Subgroups — #34187324

Order 20224.

498 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	#23771778	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098311	C₂	1	2	yes	maximal
4	C₂×#19098308	C₂	1	2	yes	maximal
5	C₇₉×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579161	C₂²	1	1	yes	characteristic
7	C₂×#14579161	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄×C₇₉⋊C₈	—	2	1	no	—
10	#19098469	—	2	1	no	—
11	#19098469	—	2	1	no	—
12	#19098469	C₄	1	4	yes	—
13	#19098429	—	2	2	no	—
14	#19098318	—	2	1	no	—
15	#19098311	—	2	4	no	—
16	#19098308	—	2	4	no	—
17	C₂×C₄⋊C₈×C₇₉	—	2	2	no	—
18	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
19	C₄³×C₇₉	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₅₈	—	2	1	no	—
21	C₂²×Dic₁₅₈	C₂×C₄	1	2	yes	—
22	C₂²×C₇₉⋊C₈	—	2	2	no	—
23	#14579161	—	2	2	no	—
24	#14579161	—	2	2	no	—
25	#14579161	—	2	2	no	—
26	#14579161	—	2	2	no	—
27	C₂×C₄⋊Dic₇₉	—	2	1	no	—
28	C₂×C₄⋊Dic₇₉	—	2	1	no	—
29	C₂×C₄⋊Dic₇₉	—	2	1	no	—
30	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
31	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
32	C₄⋊(C₇₉⋊C₈)	D₄	1	4	yes	—
33	C₄×C₇₉⋊C₈	—	4	2	no	—
34	C₇₉⋊#95	—	2	4	no	—
35	C₇₉⋊#95	—	2	4	no	—
36	C₂²×C₈×C₇₉	—	2	2	no	—
37	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₇₉	D₄	1	2	yes	—
39	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₇₉	—	2	2	no	—
43	C₄⋊C₈×C₇₉	—	2	2	no	—
44	C₄⋊C₈×C₇₉	—	4	4	no	—
45	C₂×Dic₁₅₈	—	2	4	no	—
46	C₂×Dic₁₅₈	—	4	4	no	—
47	C₂×Dic₁₅₈	—	4	2	no	—
48	C₂×Dic₁₅₈	—	4	2	no	—
49	C₂×Dic₁₅₈	—	4	1	no	—
50	C₂×Dic₁₅₈	—	4	1	no	—
51	C₂²×Dic₇₉	—	2	2	no	—
52	C₂²×Dic₇₉	—	2	2	no	—
53	C₂×C₇₉⋊C₈	—	2	4	no	—
54	C₂×C₇₉⋊C₈	—	4	2	no	—
55	C₂×C₇₉⋊C₈	—	4	2	no	—
56	C₄⋊Dic₇₉	—	2	4	no	—
57	C₄⋊Dic₇₉	—	4	2	no	—
58	C₄⋊Dic₇₉	—	4	1	no	—
59	C₄⋊Dic₇₉	—	4	1	no	—
60	C₄⋊Dic₇₉	—	4	1	no	—
61	C₄⋊Dic₇₉	—	4	1	no	—
62	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₇₉	—	4	2	no	—
68	C₂×C₈×C₇₉	—	4	4	no	—
69	C₄²×C₇₉	—	2	2	no	—
70	C₄²×C₇₉	—	2	1	no	—
71	C₄²×C₇₉	—	2	1	no	—
72	C₄²×C₇₉	—	4	2	no	—
73	C₄²×C₇₉	SD₁₆	1	4	yes	—
74	C₄²×C₇₉	Q₁₆	1	4	yes	—
75	C₄²×C₇₉	C₂×D₄	1	2	yes	—
76	C₄²×C₇₉	#41	1	2	yes	—
77	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₉	—	2	2	no	—
79	C₂×C₄×C₇₉	—	2	2	no	—
80	C₂×C₄×C₇₉	—	2	2	no	—
81	C₂×C₄×C₇₉	—	2	2	no	—
82	C₂×C₄×C₇₉	—	2	2	no	—
83	C₂×C₄×C₇₉	#103	1	4	yes	—
84	C₂×C₄×C₇₉	#103	1	4	yes	—
85	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
86	C₂×C₄×C₇₉	#121	1	2	yes	—
87	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
88	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
89	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
90	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
91	C₂×C₄×C₇₉	#135	1	2	yes	—
92	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
93	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
94	C₂×Dic₇₉	—	4	2	no	—
95	C₂×Dic₇₉	—	4	2	no	—
96	C₂×Dic₇₉	—	4	4	no	—
97	C₂×Dic₇₉	—	4	2	no	—
98	C₂×Dic₇₉	—	4	2	no	—
99	Dic₁₅₈	—	8	2	no	—
100	Dic₁₅₈	—	8	2	no	—
101	Dic₁₅₈	—	8	4	no	—
102	C₈×C₇₉	—	8	2	no	—
103	C₇₉⋊C₈	—	4	4	no	—
104	C₂²×C₇₉	#390	1	1	yes	characteristic
105	C₂²×C₇₉	#413	1	2	yes	—
106	C₂²×C₇₉	C₂×#103	1	2	yes	—
107	C₂²×C₇₉	#416	1	1	yes	characteristic
108	C₂²×C₇₉	#417	1	1	yes	characteristic
109	C₄×C₇₉	—	2	4	no	—
110	C₄×C₇₉	—	2	4	no	—
111	C₄×C₇₉	—	2	2	no	—
112	C₄×C₇₉	#463	1	2	yes	—
113	C₄×C₇₉	#464	1	2	yes	—
114	C₄×C₇₉	#487	1	2	yes	—
115	C₄×C₇₉	#492	1	1	yes	characteristic
116	C₄×C₇₉	#494	1	1	yes	characteristic
117	Dic₇₉	—	8	2	no	—
118	Dic₇₉	—	8	2	no	—
119	#9697	—	79	1	no	maximal, Sylow
120	C₂×C₇₉	#1542	1	2	yes	—
121	C₂×C₇₉	#1550	1	2	yes	—
122	C₂×C₇₉	#1956	1	1	yes	characteristic
123	C₂×C₇₉	#1967	1	1	yes	characteristic
124	C₂×C₇₉	#1983	1	1	yes	characteristic
125	#2862	—	79	1	no	—
126	#1842	D₇₉	1	1	yes	characteristic
127	C₄×C₄⋊C₈	—	79	1	no	—
128	C₂×#333	—	79	2	no	—
129	C₂×#332	—	79	2	no	—
130	C₇₉	#9697	1	1	yes	characteristic, complemented, Sylow
131	C₂×C₄⋊Q₈	—	79	1	no	—
132	C₂×C₄⋊Q₈	—	79	1	no	—
133	C₂×C₄⋊C₈	—	2	2	no	—
134	C₂×C₄⋊C₈	—	79	2	no	—
135	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
136	C₂×C₄×C₈	—	158	1	no	—
137	#391	—	79	4	no	—
138	#391	—	158	1	no	—
139	#391	—	158	1	no	—
140	#382	—	158	2	no	—
141	C₄³	C₂×D₇₉	1	1	yes	characteristic
142	#336	—	158	1	no	—
143	#333	—	158	4	no	—
144	#332	—	158	4	no	—
145	C₂²×Q₈	—	79	2	no	—
146	C₂²×Q₈	—	158	1	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	158	2	no	—
149	C₄⋊Q₈	—	158	2	no	—
150	C₄⋊Q₈	—	158	2	no	—
151	C₄⋊Q₈	—	158	2	no	—
152	C₄⋊Q₈	—	158	2	no	—
153	C₂×C₄⋊C₄	—	79	2	no	—
154	C₂×C₄⋊C₄	—	158	1	no	—
155	C₂×C₄⋊C₄	—	158	1	no	—
156	C₂×C₄⋊C₄	—	158	1	no	—
157	C₂×C₄²	C₄×D₇₉	1	2	yes	—
158	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
159	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
160	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
161	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	4	4	no	—
165	C₄⋊C₈	—	79	4	no	—
166	C₄⋊C₈	—	158	2	no	—
167	C₄×C₈	—	316	2	no	—
168	#95	—	158	4	no	—
169	#95	—	158	4	no	—
170	C₂×Q₈	—	316	4	no	—
171	C₂×Q₈	—	316	2	no	—
172	C₂×Q₈	—	316	2	no	—
173	C₂×Q₈	—	316	1	no	—
174	C₂×Q₈	—	316	1	no	—
175	C₂×Q₈	—	158	4	no	—
176	C₂²×C₄	—	158	2	no	—
177	C₂²×C₄	—	158	2	no	—
178	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
179	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
180	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	4	no	—
185	C₂×C₈	—	316	2	no	—
186	C₂×C₈	—	316	2	no	—
187	C₂×C₈	—	158	4	no	—
188	C₄⋊C₄	—	316	2	no	—
189	C₄⋊C₄	—	316	1	no	—
190	C₄⋊C₄	—	316	1	no	—
191	C₄⋊C₄	—	316	1	no	—
192	C₄⋊C₄	—	316	1	no	—
193	C₄⋊C₄	—	158	4	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	4	2	no	—
198	C₄²	#11929612	1	4	yes	—
199	C₄²	#11929614	1	4	yes	—
200	C₄²	D₄×D₇₉	1	2	yes	—
201	C₄²	#11929625	1	2	yes	—
202	C₂³	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
203	Q₈	—	632	4	no	—
204	Q₈	—	632	2	no	—
205	Q₈	—	632	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	316	2	no	—
212	C₂×C₄	—	316	2	no	—
213	C₂×C₄	—	316	4	no	—
214	C₂×C₄	—	316	2	no	—
215	C₂×C₄	—	316	2	no	—
216	C₂×C₄	#14579084	1	4	yes	—
217	C₂×C₄	#14579084	1	4	yes	—
218	C₂×C₄	#14579123	1	2	yes	—
219	C₂×C₄	#14579136	1	2	yes	—
220	C₂×C₄	#14579150	1	1	yes	characteristic
221	C₂×C₄	#14579157	1	1	yes	characteristic
222	C₂×C₄	C₂×#11929612	1	2	yes	—
223	C₂×C₄	C₂×#11929614	1	2	yes	—
224	C₂×C₄	#14579183	1	2	yes	—
225	C₂×C₄	#14579189	1	1	yes	characteristic
226	C₂×C₄	#14579194	1	1	yes	characteristic
227	C₈	—	8	2	no	—
228	C₈	—	316	4	no	—
229	C₂²	#19098464	1	1	yes	characteristic
230	C₂²	#19098545	1	2	yes	—
231	C₂²	C₂×#14579084	1	2	yes	—
232	C₂²	#19098560	1	1	yes	characteristic
233	C₂²	#19098565	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	4	no	—
236	C₄	—	2	4	no	—
237	C₄	—	632	2	no	—
238	C₄	—	632	2	no	—
239	C₄	#19098779	1	2	yes	—
240	C₄	#19098786	1	2	yes	—
241	C₄	#19098915	1	2	yes	—
242	C₄	#19098935	1	1	yes	characteristic
243	C₄	#19098942	1	1	yes	characteristic
244	C₂	#23762058	1	2	yes	—
245	C₂	#23762091	1	2	yes	—
246	C₂	#23764201	1	1	yes	characteristic
247	C₂	#23764247	1	1	yes	characteristic
248	C₂	#23764330	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1842	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1842	D₇₉
hypocenter	C₇₉	#9697
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
130. C₇₉	#9697	semidirect

Maximal subgroups

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

subgroup	index
#23771778	2
C₄×C₄⋊(C₇₉⋊C₈)	2
C₂×#19098311	2
C₂×#19098308	2
C₇₉×#1842	2
#9697	79