Subgroups — #34443628

Order 20224.

578 conjugacy classes of subgroups.

The subgroup lattice has 234 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	C₂×#19098943	C₂	1	1	yes	maximal, characteristic
3	#23765646	C₂	1	4	yes	maximal, complemented
4	#23765628	C₂	1	4	yes	maximal, complemented
5	#23765121	C₂	1	4	yes	maximal
6	C₄×M₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₁₆	—	2	1	no	—
8	C₂²×Dic₃₁₆	C₂²	1	2	yes	—
9	#19099120	—	2	1	no	—
10	C₂×#14579161	—	2	1	no	—
11	#19098968	C₂²	1	2	yes	—
12	#19098965	—	2	1	no	—
13	#19098965	C₂²	1	2	yes	—
14	#19098961	—	2	1	no	—
15	#19098943	—	2	1	no	—
16	#19098943	—	2	1	no	—
17	#19098943	C₂²	1	2	yes	—
18	#19098943	C₂²	1	2	yes	—
19	#19098601	—	2	1	no	—
20	#19098601	C₂²	1	2	yes	—
21	#19098368	C₂²	1	16	yes	—
22	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	#346×C₇₉	C₂²	1	2	yes	—
25	C₄×C₁₆×C₇₉	C₂²	1	2	yes	—
26	C₂²×Dic₁₅₈	—	2	2	no	—
27	C₂×Dic₃₁₆	—	2	8	no	—
28	C₂×Dic₃₁₆	—	2	2	no	—
29	C₂×Dic₃₁₆	—	2	2	no	—
30	C₂×Dic₃₁₆	—	2	2	no	—
31	C₂×Dic₃₁₆	—	4	4	no	—
32	C₂×Dic₃₁₆	C₂³	1	4	yes	—
33	#14579161	—	2	2	no	—
34	#14579161	—	2	2	no	—
35	#14579161	—	2	2	no	—
36	#14579161	—	4	2	no	—
37	#14579153	—	2	2	no	—
38	#14579140	—	4	2	no	—
39	C₄×Dic₁₅₈	—	4	2	no	—
40	#14579120	—	2	2	no	—
41	C₂×C₄⋊Dic₇₉	—	4	2	no	—
42	#14579095	—	2	2	no	—
43	#14579095	C₂³	1	4	yes	—
44	#14579093	—	2	2	no	—
45	#14579093	—	2	2	no	—
46	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
49	M₃₂×C₇₉	—	2	4	no	—
50	C₂×C₁₆×C₇₉	C₂³	1	4	yes	—
51	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₅₈	—	2	4	no	—
56	C₂×Dic₁₅₈	—	4	4	no	—
57	C₂×Dic₁₅₈	—	4	2	no	—
58	C₂×Dic₁₅₈	—	4	2	no	—
59	C₂×Dic₁₅₈	—	4	4	no	—
60	C₂²×Dic₇₉	—	4	2	no	—
61	Dic₃₁₆	—	2	8	no	—
62	Dic₃₁₆	—	4	4	no	—
63	Dic₃₁₆	—	4	4	no	—
64	Dic₃₁₆	—	4	4	no	—
65	C₄⋊Dic₇₉	—	2	4	no	—
66	C₄⋊Dic₇₉	—	4	4	no	—
67	C₄⋊Dic₇₉	—	4	4	no	—
68	C₄⋊Dic₇₉	—	4	2	no	—
69	Dic₇₉⋊C₄	—	4	4	no	—
70	C₂²⋊Dic₇₉	—	4	2	no	—
71	C₄×Dic₇₉	—	4	2	no	—
72	C₂²×C₄×C₇₉	D₈	1	2	yes	—
73	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
75	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₇₉	#41	1	4	yes	—
79	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
80	C₄²×C₇₉	D₈	1	2	yes	—
81	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
83	C₁₆×C₇₉	—	2	4	no	—
84	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₇₉	—	2	4	no	—
86	C₂×C₄×C₇₉	#124	1	2	yes	—
87	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
91	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
92	C₂×Dic₇₉	—	4	4	no	—
93	C₂×Dic₇₉	—	4	4	no	—
94	C₂×Dic₇₉	—	8	4	no	—
95	Dic₁₅₈	—	4	4	no	—
96	Dic₁₅₈	—	4	4	no	—
97	Dic₁₅₈	—	8	4	no	—
98	Dic₁₅₈	—	8	4	no	—
99	C₈×C₇₉	#124	1	2	yes	—
100	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
101	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
102	C₈×C₇₉	C₂×#41	1	2	yes	—
103	C₂²×C₇₉	—	2	2	no	—
104	C₂²×C₇₉	#486	1	2	yes	—
105	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
106	C₄×C₇₉	—	2	2	no	—
107	C₄×C₇₉	#486	1	2	yes	—
108	C₄×C₇₉	#510	1	4	yes	—
109	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
110	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
111	Dic₇₉	—	8	4	no	—
112	Dic₇₉	—	8	4	no	—
113	#33705	—	79	1	no	maximal, Sylow
114	C₂×C₇₉	—	2	2	no	—
115	C₂×C₇₉	C₂×#486	1	1	yes	characteristic
116	C₂×C₇₉	C₂×#510	1	2	yes	—
117	#3160	—	79	1	no	—
118	C₂×#494	—	79	1	no	—
119	#2245	—	79	4	no	—
120	#2241	—	79	4	no	—
121	#2141	—	79	4	no	—
122	C₄×M₃₂	D₇₉	1	1	yes	characteristic
123	C₇₉	#33705	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₁₆	—	79	2	no	—
125	C₂²×Q₁₆	—	158	1	no	—
126	#533	—	158	1	no	—
127	C₂×C₄⋊Q₈	—	158	1	no	—
128	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
129	#500	—	79	2	no	—
130	#499	—	79	2	no	—
131	#499	—	158	1	no	—
132	#498	—	158	1	no	—
133	#494	—	79	2	no	—
134	#494	—	79	2	no	—
135	#494	—	158	1	no	—
136	#494	—	158	1	no	—
137	#427	—	79	2	no	—
138	#427	—	158	1	no	—
139	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
140	#358	—	79	16	no	—
141	#346	C₂×D₇₉	1	2	yes	—
142	C₄×C₁₆	C₂×D₇₉	1	2	yes	—
143	C₂²×Q₈	—	158	2	no	—
144	C₂×Q₁₆	—	316	4	no	—
145	C₂×Q₁₆	—	79	4	no	—
146	C₂×Q₁₆	—	158	2	no	—
147	C₂×Q₁₆	—	158	2	no	—
148	C₂×Q₁₆	—	158	2	no	—
149	C₂×Q₁₆	—	158	8	no	—
150	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
151	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
152	C₄⋊Q₈	—	316	2	no	—
153	C₄⋊Q₈	—	158	2	no	—
154	C₄⋊Q₈	—	158	2	no	—
155	C₄⋊Q₈	—	158	2	no	—
156	#125	—	158	2	no	—
157	C₂²⋊Q₈	—	316	2	no	—
158	C₄×Q₈	—	316	2	no	—
159	#117	—	158	2	no	—
160	C₂×C₄⋊C₄	—	316	2	no	—
161	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
162	M₃₂	—	2	4	no	—
163	C₂×C₁₆	C₂²×D₇₉	1	4	yes	—
164	#107	—	79	4	no	—
165	#107	—	158	2	no	—
166	#106	—	158	2	no	—
167	#106	—	158	2	no	—
168	C₄×C₈	D₃₁₆	1	1	yes	characteristic
169	C₄×C₈	D₃₁₆	1	1	yes	characteristic
170	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
171	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
172	C₂×Q₈	—	316	2	no	—
173	C₂×Q₈	—	316	2	no	—
174	C₂×Q₈	—	316	4	no	—
175	C₂×Q₈	—	316	4	no	—
176	C₂×Q₈	—	158	4	no	—
177	C₂²×C₄	—	316	2	no	—
178	C₂²×C₄	D₆₃₂	1	2	yes	—
179	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
180	Q₁₆	—	316	4	no	—
181	Q₁₆	—	316	4	no	—
182	Q₁₆	—	316	4	no	—
183	Q₁₆	—	158	8	no	—
184	C₂×C₈	#11929613	1	4	yes	—
185	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
186	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
187	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
188	C₂×C₈	#11929624	1	4	yes	—
189	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
190	C₄⋊C₄	—	316	4	no	—
191	C₄⋊C₄	—	316	4	no	—
192	C₄⋊C₄	—	316	4	no	—
193	C₄⋊C₄	—	316	2	no	—
194	C₄⋊C₄	—	158	4	no	—
195	C₂²⋊C₄	—	316	2	no	—
196	C₄²	—	316	2	no	—
197	C₄²	D₆₃₂	1	2	yes	—
198	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
199	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
200	C₁₆	—	2	4	no	—
201	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
202	Q₈	—	316	4	no	—
203	Q₈	—	316	4	no	—
204	Q₈	—	632	4	no	—
205	Q₈	—	632	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	316	4	no	—
208	C₂×C₄	—	316	4	no	—
209	C₂×C₄	—	632	4	no	—
210	C₂×C₄	#14579149	1	2	yes	—
211	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
212	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
213	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
214	C₂×C₄	C₂×#11929613	1	2	yes	—
215	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
216	C₈	#14579149	1	2	yes	—
217	C₈	C₂×#11929613	1	2	yes	—
218	C₈	C₂×#11929613	1	2	yes	—
219	C₈	C₂×#11929624	1	2	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	#19098910	1	2	yes	—
222	C₂²	C₂²×D₆₃₂	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	632	4	no	—
225	C₄	—	632	4	no	—
226	C₄	#19098910	1	2	yes	—
227	C₄	#19099006	1	4	yes	—
228	C₄	C₂×#14579149	1	1	yes	characteristic
229	C₄	C₂²×#11929613	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	C₂×#19098910	1	1	yes	characteristic
232	C₂	C₂×#19099006	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₆₃₂
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₄×M₃₂×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂³×C₇₉	C₂×D₈
hypercenter	C₄×M₃₂	D₇₉
hypocenter	C₇₉	#33705
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #23765646	C₂	semidirect
4. #23765628	C₂	semidirect
123. C₇₉	#33705	semidirect

Maximal subgroups

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

subgroup	index
#23773914	2
C₂×#19098943	2
#23765646	2
#23765628	2
#23765121	2
C₄×M₃₂×C₇₉	2
#33705	79