Subgroups — C₄×#19098943 #34296706

Order 20224.

1086 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098943	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₁₆	C₂	1	4	yes	maximal
3	C₄×#14579161	C₂	1	2	yes	maximal
4	#23764261	C₂	1	4	yes	maximal
5	#23764202	C₂	1	2	yes	maximal
6	#23763560	C₂	1	1	yes	maximal, characteristic
7	C₄²×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₃₁₆	C₂²	1	4	yes	—
9	C₂×#14579161	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₅₈	C₂²	1	4	yes	—
11	#19098943	C₄	1	8	yes	direct product factor
12	C₄×Dic₃₁₆	—	2	16	no	—
13	C₂×#14579095	C₂²	1	4	yes	—
14	C₂×#14579086	C₂²	1	8	yes	—
15	#19098469	C₂²	1	4	yes	—
16	#19098461	—	2	4	no	—
17	#19098433	C₂²	1	2	yes	—
18	C₄×C₄⋊Dic₇₉	—	2	4	no	—
19	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
23	C₄³×C₇₉	C₂²	1	1	yes	characteristic
24	C₂²×Dic₁₅₈	C₂³	1	4	yes	—
25	C₂×Dic₃₁₆	—	2	8	no	—
26	C₂×Dic₃₁₆	—	2	8	no	—
27	C₂×Dic₃₁₆	C₂×C₄	1	16	yes	—
28	#14579161	—	2	4	no	—
29	#14579161	C₂×C₄	1	8	yes	—
30	C₄×Dic₁₅₈	—	2	16	no	—
31	C₂×C₄⋊Dic₇₉	—	2	4	no	—
32	C₂×C₄⋊Dic₇₉	—	2	4	no	—
33	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
34	C₂×Dic₇₉⋊C₄	—	2	4	no	—
35	C₂×C₄×Dic₇₉	—	2	4	no	—
36	#14579095	—	2	8	no	—
37	#14579086	—	2	16	no	—
38	C₇₉⋊#95	—	2	8	no	—
39	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
41	C₂×C₄²×C₇₉	D₄	1	2	yes	—
42	C₂×C₄²×C₇₉	D₄	1	2	yes	—
43	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₇₉	—	2	4	no	—
47	C₄×C₈×C₇₉	—	2	2	no	—
48	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
49	C₄×C₈×C₇₉	D₄	1	8	yes	—
50	C₂×Dic₁₅₈	—	2	8	no	—
51	C₂×Dic₁₅₈	—	2	8	no	—
52	C₂×Dic₁₅₈	—	2	4	no	—
53	C₂×Dic₁₅₈	C₂²×C₄	1	8	yes	—
54	C₂²×Dic₇₉	—	2	4	no	—
55	C₂²×Dic₇₉	—	2	4	no	—
56	Dic₃₁₆	—	2	32	no	—
57	C₄⋊Dic₇₉	—	2	16	no	—
58	C₄⋊Dic₇₉	—	2	8	no	—
59	C₄⋊Dic₇₉	—	4	8	no	—
60	Dic₇₉⋊C₄	—	4	8	no	—
61	C₄×Dic₇₉	—	4	8	no	—
62	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₇₉	—	2	2	no	—
68	C₂×C₈×C₇₉	—	2	2	no	—
69	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
71	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
72	C₂×C₈×C₇₉	#41	1	4	yes	—
73	C₄²×C₇₉	—	2	4	no	—
74	C₄²×C₇₉	—	2	2	no	—
75	C₄²×C₇₉	Q₁₆	1	8	yes	complemented
76	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
77	C₄²×C₇₉	C₂×D₄	1	4	yes	—
78	C₄²×C₇₉	#41	1	2	yes	—
79	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₇₉	—	2	2	no	—
81	C₂×C₄×C₇₉	—	2	2	no	—
82	C₂×C₄×C₇₉	—	2	2	no	—
83	C₂×C₄×C₇₉	C₄×D₄	1	4	yes	—
84	C₂×C₄×C₇₉	#127	1	2	yes	—
85	C₂×C₄×C₇₉	#127	1	2	yes	—
86	C₂×C₄×C₇₉	C₂×Q₁₆	1	4	yes	—
87	C₂×C₄×C₇₉	C₂×Q₁₆	1	4	yes	—
88	C₂×C₄×C₇₉	C₂×Q₁₆	1	4	yes	—
89	C₂×C₄×C₇₉	#135	1	4	yes	—
90	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₉	C₂²×D₄	1	2	yes	—
92	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
93	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
94	C₂×Dic₇₉	—	2	8	no	—
95	C₂×Dic₇₉	—	4	4	no	—
96	C₂×Dic₇₉	—	4	4	no	—
97	C₂×Dic₇₉	—	4	4	no	—
98	C₂×Dic₇₉	—	4	8	no	—
99	Dic₁₅₈	—	2	16	no	—
100	Dic₁₅₈	—	4	8	no	—
101	C₈×C₇₉	—	2	4	no	—
102	C₈×C₇₉	C₄×D₄	1	8	yes	—
103	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
104	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
105	C₂²×C₇₉	#532	1	1	yes	characteristic
106	C₂²×C₇₉	C₂²×Q₁₆	1	2	yes	—
107	C₂²×C₇₉	C₂×#135	1	2	yes	—
108	C₄×C₇₉	—	2	2	no	—
109	C₄×C₇₉	—	2	4	no	—
110	C₄×C₇₉	C₄×Q₁₆	1	8	yes	complemented
111	C₄×C₇₉	#494	1	4	yes	complemented
112	C₄×C₇₉	C₂×C₄×D₄	1	2	yes	—
113	C₄×C₇₉	#532	1	2	yes	—
114	Dic₇₉	—	4	8	no	—
115	Dic₇₉	—	8	4	no	—
116	C₄×#494	—	79	1	no	maximal, Sylow
117	C₂×C₇₉	C₄×#127	1	1	yes	characteristic
118	C₂×C₇₉	C₂×C₄×Q₁₆	1	2	yes	—
119	C₂×C₇₉	C₂×#494	1	1	yes	characteristic
120	C₂×C₇₉	#3147	1	1	yes	characteristic
121	C₂×C₇₉	#3170	1	2	yes	—
122	C₂×#494	—	79	1	no	—
123	C₂×C₄×Q₁₆	—	79	4	no	—
124	C₄×C₄⋊Q₈	—	79	2	no	—
125	#1969	—	79	4	no	—
126	#1956	—	79	2	no	—
127	#1845	—	79	1	no	—
128	C₄²×C₈	D₇₉	1	1	yes	characteristic
129	C₇₉	C₄×#494	1	1	yes	characteristic, complemented, Sylow
130	C₂²×Q₁₆	—	79	4	no	—
131	C₂×C₄⋊Q₈	—	79	2	no	—
132	C₂×C₄×Q₈	—	79	4	no	—
133	#494	—	79	8	no	—
134	C₄×Q₁₆	—	158	16	no	—
135	C₂×#107	—	79	4	no	—
136	C₂×#103	—	79	8	no	—
137	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
139	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
141	#391	—	79	4	no	—
142	#389	—	158	4	no	—
143	#384	—	79	2	no	—
144	C₄×C₄⋊C₄	—	158	4	no	—
145	C₄³	C₂×D₇₉	1	1	yes	characteristic
146	C₂²×Q₈	—	79	4	no	—
147	C₂×Q₁₆	—	79	16	no	—
148	C₂×Q₁₆	—	158	8	no	—
149	C₂×Q₁₆	—	158	8	no	—
150	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
151	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
152	C₄⋊Q₈	—	79	8	no	—
153	C₄⋊Q₈	—	158	4	no	—
154	C₄×Q₈	—	158	16	no	—
155	C₂×C₄⋊C₄	—	79	4	no	—
156	C₂×C₄⋊C₄	—	158	4	no	—
157	C₂×C₄⋊C₄	—	158	4	no	—
158	C₂×C₄⋊C₄	—	158	4	no	—
159	C₂×C₄²	—	158	4	no	—
160	C₂×C₄²	D₃₁₆	1	2	yes	—
161	C₂×C₄²	D₃₁₆	1	2	yes	—
162	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
163	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
164	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
165	#107	—	158	8	no	—
166	#103	—	158	16	no	—
167	C₄×C₈	—	2	4	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	C₄×D₇₉	1	4	yes	—
170	C₄×C₈	D₃₁₆	1	8	yes	—
171	#95	—	158	8	no	—
172	C₂×Q₈	—	79	8	no	—
173	C₂×Q₈	—	158	8	no	—
174	C₂×Q₈	—	158	8	no	—
175	C₂×Q₈	—	158	4	no	—
176	C₂²×C₄	—	158	4	no	—
177	C₂²×C₄	—	158	4	no	—
178	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
180	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
181	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
182	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
183	Q₁₆	—	158	32	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	C₂×C₄×D₇₉	1	4	yes	—
187	C₂×C₈	C₂×D₃₁₆	1	4	yes	—
188	C₂×C₈	C₂×D₃₁₆	1	4	yes	—
189	C₂×C₈	#11929624	1	4	yes	—
190	C₄⋊C₄	—	316	8	no	—
191	C₄⋊C₄	—	316	8	no	—
192	C₄⋊C₄	—	158	8	no	—
193	C₄⋊C₄	—	158	16	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	4	no	—
196	C₄²	—	316	8	no	—
197	C₄²	Dic₃₁₆	1	8	yes	complemented
198	C₄²	C₂×C₄×D₇₉	1	2	yes	—
199	C₄²	C₂×D₃₁₆	1	4	yes	—
200	C₄²	#11929624	1	2	yes	—
201	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
202	Q₈	—	316	8	no	—
203	Q₈	—	158	16	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	316	4	no	—
208	C₂×C₄	—	316	4	no	—
209	C₂×C₄	—	316	8	no	—
210	C₂×C₄	—	316	4	no	—
211	C₂×C₄	—	158	8	no	—
212	C₂×C₄	C₄×D₃₁₆	1	4	yes	—
213	C₂×C₄	#14579158	1	2	yes	—
214	C₂×C₄	#14579158	1	2	yes	—
215	C₂×C₄	C₂×Dic₃₁₆	1	4	yes	—
216	C₂×C₄	C₂×Dic₃₁₆	1	4	yes	—
217	C₂×C₄	C₂×Dic₃₁₆	1	4	yes	—
218	C₂×C₄	#14579183	1	4	yes	—
219	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
220	C₂×C₄	C₂²×D₃₁₆	1	2	yes	—
221	C₂×C₄	C₂×#11929624	1	2	yes	—
222	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
223	C₈	—	2	4	no	—
224	C₈	C₄×D₃₁₆	1	8	yes	—
225	C₂²	C₂×C₄×D₃₁₆	1	1	yes	characteristic
226	C₂²	C₂×#14579158	1	1	yes	characteristic
227	C₂²	#19099113	1	1	yes	characteristic
228	C₂²	C₂²×Dic₃₁₆	1	2	yes	—
229	C₂²	C₂×#14579183	1	2	yes	—
230	C₄	—	2	4	no	—
231	C₄	—	2	2	no	—
232	C₄	—	316	8	no	—
233	C₄	—	632	4	no	—
234	C₄	C₄×Dic₃₁₆	1	8	yes	complemented
235	C₄	#19098943	1	4	yes	direct product factor
236	C₄	C₂×C₄×D₃₁₆	1	2	yes	—
237	C₄	#19099113	1	2	yes	—
238	C₂	C₄×#14579158	1	1	yes	characteristic
239	C₂	C₂×C₄×Dic₃₁₆	1	2	yes	—
240	C₂	C₂×#19098943	1	1	yes	characteristic
241	C₂	#23773828	1	1	yes	characteristic
242	C₂	#23773986	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₄²×C₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₄²×C₈	D₇₉
hypocenter	C₇₉	C₄×#494
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
11. #19098943	C₄	direct (N × H)
75. C₄²×C₇₉	Q₁₆	semidirect
110. C₄×C₇₉	C₄×Q₁₆	semidirect
111. C₄×C₇₉	#494	semidirect
129. C₇₉	C₄×#494	semidirect
197. C₄²	Dic₃₁₆	semidirect
234. C₄	C₄×Dic₃₁₆	semidirect
235. C₄	#19098943	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×#19098943	2
C₂×C₄×Dic₃₁₆	2
C₄×#14579161	2
#23764261	2
#23764202	2
#23763560	2
C₄²×C₈×C₇₉	2
C₄×#494	79

Subgroups — C4×#19098943 #34296706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19098943 #34296706