Subgroups — C₄×#19098312 #34179213

Order 20224.

540 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₇₉⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#19098312	C₂	1	2	yes	maximal
3	#23761283	C₂	1	2	yes	maximal
4	C₄×C₄⋊Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
6	C₂×C₄×C₇₉⋊C₈	—	2	2	no	—
7	#19098318	—	2	2	no	—
8	#19098312	C₄	1	8	yes	direct product factor
9	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₇₉	C₄	1	2	yes	—
11	#391×C₇₉	C₄	1	2	yes	—
12	#389×C₇₉	—	2	2	no	—
13	#384×C₇₉	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₇₉	—	2	2	no	—
15	C₄³×C₇₉	C₂²	1	1	yes	characteristic
16	C₂²×C₇₉⋊C₈	—	2	4	no	—
17	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
18	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	8	yes	—
19	C₄×C₇₉⋊C₈	—	4	4	no	—
20	C₂²×Q₈×C₇₉	C₂×C₄	1	2	yes	—
21	C₄⋊Q₈×C₇₉	—	2	2	no	—
22	C₄⋊Q₈×C₇₉	C₂×C₄	1	4	yes	—
23	C₄×Q₈×C₇₉	—	4	4	no	—
24	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₇₉	—	2	2	no	—
29	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₉	D₄	1	2	yes	—
32	C₂×C₄²×C₇₉	D₄	1	2	yes	—
33	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
34	#95×C₇₉	—	2	4	no	—
35	C₂×C₇₉⋊C₈	—	2	8	no	—
36	C₂×C₇₉⋊C₈	—	4	4	no	—
37	C₂×C₇₉⋊C₈	—	4	4	no	—
38	C₂×Q₈×C₇₉	—	2	2	no	—
39	C₂×Q₈×C₇₉	—	4	2	no	—
40	C₂×Q₈×C₇₉	—	4	2	no	—
41	C₂×Q₈×C₇₉	C₄²	1	4	yes	—
42	C₂²×C₄×C₇₉	—	2	2	no	—
43	C₂²×C₄×C₇₉	—	2	2	no	—
44	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
49	C₄⋊C₄×C₇₉	—	2	8	no	—
50	C₄⋊C₄×C₇₉	—	4	4	no	—
51	C₄⋊C₄×C₇₉	—	4	2	no	—
52	C₄⋊C₄×C₇₉	—	4	4	no	—
53	C₄²×C₇₉	—	2	2	no	—
54	C₄²×C₇₉	—	2	4	no	—
55	C₄²×C₇₉	—	4	1	no	—
56	C₄²×C₇₉	—	4	4	no	—
57	C₄²×C₇₉	SD₁₆	1	4	yes	—
58	C₄²×C₇₉	Q₁₆	1	4	yes	—
59	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
60	C₄²×C₇₉	#41	1	2	yes	—
61	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	Q₈×C₇₉	—	4	4	no	—
63	Q₈×C₇₉	—	4	4	no	—
64	C₂×C₄×C₇₉	—	2	1	no	—
65	C₂×C₄×C₇₉	—	2	2	no	—
66	C₂×C₄×C₇₉	—	2	1	no	—
67	C₂×C₄×C₇₉	—	2	2	no	—
68	C₂×C₄×C₇₉	—	2	4	no	—
69	C₂×C₄×C₇₉	—	2	1	no	—
70	C₂×C₄×C₇₉	—	2	2	no	—
71	C₂×C₄×C₇₉	—	4	2	no	—
72	C₂×C₄×C₇₉	—	4	2	no	—
73	C₂×C₄×C₇₉	—	4	2	no	—
74	C₂×C₄×C₇₉	—	4	4	no	—
75	C₂×C₄×C₇₉	#100	1	2	yes	—
76	C₂×C₄×C₇₉	#103	1	4	yes	—
77	C₂×C₄×C₇₉	#103	1	4	yes	—
78	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
80	C₂×C₄×C₇₉	C₄×D₄	1	4	yes	—
81	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₇₉	#135	1	4	yes	—
84	C₇₉⋊C₈	—	4	8	no	—
85	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₇₉	C₂×#100	1	1	yes	characteristic
87	C₂²×C₇₉	#413	1	1	yes	characteristic
88	C₂²×C₇₉	C₂×#103	1	2	yes	—
89	C₂²×C₇₉	#416	1	2	yes	—
90	C₄×C₇₉	—	2	4	no	—
91	C₄×C₇₉	—	2	1	no	—
92	C₄×C₇₉	—	2	1	no	—
93	C₄×C₇₉	—	4	1	no	—
94	C₄×C₇₉	—	4	4	no	—
95	C₄×C₇₉	—	8	2	no	—
96	C₄×C₇₉	#333	1	4	yes	complemented
97	C₄×C₇₉	C₄×SD₁₆	1	4	yes	—
98	C₄×C₇₉	C₄×Q₁₆	1	4	yes	—
99	C₄×#333	—	79	1	no	maximal, Sylow
100	C₂×C₇₉	C₂×#333	1	1	yes	characteristic
101	C₂×C₇₉	#1542	1	2	yes	—
102	C₂×C₇₉	#1543	1	1	yes	characteristic
103	C₂×C₇₉	C₄×#100	1	1	yes	characteristic
104	C₂×C₇₉	C₄×#103	1	2	yes	—
105	C₄×C₄⋊Q₈	D₇₉	1	1	yes	characteristic
106	C₄×C₄⋊C₈	—	79	2	no	—
107	C₂×#333	—	79	2	no	—
108	#1296	—	79	2	no	—
109	C₇₉	C₄×#333	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
111	C₂×C₄×Q₈	Dic₇₉	1	2	yes	—
112	C₂×C₄⋊C₈	—	79	4	no	—
113	C₂×C₄×C₈	—	158	2	no	—
114	#391	Dic₇₉	1	2	yes	—
115	#389	—	2	2	no	—
116	#384	C₂×D₇₉	1	1	yes	characteristic
117	C₄×C₄⋊C₄	—	2	2	no	—
118	C₄³	C₂×D₇₉	1	1	yes	characteristic
119	#336	—	158	2	no	—
120	#333	—	79	8	no	—
121	C₂²×Q₈	C₂×Dic₇₉	1	2	yes	—
122	C₂²×C₈	—	158	4	no	—
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	C₄×D₇₉	1	4	yes	—
125	C₄×Q₈	—	4	4	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
130	C₂×C₄²	—	2	2	no	—
131	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
132	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
133	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
134	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
135	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
136	C₄⋊C₈	—	79	8	no	—
137	C₄⋊C₈	—	158	4	no	—
138	C₄×C₈	—	316	4	no	—
139	#95	—	2	4	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	4	2	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	C₄×Dic₇₉	1	4	yes	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
147	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
148	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
149	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
151	C₂×C₈	—	316	4	no	—
152	C₂×C₈	—	316	4	no	—
153	C₂×C₈	—	158	8	no	—
154	C₄⋊C₄	—	2	8	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	2	4	no	—
160	C₄²	—	4	1	no	—
161	C₄²	—	4	4	no	—
162	C₄²	#11929611	1	4	yes	—
163	C₄²	#11929614	1	4	yes	—
164	C₄²	C₂×C₄×D₇₉	1	2	yes	—
165	C₄²	#11929624	1	2	yes	—
166	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
167	Q₈	—	4	4	no	—
168	Q₈	—	4	4	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	#14579078	1	2	yes	—
181	C₂×C₄	#14579085	1	4	yes	—
182	C₂×C₄	#14579085	1	4	yes	—
183	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
184	C₂×C₄	#14579120	1	1	yes	characteristic
185	C₂×C₄	C₄×C₂²⋊D₇₉	1	4	yes	—
186	C₂×C₄	C₂×#11929611	1	2	yes	—
187	C₂×C₄	C₂×#11929614	1	2	yes	—
188	C₂×C₄	#14579181	1	4	yes	—
189	C₈	—	316	8	no	—
190	C₂²	C₄×C₂²⋊Dic₇₉	1	1	yes	characteristic
191	C₂²	C₂×#14579078	1	1	yes	characteristic
192	C₂²	#19098547	1	1	yes	characteristic
193	C₂²	C₂×#14579085	1	2	yes	—
194	C₂²	#19098559	1	2	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	4	1	no	—
199	C₄	—	4	4	no	—
200	C₄	—	8	2	no	—
201	C₄	#19098312	1	4	yes	direct product factor
202	C₄	C₄×#11929611	1	4	yes	—
203	C₄	C₄×#11929614	1	4	yes	—
204	C₂	C₂×#19098312	1	1	yes	characteristic
205	C₂	#23762057	1	2	yes	—
206	C₂	#23762062	1	1	yes	characteristic
207	C₂	C₄×#14579078	1	1	yes	characteristic
208	C₂	C₄×#14579085	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₄×C₄⋊Q₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×C₄⋊Q₈	D₇₉
hypocenter	C₇₉	C₄×#333
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
8. #19098312	C₄	direct (N × H)
96. C₄×C₇₉	#333	semidirect
109. C₇₉	C₄×#333	semidirect
201. C₄	#19098312	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₄⋊(C₇₉⋊C₈)	2
C₂×#19098312	2
#23761283	2
C₄×C₄⋊Q₈×C₇₉	2
C₄×#333	79

Subgroups — C4×#19098312 #34179213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19098312 #34179213