Subgroups — C₂×#23765120 #34441161

Order 20224.

1150 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579101	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098601	C₂	1	1	yes	maximal, characteristic
3	#23765120	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098367	C₂	1	4	yes	maximal, complemented
5	C₂³×Q₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579120	—	2	1	no	—
7	C₂×#14579101	—	2	4	no	—
8	C₂×#14579101	C₂²	1	4	yes	—
9	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
10	#19098601	—	2	2	no	—
11	#19098601	C₂²	1	4	yes	—
12	C₂×#14579095	C₂²	1	2	yes	—
13	C₂×#14579093	—	2	1	no	—
14	#19098367	C₂²	1	16	yes	complemented
15	C₂³×Q₈×C₇₉	—	2	1	no	—
16	C₂²×Q₁₆×C₇₉	—	2	1	no	—
17	C₂²×Q₁₆×C₇₉	—	2	2	no	—
18	C₂²×Q₁₆×C₇₉	—	2	4	no	—
19	C₂²×Q₁₆×C₇₉	C₄	1	8	yes	complemented
20	C₂²×Q₁₆×C₇₉	C₂²	1	2	yes	—
21	C₂²×Q₁₆×C₇₉	C₂²	1	4	yes	—
22	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
23	#14579120	—	2	4	no	—
24	#14579120	—	4	2	no	—
25	C₂×C₄⋊Dic₇₉	—	2	2	no	—
26	C₂×C₂²⋊Dic₇₉	—	4	1	no	—
27	C₂×C₄×Dic₇₉	—	4	1	no	—
28	#14579101	—	2	8	no	—
29	C₂×C₇₉⋊C₁₆	—	2	4	no	—
30	C₂×C₇₉⋊C₁₆	C₂³	1	4	yes	—
31	#14579095	—	2	2	no	—
32	#14579095	C₂³	1	4	yes	—
33	#14579093	—	2	4	no	—
34	C₂²×Q₈×C₇₉	—	2	8	no	—
35	C₂²×Q₈×C₇₉	—	2	2	no	—
36	C₂²×Q₈×C₇₉	—	2	4	no	—
37	C₂²×Q₈×C₇₉	—	4	1	no	—
38	C₂²×Q₈×C₇₉	—	4	2	no	—
39	C₂²×Q₈×C₇₉	—	4	4	no	—
40	C₂³×C₄×C₇₉	—	4	1	no	—
41	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
42	C₂×Q₁₆×C₇₉	—	2	16	no	—
43	C₂×Q₁₆×C₇₉	—	2	8	no	—
44	C₂×Q₁₆×C₇₉	—	2	4	no	—
45	C₂×Q₁₆×C₇₉	—	2	8	no	—
46	C₂×Q₁₆×C₇₉	—	2	4	no	—
47	C₂×Q₁₆×C₇₉	—	2	2	no	—
48	C₂×Q₁₆×C₇₉	C₂×C₄	1	8	yes	—
49	C₂×Q₁₆×C₇₉	C₂×C₄	1	16	yes	complemented
50	C₂×Q₁₆×C₇₉	C₂³	1	4	yes	—
51	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₇₉	D₄	1	2	yes	—
54	C₂²×C₈×C₇₉	D₄	1	4	yes	—
55	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
57	C₂²×Dic₇₉	—	4	2	no	—
58	C₄⋊Dic₇₉	—	2	4	no	—
59	C₄⋊Dic₇₉	—	4	2	no	—
60	C₂²⋊Dic₇₉	—	4	4	no	—
61	C₄×Dic₇₉	—	4	4	no	—
62	C₇₉⋊C₁₆	—	2	4	no	—
63	C₂⁴×C₇₉	D₈	1	1	yes	characteristic
64	C₂×Q₈×C₇₉	—	2	8	no	—
65	C₂×Q₈×C₇₉	—	2	16	no	—
66	C₂×Q₈×C₇₉	—	2	4	no	—
67	C₂×Q₈×C₇₉	—	4	16	no	—
68	C₂×Q₈×C₇₉	—	4	8	no	—
69	C₂×Q₈×C₇₉	—	4	4	no	—
70	C₂×Q₈×C₇₉	—	4	2	no	—
71	C₂×Q₈×C₇₉	—	4	8	no	—
72	C₂×Q₈×C₇₉	—	4	4	no	—
73	C₂²×C₄×C₇₉	—	4	8	no	—
74	C₂²×C₄×C₇₉	—	4	4	no	—
75	C₂²×C₄×C₇₉	—	4	2	no	—
76	C₂²×C₄×C₇₉	C₂²⋊C₄	1	4	yes	—
77	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	D₈	1	2	yes	—
79	C₂²×C₄×C₇₉	SD₁₆	1	4	yes	—
80	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
82	Q₁₆×C₇₉	—	2	8	no	—
83	Q₁₆×C₇₉	—	2	16	no	—
84	Q₁₆×C₇₉	—	2	4	no	—
85	Q₁₆×C₇₉	C₂²×C₄	1	8	yes	—
86	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
87	C₂×C₈×C₇₉	C₂²⋊C₄	1	8	yes	—
88	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
95	C₂³×C₇₉	—	2	4	no	—
96	C₂³×C₇₉	#102	1	4	yes	—
97	C₂³×C₇₉	C₂×D₈	1	2	yes	—
98	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
99	Q₈×C₇₉	—	2	8	no	—
100	Q₈×C₇₉	—	4	4	no	—
101	Q₈×C₇₉	—	4	8	no	—
102	Q₈×C₇₉	—	4	16	no	—
103	C₂×C₄×C₇₉	—	4	4	no	—
104	C₂×C₄×C₇₉	—	4	16	no	—
105	C₂×C₄×C₇₉	—	4	8	no	—
106	C₂×C₄×C₇₉	#102	1	8	yes	—
107	C₂×C₄×C₇₉	#102	1	4	yes	—
108	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₇₉	C₂×SD₁₆	1	4	yes	—
113	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
114	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
115	C₂×Dic₇₉	—	4	4	no	—
116	C₂×Dic₇₉	—	8	4	no	—
117	C₈×C₇₉	C₂×C₂²⋊C₄	1	2	yes	—
118	C₈×C₇₉	C₂×C₂²⋊C₄	1	4	yes	—
119	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
120	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
121	C₂²×C₇₉	—	2	4	no	—
122	C₂²×C₇₉	—	2	8	no	—
123	C₂²×C₇₉	C₂×#102	1	4	yes	—
124	C₂²×C₇₉	C₂×#102	1	2	yes	—
125	C₂²×C₇₉	#510	1	4	yes	—
126	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
127	C₄×C₇₉	—	4	8	no	—
128	C₄×C₇₉	C₂×#102	1	4	yes	—
129	C₄×C₇₉	C₂×#102	1	2	yes	—
130	C₄×C₇₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
131	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
132	Dic₇₉	—	8	4	no	—
133	C₂×#2141	—	79	1	no	maximal, Sylow
134	C₂×C₇₉	—	2	4	no	—
135	C₂×C₇₉	#2141	1	4	yes	complemented
136	C₂×C₇₉	C₂²×#102	1	1	yes	characteristic
137	C₂×C₇₉	C₂×#510	1	2	yes	—
138	C₂³×Q₁₆	D₇₉	1	1	yes	characteristic
139	C₂²×M₃₂	—	79	1	no	—
140	C₂×#427	—	79	1	no	—
141	#2141	—	79	8	no	—
142	C₂×#358	—	79	4	no	—
143	C₇₉	C₂×#2141	1	1	yes	characteristic, complemented, Sylow
144	C₂³×Q₈	—	2	1	no	—
145	C₂²×Q₁₆	—	2	1	no	—
146	C₂²×Q₁₆	—	2	2	no	—
147	C₂²×Q₁₆	—	2	4	no	—
148	C₂²×Q₁₆	Dic₇₉	1	8	yes	complemented
149	C₂²×Q₁₆	C₂×D₇₉	1	2	yes	—
150	C₂²×Q₁₆	C₂×D₇₉	1	4	yes	—
151	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
152	C₂×#117	—	158	1	no	—
153	C₂×M₃₂	—	79	4	no	—
154	C₂×M₃₂	—	158	4	no	—
155	C₂²×C₁₆	—	79	2	no	—
156	#427	—	79	4	no	—
157	#427	—	158	2	no	—
158	C₂×#107	—	79	2	no	—
159	C₂×#106	—	158	1	no	—
160	#358	—	79	16	no	—
161	C₂²×Q₈	—	2	8	no	—
162	C₂²×Q₈	—	2	4	no	—
163	C₂²×Q₈	—	2	2	no	—
164	C₂²×Q₈	—	4	2	no	—
165	C₂²×Q₈	—	4	1	no	—
166	C₂²×Q₈	—	4	4	no	—
167	C₂³×C₄	—	4	1	no	—
168	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
169	C₂×Q₁₆	—	2	16	no	—
170	C₂×Q₁₆	—	2	8	no	—
171	C₂×Q₁₆	—	2	4	no	—
172	C₂×Q₁₆	—	2	2	no	—
173	C₂×Q₁₆	—	2	4	no	—
174	C₂×Q₁₆	—	2	8	no	—
175	C₂×Q₁₆	C₂×Dic₇₉	1	8	yes	—
176	C₂×Q₁₆	C₂×Dic₇₉	1	16	yes	complemented
177	C₂×Q₁₆	C₂²×D₇₉	1	4	yes	—
178	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
179	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
180	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
181	C₂²×C₈	C₂²⋊D₇₉	1	4	yes	—
182	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
184	#117	—	316	2	no	—
185	#117	—	158	4	no	—
186	C₂×C₄⋊C₄	—	158	2	no	—
187	C₂×C₂²⋊C₄	—	316	1	no	—
188	C₂×C₄²	—	316	1	no	—
189	M₃₂	—	158	8	no	—
190	C₂×C₁₆	—	79	4	no	—
191	C₂×C₁₆	—	158	4	no	—
192	#107	—	79	4	no	—
193	#107	—	158	2	no	—
194	#106	—	158	4	no	—
195	C₂⁴	#11929609	1	1	yes	characteristic
196	C₂×Q₈	—	2	8	no	—
197	C₂×Q₈	—	2	16	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	4	16	no	—
200	C₂×Q₈	—	4	8	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	—	4	8	no	—
204	C₂×Q₈	—	4	4	no	—
205	C₂²×C₄	—	4	8	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	316	2	no	—
209	C₂²×C₄	C₂²⋊Dic₇₉	1	4	yes	—
210	C₂²×C₄	#11929609	1	2	yes	—
211	C₂²×C₄	#11929609	1	1	yes	characteristic
212	C₂²×C₄	#11929612	1	4	yes	—
213	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
214	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
215	Q₁₆	—	2	8	no	—
216	Q₁₆	—	2	16	no	—
217	Q₁₆	—	2	4	no	—
218	Q₁₆	C₂²×Dic₇₉	1	8	yes	—
219	C₂×C₈	C₂²⋊Dic₇₉	1	4	yes	—
220	C₂×C₈	C₂²⋊Dic₇₉	1	8	yes	—
221	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
222	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
223	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
224	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
225	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
226	C₂×C₈	C₂×C₂²⋊D₇₉	1	4	yes	—
227	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
228	C₄⋊C₄	—	316	2	no	—
229	C₄⋊C₄	—	158	4	no	—
230	C₂²⋊C₄	—	316	4	no	—
231	C₄²	—	316	4	no	—
232	C₁₆	—	158	4	no	—
233	C₂³	—	2	4	no	—
234	C₂³	#14579082	1	4	yes	—
235	C₂³	C₂×#11929609	1	2	yes	—
236	C₂³	C₂×#11929609	1	1	yes	characteristic
237	Q₈	—	2	8	no	—
238	Q₈	—	4	8	no	—
239	Q₈	—	4	16	no	—
240	Q₈	—	4	4	no	—
241	C₂×C₄	—	4	16	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	316	4	no	—
245	C₂×C₄	—	632	4	no	—
246	C₂×C₄	#14579082	1	8	yes	—
247	C₂×C₄	#14579082	1	4	yes	—
248	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	4	yes	—
249	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	2	yes	—
250	C₂×C₄	C₂×#11929609	1	2	yes	—
251	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
252	C₂×C₄	C₂×#11929612	1	4	yes	—
253	C₂×C₄	C₂×#11929612	1	2	yes	—
254	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
255	C₈	C₂×C₂²⋊Dic₇₉	1	2	yes	—
256	C₈	C₂×C₂²⋊Dic₇₉	1	4	yes	—
257	C₈	C₂³×Dic₇₉	1	1	yes	characteristic
258	C₈	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
259	C₂²	—	2	4	no	—
260	C₂²	—	2	8	no	—
261	C₂²	C₂×#14579082	1	4	yes	—
262	C₂²	C₂×#14579082	1	2	yes	—
263	C₂²	#19099004	1	4	yes	—
264	C₂²	C₂²×#11929609	1	1	yes	characteristic
265	C₄	—	4	8	no	—
266	C₄	—	632	4	no	—
267	C₄	C₂×#14579082	1	4	yes	—
268	C₄	C₂×#14579082	1	2	yes	—
269	C₄	C₂²×C₂²⋊Dic₇₉	1	1	yes	characteristic
270	C₄	C₂²×#11929612	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	#23765120	1	4	yes	direct product factor
273	C₂	C₂²×#14579082	1	1	yes	characteristic
274	C₂	C₂×#19099004	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator	C₈×C₇₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂³×Q₁₆×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂⁴×C₇₉	D₈
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	C₂³×Q₁₆	D₇₉
hypocenter	C₇₉	C₂×#2141
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #23765120	C₂	direct (N × H)
4. C₂×#19098367	C₂	semidirect
14. #19098367	C₂²	semidirect
19. C₂²×Q₁₆×C₇₉	C₄	semidirect
49. C₂×Q₁₆×C₇₉	C₂×C₄	semidirect
135. C₂×C₇₉	#2141	semidirect
143. C₇₉	C₂×#2141	semidirect
148. C₂²×Q₁₆	Dic₇₉	semidirect
176. C₂×Q₁₆	C₂×Dic₇₉	semidirect
272. C₂	#23765120	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14579101	2
C₂×#19098601	2
#23765120	2
C₂×#19098367	2
C₂³×Q₁₆×C₇₉	2
C₂×#2141	79

Subgroups — C2×#23765120 #34441161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765120 #34441161