Subgroups — #35157949

Order 20224.

1834 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23777219	C₂	1	2	yes	maximal
2	#23773946	C₂	1	6	yes	maximal, complemented
3	#23773931	C₂	1	12	yes	maximal, complemented
4	#23773914	C₂	1	3	yes	maximal, complemented
5	C₂×#19098961	C₂	1	3	yes	maximal, complemented
6	#23771995	C₂	1	4	yes	maximal
7	C₄×#131×C₇₉	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₁₅₈	—	2	2	no	—
9	#19099292	—	2	2	no	—
10	#19099270	—	2	6	no	—
11	D₄×Dic₁₅₈	—	2	6	no	—
12	#19099191	C₂²	1	12	yes	—
13	#19099120	C₂²	1	6	yes	—
14	C₂×#14579161	C₂²	1	6	yes	—
15	#19099046	C₂²	1	4	yes	—
16	#19098972	C₂²	1	24	yes	—
17	#19098968	C₂²	1	6	yes	—
18	#19098965	C₂²	1	12	yes	—
19	#19098961	C₂²	1	3	yes	—
20	#19098961	C₂²	1	12	yes	—
21	#19098961	C₂²	1	6	yes	—
22	#19098961	C₂²	1	1	yes	characteristic
23	#19098605	C₂²	1	24	yes	—
24	#19098601	C₂²	1	12	yes	—
25	C₂×#14579093	C₂²	1	12	yes	—
26	C₂×#131×C₇₉	C₂²	1	2	yes	—
27	C₄×#41×C₇₉	C₂²	1	1	yes	characteristic
28	#405×C₇₉	C₂²	1	6	yes	—
29	C₄×M₁₆×C₇₉	C₂²	1	3	yes	—
30	C₂×C₄×C₈×C₇₉	C₂²	1	3	yes	—
31	C₂²×Dic₁₅₈	—	2	6	no	—
32	#14579161	—	2	6	no	—
33	#14579161	—	2	6	no	—
34	#14579161	C₂³	1	12	yes	—
35	#14579161	C₂³	1	6	yes	—
36	#14579161	C₂³	1	2	yes	—
37	#14579160	—	2	12	no	—
38	#14579153	C₂³	1	12	yes	—
39	#14579152	—	2	12	no	—
40	#14579140	—	2	6	no	—
41	#14579140	—	2	6	no	—
42	#14579138	—	2	24	no	—
43	Q₈×Dic₇₉	—	2	4	no	—
44	C₄×Dic₁₅₈	—	2	6	no	—
45	C₄×Dic₁₅₈	—	2	6	no	—
46	D₄×Dic₇₉	—	2	12	no	—
47	#14579120	C₂³	1	12	yes	—
48	C₂×C₄⋊Dic₇₉	—	2	6	no	—
49	C₂×C₄⋊Dic₇₉	C₂³	1	12	yes	—
50	#14579095	C₂³	1	24	yes	—
51	#14579093	C₂³	1	12	yes	—
52	#14579093	C₂³	1	24	yes	—
53	#14579093	C₂³	1	4	yes	—
54	C₂×#41×C₇₉	C₂³	1	1	yes	characteristic
55	#131×C₇₉	Q₈	1	16	yes	complemented
56	C₂×M₁₆×C₇₉	C₂³	1	6	yes	—
57	C₂²×C₈×C₇₉	C₂³	1	6	yes	—
58	C₄×Q₈×C₇₉	D₄	1	2	yes	—
59	C₄×D₄×C₇₉	D₄	1	6	yes	—
60	#117×C₇₉	C₂³	1	3	yes	—
61	C₂×C₄²×C₇₉	C₂³	1	3	yes	—
62	#97×C₇₉	C₂³	1	6	yes	—
63	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₇₉	C₂³	1	6	yes	—
65	C₄×C₈×C₇₉	C₂³	1	3	yes	—
66	C₂×Dic₁₅₈	—	2	6	no	—
67	C₂×Dic₁₅₈	—	2	2	no	—
68	C₂×Dic₁₅₈	—	2	12	no	—
69	C₂×Dic₁₅₈	—	8	12	no	—
70	C₂²×Dic₇₉	—	2	12	no	—
71	C₄⋊Dic₇₉	—	2	6	no	—
72	C₄⋊Dic₇₉	—	2	12	no	—
73	C₄⋊Dic₇₉	—	2	12	no	—
74	C₄⋊Dic₇₉	—	2	2	no	—
75	C₄⋊Dic₇₉	C₂⁴	1	12	yes	—
76	C₄⋊Dic₇₉	C₂⁴	1	4	yes	—
77	Dic₇₉⋊C₄	—	2	12	no	—
78	Dic₇₉⋊C₄	—	2	12	no	—
79	C₂²⋊Dic₇₉	—	2	12	no	—
80	C₄×Dic₇₉	—	2	12	no	—
81	#41×C₇₉	C₂×Q₈	1	8	yes	—
82	C₂×Q₈×C₇₉	C₂×D₄	1	1	yes	characteristic
83	C₂×D₄×C₇₉	C₂×D₄	1	3	yes	—
84	C₂²×C₄×C₇₉	C₂×D₄	1	6	yes	—
85	C₂²×C₄×C₇₉	C₂⁴	1	3	yes	—
86	M₁₆×C₇₉	C₂×Q₈	1	24	yes	complemented
87	C₂×C₈×C₇₉	C₂×Q₈	1	24	yes	complemented
88	C₂×C₈×C₇₉	C₂⁴	1	2	yes	—
89	C₂×C₈×C₇₉	C₂⁴	1	6	yes	—
90	C₄⋊C₄×C₇₉	C₂×D₄	1	6	yes	—
91	C₂²⋊C₄×C₇₉	C₂×D₄	1	6	yes	—
92	C₄²×C₇₉	C₂×D₄	1	6	yes	—
93	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
94	C₄²×C₇₉	C₂⁴	1	3	yes	—
95	C₂³×C₇₉	C₂²×D₄	1	3	yes	—
96	Q₈×C₇₉	C₄⋊Q₈	1	4	yes	—
97	D₄×C₇₉	C₄⋊Q₈	1	12	yes	—
98	C₂×C₄×C₇₉	—	2	12	no	—
99	C₂×C₄×C₇₉	C₂²×D₄	1	6	yes	—
100	C₂×C₄×C₇₉	C₂²×D₄	1	3	yes	—
101	C₂×C₄×C₇₉	C₂²×D₄	1	2	yes	—
102	C₂×C₄×C₇₉	C₂²×Q₈	1	12	yes	—
103	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
104	C₂×Dic₇₉	—	2	4	no	—
105	C₂×Dic₇₉	—	2	12	no	—
106	C₂×Dic₇₉	—	8	12	no	—
107	Dic₁₅₈	—	4	4	no	—
108	Dic₁₅₈	—	4	12	no	—
109	Dic₁₅₈	—	8	12	no	—
110	C₈×C₇₉	C₂²×Q₈	1	4	yes	—
111	C₈×C₇₉	C₂²×Q₈	1	12	yes	—
112	C₂²×C₇₉	—	2	6	no	—
113	C₂²×C₇₉	C₂×C₄⋊Q₈	1	6	yes	—
114	C₂²×C₇₉	C₂³×D₄	1	1	yes	characteristic
115	C₄×C₇₉	—	2	6	no	—
116	C₄×C₇₉	C₂×C₄⋊Q₈	1	6	yes	—
117	C₄×C₇₉	#577	1	4	yes	—
118	C₄×C₇₉	C₂³×Q₈	1	2	yes	—
119	Dic₇₉	—	4	12	no	—
120	Dic₇₉	—	4	4	no	—
121	#60665	—	79	1	no	maximal, Sylow
122	C₂×C₇₉	—	2	6	no	—
123	C₂×C₇₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
124	C₂×C₇₉	C₂×#577	1	2	yes	—
125	#3503	—	79	2	no	—
126	#3165	—	79	6	no	—
127	#3163	—	79	12	no	—
128	#3160	—	79	3	no	—
129	C₂×#498	—	79	3	no	—
130	#2912	—	79	4	no	—
131	C₄×#131	D₇₉	1	1	yes	characteristic
132	C₇₉	#60665	1	1	yes	characteristic, complemented, Sylow
133	C₂×#131	C₂×D₇₉	1	2	yes	—
134	Q₈²	—	158	2	no	—
135	#557	—	158	2	no	—
136	#554	—	158	6	no	—
137	D₄×Q₈	—	158	6	no	—
138	#544	—	79	12	no	—
139	#533	—	79	6	no	—
140	C₂×C₄⋊Q₈	—	79	6	no	—
141	#520	—	79	4	no	—
142	C₄×#41	C₂×D₇₉	1	1	yes	characteristic
143	#501	—	79	24	no	—
144	#500	—	79	6	no	—
145	#499	—	79	12	no	—
146	#498	—	79	3	no	—
147	#498	—	79	12	no	—
148	#498	—	79	6	no	—
149	#498	—	79	1	no	—
150	#428	—	79	24	no	—
151	#427	—	79	12	no	—
152	C₂×#106	—	79	12	no	—
153	#405	C₂×D₇₉	1	6	yes	—
154	C₄×M₁₆	C₂×D₇₉	1	3	yes	—
155	C₂×C₄×C₈	C₂×D₇₉	1	3	yes	—
156	C₂×#41	C₂²×D₇₉	1	1	yes	characteristic
157	C₂²×Q₈	—	158	6	no	—
158	#131	Dic₁₅₈	1	16	yes	complemented
159	C₂×M₁₆	C₂²×D₇₉	1	6	yes	—
160	C₂²×C₈	C₂²×D₇₉	1	6	yes	—
161	C₄⋊Q₈	—	79	6	no	—
162	C₄⋊Q₈	—	79	12	no	—
163	C₄⋊Q₈	—	79	2	no	—
164	C₄⋊Q₈	—	158	12	no	—
165	C₄⋊Q₈	—	158	6	no	—
166	C₄⋊Q₈	—	158	6	no	—
167	#125	—	79	12	no	—
168	#125	—	158	12	no	—
169	C₂²⋊Q₈	—	158	6	no	—
170	C₂²⋊Q₈	—	158	6	no	—
171	C₂²⋊Q₈	—	158	24	no	—
172	C₄×Q₈	—	158	4	no	—
173	C₄×Q₈	—	158	6	no	—
174	C₄×Q₈	—	158	6	no	—
175	C₄×Q₈	D₃₁₆	1	2	yes	—
176	C₄×D₄	—	158	12	no	—
177	C₄×D₄	D₃₁₆	1	6	yes	—
178	#117	—	79	12	no	—
179	#117	C₂²×D₇₉	1	3	yes	—
180	C₂×C₄⋊C₄	—	79	12	no	—
181	C₂×C₄⋊C₄	—	158	6	no	—
182	C₂×C₄²	C₂²×D₇₉	1	3	yes	—
183	#107	—	79	24	no	—
184	#106	—	79	12	no	—
185	#106	—	79	24	no	—
186	#106	—	79	4	no	—
187	#97	C₂²×D₇₉	1	6	yes	—
188	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₇₉	1	3	yes	—
190	C₄×C₈	C₂²×D₇₉	1	6	yes	—
191	#41	C₂×Dic₁₅₈	1	8	yes	—
192	C₂×Q₈	—	632	12	no	—
193	C₂×Q₈	—	158	2	no	—
194	C₂×Q₈	—	158	6	no	—
195	C₂×Q₈	—	158	12	no	—
196	C₂×Q₈	C₂×D₃₁₆	1	1	yes	characteristic
197	C₂×D₄	C₂×D₃₁₆	1	3	yes	—
198	C₂²×C₄	—	158	12	no	—
199	C₂²×C₄	C₂×D₃₁₆	1	6	yes	—
200	C₂²×C₄	C₂³×D₇₉	1	3	yes	—
201	M₁₆	C₂×Dic₁₅₈	1	24	yes	complemented
202	C₂×C₈	C₂×Dic₁₅₈	1	24	yes	complemented
203	C₂×C₈	C₂³×D₇₉	1	2	yes	—
204	C₂×C₈	C₂³×D₇₉	1	6	yes	—
205	C₄⋊C₄	—	79	12	no	—
206	C₄⋊C₄	—	79	4	no	—
207	C₄⋊C₄	—	158	12	no	—
208	C₄⋊C₄	—	158	12	no	—
209	C₄⋊C₄	—	158	12	no	—
210	C₄⋊C₄	—	158	6	no	—
211	C₄⋊C₄	—	158	12	no	—
212	C₄⋊C₄	—	158	2	no	—
213	C₄⋊C₄	C₂×D₃₁₆	1	6	yes	—
214	C₂²⋊C₄	—	158	12	no	—
215	C₂²⋊C₄	C₂×D₃₁₆	1	6	yes	—
216	C₄²	—	158	12	no	—
217	C₄²	C₂×D₃₁₆	1	6	yes	—
218	C₄²	C₂³×D₇₉	1	1	yes	characteristic
219	C₄²	C₂³×D₇₉	1	3	yes	—
220	C₂³	C₂²×D₃₁₆	1	3	yes	—
221	Q₈	—	316	12	no	—
222	Q₈	—	316	4	no	—
223	Q₈	—	632	12	no	—
224	Q₈	#14579161	1	4	yes	—
225	D₄	#14579161	1	12	yes	—
226	C₂×C₄	—	2	12	no	—
227	C₂×C₄	—	632	12	no	—
228	C₂×C₄	—	158	4	no	—
229	C₂×C₄	—	158	12	no	—
230	C₂×C₄	C₂²×D₃₁₆	1	6	yes	—
231	C₂×C₄	C₂²×D₃₁₆	1	3	yes	—
232	C₂×C₄	C₂²×D₃₁₆	1	2	yes	—
233	C₂×C₄	C₂²×Dic₁₅₈	1	12	yes	—
234	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
235	C₈	C₂²×Dic₁₅₈	1	4	yes	—
236	C₈	C₂²×Dic₁₅₈	1	12	yes	—
237	C₂²	—	2	6	no	—
238	C₂²	C₂×#14579161	1	6	yes	—
239	C₂²	C₂³×D₃₁₆	1	1	yes	characteristic
240	C₄	—	2	6	no	—
241	C₄	—	316	12	no	—
242	C₄	—	316	4	no	—
243	C₄	C₂×#14579161	1	6	yes	—
244	C₄	#19099375	1	4	yes	—
245	C₄	C₂³×Dic₁₅₈	1	2	yes	—
246	C₂	—	2	6	no	—
247	C₂	C₂²×#14579161	1	1	yes	characteristic
248	C₂	C₂×#19099375	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₉
Fitting	C₄×#131×C₇₉	C₂
socle	C₂²×C₇₉	C₂³×D₄
prime radical	C₂×#41×C₇₉	C₂³
hypercenter	C₄×#131	D₇₉
hypocenter	C₇₉	#60665
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #23773946	C₂	semidirect
3. #23773931	C₂	semidirect
4. #23773914	C₂	semidirect
5. C₂×#19098961	C₂	semidirect
55. #131×C₇₉	Q₈	semidirect
86. M₁₆×C₇₉	C₂×Q₈	semidirect
87. C₂×C₈×C₇₉	C₂×Q₈	semidirect
132. C₇₉	#60665	semidirect
158. #131	Dic₁₅₈	semidirect
201. M₁₆	C₂×Dic₁₅₈	semidirect
202. C₂×C₈	C₂×Dic₁₅₈	semidirect

Maximal subgroups

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

subgroup	index
#23777219	2
#23773946	2
#23773931	2
#23773914	2
C₂×#19098961	2
#23771995	2
C₄×#131×C₇₉	2
#60665	79