Subgroups — C₂×#22580709 #31806104

Order 17792.

404 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555769	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984801	C₂	1	1	yes	maximal, characteristic
3	#22580709	C₂	1	4	yes	maximal, direct product factor
4	C₂×#117×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14555769	—	2	1	no	—
6	C₂×#14555769	—	2	1	no	—
7	C₂×#14555769	—	2	1	no	—
8	C₂×#14555769	—	2	1	no	—
9	C₂×#14555769	C₄	1	2	yes	complemented
10	C₂×#14555769	C₂²	1	2	yes	—
11	C₂²×C₁₃₉⋊C₈	C₄	1	2	yes	complemented
12	#17984801	—	2	1	no	—
13	#17984801	—	2	1	no	—
14	#17984801	C₄	1	2	yes	complemented
15	#17984801	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₃₉	—	2	1	no	—
17	C₂×C₂²⋊Dic₁₃₉	—	2	1	no	—
18	C₂×C₄×Dic₁₃₉	C₄	1	2	yes	complemented
19	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
20	#117×C₁₃₉	—	2	1	no	—
21	#117×C₁₃₉	—	2	1	no	—
22	#117×C₁₃₉	C₄	1	2	yes	complemented
23	#117×C₁₃₉	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
25	C₂×C₂²⋊C₄×C₁₃₉	—	2	1	no	—
26	C₂×C₄²×C₁₃₉	C₄	1	2	yes	complemented
27	C₂²×Dic₁₃₉	—	2	2	no	—
28	#14555769	—	2	2	no	—
29	#14555769	—	2	2	no	—
30	#14555769	—	2	2	no	—
31	#14555769	—	2	2	no	—
32	C₂×C₁₃₉⋊C₈	—	2	2	no	—
33	C₂×C₁₃₉⋊C₈	—	2	2	no	—
34	C₂×C₁₃₉⋊C₈	C₂×C₄	1	4	yes	complemented
35	C₄⋊Dic₁₃₉	—	2	2	no	—
36	C₄⋊Dic₁₃₉	—	2	2	no	—
37	C₂²⋊Dic₁₃₉	—	2	2	no	—
38	C₂²⋊Dic₁₃₉	—	2	2	no	—
39	C₄×Dic₁₃₉	—	2	2	no	—
40	C₄×Dic₁₃₉	C₂×C₄	1	4	yes	complemented
41	C₂⁴×C₁₃₉	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₉	—	2	2	no	—
43	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₉	Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₉	Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₁₃₉	—	2	2	no	—
58	C₄⋊C₄×C₁₃₉	—	2	2	no	—
59	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
60	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
61	C₄²×C₁₃₉	—	2	2	no	—
62	C₄²×C₁₃₉	C₂×C₄	1	4	yes	complemented
63	C₂×Dic₁₃₉	—	2	4	no	—
64	C₂×Dic₁₃₉	—	4	4	no	—
65	C₁₃₉⋊C₈	—	2	4	no	—
66	C₂³×C₁₃₉	—	4	2	no	—
67	C₂³×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂³×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂³×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂³×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂³×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
72	C₂³×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
73	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃₉	—	2	4	no	—
75	C₂×C₄×C₁₃₉	—	4	4	no	—
76	C₂×C₄×C₁₃₉	C₄²	1	1	yes	characteristic
77	C₂×C₄×C₁₃₉	C₄²	1	1	yes	characteristic
78	C₂×C₄×C₁₃₉	C₄²	1	1	yes	characteristic
79	C₂×C₄×C₁₃₉	C₄²	1	1	yes	characteristic
80	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
104	C₂²×C₁₃₉	—	2	2	no	—
105	C₂²×C₁₃₉	—	2	2	no	—
106	C₂²×C₁₃₉	—	2	2	no	—
107	C₂²×C₁₃₉	—	4	1	no	—
108	C₂²×C₁₃₉	—	4	1	no	—
109	C₂²×C₁₃₉	—	4	1	no	—
110	C₂²×C₁₃₉	—	4	1	no	—
111	C₂²×C₁₃₉	#95	1	1	yes	characteristic
112	C₂²×C₁₃₉	#95	1	1	yes	characteristic
113	C₂²×C₁₃₉	#95	1	1	yes	characteristic
114	C₂²×C₁₃₉	#95	1	1	yes	characteristic
115	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₁₃₉	—	4	4	no	—
119	C₄×C₁₃₉	#95	1	1	yes	characteristic
120	C₄×C₁₃₉	#95	1	1	yes	characteristic
121	C₄×C₁₃₉	#95	1	1	yes	characteristic
122	C₄×C₁₃₉	#95	1	1	yes	characteristic
123	C₄×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
124	C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
127	Dic₁₃₉	—	4	4	no	—
128	C₂×C₁₃₉	—	2	1	no	—
129	C₂×C₁₃₉	—	2	1	no	—
130	C₂×C₁₃₉	—	2	1	no	—
131	C₂×C₁₃₉	—	2	1	no	—
132	C₂×C₁₃₉	—	2	1	no	—
133	C₂×C₁₃₉	—	2	1	no	—
134	C₂×C₁₃₉	H(3,ℤ/4)	1	2	yes	complemented
135	C₂×C₁₃₉	C₂×#95	1	1	yes	characteristic
136	C₁₃₉	C₂×H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
137	C₂×H(3,ℤ/4)	—	139	1	no	maximal, Sylow
138	C₂²×M₁₆	—	139	1	no	—
139	C₂×#117	—	139	1	no	—
140	C₂×#117	D₁₃₉	1	1	yes	characteristic
141	H(3,ℤ/4)	—	139	4	no	—
142	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
143	C₂×M₁₆	—	278	1	no	—
144	C₂×M₁₆	—	278	1	no	—
145	C₂×M₁₆	—	278	1	no	—
146	C₂×M₁₆	—	278	1	no	—
147	C₂×M₁₆	—	139	2	no	—
148	C₂×M₁₆	—	139	2	no	—
149	C₂²×C₈	—	139	2	no	—
150	#117	—	2	1	no	—
151	#117	—	2	1	no	—
152	#117	—	278	1	no	—
153	#117	—	278	1	no	—
154	#117	—	139	2	no	—
155	#117	—	139	2	no	—
156	#117	Dic₁₃₉	1	2	yes	complemented
157	#117	C₂×D₁₃₉	1	2	yes	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	278	1	no	—
160	C₂×C₂²⋊C₄	—	2	1	no	—
161	C₂×C₂²⋊C₄	—	278	1	no	—
162	C₂×C₄²	—	139	2	no	—
163	C₂×C₄²	Dic₁₃₉	1	2	yes	complemented
164	C₂⁴	C₂²⋊D₁₃₉	1	1	yes	characteristic
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	278	2	no	—
167	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
168	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
169	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
170	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
171	C₂²×C₄	C₂×Dic₁₃₉	1	1	yes	characteristic
172	C₂²×C₄	C₂×Dic₁₃₉	1	1	yes	characteristic
173	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
174	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
175	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
176	C₂²×C₄	D₅₅₆	1	1	yes	characteristic
177	C₂²×C₄	D₅₅₆	1	1	yes	characteristic
178	C₂²×C₄	Dic₂₇₈	1	1	yes	characteristic
179	C₂²×C₄	Dic₂₇₈	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
181	M₁₆	—	278	2	no	—
182	M₁₆	—	278	2	no	—
183	M₁₆	—	278	2	no	—
184	M₁₆	—	278	2	no	—
185	C₂×C₈	—	278	2	no	—
186	C₂×C₈	—	278	2	no	—
187	C₂×C₈	—	139	4	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	278	2	no	—
191	C₄⋊C₄	—	278	2	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	278	2	no	—
195	C₂²⋊C₄	—	278	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	278	2	no	—
198	C₄²	—	139	4	no	—
199	C₄²	C₂×Dic₁₃₉	1	4	yes	complemented
200	C₂³	—	4	2	no	—
201	C₂³	(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
202	C₂³	(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
203	C₂³	C₂²⋊Dic₁₃₉	1	1	yes	characteristic
204	C₂³	C₂²⋊Dic₁₃₉	1	1	yes	characteristic
205	C₂³	Dic₁₃₉⋊C₄	1	1	yes	characteristic
206	C₂³	Dic₁₃₉⋊C₄	1	1	yes	characteristic
207	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	278	4	no	—
211	C₂×C₄	—	556	4	no	—
212	C₂×C₄	C₄×Dic₁₃₉	1	1	yes	characteristic
213	C₂×C₄	C₄×Dic₁₃₉	1	1	yes	characteristic
214	C₂×C₄	C₄×Dic₁₃₉	1	1	yes	characteristic
215	C₂×C₄	C₄×Dic₁₃₉	1	1	yes	characteristic
216	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
217	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
218	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
219	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
220	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
221	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
222	C₂×C₄	C₂²⋊Dic₁₃₉	1	1	yes	characteristic
223	C₂×C₄	C₂²⋊Dic₁₃₉	1	1	yes	characteristic
224	C₂×C₄	Dic₁₃₉⋊C₄	1	1	yes	characteristic
225	C₂×C₄	Dic₁₃₉⋊C₄	1	1	yes	characteristic
226	C₂×C₄	Dic₁₃₉⋊C₄	1	1	yes	characteristic
227	C₂×C₄	Dic₁₃₉⋊C₄	1	1	yes	characteristic
228	C₂×C₄	Dic₁₃₉⋊C₄	1	1	yes	characteristic
229	C₂×C₄	Dic₁₃₉⋊C₄	1	1	yes	characteristic
230	C₂×C₄	C₄⋊Dic₁₃₉	1	1	yes	characteristic
231	C₂×C₄	C₄⋊Dic₁₃₉	1	1	yes	characteristic
232	C₂×C₄	C₄⋊Dic₁₃₉	1	1	yes	characteristic
233	C₂×C₄	C₄⋊Dic₁₃₉	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
236	C₂×C₄	C₂²×Dic₁₃₉	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
238	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
239	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
240	C₈	—	278	4	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	C₁₃₉⋊#95	1	1	yes	characteristic
249	C₂²	C₁₃₉⋊#95	1	1	yes	characteristic
250	C₂²	C₁₃₉⋊#95	1	1	yes	characteristic
251	C₂²	C₁₃₉⋊#95	1	1	yes	characteristic
252	C₂²	C₂×(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
253	C₂²	C₂×C₂²⋊Dic₁₃₉	1	1	yes	characteristic
254	C₂²	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
255	C₄	—	4	4	no	—
256	C₄	—	556	4	no	—
257	C₄	C₁₃₉⋊#95	1	1	yes	characteristic
258	C₄	C₁₃₉⋊#95	1	1	yes	characteristic
259	C₄	C₁₃₉⋊#95	1	1	yes	characteristic
260	C₄	C₁₃₉⋊#95	1	1	yes	characteristic
261	C₄	C₂×C₄×Dic₁₃₉	1	1	yes	characteristic
262	C₄	C₂×(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
263	C₄	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
264	C₄	C₂×C₄⋊Dic₁₃₉	1	1	yes	characteristic
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	2	1	no	—
271	C₂	#22580709	1	2	yes	direct product factor
272	C₂	C₂×C₁₃₉⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₁₃₉
commutator	C₄×C₁₃₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	C₂×#117×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#95
prime radical	C₂⁴×C₁₃₉	D₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₂×#117	D₁₃₉
hypocenter	C₁₃₉	C₂×H(3,ℤ/4)
LCS	G ▷ C₄×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #22580709	C₂	direct (N × H)
9. C₂×#14555769	C₄	semidirect
11. C₂²×C₁₃₉⋊C₈	C₄	semidirect
14. #17984801	C₄	semidirect
18. C₂×C₄×Dic₁₃₉	C₄	semidirect
22. #117×C₁₃₉	C₄	semidirect
26. C₂×C₄²×C₁₃₉	C₄	semidirect
34. C₂×C₁₃₉⋊C₈	C₂×C₄	semidirect
40. C₄×Dic₁₃₉	C₂×C₄	semidirect
62. C₄²×C₁₃₉	C₂×C₄	semidirect
134. C₂×C₁₃₉	H(3,ℤ/4)	semidirect
136. C₁₃₉	C₂×H(3,ℤ/4)	semidirect
156. #117	Dic₁₃₉	semidirect
163. C₂×C₄²	Dic₁₃₉	semidirect
199. C₄²	C₂×Dic₁₃₉	semidirect
271. C₂	#22580709	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14555769	2
C₂×#17984801	2
#22580709	2
C₂×#117×C₁₃₉	2
C₂×H(3,ℤ/4)	139

Subgroups — C2×#22580709 #31806104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22580709 #31806104