Subgroups — #34258993

Order 20224.

590 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773986	C₂	1	1	yes	maximal, characteristic
2	#23772334	C₂	1	2	yes	maximal
3	#23762252	C₂	1	8	yes	maximal, complemented
4	#23762159	C₂	1	1	yes	maximal, characteristic
5	#23761856	C₂	1	2	yes	maximal
6	C₂×#322×C₇₉	C₂	1	1	yes	maximal, characteristic
7	#19099120	C₂²	1	2	yes	—
8	#19098968	C₄	1	1	yes	characteristic
9	#19098943	C₄	1	1	yes	characteristic
10	#19098914	C₄	1	2	yes	—
11	#19098804	C₄	1	4	yes	—
12	#19098693	—	2	2	no	—
13	#19098693	C₂²	1	4	yes	—
14	C₈×Dic₁₅₈	—	2	2	no	—
15	C₈×Dic₁₅₈	C₂²	1	4	yes	—
16	C₄×Dic₃₁₆	C₂²	1	4	yes	—
17	#19098589	—	2	2	no	—
18	#19098587	C₂²	1	2	yes	—
19	#19098518	—	2	2	no	—
20	#19098316	C₂²	1	4	yes	—
21	#19098295	C₂²	1	8	yes	—
22	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
25	#322×C₇₉	C₂²	1	4	yes	—
26	C₂×Dic₃₁₆	C₂×C₄	1	4	yes	—
27	C₂×#11929608	—	2	2	no	—
28	#14579161	C₂×C₄	1	2	yes	—
29	#14579161	C₂×C₄	1	2	yes	—
30	#14579153	—	2	2	no	—
31	#14579140	—	2	2	no	—
32	#14579140	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₅₈	—	2	2	no	—
34	C₄×Dic₁₅₈	C₂³	1	4	yes	—
35	#14579120	—	2	2	no	—
36	#14579095	C₂×C₄	1	4	yes	—
37	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
38	C₄⋊(C₇₉⋊C₈)	C₂³	1	4	yes	—
39	Dic₇₉⋊C₈	—	2	4	no	—
40	Dic₇₉⋊C₈	—	2	4	no	—
41	#14579086	C₂×C₄	1	8	yes	—
42	#14579071	—	2	4	no	—
43	C₈×Dic₇₉	—	2	4	no	—
44	C₂²×C₈×C₇₉	—	2	1	no	—
45	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₇₉	—	2	1	no	—
50	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₇₉	D₄	1	2	yes	—
53	C₄×C₈×C₇₉	C₂³	1	2	yes	—
54	C₄×C₈×C₇₉	C₂³	1	2	yes	—
55	C₄×C₈×C₇₉	C₂³	1	2	yes	—
56	C₂×Dic₁₅₈	—	2	2	no	—
57	C₂×Dic₁₅₈	C₂²×C₄	1	4	yes	—
58	Dic₃₁₆	—	4	4	no	—
59	#11929608	—	4	4	no	—
60	C₂×C₇₉⋊C₈	—	2	4	no	—
61	C₄⋊Dic₇₉	—	2	2	no	—
62	C₄⋊Dic₇₉	—	2	2	no	—
63	C₄⋊Dic₇₉	—	2	2	no	—
64	C₄⋊Dic₇₉	C₂²×C₄	1	4	yes	—
65	Dic₇₉⋊C₄	—	2	4	no	—
66	Dic₇₉⋊C₄	—	2	4	no	—
67	C₂²⋊Dic₇₉	—	2	4	no	—
68	C₄×Dic₇₉	—	2	4	no	—
69	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
72	C₂×C₈×C₇₉	—	2	2	no	—
73	C₂×C₈×C₇₉	—	2	2	no	—
74	C₂×C₈×C₇₉	—	4	2	no	—
75	C₂×C₈×C₇₉	—	4	2	no	—
76	C₂×C₈×C₇₉	Q₁₆	1	4	yes	complemented
77	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
81	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₇₉	#41	1	1	yes	characteristic
84	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
85	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₇₉	—	2	2	no	—
87	C₂×C₄×C₇₉	—	2	2	no	—
88	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
92	C₂×C₄×C₇₉	#135	1	2	yes	—
93	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
96	C₂×Dic₇₉	—	2	4	no	—
97	C₂×Dic₇₉	—	2	4	no	—
98	Dic₁₅₈	—	2	8	no	—
99	Dic₁₅₈	—	4	4	no	—
100	C₈×C₇₉	—	2	2	no	—
101	C₈×C₇₉	—	2	2	no	—
102	C₈×C₇₉	—	2	2	no	—
103	C₈×C₇₉	—	4	2	no	—
104	C₈×C₇₉	C₂×Q₁₆	1	4	yes	—
105	C₇₉⋊C₈	—	4	4	no	—
106	C₂²×C₇₉	—	2	1	no	—
107	C₂²×C₇₉	—	2	1	no	—
108	C₂²×C₇₉	C₄×Q₁₆	1	2	yes	—
109	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
110	C₄×C₇₉	—	2	1	no	—
111	C₄×C₇₉	—	2	1	no	—
112	C₄×C₇₉	C₄×Q₁₆	1	2	yes	—
113	C₄×C₇₉	#444	1	2	yes	—
114	C₄×C₇₉	#568	1	2	yes	—
115	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
116	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
117	Dic₇₉	—	4	4	no	—
118	Dic₇₉	—	4	4	no	—
119	#17977	—	79	1	no	maximal, Sylow
120	C₂×C₇₉	—	2	2	no	—
121	C₂×C₇₉	#2932	1	1	yes	characteristic
122	C₂×C₇₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
123	C₂×C₇₉	C₂×#444	1	1	yes	characteristic
124	#3170	—	79	1	no	—
125	#2960	—	79	2	no	—
126	#1584	—	79	8	no	—
127	#1567	—	79	1	no	—
128	#1493	—	79	2	no	—
129	C₂×#322	D₇₉	1	1	yes	characteristic
130	C₇₉	#17977	1	1	yes	characteristic, complemented, Sylow
131	#533	—	79	2	no	—
132	#500	—	79	1	no	—
133	#494	—	79	1	no	—
134	#487	—	79	2	no	—
135	#467	—	79	4	no	—
136	#446	—	79	4	no	—
137	#446	—	158	2	no	—
138	C₈×Q₈	—	79	4	no	—
139	C₈×Q₈	—	158	2	no	—
140	C₄×Q₁₆	—	79	4	no	—
141	#424	—	158	2	no	—
142	#423	—	79	2	no	—
143	#405	—	158	2	no	—
144	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
147	#335	—	79	4	no	—
148	Q₈⋊C₈	—	79	8	no	—
149	#322	C₂×D₇₉	1	4	yes	—
150	C₂×Q₁₆	—	79	4	no	—
151	C₂×M₁₆	—	158	2	no	—
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	C₄×D₇₉	1	2	yes	—
154	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
155	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
156	C₄⋊Q₈	—	79	2	no	—
157	C₄⋊Q₈	—	79	2	no	—
158	#125	—	158	2	no	—
159	C₂²⋊Q₈	—	79	4	no	—
160	C₂²⋊Q₈	—	158	2	no	—
161	C₄×Q₈	—	79	4	no	—
162	C₄×Q₈	—	158	2	no	—
163	#117	—	158	2	no	—
164	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
165	#107	—	79	4	no	—
166	C₄⋊C₈	—	79	4	no	—
167	C₄⋊C₈	—	158	4	no	—
168	C₄⋊C₈	—	158	4	no	—
169	C₄⋊C₈	—	158	2	no	—
170	#103	—	79	8	no	—
171	#97	—	158	4	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	—	158	4	no	—
174	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
175	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
176	C₄×C₈	D₃₁₆	1	2	yes	—
177	C₄×C₈	C₂²×D₇₉	1	2	yes	—
178	C₄×C₈	C₂²×D₇₉	1	2	yes	—
179	C₄×C₈	C₂²×D₇₉	1	2	yes	—
180	C₂×Q₈	—	79	4	no	—
181	C₂×Q₈	—	158	2	no	—
182	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
184	C₂²×C₄	#11929624	1	1	yes	characteristic
185	Q₁₆	—	316	4	no	—
186	M₁₆	—	316	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	158	4	no	—
192	C₂×C₈	Dic₃₁₆	1	4	yes	complemented
193	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
194	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
195	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
196	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
197	C₄⋊C₄	—	79	4	no	—
198	C₄⋊C₄	—	158	4	no	—
199	C₄⋊C₄	—	158	4	no	—
200	C₄⋊C₄	—	158	2	no	—
201	C₄⋊C₄	—	158	2	no	—
202	C₄⋊C₄	—	158	2	no	—
203	C₂²⋊C₄	—	158	4	no	—
204	C₄²	—	158	4	no	—
205	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
206	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
207	C₄²	#11929624	1	1	yes	characteristic
208	C₄²	C₂³×D₇₉	1	1	yes	characteristic
209	C₂³	C₄×D₃₁₆	1	1	yes	characteristic
210	Q₈	—	316	4	no	—
211	Q₈	—	158	8	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	158	4	no	—
215	C₂×C₄	—	158	4	no	—
216	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
217	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
218	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
219	C₂×C₄	C₂×Dic₃₁₆	1	2	yes	—
220	C₂×C₄	#14579183	1	2	yes	—
221	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
223	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	2	no	—
227	C₈	—	4	2	no	—
228	C₈	—	316	4	no	—
229	C₈	C₂×Dic₃₁₆	1	4	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	C₄×Dic₃₁₆	1	2	yes	—
233	C₂²	C₂×C₄×D₃₁₆	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	316	4	no	—
237	C₄	—	316	4	no	—
238	C₄	C₄×Dic₃₁₆	1	2	yes	—
239	C₄	#19098686	1	2	yes	—
240	C₄	#19099329	1	2	yes	—
241	C₄	C₂²×Dic₃₁₆	1	1	yes	characteristic
242	C₄	C₂×#14579183	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	#23772119	1	1	yes	characteristic
245	C₂	C₂×C₄×Dic₃₁₆	1	1	yes	characteristic
246	C₂	C₂×#19098686	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₄²	C₂³×D₇₉
Fitting	C₂×#322×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	C₂³×C₇₉	C₄×D₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	C₂×#322	D₇₉
hypocenter	C₇₉	#17977
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #23762252	C₂	semidirect
76. C₂×C₈×C₇₉	Q₁₆	semidirect
130. C₇₉	#17977	semidirect
192. C₂×C₈	Dic₃₁₆	semidirect

Maximal subgroups

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

subgroup	index
#23773986	2
#23772334	2
#23762252	2
#23762159	2
#23761856	2
C₂×#322×C₇₉	2
#17977	79