Subgroups — C₂²×#19099304 #35195139

Order 20224.

9606 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099304	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#14579158	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#14579155	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#14579146	C₂	1	3	yes	maximal, complemented
5	C₂²×#14579136	C₂	1	3	yes	maximal, complemented
6	C₂²×#14579134	C₂	1	3	yes	maximal, complemented
7	C₂²×#14579131	C₂	1	3	yes	maximal, complemented
8	C₂²×#126×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₇₉	C₂²	1	1	yes	characteristic
10	C₂³×D₃₁₆	C₂²	1	3	yes	—
11	C₂³×D₃₁₆	C₂²	1	3	yes	—
12	C₂³×D₃₁₆	C₂²	1	3	yes	complemented
13	C₂³×C₂²⋊D₇₉	C₂²	1	3	yes	—
14	C₂³×C₄×D₇₉	C₂²	1	3	yes	complemented
15	#19099304	C₂²	1	256	yes	direct product factor
16	C₂×#14579158	C₂²	1	24	yes	complemented
17	C₂×#14579155	C₂²	1	24	yes	complemented
18	C₂×#14579146	C₂²	1	72	yes	complemented
19	C₂×#14579136	C₂²	1	72	yes	complemented
20	C₂×#14579134	C₂²	1	72	yes	complemented
21	C₂×#14579131	C₂²	1	72	yes	complemented
22	C₂²×Dic₇₉⋊C₄	C₂²	1	3	yes	complemented
23	C₂²×(C₂×D₇₉)⋊C₄	C₂²	1	3	yes	—
24	C₂²×(C₂×D₇₉)⋊C₄	C₂²	1	3	yes	—
25	C₂²×(C₂×D₇₉)⋊C₄	C₂²	1	3	yes	complemented
26	C₂×#126×C₇₉	C₂²	1	24	yes	complemented
27	C₂²×C₄⋊C₄×C₇₉	C₂²	1	3	yes	complemented
28	C₂²×C₂²⋊C₄×C₇₉	C₂²	1	3	yes	—
29	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic, complemented
30	C₂⁴×D₇₉	—	4	12	no	—
31	C₂⁴×D₇₉	C₂³	1	1	yes	characteristic
32	C₂⁴×D₇₉	C₂³	1	1	yes	characteristic
33	C₂⁴×D₇₉	C₂³	1	3	yes	—
34	C₂⁴×D₇₉	C₂³	1	12	yes	—
35	C₂²×D₃₁₆	—	4	12	no	—
36	C₂²×D₃₁₆	—	4	12	no	—
37	C₂²×D₃₁₆	—	4	12	no	—
38	C₂²×D₃₁₆	C₂³	1	36	yes	—
39	C₂²×D₃₁₆	C₂³	1	36	yes	—
40	C₂²×D₃₁₆	C₂³	1	36	yes	complemented
41	C₂²×C₂²⋊D₇₉	—	4	12	no	—
42	C₂²×C₂²⋊D₇₉	C₂³	1	36	yes	—
43	C₂³×Dic₇₉	C₂³	1	3	yes	—
44	C₂²×C₄×D₇₉	—	4	12	no	—
45	C₂²×C₄×D₇₉	C₂³	1	36	yes	complemented
46	#14579158	C₂³	1	64	yes	complemented
47	#14579155	C₂³	1	64	yes	complemented
48	#14579146	C₂³	1	192	yes	complemented
49	#14579136	C₂³	1	192	yes	complemented
50	#14579134	C₂³	1	192	yes	complemented
51	#14579131	C₂³	1	192	yes	complemented
52	C₂×Dic₇₉⋊C₄	C₂³	1	36	yes	complemented
53	C₂×(C₂×D₇₉)⋊C₄	C₂³	1	36	yes	—
54	C₂×(C₂×D₇₉)⋊C₄	C₂³	1	36	yes	—
55	C₂×(C₂×D₇₉)⋊C₄	C₂³	1	36	yes	complemented
56	C₂⁵×C₇₉	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₇₉	C₂³	1	3	yes	—
58	C₂³×C₄×C₇₉	C₂³	1	3	yes	—
59	#126×C₇₉	C₂³	1	64	yes	complemented
60	C₂×C₄⋊C₄×C₇₉	C₂³	1	36	yes	complemented
61	C₂×C₂²⋊C₄×C₇₉	C₂³	1	36	yes	—
62	C₂×C₄²×C₇₉	C₂³	1	12	yes	complemented
63	C₂³×D₇₉	—	2	12	no	—
64	C₂³×D₇₉	—	2	12	no	—
65	C₂³×D₇₉	—	2	12	no	—
66	C₂³×D₇₉	—	2	12	no	—
67	C₂³×D₇₉	—	2	12	no	—
68	C₂³×D₇₉	—	4	72	no	—
69	C₂³×D₇₉	—	4	64	no	—
70	C₂³×D₇₉	C₂⁴	1	6	yes	—
71	C₂³×D₇₉	C₂⁴	1	6	yes	—
72	C₂³×D₇₉	C₂⁴	1	18	yes	—
73	C₂³×D₇₉	C₂⁴	1	16	yes	—
74	C₂×D₃₁₆	—	4	72	no	—
75	C₂×D₃₁₆	—	4	72	no	—
76	C₂×D₃₁₆	—	4	72	no	—
77	C₂×D₃₁₆	C₂⁴	1	48	yes	—
78	C₂×D₃₁₆	C₂⁴	1	48	yes	—
79	C₂×D₃₁₆	C₂⁴	1	48	yes	complemented
80	C₂×C₂²⋊D₇₉	—	4	72	no	—
81	C₂×C₂²⋊D₇₉	C₂⁴	1	48	yes	—
82	C₂²×Dic₇₉	—	2	12	no	—
83	C₂²×Dic₇₉	C₂⁴	1	18	yes	—
84	C₂×C₄×D₇₉	—	4	72	no	—
85	C₂×C₄×D₇₉	C₂⁴	1	48	yes	complemented
86	Dic₇₉⋊C₄	C₂⁴	1	48	yes	complemented
87	(C₂×D₇₉)⋊C₄	C₂⁴	1	48	yes	—
88	(C₂×D₇₉)⋊C₄	C₂⁴	1	48	yes	—
89	(C₂×D₇₉)⋊C₄	C₂⁴	1	48	yes	complemented
90	C₂⁴×C₇₉	—	2	12	no	—
91	C₂⁴×C₇₉	C₂⁴	1	6	yes	—
92	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
93	C₂²×C₄×C₇₉	—	2	12	no	—
94	C₂²×C₄×C₇₉	—	2	12	no	—
95	C₂²×C₄×C₇₉	C₂⁴	1	18	yes	—
96	C₂²×C₄×C₇₉	C₂⁴	1	18	yes	—
97	C₄⋊C₄×C₇₉	C₂⁴	1	48	yes	complemented
98	C₂²⋊C₄×C₇₉	C₂⁴	1	48	yes	—
99	C₄²×C₇₉	C₂⁴	1	16	yes	complemented
100	C₂³×C₇₉	—	2	36	no	—
101	C₂³×C₇₉	—	4	16	no	—
102	C₂³×C₇₉	H(5,𝔽₂)	1	12	yes	—
103	C₂³×C₇₉	C₂⁵	1	4	yes	—
104	C₂³×C₇₉	C₂⁵	1	3	yes	—
105	C₂²×D₇₉	—	2	36	no	—
106	C₂²×D₇₉	—	2	36	no	—
107	C₂²×D₇₉	—	2	36	no	—
108	C₂²×D₇₉	—	2	36	no	—
109	C₂²×D₇₉	—	2	36	no	—
110	C₂²×D₇₉	—	4	48	no	—
111	C₂²×D₇₉	—	4	192	no	—
112	C₂²×D₇₉	—	4	48	no	—
113	C₂²×D₇₉	—	4	16	no	—
114	C₂²×D₇₉	—	4	16	no	—
115	C₂²×D₇₉	C₂⁵	1	4	yes	—
116	C₂²×D₇₉	C₂⁵	1	4	yes	—
117	C₂²×D₇₉	C₂⁵	1	12	yes	—
118	C₂×C₄×C₇₉	—	2	36	no	—
119	C₂×C₄×C₇₉	—	2	36	no	—
120	C₂×C₄×C₇₉	C₂⁵	1	12	yes	—
121	C₂×C₄×C₇₉	C₂⁵	1	12	yes	—
122	C₂²⋊D₇₉	—	4	48	no	—
123	C₂×Dic₇₉	—	2	36	no	—
124	C₂×Dic₇₉	C₂⁵	1	12	yes	—
125	D₃₁₆	—	4	48	no	—
126	D₃₁₆	—	4	48	no	—
127	D₃₁₆	—	4	48	no	—
128	C₄×D₇₉	—	4	48	no	—
129	C₂²×C₇₉	—	2	12	no	—
130	C₂²×C₇₉	—	4	24	no	—
131	C₂²×C₇₉	#560	1	16	yes	complemented
132	C₂²×C₇₉	C₂×H(5,𝔽₂)	1	18	yes	—
133	C₂²×C₇₉	C₂⁶	1	1	yes	characteristic
134	C₂×D₇₉	—	2	12	no	—
135	C₂×D₇₉	—	2	12	no	—
136	C₂×D₇₉	—	2	12	no	—
137	C₂×D₇₉	—	2	12	no	—
138	C₂×D₇₉	—	2	12	no	—
139	C₂×D₇₉	—	4	64	no	—
140	C₂×D₇₉	—	4	72	no	—
141	C₂×D₇₉	—	4	24	no	—
142	C₂×D₇₉	—	4	24	no	—
143	C₄×C₇₉	—	2	12	no	—
144	C₄×C₇₉	—	2	12	no	—
145	Dic₇₉	—	2	12	no	—
146	C₂²×#560	—	79	1	no	maximal, Sylow
147	C₂×C₇₉	—	4	4	no	—
148	C₂×C₇₉	C₂×#560	1	12	yes	complemented
149	C₂×C₇₉	C₂²×H(5,𝔽₂)	1	3	yes	—
150	D₇₉	—	4	12	no	—
151	D₇₉	—	4	4	no	—
152	D₇₉	—	4	4	no	—
153	C₂×#560	—	79	48	no	—
154	C₂²×#127	—	79	1	no	—
155	C₂²×#126	—	79	1	no	—
156	C₂²×#126	D₇₉	1	1	yes	characteristic, complemented
157	C₂²×#123	—	79	3	no	—
158	C₂²×#121	—	79	3	no	—
159	C₂²×#121	—	79	3	no	—
160	C₂²×#120	—	79	3	no	—
161	C₇₉	C₂²×#560	1	1	yes	characteristic, complemented, Sylow
162	C₂⁶	—	79	1	no	—
163	C₂³×D₄	—	79	3	no	—
164	C₂³×D₄	—	79	3	no	—
165	C₂³×D₄	—	79	3	no	—
166	C₂³×D₄	—	79	3	no	—
167	C₂⁴×C₄	—	79	3	no	—
168	#560	—	79	256	no	—
169	C₂×#127	—	79	24	no	—
170	C₂×#126	—	79	24	no	—
171	C₂×#126	C₂×D₇₉	1	24	yes	complemented
172	C₂×#123	—	79	72	no	—
173	C₂×#121	—	79	72	no	—
174	C₂×#121	—	79	72	no	—
175	C₂×#120	—	79	72	no	—
176	C₂²×C₄⋊C₄	—	79	3	no	—
177	C₂²×C₄⋊C₄	C₂×D₇₉	1	3	yes	complemented
178	C₂²×C₂²⋊C₄	—	79	3	no	—
179	C₂²×C₂²⋊C₄	—	79	3	no	—
180	C₂²×C₂²⋊C₄	—	79	3	no	—
181	C₂²×C₂²⋊C₄	C₂×D₇₉	1	3	yes	—
182	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic, complemented
183	C₂⁵	—	316	12	no	—
184	C₂⁵	—	79	12	no	—
185	C₂⁵	—	79	1	no	—
186	C₂⁵	—	79	1	no	—
187	C₂⁵	—	79	3	no	—
188	C₂⁵	C₂²×D₇₉	1	1	yes	characteristic
189	C₂²×D₄	—	316	12	no	—
190	C₂²×D₄	—	316	12	no	—
191	C₂²×D₄	—	316	12	no	—
192	C₂²×D₄	—	316	12	no	—
193	C₂²×D₄	—	79	36	no	—
194	C₂²×D₄	—	79	36	no	—
195	C₂²×D₄	—	79	36	no	—
196	C₂²×D₄	—	79	36	no	—
197	C₂³×C₄	—	316	12	no	—
198	C₂³×C₄	—	79	36	no	—
199	C₂³×C₄	—	79	3	no	—
200	C₂³×C₄	C₂²×D₇₉	1	3	yes	—
201	C₂³×C₄	C₂²×D₇₉	1	3	yes	—
202	#127	—	79	64	no	—
203	#126	—	79	64	no	—
204	#126	C₂²×D₇₉	1	64	yes	complemented
205	#123	—	79	192	no	—
206	#121	—	79	192	no	—
207	#121	—	79	192	no	—
208	#120	—	79	192	no	—
209	C₂×C₄⋊C₄	—	79	36	no	—
210	C₂×C₄⋊C₄	C₂²×D₇₉	1	36	yes	complemented
211	C₂×C₂²⋊C₄	—	79	36	no	—
212	C₂×C₂²⋊C₄	—	79	36	no	—
213	C₂×C₂²⋊C₄	—	79	36	no	—
214	C₂×C₂²⋊C₄	C₂²×D₇₉	1	36	yes	—
215	C₂×C₄²	C₂²×D₇₉	1	12	yes	complemented
216	C₂⁴	—	2	12	no	—
217	C₂⁴	—	316	72	no	—
218	C₂⁴	—	316	64	no	—
219	C₂⁴	—	79	16	no	—
220	C₂⁴	—	79	6	no	—
221	C₂⁴	—	79	6	no	—
222	C₂⁴	—	79	18	no	—
223	C₂⁴	—	158	12	no	—
224	C₂⁴	—	158	12	no	—
225	C₂⁴	—	158	12	no	—
226	C₂⁴	—	158	12	no	—
227	C₂⁴	—	158	12	no	—
228	C₂⁴	C₂³×D₇₉	1	6	yes	—
229	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
230	C₂×D₄	—	316	72	no	—
231	C₂×D₄	—	316	72	no	—
232	C₂×D₄	—	316	72	no	—
233	C₂×D₄	—	316	72	no	—
234	C₂×D₄	—	79	48	no	—
235	C₂×D₄	—	79	48	no	—
236	C₂×D₄	—	79	48	no	—
237	C₂×D₄	—	79	48	no	—
238	C₂²×C₄	—	2	12	no	—
239	C₂²×C₄	—	2	12	no	—
240	C₂²×C₄	—	316	72	no	—
241	C₂²×C₄	—	79	48	no	—
242	C₂²×C₄	—	79	18	no	—
243	C₂²×C₄	—	158	12	no	—
244	C₂²×C₄	C₂³×D₇₉	1	18	yes	—
245	C₂²×C₄	C₂³×D₇₉	1	18	yes	—
246	C₄⋊C₄	—	79	48	no	—
247	C₄⋊C₄	C₂³×D₇₉	1	48	yes	complemented
248	C₂²⋊C₄	—	79	48	no	—
249	C₂²⋊C₄	—	79	48	no	—
250	C₂²⋊C₄	—	79	48	no	—
251	C₂²⋊C₄	C₂³×D₇₉	1	48	yes	—
252	C₄²	C₂³×D₇₉	1	16	yes	complemented
253	C₂³	—	2	36	no	—
254	C₂³	—	4	16	no	—
255	C₂³	—	316	48	no	—
256	C₂³	—	316	192	no	—
257	C₂³	—	316	16	no	—
258	C₂³	—	316	16	no	—
259	C₂³	—	316	48	no	—
260	C₂³	—	79	4	no	—
261	C₂³	—	79	4	no	—
262	C₂³	—	79	12	no	—
263	C₂³	—	158	36	no	—
264	C₂³	—	158	36	no	—
265	C₂³	—	158	36	no	—
266	C₂³	—	158	36	no	—
267	C₂³	—	158	36	no	—
268	C₂³	#14579207	1	12	yes	—
269	C₂³	C₂⁴×D₇₉	1	4	yes	—
270	C₂³	C₂⁴×D₇₉	1	3	yes	—
271	D₄	—	316	48	no	—
272	D₄	—	316	48	no	—
273	D₄	—	316	48	no	—
274	D₄	—	316	48	no	—
275	C₂×C₄	—	2	36	no	—
276	C₂×C₄	—	2	36	no	—
277	C₂×C₄	—	316	48	no	—
278	C₂×C₄	—	79	12	no	—
279	C₂×C₄	—	158	36	no	—
280	C₂×C₄	C₂⁴×D₇₉	1	12	yes	—
281	C₂×C₄	C₂⁴×D₇₉	1	12	yes	—
282	C₂²	—	2	12	no	—
283	C₂²	—	4	24	no	—
284	C₂²	—	316	64	no	—
285	C₂²	—	316	24	no	—
286	C₂²	—	316	24	no	—
287	C₂²	—	316	72	no	—
288	C₂²	—	158	12	no	—
289	C₂²	—	158	12	no	—
290	C₂²	—	158	12	no	—
291	C₂²	—	158	12	no	—
292	C₂²	—	158	12	no	—
293	C₂²	#19099304	1	16	yes	direct product factor
294	C₂²	C₂×#14579207	1	18	yes	—
295	C₂²	C₂⁵×D₇₉	1	1	yes	characteristic
296	C₄	—	2	12	no	—
297	C₄	—	2	12	no	—
298	C₄	—	158	12	no	—
299	C₂	—	4	4	no	—
300	C₂	—	316	4	no	—
301	C₂	—	316	4	no	—
302	C₂	—	316	12	no	—
303	C₂	C₂×#19099304	1	12	yes	direct product factor
304	C₂	C₂²×#14579207	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₇₉
commutator, cosocle	C₂²×C₇₉	C₂⁶
Frattini	C₂²	C₂⁵×D₇₉
Fitting	C₂²×#126×C₇₉	C₂
socle	C₂⁴×C₇₉	C₂⁴
hypercenter	C₂²×#126	D₇₉
hypocenter	C₇₉	C₂²×#560
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#126

Complements

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

N	H ≅ G/N	type
1. C₂×#19099304	C₂	direct (N × H)
2. C₂²×#14579158	C₂	semidirect
3. C₂²×#14579155	C₂	semidirect
4. C₂²×#14579146	C₂	semidirect
5. C₂²×#14579136	C₂	semidirect
6. C₂²×#14579134	C₂	semidirect
7. C₂²×#14579131	C₂	semidirect
8. C₂²×#126×C₇₉	C₂	semidirect
12. C₂³×D₃₁₆	C₂²	semidirect
14. C₂³×C₄×D₇₉	C₂²	semidirect
15. #19099304	C₂²	direct (N × H)
16. C₂×#14579158	C₂²	semidirect
17. C₂×#14579155	C₂²	semidirect
18. C₂×#14579146	C₂²	semidirect
19. C₂×#14579136	C₂²	semidirect
20. C₂×#14579134	C₂²	semidirect
21. C₂×#14579131	C₂²	semidirect
22. C₂²×Dic₇₉⋊C₄	C₂²	semidirect
25. C₂²×(C₂×D₇₉)⋊C₄	C₂²	semidirect
26. C₂×#126×C₇₉	C₂²	semidirect
27. C₂²×C₄⋊C₄×C₇₉	C₂²	semidirect
29. C₂²×C₄²×C₇₉	C₂²	semidirect
40. C₂²×D₃₁₆	C₂³	semidirect
45. C₂²×C₄×D₇₉	C₂³	semidirect
46. #14579158	C₂³	semidirect
47. #14579155	C₂³	semidirect
48. #14579146	C₂³	semidirect
49. #14579136	C₂³	semidirect
50. #14579134	C₂³	semidirect
51. #14579131	C₂³	semidirect
52. C₂×Dic₇₉⋊C₄	C₂³	semidirect
55. C₂×(C₂×D₇₉)⋊C₄	C₂³	semidirect
59. #126×C₇₉	C₂³	semidirect
60. C₂×C₄⋊C₄×C₇₉	C₂³	semidirect
62. C₂×C₄²×C₇₉	C₂³	semidirect
79. C₂×D₃₁₆	C₂⁴	semidirect
85. C₂×C₄×D₇₉	C₂⁴	semidirect
86. Dic₇₉⋊C₄	C₂⁴	semidirect
89. (C₂×D₇₉)⋊C₄	C₂⁴	semidirect
97. C₄⋊C₄×C₇₉	C₂⁴	semidirect
99. C₄²×C₇₉	C₂⁴	semidirect
131. C₂²×C₇₉	#560	semidirect
148. C₂×C₇₉	C₂×#560	semidirect
156. C₂²×#126	D₇₉	semidirect
161. C₇₉	C₂²×#560	semidirect
171. C₂×#126	C₂×D₇₉	semidirect
177. C₂²×C₄⋊C₄	C₂×D₇₉	semidirect
182. C₂²×C₄²	C₂×D₇₉	semidirect
204. #126	C₂²×D₇₉	semidirect
210. C₂×C₄⋊C₄	C₂²×D₇₉	semidirect
215. C₂×C₄²	C₂²×D₇₉	semidirect
247. C₄⋊C₄	C₂³×D₇₉	semidirect
252. C₄²	C₂³×D₇₉	semidirect
293. C₂²	#19099304	direct (N × H)
303. C₂	C₂×#19099304	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×#19099304	2
C₂²×#14579158	2
C₂²×#14579155	2
C₂²×#14579146	2
C₂²×#14579136	2
C₂²×#14579134	2
C₂²×#14579131	2
C₂²×#126×C₇₉	2
C₂²×#560	79

Subgroups — C22×#19099304 #35195139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19099304 #35195139