Subgroups — C₄×#19098386 #34203653

Order 20224.

330 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098386	C₂	1	2	yes	maximal
2	C₄×#14579097	C₂	1	1	yes	maximal, characteristic
3	C₄×#14579093	C₂	1	1	yes	maximal, characteristic
4	#23761499	C₂	1	2	yes	maximal
5	C₄×M₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579097	C₂²	1	2	yes	—
7	C₂×#14579093	C₂²	1	2	yes	—
8	C₄×#11929608	—	2	1	no	—
9	C₄×C₄⋊Dic₇₉	—	2	1	no	—
10	#19098386	C₄	1	8	yes	direct product factor
11	#19098328	—	2	1	no	—
12	#19098326	—	2	1	no	—
13	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	#346×C₇₉	C₄	1	2	yes	complemented
17	C₄×C₁₆×C₇₉	C₄	1	2	yes	complemented
18	C₂×#11929608	—	2	2	no	—
19	C₂×C₄⋊Dic₇₉	—	2	2	no	—
20	C₂×C₄×Dic₇₉	—	4	1	no	—
21	#14579097	—	2	2	no	—
22	#14579097	C₂×C₄	1	4	yes	—
23	#14579093	—	2	2	no	—
24	#14579093	C₂×C₄	1	4	yes	—
25	#14579072	—	4	1	no	—
26	C₄×C₇₉⋊C₈	—	4	1	no	—
27	C₇₉⋊#95	—	4	1	no	—
28	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
31	M₃₂×C₇₉	—	2	2	no	—
32	M₃₂×C₇₉	C₂×C₄	1	4	yes	—
33	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
38	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
39	C₂²×Dic₇₉	—	4	2	no	—
40	#11929608	—	2	4	no	—
41	#11929608	—	4	2	no	—
42	C₂×C₇₉⋊C₈	—	4	2	no	—
43	C₄⋊Dic₇₉	—	2	4	no	—
44	C₄⋊Dic₇₉	—	4	2	no	—
45	C₄×Dic₇₉	—	8	2	no	—
46	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₉	—	2	2	no	—
50	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₇₉	#41	1	2	yes	—
56	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
57	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
59	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
60	C₁₆×C₇₉	—	2	2	no	—
61	C₁₆×C₇₉	C₄²	1	4	yes	complemented
62	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₇₉	—	2	2	no	—
64	C₂×C₄×C₇₉	—	4	1	no	—
65	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
66	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
67	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
68	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
69	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#135	1	2	yes	—
73	C₂×Dic₇₉	—	4	4	no	—
74	C₂×Dic₇₉	—	8	2	no	—
75	C₂×Dic₇₉	—	8	2	no	—
76	C₈×C₇₉	—	2	1	no	—
77	C₈×C₇₉	—	2	1	no	—
78	C₈×C₇₉	C₂×C₄²	1	1	yes	characteristic
79	C₈×C₇₉	#117	1	1	yes	characteristic
80	C₈×C₇₉	C₄×Q₈	1	2	yes	—
81	C₇₉⋊C₈	—	4	4	no	—
82	C₂²×C₇₉	—	2	1	no	—
83	C₂²×C₇₉	—	2	1	no	—
84	C₂²×C₇₉	C₂×#106	1	1	yes	characteristic
85	C₂²×C₇₉	C₄×SD₁₆	1	2	yes	—
86	C₄×C₇₉	—	2	1	no	—
87	C₄×C₇₉	—	2	1	no	—
88	C₄×C₇₉	—	2	1	no	—
89	C₄×C₇₉	#365	1	2	yes	complemented
90	C₄×C₇₉	C₄×C₄⋊C₄	1	1	yes	characteristic
91	C₄×C₇₉	#427	1	1	yes	characteristic
92	C₄×C₇₉	C₄×SD₁₆	1	2	yes	—
93	Dic₇₉	—	8	4	no	—
94	C₄×#365	—	79	1	no	maximal, Sylow
95	C₂×C₇₉	—	2	2	no	—
96	C₂×C₇₉	C₄×#106	1	1	yes	characteristic
97	C₂×C₇₉	C₂×#365	1	1	yes	characteristic
98	C₂×C₇₉	#2155	1	1	yes	characteristic
99	C₂×#365	—	79	2	no	—
100	C₄×M₃₂	D₇₉	1	1	yes	characteristic
101	C₄×#108	—	79	1	no	—
102	C₄×#106	—	79	1	no	—
103	#1383	—	79	2	no	—
104	C₇₉	C₄×#365	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
106	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
107	C₂×#108	—	79	2	no	—
108	C₂×#106	—	79	2	no	—
109	C₄×M₁₆	—	158	1	no	—
110	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
111	C₄×C₄⋊C₄	—	158	1	no	—
112	#365	—	79	8	no	—
113	#346	Dic₇₉	1	2	yes	complemented
114	C₄×C₁₆	Dic₇₉	1	2	yes	complemented
115	#341	—	158	1	no	—
116	#340	—	158	1	no	—
117	C₂×M₁₆	—	158	2	no	—
118	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	158	2	no	—
121	C₂×C₄²	—	316	1	no	—
122	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
123	M₃₂	—	2	2	no	—
124	M₃₂	C₄×D₇₉	1	4	yes	—
125	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
126	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
127	#108	—	79	4	no	—
128	#108	—	158	2	no	—
129	#106	—	79	4	no	—
130	#106	—	158	2	no	—
131	#97	—	316	1	no	—
132	C₄×C₈	—	316	1	no	—
133	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
134	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
135	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
136	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
137	#95	—	316	1	no	—
138	C₂²×C₄	—	316	2	no	—
139	C₂²×C₄	#11929613	1	1	yes	characteristic
140	C₂²×C₄	#11929613	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
142	M₁₆	—	316	2	no	—
143	M₁₆	—	158	4	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	316	2	no	—
146	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
147	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
148	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
149	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
150	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
151	C₂×C₈	#11929624	1	2	yes	—
152	C₄⋊C₄	—	316	2	no	—
153	C₄⋊C₄	—	158	4	no	—
154	C₄²	—	632	2	no	—
155	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
156	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
157	C₄²	#11929613	1	1	yes	characteristic
158	C₄²	#11929613	1	1	yes	characteristic
159	C₁₆	—	2	2	no	—
160	C₁₆	C₄×Dic₇₉	1	4	yes	complemented
161	C₂³	C₂×#11929613	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	316	4	no	—
165	C₂×C₄	—	632	2	no	—
166	C₂×C₄	—	632	2	no	—
167	C₂×C₄	#14579093	1	1	yes	characteristic
168	C₂×C₄	#14579093	1	1	yes	characteristic
169	C₂×C₄	#14579093	1	1	yes	characteristic
170	C₂×C₄	#14579093	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
172	C₂×C₄	C₄×D₃₁₆	1	2	yes	—
173	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
174	C₂×C₄	#14579183	1	2	yes	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	316	4	no	—
178	C₈	C₂×C₄×Dic₇₉	1	1	yes	characteristic
179	C₈	#14579120	1	1	yes	characteristic
180	C₈	C₄×Dic₁₅₈	1	2	yes	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	C₂×#14579093	1	1	yes	characteristic
184	C₂²	C₄×#11929613	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	632	4	no	—
189	C₄	#19098386	1	2	yes	direct product factor
190	C₄	C₄×C₄⋊Dic₇₉	1	1	yes	characteristic
191	C₄	#19098601	1	1	yes	characteristic
192	C₄	C₄×#11929613	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	C₄×#14579093	1	1	yes	characteristic
195	C₂	C₂×#19098386	1	1	yes	characteristic
196	C₂	#23765173	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579093
commutator	C₈×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₄×M₃₂×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#106
prime radical	C₂³×C₇₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₄×M₃₂	D₇₉
hypocenter	C₇₉	C₄×#365
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
10. #19098386	C₄	direct (N × H)
16. #346×C₇₉	C₄	semidirect
17. C₄×C₁₆×C₇₉	C₄	semidirect
61. C₁₆×C₇₉	C₄²	semidirect
89. C₄×C₇₉	#365	semidirect
104. C₇₉	C₄×#365	semidirect
113. #346	Dic₇₉	semidirect
114. C₄×C₁₆	Dic₇₉	semidirect
160. C₁₆	C₄×Dic₇₉	semidirect
189. C₄	#19098386	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×#19098386	2
C₄×#14579097	2
C₄×#14579093	2
#23761499	2
C₄×M₃₂×C₇₉	2
C₄×#365	79

Subgroups — C4×#19098386 #34203653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19098386 #34203653