Subgroups — C₂²×#19099144 #35195033

Order 20224.

12270 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11929624	C₂	1	2	yes	maximal, complemented
2	C₂×#19099144	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#14579141	C₂	1	4	yes	maximal, complemented
4	C₂²×#14579136	C₂	1	4	yes	maximal, complemented
5	C₂²×#14579135	C₂	1	4	yes	maximal, complemented
6	C₂³×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11929624	—	2	32	no	—
8	C₂²×#11929624	C₂²	1	48	yes	complemented
9	C₂³×Dic₁₅₈	C₂²	1	2	yes	—
10	C₂³×D₃₁₆	C₂²	1	2	yes	—
11	C₂³×D₃₁₆	C₂²	1	4	yes	—
12	C₂³×C₂²⋊D₇₉	C₂²	1	4	yes	—
13	C₂³×C₄×D₇₉	C₂²	1	4	yes	—
14	#19099144	C₂²	1	256	yes	direct product factor
15	C₂×#14579141	C₂²	1	96	yes	complemented
16	C₂×#14579136	C₂²	1	96	yes	complemented
17	C₂×#14579135	C₂²	1	96	yes	complemented
18	C₂²×C₄⋊Dic₇₉	C₂²	1	4	yes	—
19	C₂²×(C₂×D₇₉)⋊C₄	C₂²	1	8	yes	—
20	C₂⁴×C₄×C₇₉	C₂²	1	2	yes	—
21	C₂⁴×C₄×C₇₉	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₇₉	C₂²	1	4	yes	—
23	C₂²×C₄⋊C₄×C₇₉	C₂²	1	24	yes	complemented
24	C₂⁴×D₇₉	C₂³	1	4	yes	—
25	C₂×#11929624	—	2	384	no	—
26	C₂×#11929624	C₂³	1	128	yes	complemented
27	C₂²×Dic₁₅₈	—	2	16	no	—
28	C₂²×Dic₁₅₈	C₂³	1	24	yes	—
29	C₂²×D₃₁₆	—	2	16	no	—
30	C₂²×D₃₁₆	—	4	16	no	—
31	C₂²×D₃₁₆	C₂³	1	24	yes	—
32	C₂²×D₃₁₆	C₂³	1	48	yes	—
33	C₂²×C₂²⋊D₇₉	—	2	32	no	—
34	C₂²×C₂²⋊D₇₉	C₂³	1	48	yes	—
35	C₂³×Dic₇₉	C₂³	1	4	yes	—
36	C₂²×C₄×D₇₉	—	2	32	no	—
37	C₂²×C₄×D₇₉	C₂³	1	48	yes	—
38	#14579141	C₂³	1	256	yes	complemented
39	#14579136	C₂³	1	256	yes	complemented
40	#14579135	C₂³	1	256	yes	complemented
41	C₂×C₄⋊Dic₇₉	C₂³	1	48	yes	—
42	C₂×(C₂×D₇₉)⋊C₄	C₂³	1	96	yes	—
43	C₂⁵×C₇₉	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₇₉	—	4	8	no	—
45	C₂³×C₄×C₇₉	D₄	1	16	yes	complemented
46	C₂³×C₄×C₇₉	C₂³	1	4	yes	—
47	C₂³×C₄×C₇₉	C₂³	1	24	yes	—
48	C₂³×C₄×C₇₉	C₂³	1	2	yes	—
49	C₂³×C₄×C₇₉	C₂³	1	12	yes	—
50	C₂×C₄⋊C₄×C₇₉	C₂³	1	48	yes	—
51	C₂×C₄⋊C₄×C₇₉	C₂³	1	64	yes	complemented
52	C₂³×D₇₉	—	2	16	no	—
53	C₂³×D₇₉	—	2	16	no	—
54	C₂³×D₇₉	—	2	16	no	—
55	C₂³×D₇₉	C₂⁴	1	24	yes	—
56	#11929624	—	2	512	no	—
57	C₂×Dic₁₅₈	—	2	96	no	—
58	C₂×Dic₁₅₈	C₂⁴	1	32	yes	—
59	C₂×D₃₁₆	—	2	96	no	—
60	C₂×D₃₁₆	—	4	96	no	—
61	C₂×D₃₁₆	C₂⁴	1	32	yes	—
62	C₂×D₃₁₆	C₂⁴	1	64	yes	—
63	C₂×C₂²⋊D₇₉	—	2	192	no	—
64	C₂×C₂²⋊D₇₉	C₂⁴	1	64	yes	—
65	C₂²×Dic₇₉	—	2	16	no	—
66	C₂²×Dic₇₉	C₂⁴	1	24	yes	—
67	C₂×C₄×D₇₉	—	2	192	no	—
68	C₂×C₄×D₇₉	C₂⁴	1	64	yes	—
69	C₄⋊Dic₇₉	C₂⁴	1	64	yes	—
70	(C₂×D₇₉)⋊C₄	C₂⁴	1	128	yes	—
71	C₂⁴×C₇₉	—	2	4	no	—
72	C₂⁴×C₇₉	—	2	4	no	—
73	C₂⁴×C₇₉	C₂×D₄	1	8	yes	—
74	C₂⁴×C₇₉	C₂⁴	1	6	yes	—
75	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	—	2	16	no	—
77	C₂²×C₄×C₇₉	—	4	48	no	—
78	C₂²×C₄×C₇₉	C₂×D₄	1	16	yes	—
79	C₂²×C₄×C₇₉	C₂×D₄	1	96	yes	complemented
80	C₂²×C₄×C₇₉	C₂⁴	1	24	yes	—
81	C₂²×C₄×C₇₉	C₂⁴	1	32	yes	—
82	C₂²×C₄×C₇₉	C₂⁴	1	12	yes	—
83	C₂²×C₄×C₇₉	C₂⁴	1	16	yes	—
84	C₄⋊C₄×C₇₉	C₂⁴	1	64	yes	—
85	C₂³×C₇₉	—	2	32	no	—
86	C₂³×C₇₉	—	2	12	no	—
87	C₂³×C₇₉	—	2	12	no	—
88	C₂³×C₇₉	C₂²×D₄	1	24	yes	—
89	C₂³×C₇₉	C₂²×D₄	1	4	yes	—
90	C₂³×C₇₉	H(5,𝔽₂)	1	4	yes	—
91	C₂³×C₇₉	2₋¹⁺⁴	1	4	yes	—
92	C₂³×C₇₉	C₂⁵	1	4	yes	—
93	C₂³×C₇₉	C₂⁵	1	3	yes	—
94	C₂²×D₇₉	—	2	48	no	—
95	C₂²×D₇₉	—	2	48	no	—
96	C₂²×D₇₉	—	2	48	no	—
97	C₂²×D₇₉	—	4	64	no	—
98	C₂²×D₇₉	C₂⁵	1	16	yes	—
99	C₂×C₄×C₇₉	—	2	48	no	—
100	C₂×C₄×C₇₉	—	4	32	no	—
101	C₂×C₄×C₇₉	C₂²×D₄	1	48	yes	—
102	C₂×C₄×C₇₉	C₂²×D₄	1	64	yes	complemented
103	C₂×C₄×C₇₉	C₂⁵	1	16	yes	—
104	C₂×C₄×C₇₉	C₂⁵	1	8	yes	—
105	C₂²⋊D₇₉	—	2	128	no	—
106	C₂×Dic₇₉	—	2	48	no	—
107	C₂×Dic₇₉	C₂⁵	1	16	yes	—
108	D₃₁₆	—	2	64	no	—
109	D₃₁₆	—	4	64	no	—
110	C₄×D₇₉	—	2	128	no	—
111	Dic₁₅₈	—	2	64	no	—
112	C₂²×C₇₉	—	2	4	no	—
113	C₂²×C₇₉	—	2	4	no	—
114	C₂²×C₇₉	—	2	48	no	—
115	C₂²×C₇₉	#537	1	16	yes	complemented
116	C₂²×C₇₉	C₂³×D₄	1	8	yes	—
117	C₂²×C₇₉	C₂³×D₄	1	6	yes	—
118	C₂²×C₇₉	C₂×H(5,𝔽₂)	1	6	yes	—
119	C₂²×C₇₉	C₂×2₋¹⁺⁴	1	6	yes	—
120	C₂²×C₇₉	C₂⁶	1	1	yes	characteristic
121	C₂×D₇₉	—	2	16	no	—
122	C₂×D₇₉	—	2	16	no	—
123	C₂×D₇₉	—	2	16	no	—
124	C₂×D₇₉	—	4	96	no	—
125	C₄×C₇₉	—	2	16	no	—
126	C₄×C₇₉	C₂³×D₄	1	16	yes	—
127	Dic₇₉	—	2	16	no	—
128	C₂²×#537	—	79	1	no	maximal, Sylow
129	C₂×C₇₉	—	2	8	no	—
130	C₂×C₇₉	C₂×#537	1	12	yes	complemented
131	C₂×C₇₉	C₂⁴×D₄	1	1	yes	characteristic
132	C₂×C₇₉	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×C₇₉	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
134	D₇₉	—	4	16	no	—
135	C₂³×#41	—	79	2	no	—
136	C₂×#537	—	79	48	no	—
137	C₂²×C₂²⋊Q₈	—	79	4	no	—
138	C₂²×#121	—	79	4	no	—
139	C₂²×#121	—	79	4	no	—
140	C₂³×C₄⋊C₄	D₇₉	1	1	yes	characteristic, complemented
141	C₇₉	C₂²×#537	1	1	yes	characteristic, complemented, Sylow
142	C₂²×#41	—	79	48	no	—
143	C₂²×#41	—	158	32	no	—
144	C₂³×Q₈	—	79	2	no	—
145	C₂³×D₄	—	79	4	no	—
146	C₂³×D₄	—	79	4	no	—
147	C₂³×D₄	—	79	2	no	—
148	C₂⁴×C₄	—	79	4	no	—
149	C₂⁴×C₄	C₂×D₇₉	1	2	yes	—
150	C₂⁴×C₄	C₂×D₇₉	1	1	yes	characteristic
151	#537	—	79	256	no	—
152	C₂×C₂²⋊Q₈	—	79	96	no	—
153	C₂×#121	—	79	96	no	—
154	C₂×#121	—	79	96	no	—
155	C₂²×C₄⋊C₄	—	79	4	no	—
156	C₂²×C₄⋊C₄	C₂×D₇₉	1	4	yes	—
157	C₂²×C₄⋊C₄	C₂×D₇₉	1	24	yes	complemented
158	C₂²×C₂²⋊C₄	—	79	8	no	—
159	C₂⁵	—	79	4	no	—
160	C₂⁵	C₂²×D₇₉	1	1	yes	characteristic
161	C₂×#41	—	79	128	no	—
162	C₂×#41	—	158	384	no	—
163	C₂²×Q₈	—	79	24	no	—
164	C₂²×Q₈	—	158	16	no	—
165	C₂²×D₄	—	316	16	no	—
166	C₂²×D₄	—	79	48	no	—
167	C₂²×D₄	—	79	48	no	—
168	C₂²×D₄	—	79	24	no	—
169	C₂²×D₄	—	158	32	no	—
170	C₂²×D₄	—	158	16	no	—
171	C₂³×C₄	—	4	8	no	—
172	C₂³×C₄	—	79	4	no	—
173	C₂³×C₄	—	79	48	no	—
174	C₂³×C₄	—	158	32	no	—
175	C₂³×C₄	D₃₁₆	1	16	yes	complemented
176	C₂³×C₄	C₂²×D₇₉	1	4	yes	—
177	C₂³×C₄	C₂²×D₇₉	1	24	yes	—
178	C₂³×C₄	C₂²×D₇₉	1	2	yes	—
179	C₂³×C₄	C₂²×D₇₉	1	12	yes	—
180	C₂²⋊Q₈	—	79	256	no	—
181	#121	—	79	256	no	—
182	#121	—	79	256	no	—
183	C₂×C₄⋊C₄	—	79	48	no	—
184	C₂×C₄⋊C₄	C₂²×D₇₉	1	48	yes	—
185	C₂×C₄⋊C₄	C₂²×D₇₉	1	64	yes	complemented
186	C₂×C₂²⋊C₄	—	79	96	no	—
187	C₂⁴	—	2	4	no	—
188	C₂⁴	—	2	4	no	—
189	C₂⁴	—	79	24	no	—
190	C₂⁴	—	158	16	no	—
191	C₂⁴	—	158	16	no	—
192	C₂⁴	—	158	16	no	—
193	C₂⁴	C₂×D₃₁₆	1	8	yes	—
194	C₂⁴	C₂³×D₇₉	1	6	yes	—
195	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
196	#41	—	158	512	no	—
197	C₂×Q₈	—	79	32	no	—
198	C₂×Q₈	—	158	96	no	—
199	C₂×D₄	—	316	96	no	—
200	C₂×D₄	—	79	64	no	—
201	C₂×D₄	—	79	64	no	—
202	C₂×D₄	—	79	32	no	—
203	C₂×D₄	—	158	192	no	—
204	C₂×D₄	—	158	96	no	—
205	C₂²×C₄	—	2	16	no	—
206	C₂²×C₄	—	4	48	no	—
207	C₂²×C₄	—	79	24	no	—
208	C₂²×C₄	—	79	64	no	—
209	C₂²×C₄	—	158	16	no	—
210	C₂²×C₄	—	158	192	no	—
211	C₂²×C₄	C₂×D₃₁₆	1	16	yes	—
212	C₂²×C₄	C₂×D₃₁₆	1	96	yes	complemented
213	C₂²×C₄	C₂³×D₇₉	1	24	yes	—
214	C₂²×C₄	C₂³×D₇₉	1	32	yes	—
215	C₂²×C₄	C₂³×D₇₉	1	12	yes	—
216	C₂²×C₄	C₂³×D₇₉	1	16	yes	—
217	C₄⋊C₄	—	79	64	no	—
218	C₄⋊C₄	C₂³×D₇₉	1	64	yes	—
219	C₂²⋊C₄	—	79	128	no	—
220	C₂³	—	2	12	no	—
221	C₂³	—	2	12	no	—
222	C₂³	—	2	32	no	—
223	C₂³	—	316	64	no	—
224	C₂³	—	79	16	no	—
225	C₂³	—	158	48	no	—
226	C₂³	—	158	48	no	—
227	C₂³	—	158	48	no	—
228	C₂³	C₂²×D₃₁₆	1	24	yes	—
229	C₂³	C₂²×D₃₁₆	1	4	yes	—
230	C₂³	#14579206	1	4	yes	—
231	C₂³	#14579209	1	4	yes	—
232	C₂³	C₂⁴×D₇₉	1	4	yes	—
233	C₂³	C₂⁴×D₇₉	1	3	yes	—
234	Q₈	—	158	64	no	—
235	D₄	—	316	64	no	—
236	D₄	—	158	128	no	—
237	D₄	—	158	64	no	—
238	C₂×C₄	—	2	48	no	—
239	C₂×C₄	—	4	32	no	—
240	C₂×C₄	—	79	16	no	—
241	C₂×C₄	—	158	48	no	—
242	C₂×C₄	—	158	128	no	—
243	C₂×C₄	C₂²×D₃₁₆	1	48	yes	—
244	C₂×C₄	C₂²×D₃₁₆	1	64	yes	complemented
245	C₂×C₄	C₂⁴×D₇₉	1	16	yes	—
246	C₂×C₄	C₂⁴×D₇₉	1	8	yes	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	2	48	no	—
250	C₂²	—	316	96	no	—
251	C₂²	—	158	16	no	—
252	C₂²	—	158	16	no	—
253	C₂²	—	158	16	no	—
254	C₂²	#19099144	1	16	yes	direct product factor
255	C₂²	C₂³×D₃₁₆	1	8	yes	—
256	C₂²	C₂³×D₃₁₆	1	6	yes	—
257	C₂²	C₂×#14579206	1	6	yes	—
258	C₂²	C₂×#14579209	1	6	yes	—
259	C₂²	C₂⁵×D₇₉	1	1	yes	characteristic
260	C₄	—	2	16	no	—
261	C₄	—	158	16	no	—
262	C₄	C₂³×D₃₁₆	1	16	yes	—
263	C₂	—	2	8	no	—
264	C₂	—	316	16	no	—
265	C₂	C₂×#19099144	1	12	yes	direct product factor
266	C₂	C₂⁴×D₃₁₆	1	1	yes	characteristic
267	C₂	C₂²×#14579206	1	1	yes	characteristic
268	C₂	C₂²×#14579209	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₇₉
commutator, cosocle	C₂²×C₇₉	C₂⁶
Frattini	C₂²	C₂⁵×D₇₉
Fitting	C₂³×C₄⋊C₄×C₇₉	C₂
socle	C₂⁴×C₇₉	C₂⁴
hypercenter	C₂³×C₄⋊C₄	D₇₉
hypocenter	C₇₉	C₂²×#537
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#11929624	C₂	semidirect
2. C₂×#19099144	C₂	direct (N × H)
3. C₂²×#14579141	C₂	semidirect
4. C₂²×#14579136	C₂	semidirect
5. C₂²×#14579135	C₂	semidirect
6. C₂³×C₄⋊C₄×C₇₉	C₂	semidirect
8. C₂²×#11929624	C₂²	semidirect
14. #19099144	C₂²	direct (N × H)
15. C₂×#14579141	C₂²	semidirect
16. C₂×#14579136	C₂²	semidirect
17. C₂×#14579135	C₂²	semidirect
23. C₂²×C₄⋊C₄×C₇₉	C₂²	semidirect
26. C₂×#11929624	C₂³	semidirect
38. #14579141	C₂³	semidirect
39. #14579136	C₂³	semidirect
40. #14579135	C₂³	semidirect
45. C₂³×C₄×C₇₉	D₄	semidirect
51. C₂×C₄⋊C₄×C₇₉	C₂³	semidirect
79. C₂²×C₄×C₇₉	C₂×D₄	semidirect
102. C₂×C₄×C₇₉	C₂²×D₄	semidirect
115. C₂²×C₇₉	#537	semidirect
130. C₂×C₇₉	C₂×#537	semidirect
140. C₂³×C₄⋊C₄	D₇₉	semidirect
141. C₇₉	C₂²×#537	semidirect
157. C₂²×C₄⋊C₄	C₂×D₇₉	semidirect
175. C₂³×C₄	D₃₁₆	semidirect
185. C₂×C₄⋊C₄	C₂²×D₇₉	semidirect
212. C₂²×C₄	C₂×D₃₁₆	semidirect
244. C₂×C₄	C₂²×D₃₁₆	semidirect
254. C₂²	#19099144	direct (N × H)
265. C₂	C₂×#19099144	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11929624	2
C₂×#19099144	2
C₂²×#14579141	2
C₂²×#14579136	2
C₂²×#14579135	2
C₂³×C₄⋊C₄×C₇₉	2
C₂²×#537	79

Subgroups — C22×#19099144 #35195033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19099144 #35195033