Subgroups — C₂×#22581465 #31819647

Order 17792.

1880 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555785	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₄×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581465	C₂	1	16	yes	maximal, direct product factor
4	C₂×#17984826	C₂	1	2	yes	maximal, complemented
5	C₂×#17984822	C₂	1	2	yes	maximal, complemented
6	C₂×#17984817	C₂	1	2	yes	maximal, complemented
7	C₂×#17984812	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₅₅₆	C₂	1	4	yes	maximal, complemented
9	C₂×#117×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₃₉	C₂²	1	1	yes	characteristic
11	C₂×#14555785	—	2	8	no	—
12	C₂×#14555785	C₂²	1	8	yes	complemented
13	C₂²×Dic₂₇₈	C₂²	1	1	yes	characteristic, complemented
14	C₂²×D₅₅₆	C₂²	1	4	yes	—
15	C₂²×D₅₅₆	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₂²⋊D₁₃₉	C₂²	1	2	yes	—
17	C₂³×Dic₁₃₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₁₃₉	—	2	8	no	—
19	C₂²×C₄×D₁₃₉	C₂²	1	2	yes	—
20	C₂²×C₄×D₁₃₉	C₂²	1	2	yes	—
21	C₂²×C₄×D₁₃₉	C₂²	1	8	yes	complemented
22	C₂²×C₄×D₁₃₉	C₂²	1	2	yes	complemented
23	#17984826	C₂²	1	16	yes	complemented
24	#17984822	C₂²	1	16	yes	complemented
25	#17984817	C₂²	1	16	yes	complemented
26	#17984812	C₂²	1	16	yes	complemented
27	C₄×D₅₅₆	C₂²	1	32	yes	complemented
28	C₂×C₄⋊Dic₁₃₉	C₂²	1	4	yes	complemented
29	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	4	yes	—
30	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	4	yes	complemented
31	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
32	#117×C₁₃₉	C₂²	1	8	yes	complemented
33	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	complemented
34	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	2	yes	—
35	C₂×C₄²×C₁₃₉	C₂²	1	2	yes	complemented
36	C₂³×D₁₃₉	—	2	8	no	—
37	C₂³×D₁₃₉	—	2	4	no	—
38	C₂³×D₁₃₉	C₂³	1	2	yes	—
39	C₂³×D₁₃₉	C₂³	1	4	yes	—
40	C₂³×D₁₃₉	C₂³	1	2	yes	—
41	#14555785	—	2	32	no	—
42	C₂×Dic₂₇₈	—	2	4	no	—
43	C₂×Dic₂₇₈	C₂³	1	4	yes	complemented
44	C₂×D₅₅₆	—	2	16	no	—
45	C₂×D₅₅₆	—	2	4	no	—
46	C₂×D₅₅₆	C₂³	1	16	yes	—
47	C₂×D₅₅₆	C₂³	1	4	yes	complemented
48	C₂×C₂²⋊D₁₃₉	—	2	8	no	—
49	C₂×C₂²⋊D₁₃₉	C₂³	1	8	yes	—
50	C₂²×Dic₁₃₉	—	2	4	no	—
51	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
52	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
53	C₂²×Dic₁₃₉	C₂³	1	4	yes	—
54	C₂×C₄×D₁₃₉	—	2	8	no	—
55	C₂×C₄×D₁₃₉	—	2	8	no	—
56	C₂×C₄×D₁₃₉	—	2	8	no	—
57	C₂×C₄×D₁₃₉	—	2	32	no	—
58	C₂×C₄×D₁₃₉	C₂³	1	8	yes	—
59	C₂×C₄×D₁₃₉	C₂³	1	8	yes	—
60	C₂×C₄×D₁₃₉	C₂³	1	8	yes	complemented
61	C₄⋊Dic₁₃₉	C₂³	1	16	yes	complemented
62	(C₂×D₁₃₉)⋊C₄	C₂³	1	16	yes	—
63	(C₂×D₁₃₉)⋊C₄	C₂³	1	16	yes	complemented
64	C₂⁴×C₁₃₉	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₉	D₄	1	8	yes	complemented
66	C₂²×C₄×C₁₃₉	C₂³	1	4	yes	—
67	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₉	C₂³	1	4	yes	—
69	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₃₉	C₂³	1	8	yes	complemented
71	C₂²⋊C₄×C₁₃₉	C₂³	1	8	yes	—
72	C₄²×C₁₃₉	C₂³	1	8	yes	complemented
73	C₂²×D₁₃₉	—	2	32	no	—
74	C₂²×D₁₃₉	—	2	8	no	—
75	C₂²×D₁₃₉	—	2	4	no	—
76	C₂²×D₁₃₉	—	2	4	no	—
77	C₂²×D₁₃₉	—	2	16	no	—
78	C₂²×D₁₃₉	—	2	4	no	—
79	C₂²×D₁₃₉	—	2	8	no	—
80	C₂²×D₁₃₉	#41	1	8	yes	complemented
81	C₂²×D₁₃₉	C₂⁴	1	4	yes	—
82	C₂²×D₁₃₉	C₂⁴	1	4	yes	—
83	Dic₂₇₈	—	2	8	no	—
84	D₅₅₆	—	2	32	no	—
85	D₅₅₆	—	2	8	no	—
86	C₂²⋊D₁₃₉	—	2	16	no	—
87	C₂×Dic₁₃₉	—	2	8	no	—
88	C₂×Dic₁₃₉	—	2	4	no	—
89	C₂×Dic₁₃₉	—	2	4	no	—
90	C₂×Dic₁₃₉	C₂⁴	1	4	yes	—
91	C₂×Dic₁₃₉	C₂⁴	1	4	yes	—
92	C₄×D₁₃₉	—	2	16	no	—
93	C₄×D₁₃₉	—	2	16	no	—
94	C₄×D₁₃₉	—	2	16	no	—
95	C₂³×C₁₃₉	—	2	4	no	—
96	C₂³×C₁₃₉	C₂×D₄	1	4	yes	—
97	C₂³×C₁₃₉	C₂⁴	1	2	yes	—
98	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₁₃₉	—	2	8	no	—
100	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
101	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
102	C₂×C₄×C₁₃₉	C₂×D₄	1	16	yes	complemented
103	C₂×C₄×C₁₃₉	C₂⁴	1	8	yes	—
104	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
105	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
106	C₂²×C₁₃₉	—	2	8	no	—
107	C₂²×C₁₃₉	—	2	4	no	—
108	C₂²×C₁₃₉	C₂²×D₄	1	4	yes	—
109	C₂²×C₁₃₉	C₂²×D₄	1	2	yes	—
110	C₂²×C₁₃₉	C₂×#41	1	4	yes	—
111	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
112	C₂×D₁₃₉	—	2	32	no	—
113	C₂×D₁₃₉	—	2	16	no	—
114	C₂×D₁₃₉	—	2	4	no	—
115	C₂×D₁₃₉	—	2	4	no	—
116	C₂×D₁₃₉	—	2	4	no	—
117	C₂×D₁₃₉	—	2	8	no	—
118	C₂×D₁₃₉	—	4	8	no	—
119	C₂×D₁₃₉	C₂×#41	1	8	yes	complemented
120	C₄×C₁₃₉	—	2	8	no	—
121	C₄×C₁₃₉	C₂²×D₄	1	4	yes	—
122	C₄×C₁₃₉	C₂²×D₄	1	4	yes	—
123	Dic₁₃₉	—	2	4	no	—
124	Dic₁₃₉	—	2	4	no	—
125	C₂×C₁₃₉	—	2	4	no	—
126	C₂×C₁₃₉	#525	1	4	yes	complemented
127	C₂×C₁₃₉	C₂³×D₄	1	1	yes	characteristic
128	C₂×C₁₃₉	C₂²×#41	1	2	yes	—
129	D₁₃₉	—	2	8	no	—
130	D₁₃₉	—	4	4	no	—
131	C₁₃₉	C₂×#525	1	1	yes	characteristic, complemented, Sylow
132	C₂×#525	—	139	1	no	maximal, Sylow
133	C₂²×#41	—	139	1	no	—
134	C₂⁴×C₄	—	139	1	no	—
135	#525	—	139	16	no	—
136	C₂×#123	—	139	2	no	—
137	C₂×C₂²⋊Q₈	—	139	2	no	—
138	C₂×#121	—	139	2	no	—
139	C₂×#120	—	139	2	no	—
140	C₂×C₄×D₄	—	139	4	no	—
141	C₂×#117	D₁₃₉	1	1	yes	characteristic, complemented
142	C₂⁵	—	139	1	no	—
143	C₂×#41	—	278	8	no	—
144	C₂×#41	—	139	8	no	—
145	C₂²×Q₈	—	139	1	no	—
146	C₂²×D₄	—	139	2	no	—
147	C₂²×D₄	—	139	4	no	—
148	C₂²×D₄	—	139	1	no	—
149	C₂³×C₄	—	278	8	no	—
150	C₂³×C₄	—	139	1	no	—
151	C₂³×C₄	—	139	2	no	—
152	C₂³×C₄	—	139	8	no	—
153	C₂³×C₄	—	139	2	no	—
154	C₂³×C₄	—	139	2	no	—
155	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
156	#123	—	139	16	no	—
157	C₂²⋊Q₈	—	139	16	no	—
158	#121	—	139	16	no	—
159	#120	—	139	16	no	—
160	C₄×D₄	—	139	32	no	—
161	#117	C₂×D₁₃₉	1	8	yes	complemented
162	C₂×C₄⋊C₄	—	139	4	no	—
163	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	complemented
164	C₂×C₂²⋊C₄	—	139	4	no	—
165	C₂×C₂²⋊C₄	—	139	4	no	—
166	C₂×C₂²⋊C₄	C₂×D₁₃₉	1	2	yes	—
167	C₂×C₄²	C₂×D₁₃₉	1	2	yes	complemented
168	C₂⁴	—	278	8	no	—
169	C₂⁴	—	278	4	no	—
170	C₂⁴	—	139	2	no	—
171	C₂⁴	—	139	4	no	—
172	C₂⁴	—	139	2	no	—
173	C₂⁴	C₂²×D₁₃₉	1	1	yes	characteristic
174	#41	—	278	32	no	—
175	C₂×Q₈	—	278	4	no	—
176	C₂×Q₈	—	139	4	no	—
177	C₂×D₄	—	278	16	no	—
178	C₂×D₄	—	278	8	no	—
179	C₂×D₄	—	278	4	no	—
180	C₂×D₄	—	139	8	no	—
181	C₂×D₄	—	139	16	no	—
182	C₂×D₄	—	139	4	no	—
183	C₂²×C₄	—	278	4	no	—
184	C₂²×C₄	—	278	8	no	—
185	C₂²×C₄	—	278	8	no	—
186	C₂²×C₄	—	278	8	no	—
187	C₂²×C₄	—	278	32	no	—
188	C₂²×C₄	—	139	2	no	—
189	C₂²×C₄	—	139	2	no	—
190	C₂²×C₄	—	139	4	no	—
191	C₂²×C₄	—	139	8	no	—
192	C₂²×C₄	—	139	8	no	—
193	C₂²×C₄	—	139	8	no	—
194	C₂²×C₄	D₅₅₆	1	8	yes	complemented
195	C₂²×C₄	C₂²×D₁₃₉	1	4	yes	—
196	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₁₃₉	1	4	yes	—
198	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
199	C₄⋊C₄	—	139	16	no	—
200	C₄⋊C₄	C₂²×D₁₃₉	1	8	yes	complemented
201	C₂²⋊C₄	—	139	16	no	—
202	C₂²⋊C₄	—	139	16	no	—
203	C₂²⋊C₄	C₂²×D₁₃₉	1	8	yes	—
204	C₄²	C₂²×D₁₃₉	1	8	yes	complemented
205	C₂³	—	2	4	no	—
206	C₂³	—	278	32	no	—
207	C₂³	—	278	16	no	—
208	C₂³	—	278	8	no	—
209	C₂³	—	278	4	no	—
210	C₂³	—	278	4	no	—
211	C₂³	—	278	4	no	—
212	C₂³	—	278	8	no	—
213	C₂³	—	139	4	no	—
214	C₂³	—	139	8	no	—
215	C₂³	—	139	4	no	—
216	C₂³	C₂×D₅₅₆	1	4	yes	—
217	C₂³	C₂³×D₁₃₉	1	2	yes	—
218	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
219	Q₈	—	278	8	no	—
220	D₄	—	278	32	no	—
221	D₄	—	278	16	no	—
222	D₄	—	278	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	278	8	no	—
225	C₂×C₄	—	278	4	no	—
226	C₂×C₄	—	278	4	no	—
227	C₂×C₄	—	278	16	no	—
228	C₂×C₄	—	278	16	no	—
229	C₂×C₄	—	278	16	no	—
230	C₂×C₄	—	139	4	no	—
231	C₂×C₄	—	139	4	no	—
232	C₂×C₄	C₂×D₅₅₆	1	4	yes	—
233	C₂×C₄	C₂×D₅₅₆	1	4	yes	—
234	C₂×C₄	C₂×D₅₅₆	1	16	yes	complemented
235	C₂×C₄	C₂³×D₁₃₉	1	8	yes	—
236	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
237	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	8	no	—
240	C₂²	—	278	32	no	—
241	C₂²	—	278	4	no	—
242	C₂²	—	278	4	no	—
243	C₂²	—	278	16	no	—
244	C₂²	—	278	4	no	—
245	C₂²	—	278	8	no	—
246	C₂²	—	556	8	no	—
247	C₂²	—	139	8	no	—
248	C₂²	C₂²×D₅₅₆	1	4	yes	—
249	C₂²	C₂²×D₅₅₆	1	2	yes	—
250	C₂²	#41×D₁₃₉	1	4	yes	—
251	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
252	C₄	—	2	8	no	—
253	C₄	—	278	4	no	—
254	C₄	—	278	4	no	—
255	C₄	C₂²×D₅₅₆	1	4	yes	—
256	C₄	C₂²×D₅₅₆	1	4	yes	—
257	C₂	—	2	4	no	—
258	C₂	—	278	8	no	—
259	C₂	—	556	4	no	—
260	C₂	#22581465	1	4	yes	direct product factor
261	C₂	C₂³×D₅₅₆	1	1	yes	characteristic
262	C₂	C₂×#41×D₁₃₉	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃₉
commutator, cosocle	C₂²×C₁₃₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₉
Fitting	C₂×#117×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂⁴
hypercenter	C₂×#117	D₁₃₉
hypocenter	C₁₃₉	C₂×#525
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. C₂²×#14555785	C₂	semidirect
2. C₂³×C₄×D₁₃₉	C₂	semidirect
3. #22581465	C₂	direct (N × H)
4. C₂×#17984826	C₂	semidirect
5. C₂×#17984822	C₂	semidirect
6. C₂×#17984817	C₂	semidirect
7. C₂×#17984812	C₂	semidirect
8. C₂×C₄×D₅₅₆	C₂	semidirect
9. C₂×#117×C₁₃₉	C₂	semidirect
12. C₂×#14555785	C₂²	semidirect
13. C₂²×Dic₂₇₈	C₂²	semidirect
15. C₂²×D₅₅₆	C₂²	semidirect
21. C₂²×C₄×D₁₃₉	C₂²	semidirect
22. C₂²×C₄×D₁₃₉	C₂²	semidirect
23. #17984826	C₂²	semidirect
24. #17984822	C₂²	semidirect
25. #17984817	C₂²	semidirect
26. #17984812	C₂²	semidirect
27. C₄×D₅₅₆	C₂²	semidirect
28. C₂×C₄⋊Dic₁₃₉	C₂²	semidirect
30. C₂×(C₂×D₁₃₉)⋊C₄	C₂²	semidirect
32. #117×C₁₃₉	C₂²	semidirect
33. C₂×C₄⋊C₄×C₁₃₉	C₂²	semidirect
35. C₂×C₄²×C₁₃₉	C₂²	semidirect
43. C₂×Dic₂₇₈	C₂³	semidirect
47. C₂×D₅₅₆	C₂³	semidirect
60. C₂×C₄×D₁₃₉	C₂³	semidirect
61. C₄⋊Dic₁₃₉	C₂³	semidirect
63. (C₂×D₁₃₉)⋊C₄	C₂³	semidirect
65. C₂²×C₄×C₁₃₉	D₄	semidirect
70. C₄⋊C₄×C₁₃₉	C₂³	semidirect
72. C₄²×C₁₃₉	C₂³	semidirect
80. C₂²×D₁₃₉	#41	semidirect
102. C₂×C₄×C₁₃₉	C₂×D₄	semidirect
119. C₂×D₁₃₉	C₂×#41	semidirect
126. C₂×C₁₃₉	#525	semidirect
131. C₁₃₉	C₂×#525	semidirect
141. C₂×#117	D₁₃₉	semidirect
161. #117	C₂×D₁₃₉	semidirect
163. C₂×C₄⋊C₄	C₂×D₁₃₉	semidirect
167. C₂×C₄²	C₂×D₁₃₉	semidirect
194. C₂²×C₄	D₅₅₆	semidirect
200. C₄⋊C₄	C₂²×D₁₃₉	semidirect
204. C₄²	C₂²×D₁₃₉	semidirect
234. C₂×C₄	C₂×D₅₅₆	semidirect
260. C₂	#22581465	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×#14555785	2
C₂³×C₄×D₁₃₉	2
#22581465	2
C₂×#17984826	2
C₂×#17984822	2
C₂×#17984817	2
C₂×#17984812	2
C₂×C₄×D₅₅₆	2
C₂×#117×C₁₃₉	2
C₂×#525	139

Subgroups — C2×#22581465 #31819647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22581465 #31819647