Subgroups — C₄×#19098612 #34286376

Order 20224.

1034 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098612	C₂	1	2	yes	maximal
2	C₂×C₄×#11929608	C₂	1	2	yes	maximal
3	#23763306	C₂	1	4	yes	maximal, complemented
4	C₄×#14579097	C₂	1	4	yes	maximal, complemented
5	#23763127	C₂	1	2	yes	maximal
6	C₂×C₄×M₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11929608	C₂²	1	4	yes	—
8	#19098612	C₄	1	16	yes	direct product factor
9	C₂×#14579097	C₂²	1	8	yes	—
10	C₄×#11929608	—	2	2	no	—
11	C₄×#11929608	—	2	2	no	—
12	C₄×#11929608	C₂²	1	4	yes	—
13	C₄×#11929608	C₂²	1	4	yes	—
14	C₂×#14579072	—	2	2	no	—
15	C₂×C₄×C₇₉⋊C₈	—	2	2	no	—
16	#19098328	C₂²	1	8	yes	—
17	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₇₉	C₄	1	8	yes	—
21	C₂×#97×C₇₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
23	C₂×#11929608	—	2	4	no	—
24	C₂×#11929608	—	2	8	no	—
25	C₂×#11929608	C₂×C₄	1	16	yes	—
26	C₂×#11929608	C₂³	1	8	yes	—
27	C₂²×C₇₉⋊C₈	—	2	4	no	—
28	#14579097	C₂×C₄	1	32	yes	complemented
29	#14579072	—	2	4	no	—
30	#14579072	—	2	4	no	—
31	C₄×C₇₉⋊C₈	—	2	4	no	—
32	C₄×C₇₉⋊C₈	—	2	4	no	—
33	C₂³×C₄×C₇₉	Q₈	1	2	yes	—
34	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₇₉	—	2	4	no	—
36	C₂×M₁₆×C₇₉	C₂×C₄	1	8	yes	—
37	C₂×M₁₆×C₇₉	C₂×C₄	1	4	yes	—
38	C₂×M₁₆×C₇₉	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
41	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₇₉	D₄	1	2	yes	—
43	C₂×C₄²×C₇₉	D₄	1	2	yes	—
44	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
45	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
47	#97×C₇₉	C₂×C₄	1	8	yes	—
48	C₄×C₈×C₇₉	C₂×C₄	1	8	yes	—
49	#11929608	—	2	16	no	—
50	#11929608	—	2	8	no	—
51	#11929608	C₂²×C₄	1	16	yes	—
52	C₂×C₇₉⋊C₈	—	2	16	no	—
53	C₂×C₇₉⋊C₈	—	2	8	no	—
54	C₂⁴×C₇₉	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	—	4	4	no	—
56	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₇₉	#41	1	4	yes	—
65	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
66	M₁₆×C₇₉	—	2	8	no	—
67	M₁₆×C₇₉	C₄²	1	16	yes	—
68	C₂×C₈×C₇₉	—	2	4	no	—
69	C₂×C₈×C₇₉	C₂²×C₄	1	8	yes	—
70	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
72	C₄²×C₇₉	C₄⋊C₄	1	4	yes	—
73	C₄²×C₇₉	C₄⋊C₄	1	4	yes	—
74	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
75	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
76	C₄²×C₇₉	C₂×D₄	1	2	yes	—
77	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
79	C₂³×C₇₉	—	4	2	no	—
80	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
81	C₂³×C₇₉	C₄×Q₈	1	4	yes	—
82	C₂³×C₇₉	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₇₉	—	2	8	no	—
84	C₂×C₄×C₇₉	—	2	4	no	—
85	C₂×C₄×C₇₉	—	4	4	no	—
86	C₂×C₄×C₇₉	—	4	2	no	—
87	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
88	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₇₉	#117	1	4	yes	—
93	C₂×C₄×C₇₉	C₄×D₄	1	8	yes	—
94	C₂×C₄×C₇₉	C₄×Q₈	1	4	yes	—
95	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
96	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
99	C₈×C₇₉	—	2	4	no	—
100	C₈×C₇₉	C₂×C₄²	1	8	yes	—
101	C₇₉⋊C₈	—	2	16	no	—
102	C₂²×C₇₉	—	2	4	no	—
103	C₂²×C₇₉	—	2	2	no	—
104	C₂²×C₇₉	—	4	4	no	—
105	C₂²×C₇₉	C₄×C₄⋊C₄	1	4	yes	—
106	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₇₉	C₂×C₄×Q₈	1	2	yes	—
108	C₄×C₇₉	—	2	4	no	—
109	C₄×C₇₉	—	2	2	no	—
110	C₄×C₇₉	C₄×C₄⋊C₄	1	4	yes	—
111	C₄×C₇₉	#430	1	4	yes	complemented
112	C₄×C₇₉	C₂²×C₄²	1	1	yes	characteristic
113	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
114	C₄×C₇₉	C₂×C₄×D₄	1	2	yes	—
115	C₄×#430	—	79	1	no	maximal, Sylow
116	C₂×C₇₉	—	2	2	no	—
117	C₂×C₇₉	—	2	4	no	—
118	C₂×C₇₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₇₉	C₂×#430	1	2	yes	—
120	C₂×#430	—	79	2	no	—
121	C₂×C₄×M₁₆	—	79	2	no	—
122	C₂×C₄×M₁₆	D₇₉	1	1	yes	characteristic
123	#1778	—	79	4	no	—
124	C₄×#108	—	79	4	no	—
125	#1738	—	79	2	no	—
126	C₇₉	C₄×#430	1	1	yes	characteristic, complemented, Sylow
127	C₂²×M₁₆	—	79	4	no	—
128	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
129	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
130	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
131	#430	—	79	16	no	—
132	C₂×#108	—	79	8	no	—
133	C₄×M₁₆	—	79	4	no	—
134	C₄×M₁₆	—	79	4	no	—
135	C₄×M₁₆	—	158	2	no	—
136	C₄×M₁₆	—	158	2	no	—
137	C₄×M₁₆	Dic₇₉	1	8	yes	—
138	C₂×#97	—	158	2	no	—
139	C₂×#97	C₂×D₇₉	1	2	yes	—
140	C₂×C₄×C₈	—	158	2	no	—
141	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
142	#341	—	79	8	no	—
143	C₂³×C₄	Dic₁₅₈	1	2	yes	—
144	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
145	C₂×M₁₆	—	2	4	no	—
146	C₂×M₁₆	—	79	16	no	—
147	C₂×M₁₆	—	79	8	no	—
148	C₂×M₁₆	—	158	4	no	—
149	C₂×M₁₆	—	158	8	no	—
150	C₂×M₁₆	C₄×D₇₉	1	8	yes	—
151	C₂×M₁₆	C₂×Dic₇₉	1	4	yes	—
152	C₂×M₁₆	C₂×Dic₇₉	1	4	yes	—
153	C₂²×C₈	—	158	4	no	—
154	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
155	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
156	C₂×C₄²	Dic₁₅₈	1	2	yes	—
157	C₂×C₄²	D₃₁₆	1	2	yes	—
158	C₂×C₄²	D₃₁₆	1	2	yes	—
159	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
160	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
161	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
162	#108	—	79	32	no	—
163	#97	—	158	4	no	—
164	#97	—	158	4	no	—
165	#97	C₂×Dic₇₉	1	8	yes	—
166	C₄×C₈	—	158	4	no	—
167	C₄×C₈	—	158	4	no	—
168	C₄×C₈	C₂×Dic₇₉	1	8	yes	—
169	C₂⁴	C₂×Dic₁₅₈	1	1	yes	characteristic
170	C₂²×C₄	—	4	4	no	—
171	C₂²×C₄	C₄⋊Dic₇₉	1	4	yes	—
172	C₂²×C₄	C₄⋊Dic₇₉	1	4	yes	—
173	C₂²×C₄	C₂×C₄×D₇₉	1	4	yes	—
174	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
175	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
176	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
177	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
178	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
179	C₂²×C₄	#11929624	1	4	yes	—
180	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
181	M₁₆	—	2	8	no	—
182	M₁₆	—	79	16	no	—
183	M₁₆	—	158	8	no	—
184	M₁₆	—	158	16	no	—
185	M₁₆	C₄×Dic₇₉	1	16	yes	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	158	16	no	—
188	C₂×C₈	—	158	8	no	—
189	C₂×C₈	C₂×C₄×D₇₉	1	8	yes	—
190	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
191	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
192	C₄²	C₄⋊Dic₇₉	1	4	yes	—
193	C₄²	C₄⋊Dic₇₉	1	4	yes	—
194	C₄²	C₂²×Dic₇₉	1	2	yes	—
195	C₄²	C₂²×Dic₇₉	1	2	yes	—
196	C₄²	C₂×D₃₁₆	1	2	yes	—
197	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
198	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
199	C₂³	—	4	2	no	—
200	C₂³	C₂×C₄⋊Dic₇₉	1	2	yes	—
201	C₂³	C₄×Dic₁₅₈	1	4	yes	—
202	C₂³	C₂²×Dic₁₅₈	1	1	yes	characteristic
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
210	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
211	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
212	C₂×C₄	#14579120	1	4	yes	—
213	C₂×C₄	C₄×D₃₁₆	1	8	yes	—
214	C₂×C₄	C₄×Dic₁₅₈	1	4	yes	—
215	C₂×C₄	C₂²×C₄×D₇₉	1	2	yes	—
216	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
217	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
218	C₂×C₄	C₂×#11929624	1	2	yes	—
219	C₈	—	2	4	no	—
220	C₈	—	158	16	no	—
221	C₈	C₂×C₄×Dic₇₉	1	8	yes	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	4	4	no	—
225	C₂²	C₄×C₄⋊Dic₇₉	1	4	yes	—
226	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
227	C₂²	C₂×C₄×Dic₁₅₈	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	C₄×C₄⋊Dic₇₉	1	4	yes	—
231	C₄	#19098612	1	4	yes	direct product factor
232	C₄	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
233	C₄	C₂×#14579120	1	1	yes	characteristic
234	C₄	C₂×C₄×D₃₁₆	1	2	yes	—
235	C₂	—	2	4	no	—
236	C₂	—	2	2	no	—
237	C₂	C₂×C₄×C₄⋊Dic₇₉	1	1	yes	characteristic
238	C₂	C₂×#19098612	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₃₁₆
commutator	C₄×C₇₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×C₄×M₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₇₉	C₂×Q₈
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₇₉
hypocenter	C₇₉	C₄×#430
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #23763306	C₂	semidirect
4. C₄×#14579097	C₂	semidirect
8. #19098612	C₄	direct (N × H)
28. #14579097	C₂×C₄	semidirect
111. C₄×C₇₉	#430	semidirect
126. C₇₉	C₄×#430	semidirect
231. C₄	#19098612	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×#19098612	2
C₂×C₄×#11929608	2
#23763306	2
C₄×#14579097	2
#23763127	2
C₂×C₄×M₁₆×C₇₉	2
C₄×#430	79

Subgroups — C4×#19098612 #34286376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19098612 #34286376