Subgroups — C₂×#23765552 #34442898

Order 20224.

574 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×#19098968	C₂	1	1	yes	maximal, characteristic
2	#23765552	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19098388	C₂	1	2	yes	maximal
4	C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
5	C₂×#19098367	C₂	1	2	yes	maximal
6	C₂×C₄×Q₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#14579161	—	2	1	no	—
8	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
9	#19098968	—	2	2	no	—
10	#19098968	C₂²	1	4	yes	—
11	C₂×#14579095	—	2	1	no	—
12	C₂×#14579095	C₂²	1	2	yes	—
13	#19098388	C₂²	1	8	yes	—
14	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	4	yes	—
15	#19098367	C₂²	1	8	yes	—
16	C₂²×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₇₉	—	2	1	no	—
18	C₄×Q₁₆×C₇₉	—	2	2	no	—
19	C₄×Q₁₆×C₇₉	C₂²	1	4	yes	—
20	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×#103×C₇₉	—	2	1	no	—
22	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₅₈	—	4	1	no	—
24	#14579161	—	2	4	no	—
25	#14579161	—	4	2	no	—
26	C₂×C₄⋊Dic₇₉	—	2	2	no	—
27	C₂×C₄⋊Dic₇₉	—	4	1	no	—
28	C₂×C₇₉⋊C₁₆	—	2	4	no	—
29	C₂×C₇₉⋊C₁₆	C₂³	1	4	yes	—
30	#14579095	—	2	2	no	—
31	#14579095	—	2	4	no	—
32	#14579095	C₂³	1	4	yes	—
33	C₂²×Q₈×C₇₉	—	2	1	no	—
34	C₂×Q₁₆×C₇₉	—	2	1	no	—
35	C₂×Q₁₆×C₇₉	—	2	2	no	—
36	C₂×Q₁₆×C₇₉	Q₈	1	4	yes	complemented
37	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
38	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
40	C₄×Q₈×C₇₉	—	2	4	no	—
41	C₄×Q₈×C₇₉	—	4	2	no	—
42	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
43	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
44	C₂×C₄²×C₇₉	—	4	1	no	—
45	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
46	#107×C₇₉	—	2	1	no	—
47	#107×C₇₉	C₂³	1	2	yes	—
48	#103×C₇₉	—	2	4	no	—
49	C₄×C₈×C₇₉	D₄	1	2	yes	—
50	C₄×C₈×C₇₉	C₂³	1	2	yes	—
51	C₂×Dic₁₅₈	—	4	4	no	—
52	C₂×Dic₁₅₈	—	8	4	no	—
53	C₂²×Dic₇₉	—	4	2	no	—
54	C₄⋊Dic₇₉	—	2	4	no	—
55	C₄⋊Dic₇₉	—	4	2	no	—
56	C₄⋊Dic₇₉	—	4	4	no	—
57	C₇₉⋊C₁₆	—	2	4	no	—
58	C₂×Q₈×C₇₉	—	2	4	no	—
59	C₂×Q₈×C₇₉	—	2	2	no	—
60	C₂×Q₈×C₇₉	—	4	1	no	—
61	C₂×Q₈×C₇₉	—	4	2	no	—
62	C₂²×C₄×C₇₉	—	4	1	no	—
63	C₂²×C₄×C₇₉	—	4	1	no	—
64	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₇₉	—	2	4	no	—
68	Q₁₆×C₇₉	—	2	2	no	—
69	Q₁₆×C₇₉	C₂×Q₈	1	4	yes	complemented
70	C₂×C₈×C₇₉	—	2	2	no	—
71	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₇₉	#41	1	2	yes	—
75	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
76	C₄⋊C₄×C₇₉	—	2	2	no	—
77	C₄⋊C₄×C₇₉	—	4	1	no	—
78	C₄⋊C₄×C₇₉	—	4	4	no	—
79	C₄²×C₇₉	—	4	4	no	—
80	C₄²×C₇₉	D₈	1	2	yes	—
81	C₄²×C₇₉	C₂×D₄	1	2	yes	—
82	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
83	Q₈×C₇₉	—	2	4	no	—
84	Q₈×C₇₉	—	4	2	no	—
85	Q₈×C₇₉	—	4	4	no	—
86	C₂×C₄×C₇₉	—	2	2	no	—
87	C₂×C₄×C₇₉	—	4	2	no	—
88	C₂×C₄×C₇₉	—	4	4	no	—
89	C₂×C₄×C₇₉	—	4	2	no	—
90	C₂×C₄×C₇₉	—	8	2	no	—
91	C₂×C₄×C₇₉	Q₃₂	1	4	yes	—
92	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
93	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₇₉	#137	1	2	yes	—
97	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₇₉	—	4	4	no	—
99	C₂×Dic₇₉	—	8	4	no	—
100	Dic₁₅₈	—	8	8	no	—
101	C₈×C₇₉	—	2	2	no	—
102	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
103	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
104	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
105	C₂²×C₇₉	#474	1	2	yes	—
106	C₂²×C₇₉	C₂×Q₃₂	1	2	yes	—
107	C₂²×C₇₉	#509	1	2	yes	—
108	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
109	C₄×C₇₉	—	2	2	no	—
110	C₄×C₇₉	—	4	4	no	—
111	C₄×C₇₉	—	8	2	no	—
112	C₄×C₇₉	#474	1	2	yes	—
113	C₄×C₇₉	C₂×Q₃₂	1	4	yes	—
114	C₄×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
115	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
116	Dic₇₉	—	8	4	no	—
117	C₂×#2227	—	79	1	no	maximal, Sylow
118	C₂×C₇₉	#2227	1	4	yes	complemented
119	C₂×C₇₉	C₂×#474	1	1	yes	characteristic
120	C₂×C₇₉	C₂²×Q₃₂	1	1	yes	characteristic
121	C₂×C₇₉	C₂×#509	1	1	yes	characteristic
122	C₂×#500	—	79	1	no	—
123	C₂×C₄×Q₁₆	D₇₉	1	1	yes	characteristic
124	#2227	—	79	8	no	—
125	C₂×#366	—	79	2	no	—
126	C₂×C₄⋊C₁₆	—	79	1	no	—
127	C₂×#358	—	79	2	no	—
128	C₇₉	C₂×#2227	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
130	C₂×C₄⋊Q₈	—	158	1	no	—
131	C₂×C₄×Q₈	—	2	1	no	—
132	C₂²×C₁₆	—	79	2	no	—
133	#500	—	79	4	no	—
134	#500	—	158	2	no	—
135	C₄×Q₁₆	—	2	2	no	—
136	C₄×Q₁₆	C₂×D₇₉	1	4	yes	—
137	C₂×#107	—	79	2	no	—
138	C₂×#107	—	158	1	no	—
139	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
140	C₂×#103	—	2	1	no	—
141	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
142	#366	—	79	8	no	—
143	C₄⋊C₁₆	—	79	4	no	—
144	#358	—	79	8	no	—
145	C₂²×Q₈	—	2	1	no	—
146	C₂²×Q₈	—	316	1	no	—
147	C₂×Q₁₆	—	2	1	no	—
148	C₂×Q₁₆	—	2	2	no	—
149	C₂×Q₁₆	Dic₁₅₈	1	4	yes	complemented
150	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
151	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
153	C₄⋊Q₈	—	316	2	no	—
154	C₄⋊Q₈	—	158	4	no	—
155	C₄×Q₈	—	2	4	no	—
156	C₄×Q₈	—	4	2	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	—	4	1	no	—
159	C₂×C₄⋊C₄	—	316	1	no	—
160	C₂×C₄⋊C₄	—	158	2	no	—
161	C₂×C₄²	—	4	1	no	—
162	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
163	C₂×C₁₆	—	79	4	no	—
164	C₂×C₁₆	—	158	4	no	—
165	#107	—	2	1	no	—
166	#107	—	79	4	no	—
167	#107	—	158	2	no	—
168	#107	—	158	4	no	—
169	#107	C₂²×D₇₉	1	2	yes	—
170	#103	—	2	4	no	—
171	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
172	C₄×C₈	C₂²×D₇₉	1	2	yes	—
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	—	4	1	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	—	316	4	no	—
178	C₂×Q₈	—	632	4	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	—	316	2	no	—
182	C₂²×C₄	#11929609	1	1	yes	characteristic
183	C₂²×C₄	#11929609	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
185	Q₁₆	—	2	2	no	—
186	Q₁₆	—	2	4	no	—
187	Q₁₆	C₂×Dic₁₅₈	1	4	yes	complemented
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
190	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
191	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
192	C₂×C₈	#11929624	1	2	yes	—
193	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	4	1	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	316	4	no	—
198	C₄⋊C₄	—	316	2	no	—
199	C₄⋊C₄	—	158	4	no	—
200	C₄²	—	4	4	no	—
201	C₄²	#11929609	1	2	yes	—
202	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
203	C₁₆	—	158	4	no	—
204	C₂³	C₂×#11929609	1	1	yes	characteristic
205	Q₈	—	2	4	no	—
206	Q₈	—	4	4	no	—
207	Q₈	—	4	2	no	—
208	Q₈	—	632	8	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	316	4	no	—
215	C₂×C₄	—	632	4	no	—
216	C₂×C₄	#14579107	1	4	yes	—
217	C₂×C₄	#14579140	1	2	yes	—
218	C₂×C₄	C₂×#11929609	1	2	yes	—
219	C₂×C₄	C₂×#11929609	1	2	yes	—
220	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
221	C₂×C₄	#14579191	1	2	yes	—
222	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	#14579140	1	2	yes	—
225	C₈	C₂²×Dic₁₅₈	1	1	yes	characteristic
226	C₈	C₂×#11929624	1	1	yes	characteristic
227	C₂²	#19098843	1	2	yes	—
228	C₂²	C₂×#14579107	1	2	yes	—
229	C₂²	#19098998	1	2	yes	—
230	C₂²	C₂²×#11929609	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	4	4	no	—
233	C₄	—	8	2	no	—
234	C₄	—	632	4	no	—
235	C₄	#19098843	1	2	yes	—
236	C₄	C₂×#14579107	1	4	yes	—
237	C₄	C₂×#14579140	1	1	yes	characteristic
238	C₄	C₂×#14579191	1	1	yes	characteristic
239	C₂	#23765552	1	4	yes	direct product factor
240	C₂	C₂×#19098843	1	1	yes	characteristic
241	C₂	C₂²×#14579107	1	1	yes	characteristic
242	C₂	C₂×#19098998	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×C₄×Q₁₆×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
hypercenter	C₂×C₄×Q₁₆	D₇₉
hypocenter	C₇₉	C₂×#2227
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #23765552	C₂	direct (N × H)
36. C₂×Q₁₆×C₇₉	Q₈	semidirect
69. Q₁₆×C₇₉	C₂×Q₈	semidirect
118. C₂×C₇₉	#2227	semidirect
128. C₇₉	C₂×#2227	semidirect
149. C₂×Q₁₆	Dic₁₅₈	semidirect
187. Q₁₆	C₂×Dic₁₅₈	semidirect
239. C₂	#23765552	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×#19098968	2
#23765552	2
C₂×#19098388	2
C₂×C₄⋊(C₇₉⋊C₁₆)	2
C₂×#19098367	2
C₂×C₄×Q₁₆×C₇₉	2
C₂×#2227	79

Subgroups — C2×#23765552 #34442898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765552 #34442898