Subgroups — C₂×#23765205 #34441342

Order 20224.

610 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098517	C₂	1	1	yes	maximal, characteristic
2	#23765205	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₇₉⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×(C₂×D₇₉)⋊C₁₆	C₂	1	2	yes	maximal
5	C₂×#19098339	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11929607	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₇₉	C₂²	1	1	yes	characteristic
9	C₂×#14579118	C₄	1	1	yes	characteristic
10	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
11	#19098517	C₄	1	4	yes	—
12	#19098517	C₂²	1	4	yes	—
13	C₂×#14579072	C₄	1	1	yes	characteristic
14	C₂×#14579071	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
16	Dic₇₉⋊C₁₆	C₂²	1	8	yes	complemented
17	(C₂×D₇₉)⋊C₁₆	C₂²	1	8	yes	—
18	#19098339	C₂²	1	4	yes	complemented
19	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₄×C₁₆×C₇₉	C₂²	1	4	yes	complemented
22	C₂²×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂×#11929607	—	2	4	no	—
24	C₂×#11929607	C₂×C₄	1	4	yes	—
25	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₇₉	—	2	4	no	—
28	C₂×C₈×D₇₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×D₇₉	C₂³	1	2	yes	—
30	#14579118	C₈	1	4	yes	—
31	#14579118	C₂×C₄	1	4	yes	—
32	C₂×Dic₇₉⋊C₄	C₈	1	2	yes	—
33	C₂×(C₂×D₇₉)⋊C₄	C₈	1	2	yes	—
34	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₇₉⋊C₁₆	—	2	4	no	—
36	C₂×C₇₉⋊C₁₆	C₂³	1	4	yes	—
37	#14579072	C₂×C₄	1	4	yes	—
38	#14579071	C₂×C₄	1	4	yes	—
39	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
40	C₈×Dic₇₉	C₂³	1	2	yes	—
41	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₆×C₇₉	—	2	4	no	—
45	C₂×C₁₆×C₇₉	C₂³	1	4	yes	—
46	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₇₉	C₂³	1	2	yes	—
48	C₂³×D₇₉	C₂×C₈	1	1	yes	characteristic
49	C₂²×Dic₇₉	C₂×C₈	1	2	yes	—
50	C₂²×Dic₇₉	C₂×C₈	1	1	yes	characteristic
51	C₂×C₄×D₇₉	—	2	2	no	—
52	C₂×C₄×D₇₉	—	2	2	no	—
53	C₂×C₄×D₇₉	C₂×C₈	1	2	yes	—
54	C₂×C₄×D₇₉	C₂²×C₄	1	2	yes	—
55	#11929607	—	2	8	no	—
56	C₂×C₇₉⋊C₈	—	2	2	no	—
57	C₂×C₇₉⋊C₈	—	2	2	no	—
58	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
59	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
60	C₈×D₇₉	—	2	4	no	—
61	C₈×D₇₉	—	2	4	no	—
62	Dic₇₉⋊C₄	C₂×C₈	1	8	yes	—
63	(C₂×D₇₉)⋊C₄	C₂×C₈	1	8	yes	—
64	C₄×Dic₇₉	C₂×C₈	1	2	yes	—
65	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
66	C₇₉⋊C₁₆	—	2	4	no	—
67	C₂²×C₄×C₇₉	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₉	M₁₆	1	4	yes	—
70	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₇₉	#41	1	4	yes	—
73	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
74	C₄²×C₇₉	C₂×C₈	1	2	yes	—
75	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
76	C₁₆×C₇₉	—	2	4	no	—
77	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
78	C₂²×D₇₉	—	2	2	no	—
79	C₂²×D₇₉	—	2	4	no	—
80	C₂²×D₇₉	C₂²×C₈	1	2	yes	—
81	C₂×C₄×C₇₉	—	2	4	no	—
82	C₂×C₄×C₇₉	#117	1	2	yes	—
83	C₂×C₄×C₇₉	C₂²×C₈	1	4	yes	—
84	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
87	C₂×Dic₇₉	—	2	2	no	—
88	C₂×Dic₇₉	—	2	4	no	—
89	C₂×Dic₇₉	C₂²×C₈	1	4	yes	—
90	C₂×Dic₇₉	C₂²×C₈	1	2	yes	—
91	C₄×D₇₉	—	2	2	no	—
92	C₄×D₇₉	—	2	2	no	—
93	C₄×D₇₉	—	2	4	no	—
94	C₈×C₇₉	#117	1	2	yes	—
95	C₈×C₇₉	C₂×M₁₆	1	4	yes	—
96	C₈×C₇₉	C₂×#41	1	2	yes	—
97	C₇₉⋊C₈	—	2	2	no	—
98	C₇₉⋊C₈	—	2	2	no	—
99	C₂²×C₇₉	#431	1	2	yes	—
100	C₂²×C₇₉	#504	1	4	yes	—
101	C₂²×C₇₉	C₂³×C₈	1	1	yes	characteristic
102	C₂×D₇₉	—	2	4	no	—
103	C₂×D₇₉	—	2	2	no	—
104	C₂×D₇₉	—	4	4	no	—
105	C₄×C₇₉	—	2	4	no	—
106	C₄×C₇₉	#431	1	2	yes	—
107	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
108	C₄×C₇₉	C₂²×M₁₆	1	1	yes	characteristic
109	Dic₇₉	—	2	4	no	—
110	Dic₇₉	—	2	2	no	—
111	C₂×#2164	—	79	1	no	maximal, Sylow
112	C₂×C₇₉	#2164	1	4	yes	complemented
113	C₂×C₇₉	C₂×#431	1	1	yes	characteristic
114	C₂×C₇₉	C₂×#504	1	2	yes	—
115	D₇₉	—	4	2	no	—
116	C₂×#405	—	79	1	no	—
117	#2164	—	79	8	no	—
118	C₂×C₄⋊C₁₆	—	79	2	no	—
119	C₂×C₂²⋊C₁₆	—	79	2	no	—
120	C₂×#346	—	79	1	no	—
121	C₂×C₄×C₁₆	D₇₉	1	1	yes	characteristic, complemented
122	C₇₉	C₂×#2164	1	1	yes	characteristic, complemented, Sylow
123	C₂²×M₁₆	—	79	1	no	—
124	C₂³×C₈	—	79	1	no	—
125	C₂×#117	—	79	1	no	—
126	C₂²×C₁₆	—	79	2	no	—
127	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
128	#405	—	79	4	no	—
129	#405	—	79	4	no	—
130	C₂×#97	—	79	1	no	—
131	C₂×#97	—	79	1	no	—
132	C₂×C₄×C₈	—	79	1	no	—
133	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
134	C₄⋊C₁₆	—	79	8	no	—
135	C₂²⋊C₁₆	—	79	8	no	—
136	#346	—	79	4	no	—
137	C₄×C₁₆	C₂×D₇₉	1	4	yes	complemented
138	C₂³×C₄	—	79	1	no	—
139	C₂×M₁₆	—	79	4	no	—
140	C₂×M₁₆	—	158	4	no	—
141	C₂²×C₈	—	79	1	no	—
142	C₂²×C₈	—	79	1	no	—
143	C₂²×C₈	—	79	2	no	—
144	C₂²×C₈	—	79	2	no	—
145	C₂²×C₈	—	158	4	no	—
146	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
147	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
148	#117	—	79	4	no	—
149	#117	—	79	4	no	—
150	C₂×C₄⋊C₄	—	79	2	no	—
151	C₂×C₂²⋊C₄	—	79	2	no	—
152	C₂×C₄²	—	79	1	no	—
153	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
154	C₂×C₁₆	—	2	4	no	—
155	C₂×C₁₆	—	79	4	no	—
156	C₂×C₁₆	—	158	4	no	—
157	C₂×C₁₆	C₂²×D₇₉	1	4	yes	—
158	#97	—	79	4	no	—
159	#97	—	79	4	no	—
160	C₄×C₈	—	79	2	no	—
161	C₄×C₈	—	79	2	no	—
162	C₄×C₈	C₄×D₇₉	1	2	yes	—
163	C₄×C₈	C₂²×D₇₉	1	2	yes	—
164	C₂⁴	—	79	1	no	—
165	C₂²×C₄	—	79	2	no	—
166	C₂²×C₄	—	79	1	no	—
167	C₂²×C₄	—	79	2	no	—
168	C₂²×C₄	—	79	2	no	—
169	C₂²×C₄	—	158	2	no	—
170	C₂²×C₄	—	158	2	no	—
171	C₂²×C₄	C₈×D₇₉	1	2	yes	—
172	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
173	M₁₆	—	158	8	no	—
174	C₂×C₈	—	79	2	no	—
175	C₂×C₈	—	79	2	no	—
176	C₂×C₈	—	158	2	no	—
177	C₂×C₈	—	158	2	no	—
178	C₂×C₈	—	158	4	no	—
179	C₂×C₈	—	158	4	no	—
180	C₂×C₈	#11929607	1	4	yes	—
181	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
182	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
183	C₂×C₈	#11929624	1	4	yes	—
184	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
185	C₄⋊C₄	—	79	8	no	—
186	C₂²⋊C₄	—	79	8	no	—
187	C₄²	—	79	2	no	—
188	C₄²	—	79	2	no	—
189	C₄²	C₈×D₇₉	1	2	yes	—
190	C₄²	C₂×C₄×D₇₉	1	2	yes	—
191	C₁₆	—	2	4	no	—
192	C₁₆	—	158	4	no	—
193	C₂³	—	79	2	no	—
194	C₂³	—	158	4	no	—
195	C₂³	—	158	2	no	—
196	C₂³	C₂×C₈×D₇₉	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	79	4	no	—
199	C₂×C₄	—	79	2	no	—
200	C₂×C₄	—	158	4	no	—
201	C₂×C₄	—	158	2	no	—
202	C₂×C₄	—	158	2	no	—
203	C₂×C₄	—	158	2	no	—
204	C₂×C₄	—	158	4	no	—
205	C₂×C₄	#14579118	1	2	yes	—
206	C₂×C₄	C₂×C₈×D₇₉	1	4	yes	—
207	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
208	C₂×C₄	C₂×#11929607	1	2	yes	—
209	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
210	C₈	—	158	2	no	—
211	C₈	—	158	2	no	—
212	C₈	#14579118	1	2	yes	—
213	C₈	C₂×#11929607	1	4	yes	—
214	C₈	C₂×#11929624	1	2	yes	—
215	C₂²	—	316	4	no	—
216	C₂²	—	158	2	no	—
217	C₂²	—	158	4	no	—
218	C₂²	#19098614	1	2	yes	—
219	C₂²	#19098979	1	4	yes	—
220	C₂²	C₂²×C₈×D₇₉	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	158	2	no	—
223	C₄	—	158	4	no	—
224	C₄	#19098614	1	2	yes	—
225	C₄	C₂×#14579118	1	1	yes	characteristic
226	C₄	C₂²×#11929607	1	1	yes	characteristic
227	C₂	—	316	2	no	—
228	C₂	#23765205	1	4	yes	direct product factor
229	C₂	C₂×#19098614	1	1	yes	characteristic
230	C₂	C₂×#19098979	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×C₄×C₁₆×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×C₈
prime radical	C₂³×D₇₉	C₂×C₈
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₇₉
hypocenter	C₇₉	C₂×#2164
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #23765205	C₂	direct (N × H)
3. C₂×Dic₇₉⋊C₁₆	C₂	semidirect
5. C₂×#19098339	C₂	semidirect
6. C₂×C₄×C₁₆×C₇₉	C₂	semidirect
16. Dic₇₉⋊C₁₆	C₂²	semidirect
18. #19098339	C₂²	semidirect
21. C₄×C₁₆×C₇₉	C₂²	semidirect
112. C₂×C₇₉	#2164	semidirect
121. C₂×C₄×C₁₆	D₇₉	semidirect
122. C₇₉	C₂×#2164	semidirect
137. C₄×C₁₆	C₂×D₇₉	semidirect
228. C₂	#23765205	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×#19098517	2
#23765205	2
C₂×Dic₇₉⋊C₁₆	2
C₂×(C₂×D₇₉)⋊C₁₆	2
C₂×#19098339	2
C₂×C₄×C₁₆×C₇₉	2
C₂×#2164	79

Subgroups — C2×#23765205 #34441342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765205 #34441342