Subgroups — #35157965

Order 20224.

1834 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23777219	C₂	1	2	yes	maximal
2	#23773946	C₂	1	6	yes	maximal, complemented
3	#23773939	C₂	1	12	yes	maximal, complemented
4	#23773914	C₂	1	3	yes	maximal, complemented
5	C₂×#19098968	C₂	1	3	yes	maximal, complemented
6	#23772000	C₂	1	4	yes	maximal
7	C₄×#131×C₇₉	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₁₅₈	—	2	2	no	—
9	#19099292	—	2	2	no	—
10	#19099270	—	2	6	no	—
11	D₄×Dic₁₅₈	—	2	6	no	—
12	#19099191	C₂²	1	12	yes	—
13	#19099120	C₂²	1	6	yes	—
14	C₂×#14579161	C₂²	1	6	yes	—
15	#19099046	C₂²	1	4	yes	—
16	#19098972	C₂²	1	24	yes	—
17	#19098968	C₂²	1	3	yes	—
18	#19098968	C₂²	1	12	yes	—
19	#19098968	C₂²	1	6	yes	—
20	#19098968	C₂²	1	1	yes	characteristic
21	#19098965	C₂²	1	12	yes	—
22	#19098961	C₂²	1	6	yes	—
23	#19098605	C₂²	1	24	yes	—
24	#19098601	C₂²	1	12	yes	—
25	C₂×#14579095	C₂²	1	12	yes	—
26	C₂×#131×C₇₉	C₂²	1	2	yes	—
27	C₄×#41×C₇₉	C₂²	1	1	yes	characteristic
28	#405×C₇₉	C₂²	1	6	yes	—
29	C₄×M₁₆×C₇₉	C₂²	1	3	yes	—
30	C₂×C₄×C₈×C₇₉	C₂²	1	3	yes	—
31	C₂²×Dic₁₅₈	—	2	6	no	—
32	#14579161	—	2	6	no	—
33	#14579161	—	2	6	no	—
34	#14579161	C₂³	1	12	yes	—
35	#14579161	C₂³	1	2	yes	—
36	#14579161	C₂³	1	6	yes	—
37	#14579160	—	2	12	no	—
38	#14579153	C₂³	1	12	yes	—
39	#14579152	—	2	12	no	—
40	#14579140	—	2	6	no	—
41	#14579140	—	2	6	no	—
42	#14579138	—	2	24	no	—
43	Q₈×Dic₇₉	—	2	4	no	—
44	C₄×Dic₁₅₈	—	2	6	no	—
45	C₄×Dic₁₅₈	—	2	6	no	—
46	D₄×Dic₇₉	—	2	12	no	—
47	#14579120	C₂³	1	12	yes	—
48	C₂×C₄⋊Dic₇₉	—	2	6	no	—
49	C₂×C₄⋊Dic₇₉	C₂³	1	12	yes	—
50	#14579095	C₂³	1	4	yes	—
51	#14579095	C₂³	1	12	yes	—
52	#14579095	C₂³	1	24	yes	—
53	#14579093	C₂³	1	24	yes	—
54	C₂×#41×C₇₉	C₂³	1	1	yes	characteristic
55	#131×C₇₉	Q₈	1	16	yes	complemented
56	C₂×M₁₆×C₇₉	C₂³	1	6	yes	—
57	C₂²×C₈×C₇₉	C₂³	1	6	yes	—
58	C₄×Q₈×C₇₉	D₄	1	1	yes	characteristic
59	C₄×Q₈×C₇₉	D₄	1	1	yes	characteristic
60	C₄×D₄×C₇₉	D₄	1	3	yes	—
61	C₄×D₄×C₇₉	D₄	1	3	yes	—
62	#117×C₇₉	C₂³	1	3	yes	—
63	C₂×C₄²×C₇₉	C₂³	1	3	yes	—
64	#97×C₇₉	C₂³	1	6	yes	—
65	C₄×C₈×C₇₉	C₂³	1	3	yes	—
66	C₄×C₈×C₇₉	C₂³	1	6	yes	—
67	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
68	C₂×Dic₁₅₈	—	2	6	no	—
69	C₂×Dic₁₅₈	—	2	2	no	—
70	C₂×Dic₁₅₈	—	2	12	no	—
71	C₂×Dic₁₅₈	—	8	12	no	—
72	C₂²×Dic₇₉	—	2	12	no	—
73	C₄⋊Dic₇₉	—	2	6	no	—
74	C₄⋊Dic₇₉	—	2	12	no	—
75	C₄⋊Dic₇₉	—	2	12	no	—
76	C₄⋊Dic₇₉	—	2	2	no	—
77	C₄⋊Dic₇₉	C₂⁴	1	4	yes	—
78	C₄⋊Dic₇₉	C₂⁴	1	12	yes	—
79	Dic₇₉⋊C₄	—	2	12	no	—
80	Dic₇₉⋊C₄	—	2	12	no	—
81	C₂²⋊Dic₇₉	—	2	12	no	—
82	C₄×Dic₇₉	—	2	12	no	—
83	#41×C₇₉	C₂×Q₈	1	8	yes	—
84	C₂×Q₈×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₂×D₄×C₇₉	C₂×D₄	1	3	yes	—
86	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
87	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
88	C₂²×C₄×C₇₉	C₂⁴	1	3	yes	—
89	M₁₆×C₇₉	C₂×Q₈	1	24	yes	complemented
90	C₂×C₈×C₇₉	C₂×Q₈	1	24	yes	complemented
91	C₂×C₈×C₇₉	C₂⁴	1	2	yes	—
92	C₂×C₈×C₇₉	C₂⁴	1	6	yes	—
93	C₄⋊C₄×C₇₉	C₂×D₄	1	3	yes	—
94	C₄⋊C₄×C₇₉	C₂×D₄	1	3	yes	—
95	C₂²⋊C₄×C₇₉	C₂×D₄	1	6	yes	—
96	C₄²×C₇₉	C₂×D₄	1	3	yes	—
97	C₄²×C₇₉	C₂×D₄	1	3	yes	—
98	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
99	C₄²×C₇₉	C₂⁴	1	3	yes	—
100	C₂³×C₇₉	C₂²×D₄	1	3	yes	—
101	Q₈×C₇₉	C₄⋊Q₈	1	4	yes	—
102	D₄×C₇₉	C₄⋊Q₈	1	12	yes	—
103	C₂×C₄×C₇₉	—	2	6	no	—
104	C₂×C₄×C₇₉	—	2	6	no	—
105	C₂×C₄×C₇₉	C₂²×D₄	1	3	yes	—
106	C₂×C₄×C₇₉	C₂²×D₄	1	3	yes	—
107	C₂×C₄×C₇₉	C₂²×D₄	1	3	yes	—
108	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₂²×Q₈	1	12	yes	—
111	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
112	C₂×Dic₇₉	—	2	4	no	—
113	C₂×Dic₇₉	—	2	12	no	—
114	C₂×Dic₇₉	—	8	12	no	—
115	Dic₁₅₈	—	4	12	no	—
116	Dic₁₅₈	—	4	4	no	—
117	Dic₁₅₈	—	8	12	no	—
118	C₈×C₇₉	C₂²×Q₈	1	4	yes	—
119	C₈×C₇₉	C₂²×Q₈	1	12	yes	—
120	C₂²×C₇₉	—	2	3	no	—
121	C₂²×C₇₉	—	2	3	no	—
122	C₂²×C₇₉	C₂×C₄⋊Q₈	1	6	yes	—
123	C₂²×C₇₉	C₂³×D₄	1	1	yes	characteristic
124	C₄×C₇₉	—	2	3	no	—
125	C₄×C₇₉	—	2	3	no	—
126	C₄×C₇₉	C₂×C₄⋊Q₈	1	6	yes	—
127	C₄×C₇₉	#576	1	2	yes	—
128	C₄×C₇₉	#578	1	2	yes	—
129	C₄×C₇₉	C₂³×Q₈	1	2	yes	—
130	Dic₇₉	—	4	12	no	—
131	Dic₇₉	—	4	4	no	—
132	#60667	—	79	1	no	maximal, Sylow
133	C₂×C₇₉	—	2	6	no	—
134	C₂×C₇₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
135	C₂×C₇₉	C₂×#576	1	1	yes	characteristic
136	C₂×C₇₉	C₂×#578	1	1	yes	characteristic
137	#3503	—	79	2	no	—
138	#3165	—	79	6	no	—
139	#3164	—	79	12	no	—
140	#3160	—	79	3	no	—
141	C₂×#500	—	79	3	no	—
142	#2913	—	79	4	no	—
143	C₄×#131	D₇₉	1	1	yes	characteristic
144	C₇₉	#60667	1	1	yes	characteristic, complemented, Sylow
145	C₂×#131	C₂×D₇₉	1	2	yes	—
146	Q₈²	—	158	2	no	—
147	#557	—	158	2	no	—
148	#554	—	158	6	no	—
149	D₄×Q₈	—	158	6	no	—
150	#544	—	79	12	no	—
151	#533	—	79	6	no	—
152	C₂×C₄⋊Q₈	—	79	6	no	—
153	#520	—	79	4	no	—
154	C₄×#41	C₂×D₇₉	1	1	yes	characteristic
155	#501	—	79	24	no	—
156	#500	—	79	3	no	—
157	#500	—	79	12	no	—
158	#500	—	79	6	no	—
159	#500	—	79	1	no	—
160	#499	—	79	12	no	—
161	#498	—	79	6	no	—
162	#428	—	79	24	no	—
163	#427	—	79	12	no	—
164	C₂×#107	—	79	12	no	—
165	#405	C₂×D₇₉	1	6	yes	—
166	C₄×M₁₆	C₂×D₇₉	1	3	yes	—
167	C₂×C₄×C₈	C₂×D₇₉	1	3	yes	—
168	C₂×#41	C₂²×D₇₉	1	1	yes	characteristic
169	C₂²×Q₈	—	158	6	no	—
170	#131	Dic₁₅₈	1	16	yes	complemented
171	C₂×M₁₆	C₂²×D₇₉	1	6	yes	—
172	C₂²×C₈	C₂²×D₇₉	1	6	yes	—
173	C₄⋊Q₈	—	79	6	no	—
174	C₄⋊Q₈	—	79	12	no	—
175	C₄⋊Q₈	—	79	2	no	—
176	C₄⋊Q₈	—	158	12	no	—
177	C₄⋊Q₈	—	158	6	no	—
178	C₄⋊Q₈	—	158	6	no	—
179	#125	—	79	12	no	—
180	#125	—	158	12	no	—
181	C₂²⋊Q₈	—	158	6	no	—
182	C₂²⋊Q₈	—	158	6	no	—
183	C₂²⋊Q₈	—	158	24	no	—
184	C₄×Q₈	—	158	4	no	—
185	C₄×Q₈	—	158	6	no	—
186	C₄×Q₈	—	158	6	no	—
187	C₄×Q₈	D₃₁₆	1	1	yes	characteristic
188	C₄×Q₈	D₃₁₆	1	1	yes	characteristic
189	C₄×D₄	—	158	12	no	—
190	C₄×D₄	D₃₁₆	1	3	yes	—
191	C₄×D₄	D₃₁₆	1	3	yes	—
192	#117	—	79	12	no	—
193	#117	C₂²×D₇₉	1	3	yes	—
194	C₂×C₄⋊C₄	—	79	12	no	—
195	C₂×C₄⋊C₄	—	158	6	no	—
196	C₂×C₄²	C₂²×D₇₉	1	3	yes	—
197	#107	—	79	12	no	—
198	#107	—	79	24	no	—
199	#107	—	79	4	no	—
200	#106	—	79	24	no	—
201	#97	C₂²×D₇₉	1	6	yes	—
202	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
203	C₄×C₈	C₂²×D₇₉	1	3	yes	—
204	C₄×C₈	C₂²×D₇₉	1	6	yes	—
205	#41	C₂×Dic₁₅₈	1	8	yes	—
206	C₂×Q₈	—	632	12	no	—
207	C₂×Q₈	—	158	2	no	—
208	C₂×Q₈	—	158	6	no	—
209	C₂×Q₈	—	158	12	no	—
210	C₂×Q₈	C₂×D₃₁₆	1	1	yes	characteristic
211	C₂×D₄	C₂×D₃₁₆	1	3	yes	—
212	C₂²×C₄	—	158	12	no	—
213	C₂²×C₄	C₂×D₃₁₆	1	3	yes	—
214	C₂²×C₄	C₂×D₃₁₆	1	3	yes	—
215	C₂²×C₄	C₂³×D₇₉	1	3	yes	—
216	M₁₆	C₂×Dic₁₅₈	1	24	yes	complemented
217	C₂×C₈	C₂×Dic₁₅₈	1	24	yes	complemented
218	C₂×C₈	C₂³×D₇₉	1	2	yes	—
219	C₂×C₈	C₂³×D₇₉	1	6	yes	—
220	C₄⋊C₄	—	79	4	no	—
221	C₄⋊C₄	—	79	12	no	—
222	C₄⋊C₄	—	158	12	no	—
223	C₄⋊C₄	—	158	12	no	—
224	C₄⋊C₄	—	158	12	no	—
225	C₄⋊C₄	—	158	6	no	—
226	C₄⋊C₄	—	158	12	no	—
227	C₄⋊C₄	—	158	2	no	—
228	C₄⋊C₄	C₂×D₃₁₆	1	3	yes	—
229	C₄⋊C₄	C₂×D₃₁₆	1	3	yes	—
230	C₂²⋊C₄	—	158	12	no	—
231	C₂²⋊C₄	C₂×D₃₁₆	1	6	yes	—
232	C₄²	—	158	12	no	—
233	C₄²	C₂×D₃₁₆	1	3	yes	—
234	C₄²	C₂×D₃₁₆	1	3	yes	—
235	C₄²	C₂³×D₇₉	1	1	yes	characteristic
236	C₄²	C₂³×D₇₉	1	3	yes	—
237	C₂³	C₂²×D₃₁₆	1	3	yes	—
238	Q₈	—	316	12	no	—
239	Q₈	—	316	4	no	—
240	Q₈	—	632	12	no	—
241	Q₈	#14579161	1	4	yes	—
242	D₄	#14579161	1	12	yes	—
243	C₂×C₄	—	2	6	no	—
244	C₂×C₄	—	2	6	no	—
245	C₂×C₄	—	632	12	no	—
246	C₂×C₄	—	158	4	no	—
247	C₂×C₄	—	158	12	no	—
248	C₂×C₄	C₂²×D₃₁₆	1	3	yes	—
249	C₂×C₄	C₂²×D₃₁₆	1	3	yes	—
250	C₂×C₄	C₂²×D₃₁₆	1	3	yes	—
251	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
253	C₂×C₄	C₂²×Dic₁₅₈	1	12	yes	—
254	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
255	C₈	C₂²×Dic₁₅₈	1	4	yes	—
256	C₈	C₂²×Dic₁₅₈	1	12	yes	—
257	C₂²	—	2	3	no	—
258	C₂²	—	2	3	no	—
259	C₂²	C₂×#14579161	1	6	yes	—
260	C₂²	C₂³×D₃₁₆	1	1	yes	characteristic
261	C₄	—	2	3	no	—
262	C₄	—	2	3	no	—
263	C₄	—	316	12	no	—
264	C₄	—	316	4	no	—
265	C₄	C₂×#14579161	1	6	yes	—
266	C₄	#19099368	1	2	yes	—
267	C₄	#19099380	1	2	yes	—
268	C₄	C₂³×Dic₁₅₈	1	2	yes	—
269	C₂	—	2	6	no	—
270	C₂	C₂²×#14579161	1	1	yes	characteristic
271	C₂	C₂×#19099368	1	1	yes	characteristic
272	C₂	C₂×#19099380	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₉
Fitting	C₄×#131×C₇₉	C₂
socle	C₂²×C₇₉	C₂³×D₄
prime radical	C₂×#41×C₇₉	C₂³
hypercenter	C₄×#131	D₇₉
hypocenter	C₇₉	#60667
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #23773946	C₂	semidirect
3. #23773939	C₂	semidirect
4. #23773914	C₂	semidirect
5. C₂×#19098968	C₂	semidirect
55. #131×C₇₉	Q₈	semidirect
89. M₁₆×C₇₉	C₂×Q₈	semidirect
90. C₂×C₈×C₇₉	C₂×Q₈	semidirect
144. C₇₉	#60667	semidirect
170. #131	Dic₁₅₈	semidirect
216. M₁₆	C₂×Dic₁₅₈	semidirect
217. C₂×C₈	C₂×Dic₁₅₈	semidirect

Maximal subgroups

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

subgroup	index
#23777219	2
#23773946	2
#23773939	2
#23773914	2
C₂×#19098968	2
#23772000	2
C₄×#131×C₇₉	2
#60667	79