Subgroups — #34198677

Order 20224.

394 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765199	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#14579095	C₂	1	1	yes	maximal, characteristic, complemented
3	#23761409	C₂	1	4	yes	maximal, complemented
4	C₇₉×#3169	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579095	—	2	1	no	—
6	C₄×C₄⋊Dic₇₉	—	2	1	no	—
7	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
8	C₂²⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
9	C₄×C₇₉⋊C₁₆	—	2	1	no	—
10	#19098326	C₂²	1	2	yes	—
11	#532×C₇₉	—	2	1	no	—
12	#500×C₇₉	C₄	1	1	yes	characteristic, complemented
13	#493×C₇₉	C₄	1	1	yes	characteristic, complemented
14	#486×C₇₉	—	2	1	no	—
15	#466×C₇₉	—	2	1	no	—
16	#466×C₇₉	C₂²	1	2	yes	—
17	C₄×D₈×C₇₉	—	2	1	no	—
18	C₄×D₈×C₇₉	C₄	1	2	yes	complemented
19	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₇₉	—	2	2	no	—
21	C₂×C₄×Dic₇₉	—	4	1	no	—
22	C₂×C₇₉⋊C₁₆	—	2	2	no	—
23	#14579095	—	2	4	no	—
24	C₇₉⋊#95	—	4	1	no	—
25	C₂×#41×C₇₉	—	4	1	no	—
26	C₂×D₈×C₇₉	—	2	1	no	—
27	C₂×D₈×C₇₉	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₄⋊Q₈×C₇₉	—	2	1	no	—
31	#127×C₇₉	—	2	1	no	—
32	#124×C₇₉	—	4	1	no	—
33	#121×C₇₉	—	2	2	no	—
34	#121×C₇₉	—	4	1	no	—
35	C₄×D₄×C₇₉	—	2	2	no	—
36	C₄×D₄×C₇₉	—	4	1	no	—
37	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
38	#107×C₇₉	—	2	1	no	—
39	#107×C₇₉	C₂×C₄	1	2	yes	—
40	#102×C₇₉	—	2	2	no	—
41	#102×C₇₉	—	2	2	no	—
42	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
46	C₂²×Dic₇₉	—	4	2	no	—
47	C₄⋊Dic₇₉	—	2	4	no	—
48	C₄⋊Dic₇₉	—	4	2	no	—
49	C₄×Dic₇₉	—	8	2	no	—
50	C₇₉⋊C₁₆	—	4	2	no	—
51	#41×C₇₉	—	8	2	no	—
52	#41×C₇₉	—	8	2	no	—
53	C₂×Q₈×C₇₉	—	4	1	no	—
54	C₂×D₄×C₇₉	—	2	2	no	—
55	C₂×D₄×C₇₉	—	4	2	no	—
56	C₂×D₄×C₇₉	—	4	1	no	—
57	C₂×D₄×C₇₉	—	4	1	no	—
58	C₂²×C₄×C₇₉	—	4	2	no	—
59	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
62	D₈×C₇₉	—	4	2	no	—
63	D₈×C₇₉	—	4	2	no	—
64	C₂×C₈×C₇₉	—	2	2	no	—
65	C₂×C₈×C₇₉	—	4	1	no	—
66	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₄⋊C₄×C₇₉	—	2	2	no	—
71	C₄⋊C₄×C₇₉	—	4	1	no	—
72	C₂²⋊C₄×C₇₉	—	4	2	no	—
73	C₂²⋊C₄×C₇₉	—	4	2	no	—
74	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
77	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
78	C₂³×C₇₉	—	4	2	no	—
79	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
80	Q₈×C₇₉	—	8	2	no	—
81	D₄×C₇₉	—	4	2	no	—
82	D₄×C₇₉	—	4	2	no	—
83	D₄×C₇₉	—	8	2	no	—
84	D₄×C₇₉	—	8	2	no	—
85	D₄×C₇₉	—	8	2	no	—
86	D₄×C₇₉	—	8	2	no	—
87	C₂×C₄×C₇₉	—	2	1	no	—
88	C₂×C₄×C₇₉	—	2	1	no	—
89	C₂×C₄×C₇₉	—	4	1	no	—
90	C₂×C₄×C₇₉	—	4	2	no	—
91	C₂×C₄×C₇₉	—	8	4	no	—
92	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
93	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
94	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
95	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
96	C₂×C₄×C₇₉	D₁₆	1	2	yes	—
97	C₂×C₄×C₇₉	SD₃₂	1	2	yes	—
98	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×Dic₇₉	—	4	4	no	—
101	C₂×Dic₇₉	—	8	2	no	—
102	C₂×Dic₇₉	—	8	2	no	—
103	C₈×C₇₉	—	2	1	no	—
104	C₈×C₇₉	—	2	1	no	—
105	C₈×C₇₉	—	2	1	no	—
106	C₈×C₇₉	#104	1	2	yes	—
107	C₂²×C₇₉	—	2	1	no	—
108	C₂²×C₇₉	—	2	1	no	—
109	C₂²×C₇₉	—	8	2	no	—
110	C₂²×C₇₉	—	8	2	no	—
111	C₂²×C₇₉	—	8	2	no	—
112	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
113	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
114	C₂²×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
115	C₄×C₇₉	—	2	1	no	—
116	C₄×C₇₉	—	2	1	no	—
117	C₄×C₇₉	—	2	1	no	—
118	C₄×C₇₉	—	8	2	no	—
119	C₄×C₇₉	#357	1	2	yes	—
120	C₄×C₇₉	#357	1	2	yes	—
121	C₄×C₇₉	#410	1	1	yes	characteristic
122	C₄×C₇₉	C₂×#104	1	1	yes	characteristic
123	Dic₇₉	—	8	4	no	—
124	#11742	—	79	1	no	maximal, Sylow
125	C₂×C₇₉	—	2	1	no	—
126	C₂×C₇₉	—	2	1	no	—
127	C₂×C₇₉	—	16	2	no	—
128	C₂×C₇₉	#1502	1	1	yes	characteristic
129	C₂×C₇₉	C₂×#357	1	1	yes	characteristic
130	C₂×C₇₉	#2145	1	1	yes	characteristic
131	#3169	D₇₉	1	1	yes	characteristic
132	#2162	—	79	1	no	—
133	C₄×#107	—	79	1	no	—
134	#1347	—	79	4	no	—
135	C₇₉	#11742	1	1	yes	characteristic, complemented, Sylow
136	#532	—	2	1	no	—
137	#500	Dic₇₉	1	1	yes	characteristic, complemented
138	#493	Dic₇₉	1	1	yes	characteristic, complemented
139	#486	—	2	1	no	—
140	#466	—	2	1	no	—
141	#466	C₂×D₇₉	1	2	yes	—
142	C₄×D₈	—	2	1	no	—
143	C₄×D₈	Dic₇₉	1	2	yes	complemented
144	C₂×#107	—	158	1	no	—
145	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
146	C₄×C₄⋊C₄	—	158	1	no	—
147	C₄⋊C₁₆	—	158	1	no	—
148	C₂²⋊C₁₆	—	79	2	no	—
149	C₄×C₁₆	—	158	1	no	—
150	#340	—	79	2	no	—
151	C₂×#41	—	4	1	no	—
152	C₂×D₈	—	2	1	no	—
153	C₂×D₈	C₂×Dic₇₉	1	2	yes	—
154	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
156	C₄⋊Q₈	—	2	1	no	—
157	#127	—	2	1	no	—
158	#124	—	4	1	no	—
159	#121	—	2	2	no	—
160	#121	—	4	1	no	—
161	C₄×D₄	—	2	2	no	—
162	C₄×D₄	—	4	1	no	—
163	C₂×C₄⋊C₄	—	158	2	no	—
164	C₂×C₄²	—	316	1	no	—
165	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
166	C₂×C₁₆	—	158	2	no	—
167	#107	—	2	1	no	—
168	#107	—	158	4	no	—
169	#107	C₂×Dic₇₉	1	2	yes	—
170	#102	—	2	2	no	—
171	#102	—	2	2	no	—
172	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
173	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
175	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
176	#95	—	316	1	no	—
177	#41	—	8	2	no	—
178	#41	—	8	2	no	—
179	C₂×Q₈	—	4	1	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂×D₄	—	4	1	no	—
183	C₂×D₄	—	4	1	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	316	2	no	—
186	C₂²×C₄	#11929609	1	1	yes	characteristic
187	C₂²×C₄	#11929609	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
189	D₈	—	4	2	no	—
190	D₈	—	4	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	4	1	no	—
193	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
194	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
195	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
196	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	4	1	no	—
199	C₄⋊C₄	—	316	2	no	—
200	C₄⋊C₄	—	158	4	no	—
201	C₂²⋊C₄	—	4	2	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₄²	—	632	2	no	—
204	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
205	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
206	C₄²	#11929612	1	1	yes	characteristic
207	C₄²	#11929612	1	1	yes	characteristic
208	C₁₆	—	316	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	C₂×#11929609	1	1	yes	characteristic
211	Q₈	—	8	2	no	—
212	D₄	—	4	2	no	—
213	D₄	—	4	2	no	—
214	D₄	—	8	2	no	—
215	D₄	—	8	2	no	—
216	D₄	—	8	2	no	—
217	D₄	—	8	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	4	no	—
223	C₂×C₄	—	316	4	no	—
224	C₂×C₄	—	632	2	no	—
225	C₂×C₄	—	632	2	no	—
226	C₂×C₄	#14579082	1	1	yes	characteristic
227	C₂×C₄	#14579082	1	1	yes	characteristic
228	C₂×C₄	#14579082	1	1	yes	characteristic
229	C₂×C₄	#14579082	1	1	yes	characteristic
230	C₂×C₄	#14579102	1	2	yes	—
231	C₂×C₄	#14579105	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
233	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	#14579088	1	2	yes	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	8	2	no	—
243	C₂²	C₂×#14579082	1	1	yes	characteristic
244	C₂²	C₂×#14579102	1	1	yes	characteristic
245	C₂²	C₂×#14579105	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	8	2	no	—
250	C₄	—	632	4	no	—
251	C₄	#19098364	1	2	yes	—
252	C₄	#19098364	1	2	yes	—
253	C₄	#19098537	1	1	yes	characteristic
254	C₄	C₂×#14579088	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	—	16	2	no	—
258	C₂	#23761895	1	1	yes	characteristic
259	C₂	C₂×#19098364	1	1	yes	characteristic
260	C₂	#23765136	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579082
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting, prime radical	C₇₉×#3169	C₂
socle	C₂²×C₇₉	C₂×#102
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#3169	D₇₉
hypocenter	C₇₉	#11742
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #23765199	C₂	semidirect
2. C₄×#14579095	C₂	semidirect
3. #23761409	C₂	semidirect
12. #500×C₇₉	C₄	semidirect
13. #493×C₇₉	C₄	semidirect
18. C₄×D₈×C₇₉	C₄	semidirect
135. C₇₉	#11742	semidirect
137. #500	Dic₇₉	semidirect
138. #493	Dic₇₉	semidirect
143. C₄×D₈	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23765199	2
C₄×#14579095	2
#23761409	2
C₇₉×#3169	2
#11742	79