Subgroups — #34186973

Order 20224.

298 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23764175	C₂	1	2	yes	maximal
2	#23762159	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098316	C₂	1	1	yes	maximal, characteristic
4	#23761296	C₂	1	2	yes	maximal
5	C₈×M₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579096	—	2	2	no	—
7	#19098587	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
9	#19098327	—	2	2	no	—
10	#19098316	—	2	1	no	—
11	#19098316	—	2	1	no	—
12	#19098316	C₄	1	2	yes	—
13	#19098316	C₄	1	2	yes	—
14	#431×C₇₉	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	#322×C₇₉	—	2	1	no	—
18	C₈²×C₇₉	—	2	1	no	—
19	C₂×#11929608	—	2	2	no	—
20	C₂²×C₇₉⋊C₈	—	2	2	no	—
21	#14579096	—	4	4	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
26	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
27	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₇₉	—	2	1	no	—
32	C₂²⋊C₈×C₇₉	Q₈	1	2	yes	—
33	#97×C₇₉	—	2	1	no	—
34	C₄×C₈×C₇₉	—	2	1	no	—
35	C₄×C₈×C₇₉	—	2	1	no	—
36	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
39	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
40	#11929608	—	4	2	no	—
41	#11929608	—	4	2	no	—
42	C₂×C₇₉⋊C₈	—	2	4	no	—
43	C₂×C₇₉⋊C₈	—	4	2	no	—
44	C₂×C₇₉⋊C₈	—	4	2	no	—
45	C₂×C₇₉⋊C₈	—	4	2	no	—
46	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
49	M₁₆×C₇₉	—	2	1	no	—
50	M₁₆×C₇₉	—	2	1	no	—
51	M₁₆×C₇₉	D₈	1	2	yes	—
52	M₁₆×C₇₉	Q₁₆	1	2	yes	—
53	C₂×C₈×C₇₉	—	2	2	no	—
54	C₂×C₈×C₇₉	—	2	1	no	—
55	C₂×C₈×C₇₉	—	2	1	no	—
56	C₂×C₈×C₇₉	—	2	2	no	—
57	C₂×C₈×C₇₉	—	4	1	no	—
58	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
62	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₇₉	#41	1	1	yes	characteristic
65	C₄²×C₇₉	#41	1	1	yes	characteristic
66	C₂³×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₉	—	2	1	no	—
68	C₂×C₄×C₇₉	—	2	1	no	—
69	C₂×C₄×C₇₉	—	4	1	no	—
70	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₉	#125	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
79	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
80	C₈×C₇₉	—	2	1	no	—
81	C₈×C₇₉	—	2	1	no	—
82	C₈×C₇₉	—	2	1	no	—
83	C₈×C₇₉	—	2	2	no	—
84	C₈×C₇₉	—	2	2	no	—
85	C₈×C₇₉	—	4	2	no	—
86	C₈×C₇₉	#108	1	2	yes	complemented
87	C₇₉⋊C₈	—	4	4	no	—
88	C₇₉⋊C₈	—	8	2	no	—
89	C₂²×C₇₉	—	2	1	no	—
90	C₂²×C₇₉	—	2	1	no	—
91	C₂²×C₇₉	#389	1	1	yes	characteristic
92	C₂²×C₇₉	#499	1	1	yes	characteristic
93	C₂²×C₇₉	#500	1	1	yes	characteristic
94	C₄×C₇₉	—	2	1	no	—
95	C₄×C₇₉	—	2	1	no	—
96	C₄×C₇₉	—	2	1	no	—
97	C₄×C₇₉	C₂×#108	1	1	yes	characteristic
98	C₄×C₇₉	#430	1	1	yes	characteristic
99	C₄×C₇₉	C₄×D₈	1	1	yes	characteristic
100	C₄×C₇₉	C₄×Q₁₆	1	1	yes	characteristic
101	C₄×C₇₉	#466	1	1	yes	characteristic
102	C₄×C₇₉	#467	1	1	yes	characteristic
103	#9624	—	79	1	no	maximal, Sylow
104	C₂×C₇₉	—	2	1	no	—
105	C₂×C₇₉	—	2	1	no	—
106	C₂×C₇₉	#1942	1	1	yes	characteristic
107	C₂×C₇₉	#1950	1	1	yes	characteristic
108	C₂×C₇₉	#1953	1	1	yes	characteristic
109	#1950	—	79	2	no	—
110	#1567	—	79	1	no	—
111	C₂×#335	—	79	1	no	—
112	C₈×M₁₆	D₇₉	1	1	yes	characteristic
113	#1301	—	79	2	no	—
114	C₇₉	#9624	1	1	yes	characteristic, complemented, Sylow
115	#431	C₂×D₇₉	1	1	yes	characteristic
116	C₂×#108	—	158	2	no	—
117	#423	—	79	2	no	—
118	C₂×C₄⋊C₈	—	79	2	no	—
119	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
121	#341	—	158	2	no	—
122	#335	—	79	2	no	—
123	#335	—	79	2	no	—
124	#335	—	158	1	no	—
125	#335	—	158	1	no	—
126	#322	—	2	1	no	—
127	C₈²	—	2	1	no	—
128	C₂×M₁₆	—	158	2	no	—
129	C₂×M₁₆	D₃₁₆	1	1	yes	characteristic
130	C₂×M₁₆	D₃₁₆	1	1	yes	characteristic
131	C₂²×C₈	—	158	2	no	—
132	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
133	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
134	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
135	#108	—	316	4	no	—
136	C₄⋊C₈	—	2	1	no	—
137	C₄⋊C₈	—	79	4	no	—
138	C₄⋊C₈	—	158	2	no	—
139	C₄⋊C₈	—	158	2	no	—
140	C₄⋊C₈	—	158	2	no	—
141	C₂²⋊C₈	Dic₁₅₈	1	2	yes	—
142	#97	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
146	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
150	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
151	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
152	M₁₆	—	2	1	no	—
153	M₁₆	—	2	1	no	—
154	M₁₆	—	316	2	no	—
155	M₁₆	—	316	2	no	—
156	M₁₆	D₆₃₂	1	2	yes	—
157	M₁₆	Dic₃₁₆	1	2	yes	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	—	316	2	no	—
164	C₂×C₈	—	316	2	no	—
165	C₂×C₈	—	316	2	no	—
166	C₂×C₈	—	158	4	no	—
167	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
168	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
169	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
170	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
171	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
172	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
173	C₄²	#11929624	1	1	yes	characteristic
174	C₄²	#11929624	1	1	yes	characteristic
175	C₂³	#14579161	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
180	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
181	C₂×C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
182	C₂×C₄	#14579135	1	1	yes	characteristic
183	C₂×C₄	#14579140	1	1	yes	characteristic
184	C₂×C₄	#14579153	1	1	yes	characteristic
185	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
186	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
187	C₂×C₄	#14579183	1	1	yes	characteristic
188	C₂×C₄	#14579183	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	2	2	no	—
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	4	2	no	—
195	C₈	—	316	4	no	—
196	C₈	—	632	2	no	—
197	C₈	#14579096	1	2	yes	complemented
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#19098461	1	1	yes	characteristic
201	C₂²	#19098965	1	1	yes	characteristic
202	C₂²	#19098968	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	C₂×#14579096	1	1	yes	characteristic
207	C₄	#19098611	1	1	yes	characteristic
208	C₄	C₄×D₆₃₂	1	1	yes	characteristic
209	C₄	C₄×Dic₃₁₆	1	1	yes	characteristic
210	C₄	#19098799	1	1	yes	characteristic
211	C₄	#19098804	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#23764133	1	1	yes	characteristic
215	C₂	#23764175	1	1	yes	characteristic
216	C₂	#23764190	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579135
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₈×M₁₆×C₇₉	C₂
socle	C₂²×C₇₉	#389
prime radical	C₂³×C₇₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₈×M₁₆	D₇₉
hypocenter	C₇₉	#9624
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
86. C₈×C₇₉	#108	semidirect
114. C₇₉	#9624	semidirect
197. C₈	#14579096	semidirect

Maximal subgroups

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

subgroup	index
#23764175	2
#23762159	2
C₂×#19098316	2
#23761296	2
C₈×M₁₆×C₇₉	2
#9624	79