1	6	yes	—
258	C₄	C₂²×C₄×D₁₃₉	1	2	yes	—
259	C₄	#17984889	1	2	yes	direct product factor
260	C₂	—	2	6	no	—
261	C₂	—	278	4	no	—
262	C₂	C₂³×C₄×D₁₃₉	1	1	yes	characteristic
263	C₂	C₂×#17984889	1	1	yes	characteristic
264	C₂	#22581789	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₃₉
commutator	C₂×C₁₃₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₃₉
Fitting	C₄×#41×C₁₃₉	C₂
socle, cosocle	C₂²×C₁₃₉	C₂⁵
prime radical	C₂×#17984889	C₂
hypercenter	C₄×#41	D₁₃₉
hypocenter	C₁₃₉	C₄×2₋¹⁺⁴
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. #22581436	C₂	semidirect
3. #22581433	C₂	semidirect
4. #22581432	C₂	semidirect
5. #22581430	C₂	semidirect
6. #22581429	C₂	semidirect
7. C₄×#14555785	C₂	semidirect
8. C₄×#14555784	C₂	semidirect
9. C₂×C₄×Dic₂₇₈	C₂	semidirect
10. C₄×Q₈×D₁₃₉	C₂	semidirect
11. C₄×#41×C₁₃₉	C₂	semidirect
12. #17984889	C₄	direct (N × H)
47. #14555785	C₂×C₄	semidirect
48. #14555784	C₂×C₄	semidirect
49. C₂×Dic₂₇₈	C₂×C₄	semidirect
52. Q₈×D₁₃₉	C₂×C₄	semidirect
73. #41×C₁₃₉	C₂×C₄	semidirect
123. C₄×C₁₃₉	2₋¹⁺⁴	semidirect
132. C₁₃₉	C₄×2₋¹⁺⁴	semidirect
142. C₄×#41	D₁₃₉	semidirect
182. #41	C₄×D₁₃₉	semidirect
259. C₄	#17984889	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17984889	2
#22581436	2
#22581433	2
#22581432	2
#22581430	2
#22581429	2
C₄×#14555785	2
C₄×#14555784	2
C₂×C₄×Dic₂₇₈	2
C₄×Q₈×D₁₃₉	2
C₄×#41×C₁₃₉	2
C₄×2₋¹⁺⁴	139

Subgroups — C4×#17984889 #31819614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17984889 #31819614