Subgroups — C₂²×#19099168 #35195046

Order 20224.

8838 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099168	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#14579158	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#14579153	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#14579146	C₂	1	4	yes	maximal, complemented
5	C₂²×#14579135	C₂	1	2	yes	maximal, complemented
6	C₂²×#14579134	C₂	1	4	yes	maximal, complemented
7	C₂²×C₄×D₃₁₆	C₂	1	2	yes	maximal, complemented
8	C₂²×#117×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₃₁₆	C₂²	1	2	yes	—
10	C₂³×D₃₁₆	C₂²	1	4	yes	—
11	C₂³×C₂²⋊D₇₉	C₂²	1	4	yes	—
12	C₂³×C₄×D₇₉	C₂²	1	4	yes	—
13	#19099168	C₂²	1	256	yes	direct product factor
14	C₂×#14579158	C₂²	1	24	yes	complemented
15	C₂×#14579153	C₂²	1	24	yes	complemented
16	C₂×#14579146	C₂²	1	96	yes	complemented
17	C₂×#14579135	C₂²	1	48	yes	complemented
18	C₂×#14579134	C₂²	1	96	yes	complemented
19	C₂×C₄×D₃₁₆	C₂²	1	48	yes	complemented
20	C₂²×C₄⋊Dic₇₉	C₂²	1	2	yes	—
21	C₂²×Dic₇₉⋊C₄	C₂²	1	4	yes	—
22	C₂²×(C₂×D₇₉)⋊C₄	C₂²	1	4	yes	—
23	C₂²×(C₂×D₇₉)⋊C₄	C₂²	1	4	yes	—
24	C₂⁴×C₄×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×#117×C₇₉	C₂²	1	24	yes	complemented
26	C₂²×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₇₉	C₂²	1	2	yes	—
28	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
30	C₂⁴×D₇₉	C₂³	1	4	yes	—
31	C₂²×D₃₁₆	—	2	16	no	—
32	C₂²×D₃₁₆	—	4	16	no	—
33	C₂²×D₃₁₆	C₂³	1	24	yes	—
34	C₂²×D₃₁₆	C₂³	1	48	yes	—
35	C₂²×C₂²⋊D₇₉	—	4	16	no	—
36	C₂²×C₂²⋊D₇₉	C₂³	1	48	yes	—
37	C₂³×Dic₇₉	C₂³	1	4	yes	—
38	C₂²×C₄×D₇₉	—	4	16	no	—
39	C₂²×C₄×D₇₉	C₂³	1	48	yes	—
40	#14579158	C₂³	1	64	yes	complemented
41	#14579153	C₂³	1	64	yes	complemented
42	#14579146	C₂³	1	256	yes	complemented
43	#14579135	C₂³	1	128	yes	complemented
44	#14579134	C₂³	1	256	yes	complemented
45	C₄×D₃₁₆	C₂³	1	128	yes	complemented
46	C₂×C₄⋊Dic₇₉	C₂³	1	24	yes	—
47	C₂×Dic₇₉⋊C₄	C₂³	1	48	yes	—
48	C₂×(C₂×D₇₉)⋊C₄	C₂³	1	48	yes	—
49	C₂×(C₂×D₇₉)⋊C₄	C₂³	1	48	yes	—
50	C₂⁵×C₇₉	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₇₉	—	2	8	no	—
52	C₂³×C₄×C₇₉	C₂³	1	2	yes	—
53	C₂³×C₄×C₇₉	C₂³	1	2	yes	—
54	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₇₉	C₂³	1	12	yes	—
56	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
57	#117×C₇₉	C₂³	1	64	yes	complemented
58	C₂×C₄⋊C₄×C₇₉	C₂³	1	24	yes	—
59	C₂×C₂²⋊C₄×C₇₉	C₂³	1	24	yes	—
60	C₂×C₄²×C₇₉	C₂³	1	12	yes	—
61	C₂×C₄²×C₇₉	C₂³	1	12	yes	—
62	C₂³×D₇₉	—	2	16	no	—
63	C₂³×D₇₉	—	2	16	no	—
64	C₂³×D₇₉	—	2	16	no	—
65	C₂³×D₇₉	C₂⁴	1	24	yes	—
66	C₂×D₃₁₆	—	2	96	no	—
67	C₂×D₃₁₆	—	4	96	no	—
68	C₂×D₃₁₆	C₂⁴	1	32	yes	—
69	C₂×D₃₁₆	C₂⁴	1	64	yes	—
70	C₂×C₂²⋊D₇₉	—	4	96	no	—
71	C₂×C₂²⋊D₇₉	C₂⁴	1	64	yes	—
72	C₂²×Dic₇₉	—	2	16	no	—
73	C₂²×Dic₇₉	C₂⁴	1	24	yes	—
74	C₂×C₄×D₇₉	—	4	96	no	—
75	C₂×C₄×D₇₉	C₂⁴	1	64	yes	—
76	C₄⋊Dic₇₉	C₂⁴	1	32	yes	—
77	Dic₇₉⋊C₄	C₂⁴	1	64	yes	—
78	(C₂×D₇₉)⋊C₄	C₂⁴	1	64	yes	—
79	(C₂×D₇₉)⋊C₄	C₂⁴	1	64	yes	—
80	C₂⁴×C₇₉	—	2	4	no	—
81	C₂⁴×C₇₉	—	2	8	no	—
82	C₂⁴×C₇₉	C₂⁴	1	6	yes	—
83	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	—	2	4	no	—
85	C₂²×C₄×C₇₉	—	2	8	no	—
86	C₂²×C₄×C₇₉	—	2	8	no	—
87	C₂²×C₄×C₇₉	—	2	48	no	—
88	C₂²×C₄×C₇₉	#41	1	8	yes	—
89	C₂²×C₄×C₇₉	C₂⁴	1	12	yes	—
90	C₂²×C₄×C₇₉	C₂⁴	1	12	yes	—
91	C₂²×C₄×C₇₉	C₂⁴	1	6	yes	—
92	C₂²×C₄×C₇₉	C₂⁴	1	16	yes	—
93	C₂²×C₄×C₇₉	C₂⁴	1	6	yes	—
94	C₄⋊C₄×C₇₉	C₂⁴	1	32	yes	—
95	C₂²⋊C₄×C₇₉	C₂⁴	1	32	yes	—
96	C₄²×C₇₉	C₂⁴	1	16	yes	—
97	C₄²×C₇₉	C₂⁴	1	16	yes	—
98	C₂³×C₇₉	—	2	12	no	—
99	C₂³×C₇₉	—	2	24	no	—
100	C₂³×C₇₉	—	4	16	no	—
101	C₂³×C₇₉	C₂×#41	1	4	yes	—
102	C₂³×C₇₉	H(5,𝔽₂)	1	8	yes	—
103	C₂³×C₇₉	C₂⁵	1	4	yes	—
104	C₂³×C₇₉	C₂⁵	1	3	yes	—
105	C₂²×D₇₉	—	2	48	no	—
106	C₂²×D₇₉	—	2	48	no	—
107	C₂²×D₇₉	—	2	48	no	—
108	C₂²×D₇₉	—	4	64	no	—
109	C₂²×D₇₉	C₂⁵	1	16	yes	—
110	C₂×C₄×C₇₉	—	2	12	no	—
111	C₂×C₄×C₇₉	—	2	24	no	—
112	C₂×C₄×C₇₉	—	2	24	no	—
113	C₂×C₄×C₇₉	—	2	32	no	—
114	C₂×C₄×C₇₉	C₂×#41	1	24	yes	—
115	C₂×C₄×C₇₉	C₂⁵	1	4	yes	—
116	C₂×C₄×C₇₉	C₂⁵	1	8	yes	—
117	C₂×C₄×C₇₉	C₂⁵	1	8	yes	—
118	C₂×C₄×C₇₉	C₂⁵	1	4	yes	—
119	C₂²⋊D₇₉	—	4	64	no	—
120	C₂×Dic₇₉	—	2	48	no	—
121	C₂×Dic₇₉	C₂⁵	1	16	yes	—
122	D₃₁₆	—	2	64	no	—
123	D₃₁₆	—	4	64	no	—
124	C₄×D₇₉	—	4	64	no	—
125	C₂²×C₇₉	—	2	4	no	—
126	C₂²×C₇₉	—	2	8	no	—
127	C₂²×C₇₉	—	4	24	no	—
128	C₂²×C₇₉	#540	1	16	yes	complemented
129	C₂²×C₇₉	C₂²×#41	1	6	yes	—
130	C₂²×C₇₉	C₂×H(5,𝔽₂)	1	12	yes	—
131	C₂²×C₇₉	C₂⁶	1	1	yes	characteristic
132	C₂×D₇₉	—	2	16	no	—
133	C₂×D₇₉	—	2	16	no	—
134	C₂×D₇₉	—	2	16	no	—
135	C₂×D₇₉	—	4	96	no	—
136	C₄×C₇₉	—	2	4	no	—
137	C₄×C₇₉	—	2	8	no	—
138	C₄×C₇₉	—	2	8	no	—
139	C₄×C₇₉	C₂²×#41	1	8	yes	—
140	Dic₇₉	—	2	16	no	—
141	C₂²×#540	—	79	1	no	maximal, Sylow
142	C₂×C₇₉	—	4	4	no	—
143	C₂×C₇₉	C₂×#540	1	12	yes	complemented
144	C₂×C₇₉	C₂³×#41	1	1	yes	characteristic
145	C₂×C₇₉	C₂²×H(5,𝔽₂)	1	2	yes	—
146	D₇₉	—	4	16	no	—
147	C₂×#540	—	79	48	no	—
148	C₂²×#127	—	79	1	no	—
149	C₂²×#125	—	79	1	no	—
150	C₂²×#123	—	79	4	no	—
151	C₂²×#121	—	79	2	no	—
152	C₂²×#121	—	79	4	no	—
153	C₂²×C₄×D₄	—	79	2	no	—
154	C₂²×#117	D₇₉	1	1	yes	characteristic, complemented
155	C₇₉	C₂²×#540	1	1	yes	characteristic, complemented, Sylow
156	C₂³×D₄	—	79	4	no	—
157	C₂³×D₄	—	79	4	no	—
158	C₂³×D₄	—	79	2	no	—
159	C₂⁴×C₄	—	79	4	no	—
160	C₂⁴×C₄	C₂×D₇₉	1	1	yes	characteristic
161	#540	—	79	256	no	—
162	C₂×#127	—	79	24	no	—
163	C₂×#125	—	79	24	no	—
164	C₂×#123	—	79	96	no	—
165	C₂×#121	—	79	48	no	—
166	C₂×#121	—	79	96	no	—
167	C₂×C₄×D₄	—	79	48	no	—
168	C₂×#117	C₂×D₇₉	1	24	yes	complemented
169	C₂²×C₄⋊C₄	—	79	2	no	—
170	C₂²×C₄⋊C₄	—	79	4	no	—
171	C₂²×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
172	C₂²×C₂²⋊C₄	—	79	4	no	—
173	C₂²×C₂²⋊C₄	—	79	4	no	—
174	C₂²×C₂²⋊C₄	C₂×D₇₉	1	2	yes	—
175	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
176	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
177	C₂⁵	—	79	4	no	—
178	C₂⁵	C₂²×D₇₉	1	1	yes	characteristic
179	C₂²×D₄	—	316	16	no	—
180	C₂²×D₄	—	316	16	no	—
181	C₂²×D₄	—	79	48	no	—
182	C₂²×D₄	—	79	48	no	—
183	C₂²×D₄	—	79	24	no	—
184	C₂²×D₄	—	158	16	no	—
185	C₂³×C₄	—	2	8	no	—
186	C₂³×C₄	—	316	16	no	—
187	C₂³×C₄	—	79	48	no	—
188	C₂³×C₄	—	79	4	no	—
189	C₂³×C₄	C₂²×D₇₉	1	12	yes	—
190	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
191	C₂³×C₄	C₂²×D₇₉	1	2	yes	—
192	C₂³×C₄	C₂²×D₇₉	1	2	yes	—
193	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
194	#127	—	79	64	no	—
195	#125	—	79	64	no	—
196	#123	—	79	256	no	—
197	#121	—	79	128	no	—
198	#121	—	79	256	no	—
199	C₄×D₄	—	79	128	no	—
200	#117	C₂²×D₇₉	1	64	yes	complemented
201	C₂×C₄⋊C₄	—	79	24	no	—
202	C₂×C₄⋊C₄	—	79	48	no	—
203	C₂×C₄⋊C₄	C₂²×D₇₉	1	24	yes	—
204	C₂×C₂²⋊C₄	—	79	48	no	—
205	C₂×C₂²⋊C₄	—	79	48	no	—
206	C₂×C₂²⋊C₄	C₂²×D₇₉	1	24	yes	—
207	C₂×C₄²	C₂²×D₇₉	1	12	yes	—
208	C₂×C₄²	C₂²×D₇₉	1	12	yes	—
209	C₂⁴	—	2	4	no	—
210	C₂⁴	—	2	8	no	—
211	C₂⁴	—	79	24	no	—
212	C₂⁴	—	158	16	no	—
213	C₂⁴	—	158	16	no	—
214	C₂⁴	—	158	16	no	—
215	C₂⁴	C₂³×D₇₉	1	6	yes	—
216	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
217	C₂×D₄	—	316	96	no	—
218	C₂×D₄	—	316	96	no	—
219	C₂×D₄	—	79	64	no	—
220	C₂×D₄	—	79	64	no	—
221	C₂×D₄	—	79	32	no	—
222	C₂×D₄	—	158	96	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	—	2	48	no	—
227	C₂²×C₄	—	316	96	no	—
228	C₂²×C₄	—	79	64	no	—
229	C₂²×C₄	—	79	24	no	—
230	C₂²×C₄	—	158	16	no	—
231	C₂²×C₄	#11929624	1	8	yes	—
232	C₂²×C₄	C₂³×D₇₉	1	12	yes	—
233	C₂²×C₄	C₂³×D₇₉	1	12	yes	—
234	C₂²×C₄	C₂³×D₇₉	1	6	yes	—
235	C₂²×C₄	C₂³×D₇₉	1	16	yes	—
236	C₂²×C₄	C₂³×D₇₉	1	6	yes	—
237	C₄⋊C₄	—	79	64	no	—
238	C₄⋊C₄	—	79	32	no	—
239	C₄⋊C₄	C₂³×D₇₉	1	32	yes	—
240	C₂²⋊C₄	—	79	64	no	—
241	C₂²⋊C₄	—	79	64	no	—
242	C₂²⋊C₄	C₂³×D₇₉	1	32	yes	—
243	C₄²	C₂³×D₇₉	1	16	yes	—
244	C₄²	C₂³×D₇₉	1	16	yes	—
245	C₂³	—	2	24	no	—
246	C₂³	—	2	12	no	—
247	C₂³	—	4	16	no	—
248	C₂³	—	316	64	no	—
249	C₂³	—	79	16	no	—
250	C₂³	—	158	48	no	—
251	C₂³	—	158	48	no	—
252	C₂³	—	158	48	no	—
253	C₂³	C₂×#11929624	1	4	yes	—
254	C₂³	#14579207	1	8	yes	—
255	C₂³	C₂⁴×D₇₉	1	4	yes	—
256	C₂³	C₂⁴×D₇₉	1	3	yes	—
257	D₄	—	316	64	no	—
258	D₄	—	316	64	no	—
259	D₄	—	158	64	no	—
260	C₂×C₄	—	2	24	no	—
261	C₂×C₄	—	2	24	no	—
262	C₂×C₄	—	2	12	no	—
263	C₂×C₄	—	2	32	no	—
264	C₂×C₄	—	316	64	no	—
265	C₂×C₄	—	79	16	no	—
266	C₂×C₄	—	158	48	no	—
267	C₂×C₄	C₂×#11929624	1	24	yes	—
268	C₂×C₄	C₂⁴×D₇₉	1	8	yes	—
269	C₂×C₄	C₂⁴×D₇₉	1	8	yes	—
270	C₂×C₄	C₂⁴×D₇₉	1	4	yes	—
271	C₂×C₄	C₂⁴×D₇₉	1	4	yes	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	4	24	no	—
275	C₂²	—	316	96	no	—
276	C₂²	—	158	16	no	—
277	C₂²	—	158	16	no	—
278	C₂²	—	158	16	no	—
279	C₂²	#19099168	1	16	yes	direct product factor
280	C₂²	C₂²×#11929624	1	6	yes	—
281	C₂²	C₂×#14579207	1	12	yes	—
282	C₂²	C₂⁵×D₇₉	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	2	8	no	—
285	C₄	—	2	8	no	—
286	C₄	—	158	16	no	—
287	C₄	C₂²×#11929624	1	8	yes	—
288	C₂	—	4	4	no	—
289	C₂	—	316	16	no	—
290	C₂	C₂×#19099168	1	12	yes	direct product factor
291	C₂	C₂³×#11929624	1	1	yes	characteristic
292	C₂	C₂²×#14579207	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₇₉
commutator, cosocle	C₂²×C₇₉	C₂⁶
Frattini	C₂²	C₂⁵×D₇₉
Fitting	C₂²×#117×C₇₉	C₂
socle	C₂⁴×C₇₉	C₂⁴
hypercenter	C₂²×#117	D₇₉
hypocenter	C₇₉	C₂²×#540
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
1. C₂×#19099168	C₂	direct (N × H)
2. C₂²×#14579158	C₂	semidirect
3. C₂²×#14579153	C₂	semidirect
4. C₂²×#14579146	C₂	semidirect
5. C₂²×#14579135	C₂	semidirect
6. C₂²×#14579134	C₂	semidirect
7. C₂²×C₄×D₃₁₆	C₂	semidirect
8. C₂²×#117×C₇₉	C₂	semidirect
13. #19099168	C₂²	direct (N × H)
14. C₂×#14579158	C₂²	semidirect
15. C₂×#14579153	C₂²	semidirect
16. C₂×#14579146	C₂²	semidirect
17. C₂×#14579135	C₂²	semidirect
18. C₂×#14579134	C₂²	semidirect
19. C₂×C₄×D₃₁₆	C₂²	semidirect
25. C₂×#117×C₇₉	C₂²	semidirect
40. #14579158	C₂³	semidirect
41. #14579153	C₂³	semidirect
42. #14579146	C₂³	semidirect
43. #14579135	C₂³	semidirect
44. #14579134	C₂³	semidirect
45. C₄×D₃₁₆	C₂³	semidirect
57. #117×C₇₉	C₂³	semidirect
128. C₂²×C₇₉	#540	semidirect
143. C₂×C₇₉	C₂×#540	semidirect
154. C₂²×#117	D₇₉	semidirect
155. C₇₉	C₂²×#540	semidirect
168. C₂×#117	C₂×D₇₉	semidirect
200. #117	C₂²×D₇₉	semidirect
279. C₂²	#19099168	direct (N × H)
290. C₂	C₂×#19099168	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×#19099168	2
C₂²×#14579158	2
C₂²×#14579153	2
C₂²×#14579146	2
C₂²×#14579135	2
C₂²×#14579134	2
C₂²×C₄×D₃₁₆	2
C₂²×#117×C₇₉	2
C₂²×#540	79

Subgroups — C22×#19099168 #35195046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19099168 #35195046