Subgroups — #34186883

Order 20224.

306 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579097	C₂	1	1	yes	maximal, characteristic
2	#23762150	C₂	1	2	yes	maximal
3	#23761683	C₂	1	1	yes	maximal, characteristic
4	#23761250	C₂	1	2	yes	maximal
5	C₂×#335×C₇₉	C₂	1	1	yes	maximal, characteristic
6	#19098616	C₂²	1	1	yes	characteristic
7	#19098616	C₂²	1	1	yes	characteristic
8	C₂×#14579097	C₄	1	1	yes	characteristic
9	C₂×#14579097	C₄	1	1	yes	characteristic
10	C₄×#11929608	C₂²	1	2	yes	—
11	#19098328	C₄	1	2	yes	—
12	#19098315	—	2	2	no	—
13	#19098313	—	2	2	no	—
14	#19098284	—	2	1	no	—
15	C₈×C₇₉⋊C₈	—	2	1	no	—
16	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	#335×C₇₉	—	2	1	no	—
19	#335×C₇₉	—	2	1	no	—
20	C₂×#11929608	C₂×C₄	1	2	yes	—
21	C₂×#11929608	C₂×C₄	1	2	yes	—
22	#14579097	—	2	2	no	—
23	#14579097	C₈	1	4	yes	complemented
24	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
25	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
26	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
27	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
28	C₂²⋊(C₇₉⋊C₈)	Q₈	1	2	yes	—
29	#14579072	—	2	2	no	—
30	C₄×C₇₉⋊C₈	—	2	2	no	—
31	C₄×C₇₉⋊C₈	—	2	1	no	—
32	C₄×C₇₉⋊C₈	—	2	1	no	—
33	C₂²×C₈×C₇₉	—	2	2	no	—
34	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₇₉	—	2	2	no	—
38	C₄⋊C₈×C₇₉	—	2	2	no	—
39	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
40	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
42	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
43	#11929608	—	2	2	no	—
44	#11929608	—	2	2	no	—
45	#11929608	—	2	2	no	—
46	#11929608	C₂×C₈	1	4	yes	—
47	C₂×C₇₉⋊C₈	—	2	2	no	—
48	C₂×C₇₉⋊C₈	—	2	2	no	—
49	C₂×C₇₉⋊C₈	—	2	1	no	—
50	C₂×C₇₉⋊C₈	—	2	1	no	—
51	C₂×C₇₉⋊C₈	—	2	2	no	—
52	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
55	C₂×C₈×C₇₉	—	2	1	no	—
56	C₂×C₈×C₇₉	—	2	1	no	—
57	C₂×C₈×C₇₉	—	4	2	no	—
58	C₂×C₈×C₇₉	—	8	2	no	—
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	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₇₉	M₁₆	1	2	yes	—
65	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₇₉	#41	1	1	yes	characteristic
69	C₂³×C₇₉	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₉	—	2	1	no	—
71	C₂×C₄×C₇₉	—	2	1	no	—
72	C₂×C₄×C₇₉	—	4	1	no	—
73	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₇₉	#125	1	1	yes	characteristic
78	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
82	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
83	C₈×C₇₉	—	2	1	no	—
84	C₈×C₇₉	—	2	1	no	—
85	C₈×C₇₉	—	8	2	no	—
86	C₈×C₇₉	C₄⋊C₈	1	2	yes	—
87	C₈×C₇₉	C₄⋊C₈	1	2	yes	—
88	C₇₉⋊C₈	—	2	4	no	—
89	C₇₉⋊C₈	—	4	2	no	—
90	C₇₉⋊C₈	—	4	2	no	—
91	C₇₉⋊C₈	—	4	1	no	—
92	C₇₉⋊C₈	—	4	1	no	—
93	C₂²×C₇₉	—	2	1	no	—
94	C₂²×C₇₉	—	2	1	no	—
95	C₂²×C₇₉	#389	1	1	yes	characteristic
96	C₂²×C₇₉	C₈×Q₈	1	1	yes	characteristic
97	C₂²×C₇₉	#446	1	1	yes	characteristic
98	C₄×C₇₉	—	2	1	no	—
99	C₄×C₇₉	—	2	1	no	—
100	C₄×C₇₉	—	2	1	no	—
101	C₄×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₄×C₇₉	#424	1	1	yes	characteristic
103	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
104	C₄×C₇₉	#436	1	1	yes	characteristic
105	C₄×C₇₉	#472	1	1	yes	characteristic
106	C₄×C₇₉	#473	1	1	yes	characteristic
107	#9608	—	79	1	no	maximal, Sylow
108	C₂×C₇₉	—	2	1	no	—
109	C₂×C₇₉	—	2	1	no	—
110	C₂×C₇₉	#1939	1	1	yes	characteristic
111	C₂×C₇₉	#1952	1	1	yes	characteristic
112	C₂×C₇₉	#1953	1	1	yes	characteristic
113	C₄×#108	—	79	1	no	—
114	#1565	—	79	2	no	—
115	C₂×#335	D₇₉	1	1	yes	characteristic
116	#1450	—	79	1	no	—
117	#1283	—	79	2	no	—
118	C₇₉	#9608	1	1	yes	characteristic, complemented, Sylow
119	#431	—	79	1	no	—
120	#431	—	79	1	no	—
121	C₂×#108	—	79	1	no	—
122	C₂×#108	—	79	1	no	—
123	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
124	C₄×M₁₆	—	79	2	no	—
125	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
126	#341	—	79	2	no	—
127	#335	—	2	1	no	—
128	#335	—	2	1	no	—
129	#335	—	158	2	no	—
130	#334	—	158	2	no	—
131	#322	—	158	1	no	—
132	C₈²	—	158	1	no	—
133	C₂×M₁₆	—	79	2	no	—
134	C₂×M₁₆	—	79	2	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
137	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
138	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
139	#108	—	79	4	no	—
140	#108	—	158	2	no	—
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	79	1	no	—
144	C₄⋊C₈	—	79	1	no	—
145	C₄⋊C₈	—	158	1	no	—
146	C₂²⋊C₈	—	79	2	no	—
147	C₂²⋊C₈	—	158	1	no	—
148	#97	—	158	2	no	—
149	C₄×C₈	—	158	2	no	—
150	C₄×C₈	—	158	1	no	—
151	C₄×C₈	—	158	1	no	—
152	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
153	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
157	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
158	C₂²×C₄	#11929623	1	1	yes	characteristic
159	M₁₆	—	79	4	no	—
160	M₁₆	—	158	2	no	—
161	M₁₆	—	158	2	no	—
162	M₁₆	—	158	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	8	2	no	—
167	C₂×C₈	—	158	2	no	—
168	C₂×C₈	—	158	1	no	—
169	C₂×C₈	—	158	1	no	—
170	C₂×C₈	—	158	2	no	—
171	C₂×C₈	—	158	2	no	—
172	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
173	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
174	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
175	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
176	C₂×C₈	C₈×D₇₉	1	2	yes	—
177	C₂×C₈	#11929607	1	2	yes	—
178	C₄²	D₄×D₇₉	1	1	yes	characteristic
179	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
180	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
181	C₄²	#11929625	1	1	yes	characteristic
182	C₂³	#14579126	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
187	C₂×C₄	#14579123	1	1	yes	characteristic
188	C₂×C₄	#14579132	1	1	yes	characteristic
189	C₂×C₄	#14579137	1	1	yes	characteristic
190	C₂×C₄	#14579152	1	1	yes	characteristic
191	C₂×C₄	#14579160	1	1	yes	characteristic
192	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
193	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
194	C₂×C₄	#14579169	1	1	yes	characteristic
195	C₂×C₄	#14579169	1	1	yes	characteristic
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	—	8	2	no	—
199	C₈	—	316	2	no	—
200	C₈	—	316	2	no	—
201	C₈	—	316	1	no	—
202	C₈	—	316	1	no	—
203	C₈	—	158	4	no	—
204	C₈	Dic₇₉⋊C₈	1	2	yes	—
205	C₈	Dic₇₉⋊C₈	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#19098459	1	1	yes	characteristic
209	C₂²	#19098687	1	1	yes	characteristic
210	C₂²	#19098690	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	C₂×Dic₇₉⋊C₈	1	1	yes	characteristic
215	C₄	#19098591	1	1	yes	characteristic
216	C₄	#19098628	1	1	yes	characteristic
217	C₄	#19098640	1	1	yes	characteristic
218	C₄	#19098830	1	1	yes	characteristic
219	C₄	#19098835	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#23764116	1	1	yes	characteristic
223	C₂	#23764183	1	1	yes	characteristic
224	C₂	#23764188	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579132
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#335×C₇₉	C₂
socle	C₂²×C₇₉	#389
prime radical	C₂³×C₇₉	C₄×Q₈
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#335	D₇₉
hypocenter	C₇₉	#9608
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
23. #14579097	C₈	semidirect
118. C₇₉	#9608	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579097	2
#23762150	2
#23761683	2
#23761250	2
C₂×#335×C₇₉	2
#9608	79