Subgroups — C₂²×#19098674 #35152313

Order 20224.

3654 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₆₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14579123	C₂	1	2	yes	maximal, complemented
3	C₂×#19098674	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#14579081	C₂	1	2	yes	maximal, complemented
5	C₂²×#14579071	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#106×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₃₁₆	C₂²	1	2	yes	—
8	C₂³×C₄×D₇₉	—	2	2	no	—
9	C₂²×D₆₃₂	C₄	1	16	yes	complemented
10	C₂²×D₆₃₂	C₂²	1	12	yes	—
11	C₂³×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#14579123	—	2	12	no	—
13	C₂×#14579123	C₂²	1	24	yes	complemented
14	C₂²×(C₂×D₇₉)⋊C₄	—	2	2	no	—
15	C₂²×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
16	#19098674	C₂²	1	64	yes	direct product factor
17	C₂×#14579081	C₂²	1	24	yes	complemented
18	C₂×#14579071	C₂²	1	12	yes	complemented
19	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
21	C₂×#106×C₇₉	C₂²	1	12	yes	complemented
22	C₂⁴×D₇₉	—	2	2	no	—
23	C₂²×D₃₁₆	—	2	6	no	—
24	C₂²×D₃₁₆	—	2	8	no	—
25	C₂²×D₃₁₆	C₂×C₄	1	16	yes	complemented
26	C₂²×D₃₁₆	C₂³	1	12	yes	—
27	C₂³×Dic₇₉	D₄	1	2	yes	—
28	C₂²×C₄×D₇₉	—	2	24	no	—
29	C₂²×C₄×D₇₉	—	4	16	no	—
30	C₂×D₆₃₂	C₂×C₄	1	96	yes	complemented
31	C₂×D₆₃₂	C₂³	1	16	yes	—
32	C₂²×C₇₉⋊C₈	—	2	4	no	—
33	C₂²×C₇₉⋊C₈	C₂³	1	6	yes	—
34	#14579123	—	2	48	no	—
35	#14579123	C₂³	1	32	yes	complemented
36	C₂×(C₂×D₇₉)⋊C₄	—	2	24	no	—
37	C₂×C₄×Dic₇₉	D₄	1	6	yes	—
38	C₂×C₄×Dic₇₉	C₂³	1	6	yes	—
39	#14579081	C₂³	1	32	yes	complemented
40	#14579071	C₂³	1	16	yes	complemented
41	C₂³×C₄×C₇₉	—	2	2	no	—
42	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₇₉	C₂×C₄	1	8	yes	complemented
44	C₂²×C₈×C₇₉	C₂³	1	6	yes	—
45	C₂×C₄⋊C₄×C₇₉	—	2	6	no	—
46	C₂×C₄⋊C₄×C₇₉	C₂³	1	12	yes	—
47	#106×C₇₉	C₂³	1	16	yes	complemented
48	C₂³×D₇₉	—	2	16	no	—
49	C₂³×D₇₉	—	2	12	no	—
50	C₂³×D₇₉	—	4	8	no	—
51	C₂³×D₇₉	—	4	8	no	—
52	C₂×D₃₁₆	—	2	48	no	—
53	C₂×D₃₁₆	—	2	12	no	—
54	C₂×D₃₁₆	—	2	24	no	—
55	C₂×D₃₁₆	C₂²×C₄	1	48	yes	complemented
56	C₂×D₃₁₆	C₂⁴	1	8	yes	—
57	C₂²×Dic₇₉	—	2	8	no	—
58	C₂²×Dic₇₉	C₂×D₄	1	12	yes	—
59	C₂×C₄×D₇₉	—	2	32	no	—
60	C₂×C₄×D₇₉	—	4	96	no	—
61	D₆₃₂	C₂²×C₄	1	64	yes	complemented
62	C₂×C₇₉⋊C₈	—	2	12	no	—
63	C₂×C₇₉⋊C₈	C₂⁴	1	4	yes	—
64	(C₂×D₇₉)⋊C₄	—	2	32	no	—
65	C₄×Dic₇₉	C₂×D₄	1	12	yes	—
66	C₄×Dic₇₉	C₂⁴	1	4	yes	—
67	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	—	2	12	no	—
69	C₂²×C₄×C₇₉	—	4	8	no	—
70	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₇₉	#41	1	4	yes	—
73	C₂²×C₄×C₇₉	C₂⁴	1	3	yes	—
74	C₂×C₈×C₇₉	C₂²×C₄	1	24	yes	complemented
75	C₂×C₈×C₇₉	C₂⁴	1	4	yes	—
76	C₄⋊C₄×C₇₉	—	2	12	no	—
77	C₄⋊C₄×C₇₉	C₂⁴	1	8	yes	—
78	C₂³×C₇₉	C₄×D₄	1	4	yes	—
79	C₂³×C₇₉	AGL(1,ℤ/8)	1	8	yes	—
80	C₂³×C₇₉	C₂²×D₄	1	3	yes	—
81	C₂²×D₇₉	—	2	48	no	—
82	C₂²×D₇₉	—	2	8	no	—
83	C₂²×D₇₉	—	4	24	no	—
84	C₂²×D₇₉	—	4	24	no	—
85	C₂²×D₇₉	—	4	64	no	—
86	C₂×C₄×C₇₉	—	2	8	no	—
87	C₂×C₄×C₇₉	—	4	24	no	—
88	C₂×C₄×C₇₉	C₄×D₄	1	12	yes	—
89	C₂×C₄×C₇₉	C₂³×C₄	1	6	yes	—
90	C₂×C₄×C₇₉	C₂²×D₄	1	3	yes	—
91	C₂×C₄×C₇₉	C₂×#41	1	6	yes	—
92	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
93	C₂×Dic₇₉	—	2	24	no	—
94	C₂×Dic₇₉	C₂²×D₄	1	8	yes	—
95	D₃₁₆	—	2	48	no	—
96	D₃₁₆	—	2	8	no	—
97	D₃₁₆	C₂³×C₄	1	16	yes	complemented
98	C₄×D₇₉	—	4	64	no	—
99	C₈×C₇₉	C₂³×C₄	1	8	yes	complemented
100	C₇₉⋊C₈	—	2	4	no	—
101	C₂²×C₇₉	#442	1	16	yes	complemented
102	C₂²×C₇₉	C₂×C₄×D₄	1	6	yes	—
103	C₂²×C₇₉	C₂×AGL(1,ℤ/8)	1	12	yes	—
104	C₂²×C₇₉	C₂³×D₄	1	1	yes	characteristic
105	C₂×D₇₉	—	2	16	no	—
106	C₂×D₇₉	—	4	96	no	—
107	C₂×D₇₉	—	4	8	no	—
108	C₂×D₇₉	—	4	8	no	—
109	C₄×C₇₉	—	4	8	no	—
110	C₄×C₇₉	C₂×C₄×D₄	1	6	yes	—
111	C₄×C₇₉	C₂⁴×C₄	1	1	yes	characteristic
112	C₄×C₇₉	C₂²×#41	1	1	yes	characteristic
113	Dic₇₉	—	2	8	no	—
114	C₂²×#442	—	79	1	no	maximal, Sylow
115	C₂×C₇₉	C₂×#442	1	12	yes	complemented
116	C₂×C₇₉	C₂²×C₄×D₄	1	1	yes	characteristic
117	C₂×C₇₉	C₂²×AGL(1,ℤ/8)	1	2	yes	—
118	D₇₉	—	4	16	no	—
119	C₂³×D₈	—	79	1	no	—
120	C₂²×C₄×D₄	—	79	2	no	—
121	C₂×#442	—	79	24	no	—
122	C₂²×#106	D₇₉	1	1	yes	characteristic, complemented
123	C₂²×#102	—	79	2	no	—
124	C₂²×#97	—	79	1	no	—
125	C₇₉	C₂²×#442	1	1	yes	characteristic, complemented, Sylow
126	C₂³×D₄	—	79	2	no	—
127	C₂⁴×C₄	—	158	2	no	—
128	C₂²×D₈	—	79	12	no	—
129	C₂²×D₈	—	79	16	no	—
130	C₂³×C₈	—	79	1	no	—
131	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
132	C₂×C₄×D₄	—	79	24	no	—
133	C₂×C₄×D₄	—	158	12	no	—
134	C₂²×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
135	C₂²×C₂²⋊C₄	—	158	2	no	—
136	C₂²×C₄²	—	79	1	no	—
137	#442	—	79	64	no	—
138	C₂×#106	C₂×D₇₉	1	12	yes	complemented
139	C₂×#102	—	79	24	no	—
140	C₂×#97	—	79	12	no	—
141	C₂⁵	—	158	2	no	—
142	C₂²×D₄	—	79	16	no	—
143	C₂²×D₄	—	79	12	no	—
144	C₂²×D₄	—	158	6	no	—
145	C₂²×D₄	—	158	8	no	—
146	C₂³×C₄	—	2	2	no	—
147	C₂³×C₄	—	316	16	no	—
148	C₂³×C₄	—	79	2	no	—
149	C₂³×C₄	—	158	24	no	—
150	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
151	C₂×D₈	—	79	96	no	—
152	C₂×D₈	—	79	16	no	—
153	C₂²×C₈	—	79	6	no	—
154	C₂²×C₈	—	158	4	no	—
155	C₂²×C₈	C₄×D₇₉	1	8	yes	complemented
156	C₂²×C₈	C₂²×D₇₉	1	6	yes	—
157	C₄×D₄	—	79	32	no	—
158	C₄×D₄	—	158	48	no	—
159	C₂×C₄⋊C₄	—	2	6	no	—
160	C₂×C₄⋊C₄	C₂²×D₇₉	1	12	yes	—
161	C₂×C₂²⋊C₄	—	158	24	no	—
162	C₂×C₄²	—	79	6	no	—
163	C₂×C₄²	—	79	6	no	—
164	#106	C₂²×D₇₉	1	16	yes	complemented
165	#102	—	79	32	no	—
166	#97	—	79	16	no	—
167	C₂⁴	—	316	8	no	—
168	C₂⁴	—	316	8	no	—
169	C₂⁴	—	158	16	no	—
170	C₂⁴	—	158	12	no	—
171	C₂⁴	D₄×D₇₉	1	1	yes	characteristic
172	C₂×D₄	—	79	8	no	—
173	C₂×D₄	—	79	48	no	—
174	C₂×D₄	—	158	24	no	—
175	C₂×D₄	—	158	48	no	—
176	C₂×D₄	—	158	12	no	—
177	C₂²×C₄	—	2	12	no	—
178	C₂²×C₄	—	4	8	no	—
179	C₂²×C₄	—	316	96	no	—
180	C₂²×C₄	—	79	12	no	—
181	C₂²×C₄	—	158	32	no	—
182	C₂²×C₄	—	158	8	no	—
183	C₂²×C₄	C₂×C₄×D₇₉	1	4	yes	—
184	C₂²×C₄	D₄×D₇₉	1	3	yes	—
185	C₂²×C₄	#11929625	1	4	yes	—
186	C₂²×C₄	C₂³×D₇₉	1	3	yes	—
187	D₈	—	79	64	no	—
188	C₂×C₈	—	79	4	no	—
189	C₂×C₈	—	158	12	no	—
190	C₂×C₈	C₂×C₄×D₇₉	1	24	yes	complemented
191	C₂×C₈	C₂³×D₇₉	1	4	yes	—
192	C₄⋊C₄	—	2	12	no	—
193	C₄⋊C₄	C₂³×D₇₉	1	8	yes	—
194	C₂²⋊C₄	—	158	32	no	—
195	C₄²	—	79	4	no	—
196	C₄²	—	79	12	no	—
197	C₂³	—	316	64	no	—
198	C₂³	—	316	24	no	—
199	C₂³	—	316	24	no	—
200	C₂³	—	158	48	no	—
201	C₂³	—	158	8	no	—
202	C₂³	#14579123	1	4	yes	—
203	C₂³	#14579188	1	8	yes	—
204	C₂³	C₂×D₄×D₇₉	1	3	yes	—
205	D₄	—	79	16	no	—
206	D₄	—	158	48	no	—
207	D₄	—	158	8	no	—
208	C₂×C₄	—	2	8	no	—
209	C₂×C₄	—	4	24	no	—
210	C₂×C₄	—	316	64	no	—
211	C₂×C₄	—	79	8	no	—
212	C₂×C₄	—	158	24	no	—
213	C₂×C₄	#14579123	1	12	yes	—
214	C₂×C₄	C₂²×C₄×D₇₉	1	6	yes	—
215	C₂×C₄	C₂×D₄×D₇₉	1	3	yes	—
216	C₂×C₄	C₂×#11929625	1	6	yes	—
217	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
218	C₈	—	158	4	no	—
219	C₈	C₂²×C₄×D₇₉	1	8	yes	complemented
220	C₂²	—	316	96	no	—
221	C₂²	—	316	8	no	—
222	C₂²	—	316	8	no	—
223	C₂²	—	158	16	no	—
224	C₂²	#19098674	1	16	yes	direct product factor
225	C₂²	C₂×#14579123	1	6	yes	—
226	C₂²	C₂×#14579188	1	12	yes	—
227	C₂²	C₂²×D₄×D₇₉	1	1	yes	characteristic
228	C₄	—	4	8	no	—
229	C₄	—	158	8	no	—
230	C₄	C₂×#14579123	1	6	yes	—
231	C₄	C₂³×C₄×D₇₉	1	1	yes	characteristic
232	C₄	C₂²×#11929625	1	1	yes	characteristic
233	C₂	—	316	16	no	—
234	C₂	C₂×#19098674	1	12	yes	direct product factor
235	C₂	C₂²×#14579123	1	1	yes	characteristic
236	C₂	C₂²×#14579188	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₇₉
commutator	C₄×C₇₉	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇₉
Fitting	C₂²×#106×C₇₉	C₂
socle	C₂⁴×C₇₉	C₂×D₄
prime radical	C₂³×D₆₃₂	C₂
cosocle	C₂×C₄×C₇₉	C₂⁵
hypercenter	C₂²×#106	D₇₉
hypocenter	C₇₉	C₂²×#442
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
2. C₂²×#14579123	C₂	semidirect
3. C₂×#19098674	C₂	direct (N × H)
4. C₂²×#14579081	C₂	semidirect
5. C₂²×#14579071	C₂	semidirect
6. C₂²×#106×C₇₉	C₂	semidirect
9. C₂²×D₆₃₂	C₄	semidirect
13. C₂×#14579123	C₂²	semidirect
16. #19098674	C₂²	direct (N × H)
17. C₂×#14579081	C₂²	semidirect
18. C₂×#14579071	C₂²	semidirect
21. C₂×#106×C₇₉	C₂²	semidirect
25. C₂²×D₃₁₆	C₂×C₄	semidirect
30. C₂×D₆₃₂	C₂×C₄	semidirect
35. #14579123	C₂³	semidirect
39. #14579081	C₂³	semidirect
40. #14579071	C₂³	semidirect
43. C₂²×C₈×C₇₉	C₂×C₄	semidirect
47. #106×C₇₉	C₂³	semidirect
55. C₂×D₃₁₆	C₂²×C₄	semidirect
61. D₆₃₂	C₂²×C₄	semidirect
74. C₂×C₈×C₇₉	C₂²×C₄	semidirect
97. D₃₁₆	C₂³×C₄	semidirect
99. C₈×C₇₉	C₂³×C₄	semidirect
101. C₂²×C₇₉	#442	semidirect
115. C₂×C₇₉	C₂×#442	semidirect
122. C₂²×#106	D₇₉	semidirect
125. C₇₉	C₂²×#442	semidirect
138. C₂×#106	C₂×D₇₉	semidirect
155. C₂²×C₈	C₄×D₇₉	semidirect
164. #106	C₂²×D₇₉	semidirect
190. C₂×C₈	C₂×C₄×D₇₉	semidirect
219. C₈	C₂²×C₄×D₇₉	semidirect
224. C₂²	#19098674	direct (N × H)
234. C₂	C₂×#19098674	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×D₆₃₂	2
C₂²×#14579123	2
C₂×#19098674	2
C₂²×#14579081	2
C₂²×#14579071	2
C₂²×#106×C₇₉	2
C₂²×#442	79

Subgroups — C22×#19098674 #35152313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19098674 #35152313