Subgroups — #34186984

Order 20224.

290 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23764176	C₂	1	2	yes	maximal
2	#23762159	C₂	1	1	yes	maximal, characteristic
3	#23761689	C₂	1	1	yes	maximal, characteristic
4	#23761297	C₂	1	2	yes	maximal
5	C₂×#334×C₇₉	C₂	1	1	yes	maximal, characteristic
6	#19098616	C₂²	1	1	yes	characteristic
7	C₂×#14579097	—	2	2	no	—
8	#19098587	C₂²	1	2	yes	—
9	C₄×#11929608	C₂²	1	1	yes	characteristic
10	#19098328	—	2	2	no	—
11	#19098316	—	2	1	no	—
12	#19098316	—	2	1	no	—
13	#19098285	—	2	1	no	—
14	#19098284	—	2	1	no	—
15	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	#334×C₇₉	C₄	1	2	yes	—
18	#334×C₇₉	C₄	1	2	yes	—
19	C₂×#11929608	—	2	2	no	—
20	C₂×#11929608	D₄	1	1	yes	characteristic
21	C₂×#11929608	D₄	1	1	yes	characteristic
22	#14579097	—	4	4	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
26	C₂²⋊(C₇₉⋊C₈)	Q₈	1	2	yes	—
27	#14579072	—	2	1	no	—
28	C₄×C₇₉⋊C₈	—	2	1	no	—
29	C₄×C₇₉⋊C₈	—	2	1	no	—
30	C₂²×C₈×C₇₉	—	2	2	no	—
31	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
32	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₇₉	—	2	2	no	—
35	C₄⋊C₈×C₇₉	C₂×C₄	1	4	yes	—
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	—	2	2	no	—
41	#11929608	—	2	2	no	—
42	#11929608	—	4	2	no	—
43	#11929608	—	4	2	no	—
44	C₂×C₇₉⋊C₈	—	2	1	no	—
45	C₂×C₇₉⋊C₈	—	2	1	no	—
46	C₂×C₇₉⋊C₈	—	2	2	no	—
47	C₂×C₇₉⋊C₈	—	4	2	no	—
48	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₇₉	—	2	2	no	—
52	C₂×C₈×C₇₉	—	2	4	no	—
53	C₂×C₈×C₇₉	—	4	1	no	—
54	C₂×C₈×C₇₉	—	4	2	no	—
55	C₂×C₈×C₇₉	—	4	2	no	—
56	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
60	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₇₉	#41	1	1	yes	characteristic
63	C₄²×C₇₉	#41	1	1	yes	characteristic
64	C₂³×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₇₉	—	2	1	no	—
66	C₂×C₄×C₇₉	—	2	1	no	—
67	C₂×C₄×C₇₉	—	4	1	no	—
68	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₇₉	#125	1	1	yes	characteristic
74	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×C₄×C₇₉	#137	1	2	yes	—
76	C₈×C₇₉	—	2	1	no	—
77	C₈×C₇₉	—	2	1	no	—
78	C₈×C₇₉	—	2	1	no	—
79	C₈×C₇₉	—	4	4	no	—
80	C₈×C₇₉	#108	1	2	yes	complemented
81	C₇₉⋊C₈	—	4	2	no	—
82	C₇₉⋊C₈	—	4	1	no	—
83	C₇₉⋊C₈	—	4	1	no	—
84	C₇₉⋊C₈	—	8	2	no	—
85	C₂²×C₇₉	—	2	1	no	—
86	C₂²×C₇₉	—	2	1	no	—
87	C₂²×C₇₉	#389	1	1	yes	characteristic
88	C₂²×C₇₉	#501	1	2	yes	—
89	C₄×C₇₉	—	2	1	no	—
90	C₄×C₇₉	—	2	1	no	—
91	C₄×C₇₉	—	2	1	no	—
92	C₄×C₇₉	C₂×#108	1	1	yes	characteristic
93	C₄×C₇₉	#430	1	1	yes	characteristic
94	C₄×C₇₉	#440	1	2	yes	—
95	C₄×C₇₉	#469	1	1	yes	characteristic
96	C₄×C₇₉	#470	1	1	yes	characteristic
97	#9626	—	79	1	no	maximal, Sylow
98	C₂×C₇₉	—	2	2	no	—
99	C₂×C₇₉	#1944	1	1	yes	characteristic
100	C₂×C₇₉	#1950	1	1	yes	characteristic
101	C₂×C₇₉	#1953	1	1	yes	characteristic
102	#1950	—	79	2	no	—
103	#1567	—	79	1	no	—
104	C₂×#334	D₇₉	1	1	yes	characteristic
105	#1451	—	79	1	no	—
106	#1301	—	79	2	no	—
107	C₇₉	#9626	1	1	yes	characteristic, complemented, Sylow
108	#431	—	79	1	no	—
109	C₂×#108	—	158	2	no	—
110	#423	—	79	2	no	—
111	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
112	C₄×M₁₆	—	79	1	no	—
113	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
114	#341	—	158	2	no	—
115	#335	—	158	1	no	—
116	#335	—	158	1	no	—
117	#334	Dic₇₉	1	2	yes	—
118	#334	Dic₇₉	1	2	yes	—
119	#322	—	158	1	no	—
120	#322	—	158	1	no	—
121	C₂×M₁₆	—	79	1	no	—
122	C₂×M₁₆	—	79	1	no	—
123	C₂×M₁₆	—	158	2	no	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
126	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
127	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
128	#108	—	316	4	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	158	2	no	—
131	C₄⋊C₈	—	158	1	no	—
132	C₄⋊C₈	—	158	2	no	—
133	C₄⋊C₈	C₂×Dic₇₉	1	4	yes	—
134	C₂²⋊C₈	—	79	2	no	—
135	#97	—	158	1	no	—
136	C₄×C₈	—	158	1	no	—
137	C₄×C₈	—	158	1	no	—
138	C₄×C₈	D₃₁₆	1	1	yes	characteristic
139	C₄×C₈	D₃₁₆	1	1	yes	characteristic
140	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
141	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
142	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
143	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
144	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
145	M₁₆	—	316	2	no	—
146	M₁₆	—	316	2	no	—
147	M₁₆	—	158	2	no	—
148	M₁₆	—	158	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	316	2	no	—
155	C₂×C₈	—	158	1	no	—
156	C₂×C₈	—	158	1	no	—
157	C₂×C₈	—	158	2	no	—
158	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
159	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
160	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
161	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
162	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
163	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
164	C₄²	#11929623	1	1	yes	characteristic
165	C₄²	#11929625	1	1	yes	characteristic
166	C₂³	#14579160	1	1	yes	characteristic
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
171	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
172	C₂×C₄	Q₈×Dic₇₉	1	1	yes	characteristic
173	C₂×C₄	#14579136	1	1	yes	characteristic
174	C₂×C₄	#14579141	1	1	yes	characteristic
175	C₂×C₄	#14579152	1	1	yes	characteristic
176	C₂×C₄	#14579188	1	2	yes	—
177	C₂×C₄	#14579190	1	2	yes	—
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	—	4	4	no	—
182	C₈	—	316	2	no	—
183	C₈	—	316	1	no	—
184	C₈	—	316	1	no	—
185	C₈	—	632	2	no	—
186	C₈	#14579097	1	2	yes	complemented
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	#19098462	1	1	yes	characteristic
190	C₂²	#19098970	1	2	yes	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	C₂×#14579097	1	1	yes	characteristic
195	C₄	#19098612	1	1	yes	characteristic
196	C₄	#19098664	1	2	yes	—
197	C₄	#19098817	1	1	yes	characteristic
198	C₄	#19098822	1	1	yes	characteristic
199	C₂	—	2	2	no	—
200	C₂	#23764144	1	1	yes	characteristic
201	C₂	#23764176	1	1	yes	characteristic
202	C₂	#23764191	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579136
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#334×C₇₉	C₂
socle	C₂²×C₇₉	#389
prime radical	C₂³×C₇₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#334	D₇₉
hypocenter	C₇₉	#9626
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
80. C₈×C₇₉	#108	semidirect
107. C₇₉	#9626	semidirect
186. C₈	#14579097	semidirect

Maximal subgroups

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

subgroup	index
#23764176	2
#23762159	2
#23761689	2
#23761297	2
C₂×#334×C₇₉	2
#9626	79