Subgroups — #34206836

Order 20224.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773914	C₂	1	1	yes	maximal, characteristic
2	C₄×#14579097	C₂	1	1	yes	maximal, characteristic
3	#23761496	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#19099120	—	2	1	no	—
6	#19098968	C₄	1	1	yes	characteristic
7	#19098968	C₄	1	1	yes	characteristic
8	#19098965	—	2	1	no	—
9	#19098965	C₄	1	2	yes	—
10	#19098961	—	2	1	no	—
11	C₂×#14579097	C₂²	1	2	yes	—
12	#19098601	—	2	1	no	—
13	#19098601	C₂²	1	2	yes	—
14	C₄×#11929608	—	2	1	no	—
15	#19098328	—	2	1	no	—
16	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₁₆×C₇₉	—	2	1	no	—
19	C₄⋊C₁₆×C₇₉	—	2	1	no	—
20	C₂×#11929608	—	2	2	no	—
21	#14579161	—	2	1	no	—
22	#14579161	—	2	1	no	—
23	#14579153	—	2	2	no	—
24	#14579140	—	4	1	no	—
25	#14579140	—	4	1	no	—
26	C₄×Dic₁₅₈	—	4	1	no	—
27	C₄×Dic₁₅₈	—	4	1	no	—
28	#14579120	—	2	2	no	—
29	#14579097	—	2	2	no	—
30	#14579097	Q₈	1	4	yes	complemented
31	#14579095	—	2	1	no	—
32	#14579095	—	2	1	no	—
33	#14579095	C₂×C₄	1	2	yes	—
34	#14579095	C₂×C₄	1	2	yes	—
35	#14579093	—	2	2	no	—
36	#14579093	—	2	2	no	—
37	#14579072	—	4	1	no	—
38	C₄×C₇₉⋊C₈	—	4	1	no	—
39	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₇₉	—	2	2	no	—
43	C₂×C₁₆×C₇₉	—	2	4	no	—
44	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
48	C₂×Dic₁₅₈	—	4	1	no	—
49	C₂×Dic₁₅₈	—	4	1	no	—
50	#11929608	—	2	4	no	—
51	#11929608	—	4	2	no	—
52	C₂×C₇₉⋊C₈	—	4	2	no	—
53	C₄⋊Dic₇₉	—	2	2	no	—
54	C₄⋊Dic₇₉	—	2	2	no	—
55	C₄⋊Dic₇₉	—	4	1	no	—
56	C₄⋊Dic₇₉	—	4	1	no	—
57	Dic₇₉⋊C₄	—	4	2	no	—
58	Dic₇₉⋊C₄	—	4	2	no	—
59	C₂²⋊Dic₇₉	—	4	2	no	—
60	C₄×Dic₇₉	—	4	2	no	—
61	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₇₉	—	2	2	no	—
65	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₇₉	#41	1	2	yes	—
71	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₇₉	D₈	1	1	yes	characteristic
74	C₄²×C₇₉	D₈	1	1	yes	characteristic
75	C₁₆×C₇₉	—	4	2	no	—
76	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₉	—	2	2	no	—
78	C₂×C₄×C₇₉	—	4	1	no	—
79	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
80	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
81	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
82	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
83	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
85	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₇₉	#137	1	2	yes	—
87	C₂×Dic₇₉	—	4	2	no	—
88	C₂×Dic₇₉	—	4	2	no	—
89	Dic₁₅₈	—	8	2	no	—
90	Dic₁₅₈	—	8	2	no	—
91	C₈×C₇₉	—	2	1	no	—
92	C₈×C₇₉	—	2	1	no	—
93	C₈×C₇₉	C₄×Q₈	1	2	yes	—
94	C₈×C₇₉	#124	1	1	yes	characteristic
95	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
96	C₇₉⋊C₈	—	4	4	no	—
97	C₂²×C₇₉	—	2	1	no	—
98	C₂²×C₇₉	—	2	1	no	—
99	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
100	C₂²×C₇₉	#475	1	2	yes	—
101	C₄×C₇₉	—	2	1	no	—
102	C₄×C₇₉	—	2	1	no	—
103	C₄×C₇₉	—	2	1	no	—
104	C₄×C₇₉	#359	1	2	yes	—
105	C₄×C₇₉	#391	1	1	yes	characteristic
106	C₄×C₇₉	#419	1	1	yes	characteristic
107	C₄×C₇₉	#474	1	2	yes	—
108	Dic₇₉	—	8	2	no	—
109	Dic₇₉	—	8	2	no	—
110	#13274	—	79	1	no	maximal, Sylow
111	C₂×C₇₉	—	2	2	no	—
112	C₂×C₇₉	#1988	1	1	yes	characteristic
113	C₂×C₇₉	C₂×#359	1	1	yes	characteristic
114	C₂×C₇₉	#2143	1	1	yes	characteristic
115	#3160	—	79	1	no	—
116	C₂×C₄⋊C₁₆	D₇₉	1	1	yes	characteristic
117	C₄×#108	—	79	1	no	—
118	#1382	—	79	4	no	—
119	C₇₉	#13274	1	1	yes	characteristic, complemented, Sylow
120	#533	—	158	1	no	—
121	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
122	#500	—	79	1	no	—
123	#500	—	79	1	no	—
124	#499	—	79	2	no	—
125	#499	—	158	1	no	—
126	#498	—	158	1	no	—
127	C₂×#108	—	79	2	no	—
128	#427	—	79	2	no	—
129	#427	—	158	1	no	—
130	C₄×M₁₆	—	158	1	no	—
131	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
132	C₄⋊C₁₆	—	2	1	no	—
133	C₄⋊C₁₆	—	2	1	no	—
134	#341	—	158	1	no	—
135	C₂×M₁₆	—	158	2	no	—
136	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
138	C₄⋊Q₈	—	158	1	no	—
139	C₄⋊Q₈	—	158	1	no	—
140	#125	—	158	2	no	—
141	C₂²⋊Q₈	—	316	1	no	—
142	C₂²⋊Q₈	—	316	1	no	—
143	C₄×Q₈	—	316	1	no	—
144	C₄×Q₈	—	316	1	no	—
145	#117	—	158	2	no	—
146	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	—	2	4	no	—
149	#108	—	79	4	no	—
150	#108	—	158	2	no	—
151	#107	—	79	2	no	—
152	#107	—	79	2	no	—
153	#107	—	158	1	no	—
154	#107	—	158	1	no	—
155	#106	—	158	2	no	—
156	#106	—	158	2	no	—
157	#97	—	316	1	no	—
158	C₄×C₈	—	316	1	no	—
159	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
160	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
163	C₂×Q₈	—	316	1	no	—
164	C₂×Q₈	—	316	1	no	—
165	C₂²×C₄	#11929613	1	1	yes	characteristic
166	C₂²×C₄	#11929613	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
168	M₁₆	—	316	2	no	—
169	M₁₆	—	158	4	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	316	2	no	—
172	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
173	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
174	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
175	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
176	C₂×C₈	Q₈×D₇₉	1	2	yes	—
177	C₂×C₈	#11929623	1	2	yes	—
178	C₄⋊C₄	—	316	2	no	—
179	C₄⋊C₄	—	316	2	no	—
180	C₄⋊C₄	—	316	1	no	—
181	C₄⋊C₄	—	316	1	no	—
182	C₄⋊C₄	—	158	2	no	—
183	C₄⋊C₄	—	158	2	no	—
184	C₂²⋊C₄	—	316	2	no	—
185	C₄²	—	316	2	no	—
186	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
187	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
188	C₄²	D₆₃₂	1	1	yes	characteristic
189	C₄²	D₆₃₂	1	1	yes	characteristic
190	C₁₆	—	4	2	no	—
191	C₂³	C₂×#11929613	1	1	yes	characteristic
192	Q₈	—	632	2	no	—
193	Q₈	—	632	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	316	2	no	—
197	C₂×C₄	—	316	2	no	—
198	C₂×C₄	#14579083	1	1	yes	characteristic
199	C₂×C₄	#14579083	1	1	yes	characteristic
200	C₂×C₄	#14579083	1	1	yes	characteristic
201	C₂×C₄	#14579083	1	1	yes	characteristic
202	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
203	C₂×C₄	#14579141	1	2	yes	—
204	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
205	C₂×C₄	#14579194	1	2	yes	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	316	4	no	—
209	C₈	#14579126	1	2	yes	—
210	C₈	#14579148	1	1	yes	characteristic
211	C₈	#14579159	1	1	yes	characteristic
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	C₂×#14579083	1	1	yes	characteristic
215	C₂²	#19098849	1	2	yes	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	632	2	no	—
220	C₄	—	632	2	no	—
221	C₄	#19098371	1	2	yes	—
222	C₄	#19098468	1	1	yes	characteristic
223	C₄	#19098573	1	1	yes	characteristic
224	C₄	#19098844	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	#23764351	1	1	yes	characteristic
227	C₂	C₂×#19098371	1	1	yes	characteristic
228	C₂	#23765129	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₃₂
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×C₄⋊C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#102
prime radical	C₂³×C₇₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₇₉
hypocenter	C₇₉	#13274
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
30. #14579097	Q₈	semidirect
119. C₇₉	#13274	semidirect

Maximal subgroups

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

subgroup	index
#23773914	2
C₄×#14579097	2
#23761496	2
C₂×C₄⋊C₁₆×C₇₉	2
#13274	79