Subgroups — #35197344

Order 20224.

6578 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23777208	C₂	1	24	yes	maximal, complemented
2	#23777205	C₂	1	24	yes	maximal, complemented
3	#23776805	C₂	1	3	yes	maximal
4	#23776802	C₂	1	8	yes	maximal
5	C₂×#19099040	C₂	1	3	yes	maximal
6	C₇₉×#3458	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₅₈	C₂²	1	1	yes	characteristic
8	Q₈×Dic₁₅₈	C₂²	1	16	yes	complemented
9	#19099293	C₂²	1	48	yes	complemented
10	#19099291	C₂²	1	48	yes	complemented
11	#19099290	C₂²	1	96	yes	complemented
12	#19099288	C₂²	1	48	yes	complemented
13	#19099194	C₂²	1	12	yes	—
14	#19099192	C₂²	1	96	yes	—
15	#19099191	C₂²	1	24	yes	—
16	#19099189	C₂²	1	12	yes	—
17	#19099120	C₂²	1	12	yes	—
18	#19099116	C₂²	1	24	yes	—
19	#19099115	C₂²	1	96	yes	—
20	#19099040	C₂²	1	24	yes	—
21	#19099040	C₂²	1	24	yes	—
22	C₂×C₄×Dic₁₅₈	C₂²	1	12	yes	—
23	C₂×#14579126	C₂²	1	12	yes	—
24	C₂×#14579120	C₂²	1	3	yes	—
25	C₂×#14579117	C₂²	1	12	yes	—
26	#544×C₇₉	C₂²	1	12	yes	—
27	#543×C₇₉	C₂²	1	4	yes	—
28	#533×C₇₉	C₂²	1	12	yes	—
29	C₂×#117×C₇₉	C₂²	1	3	yes	—
30	C₂²×Dic₁₅₈	Q₈	1	16	yes	complemented
31	C₂²×Dic₁₅₈	C₂³	1	6	yes	—
32	C₂²×Dic₁₅₈	C₂³	1	6	yes	—
33	C₂³×Dic₇₉	C₂³	1	2	yes	—
34	#14579161	C₂³	1	24	yes	—
35	#14579160	C₂³	1	48	yes	—
36	#14579160	C₂³	1	96	yes	—
37	#14579159	C₂³	1	96	yes	—
38	#14579153	C₂³	1	24	yes	—
39	#14579152	C₂³	1	48	yes	—
40	#14579151	C₂³	1	192	yes	—
41	#14579140	C₂³	1	48	yes	—
42	#14579138	C₂³	1	96	yes	—
43	Q₈×Dic₇₉	C₂³	1	32	yes	—
44	C₄×Dic₁₅₈	C₂³	1	48	yes	—
45	C₄×Dic₁₅₈	C₂³	1	48	yes	—
46	C₄×Dic₁₅₈	C₂³	1	48	yes	—
47	#14579126	C₂³	1	96	yes	—
48	#14579126	C₂³	1	96	yes	—
49	#14579120	C₂³	1	12	yes	—
50	#14579120	C₂³	1	12	yes	—
51	#14579117	C₂³	1	96	yes	—
52	C₂×C₄⋊Dic₇₉	C₂³	1	6	yes	—
53	C₂×Dic₇₉⋊C₄	C₂³	1	24	yes	—
54	C₂×C₂²⋊Dic₇₉	C₂³	1	6	yes	—
55	C₂×C₄×Dic₇₉	C₂³	1	24	yes	—
56	C₂×C₄×Dic₇₉	C₂³	1	6	yes	—
57	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
58	C₄⋊Q₈×C₇₉	C₂³	1	24	yes	—
59	#125×C₇₉	C₂³	1	24	yes	—
60	C₂²⋊Q₈×C₇₉	C₂³	1	48	yes	—
61	C₄×Q₈×C₇₉	C₂³	1	16	yes	—
62	#117×C₇₉	C₂³	1	12	yes	—
63	#117×C₇₉	C₂³	1	12	yes	—
64	C₂×C₄⋊C₄×C₇₉	C₂³	1	6	yes	—
65	C₂×C₂²⋊C₄×C₇₉	C₂³	1	6	yes	—
66	C₂×C₄²×C₇₉	C₂³	1	6	yes	—
67	C₂×Dic₁₅₈	C₂×Q₈	1	48	yes	—
68	C₂×Dic₁₅₈	C₂×Q₈	1	48	yes	—
69	C₂×Dic₁₅₈	C₂⁴	1	48	yes	—
70	C₂×Dic₁₅₈	C₂⁴	1	12	yes	—
71	C₂×Dic₁₅₈	C₂⁴	1	4	yes	—
72	C₂²×Dic₇₉	C₂×Q₈	1	16	yes	—
73	C₂²×Dic₇₉	C₂⁴	1	12	yes	—
74	C₂²×Dic₇₉	C₂⁴	1	12	yes	—
75	C₄⋊Dic₇₉	C₂⁴	1	12	yes	—
76	C₄⋊Dic₇₉	C₂⁴	1	12	yes	—
77	C₄⋊Dic₇₉	C₂⁴	1	48	yes	—
78	Dic₇₉⋊C₄	C₂⁴	1	96	yes	—
79	Dic₇₉⋊C₄	C₂⁴	1	96	yes	—
80	C₂²⋊Dic₇₉	C₂⁴	1	24	yes	—
81	C₄×Dic₇₉	C₂⁴	1	96	yes	—
82	C₄×Dic₇₉	C₂⁴	1	24	yes	—
83	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
84	C₂×Q₈×C₇₉	C₂⁴	1	16	yes	—
85	C₂²×C₄×C₇₉	C₂×Q₈	1	8	yes	—
86	C₂²×C₄×C₇₉	C₂⁴	1	12	yes	—
87	C₂²×C₄×C₇₉	C₂⁴	1	3	yes	—
88	C₂²×C₄×C₇₉	C₂⁴	1	3	yes	—
89	C₄⋊C₄×C₇₉	C₂⁴	1	24	yes	—
90	C₄⋊C₄×C₇₉	C₂⁴	1	48	yes	—
91	C₂²⋊C₄×C₇₉	C₂⁴	1	24	yes	—
92	C₄²×C₇₉	C₂⁴	1	12	yes	—
93	C₄²×C₇₉	C₂⁴	1	12	yes	—
94	C₂³×C₇₉	—	4	2	no	—
95	C₂³×C₇₉	C₂²×Q₈	1	4	yes	—
96	C₂³×C₇₉	C₂⁵	1	3	yes	—
97	Q₈×C₇₉	—	4	16	no	—
98	C₂×C₄×C₇₉	—	4	12	no	—
99	C₂×C₄×C₇₉	C₂²×Q₈	1	12	yes	—
100	C₂×C₄×C₇₉	C₂²×Q₈	1	12	yes	—
101	C₂×C₄×C₇₉	C₂⁵	1	24	yes	—
102	C₂×C₄×C₇₉	C₂⁵	1	3	yes	—
103	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
104	C₂×Dic₇₉	—	4	12	no	—
105	C₂×Dic₇₉	C₂²×Q₈	1	48	yes	—
106	C₂×Dic₇₉	C₂⁵	1	24	yes	—
107	C₂×Dic₇₉	C₂⁵	1	8	yes	—
108	Dic₁₅₈	—	4	48	no	—
109	Dic₁₅₈	C₂²×Q₈	1	48	yes	—
110	Dic₁₅₈	C₂²×Q₈	1	16	yes	—
111	C₂²×C₇₉	—	2	6	no	—
112	C₂²×C₇₉	—	4	4	no	—
113	C₂²×C₇₉	C₂³×Q₈	1	6	yes	—
114	C₂²×C₇₉	C₂⁶	1	1	yes	characteristic
115	C₄×C₇₉	—	2	24	no	—
116	C₄×C₇₉	C₂³×Q₈	1	6	yes	—
117	C₄×C₇₉	C₂³×Q₈	1	2	yes	—
118	Dic₇₉	—	2	24	no	—
119	Dic₇₉	C₂³×Q₈	1	16	yes	—
120	#62849	—	79	1	no	maximal, Sylow
121	C₂×C₇₉	—	2	6	no	—
122	C₂×C₇₉	C₂⁴×Q₈	1	1	yes	characteristic
123	C₂×C₇₉	2₋¹⁺⁶	1	2	yes	—
124	#3502	—	79	24	no	—
125	#3502	—	79	24	no	—
126	#3458	—	79	8	no	—
127	#3458	—	79	3	no	—
128	#3458	D₇₉	1	1	yes	characteristic
129	C₂×#519	—	79	3	no	—
130	C₇₉	#62849	1	1	yes	characteristic, complemented, Sylow
131	C₂³×Q₈	—	79	1	no	—
132	Q₈²	—	79	48	no	—
133	Q₈²	—	79	16	no	—
134	#557	—	79	48	no	—
135	#557	—	79	96	no	—
136	#557	—	79	48	no	—
137	#544	—	79	96	no	—
138	#544	—	79	24	no	—
139	#544	—	79	12	no	—
140	#544	C₂×D₇₉	1	12	yes	—
141	#543	—	79	12	no	—
142	#543	C₂×D₇₉	1	4	yes	—
143	#533	—	79	96	no	—
144	#533	—	79	24	no	—
145	#533	—	79	12	no	—
146	#533	C₂×D₇₉	1	12	yes	—
147	#519	—	79	24	no	—
148	#519	—	79	24	no	—
149	C₂×C₄×Q₈	—	79	12	no	—
150	C₂×C₄×Q₈	—	79	12	no	—
151	C₂×#117	—	79	3	no	—
152	C₂×#117	—	79	12	no	—
153	C₂×#117	C₂×D₇₉	1	3	yes	—
154	C₂²×Q₈	—	79	16	no	—
155	C₂²×Q₈	—	79	6	no	—
156	C₂²×Q₈	—	79	6	no	—
157	C₂³×C₄	—	79	2	no	—
158	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
159	C₄⋊Q₈	—	79	96	no	—
160	C₄⋊Q₈	—	79	96	no	—
161	C₄⋊Q₈	—	79	48	no	—
162	C₄⋊Q₈	—	79	24	no	—
163	C₄⋊Q₈	C₂²×D₇₉	1	24	yes	—
164	#125	—	79	192	no	—
165	#125	—	79	48	no	—
166	#125	—	79	24	no	—
167	#125	C₂²×D₇₉	1	24	yes	—
168	C₂²⋊Q₈	—	79	48	no	—
169	C₂²⋊Q₈	—	79	96	no	—
170	C₂²⋊Q₈	C₂²×D₇₉	1	48	yes	—
171	C₄×Q₈	—	79	32	no	—
172	C₄×Q₈	—	79	96	no	—
173	C₄×Q₈	—	79	96	no	—
174	C₄×Q₈	—	79	48	no	—
175	C₄×Q₈	—	79	48	no	—
176	C₄×Q₈	—	79	48	no	—
177	C₄×Q₈	C₂²×D₇₉	1	16	yes	—
178	#117	—	79	12	no	—
179	#117	—	79	12	no	—
180	#117	—	79	96	no	—
181	#117	C₂²×D₇₉	1	12	yes	—
182	#117	C₂²×D₇₉	1	12	yes	—
183	C₂×C₄⋊C₄	—	79	24	no	—
184	C₂×C₄⋊C₄	—	79	6	no	—
185	C₂×C₄⋊C₄	C₂²×D₇₉	1	6	yes	—
186	C₂×C₂²⋊C₄	—	79	6	no	—
187	C₂×C₂²⋊C₄	C₂²×D₇₉	1	6	yes	—
188	C₂×C₄²	—	79	6	no	—
189	C₂×C₄²	—	79	24	no	—
190	C₂×C₄²	C₂²×D₇₉	1	6	yes	—
191	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
192	C₂×Q₈	—	79	48	no	—
193	C₂×Q₈	—	79	48	no	—
194	C₂×Q₈	—	79	48	no	—
195	C₂×Q₈	—	79	12	no	—
196	C₂×Q₈	—	79	4	no	—
197	C₂×Q₈	C₂³×D₇₉	1	16	yes	—
198	C₂²×C₄	—	79	12	no	—
199	C₂²×C₄	—	79	16	no	—
200	C₂²×C₄	—	79	12	no	—
201	C₂²×C₄	Q₈×D₇₉	1	8	yes	—
202	C₂²×C₄	C₂³×D₇₉	1	12	yes	—
203	C₂²×C₄	C₂³×D₇₉	1	3	yes	—
204	C₂²×C₄	C₂³×D₇₉	1	3	yes	—
205	C₄⋊C₄	—	79	96	no	—
206	C₄⋊C₄	—	79	96	no	—
207	C₄⋊C₄	—	79	12	no	—
208	C₄⋊C₄	—	79	12	no	—
209	C₄⋊C₄	—	79	48	no	—
210	C₄⋊C₄	C₂³×D₇₉	1	24	yes	—
211	C₄⋊C₄	C₂³×D₇₉	1	48	yes	—
212	C₂²⋊C₄	—	79	24	no	—
213	C₂²⋊C₄	C₂³×D₇₉	1	24	yes	—
214	C₄²	—	79	96	no	—
215	C₄²	—	79	24	no	—
216	C₄²	C₂³×D₇₉	1	12	yes	—
217	C₄²	C₂³×D₇₉	1	12	yes	—
218	C₂³	—	4	2	no	—
219	C₂³	C₂×Q₈×D₇₉	1	4	yes	—
220	C₂³	C₂⁴×D₇₉	1	3	yes	—
221	Q₈	—	4	16	no	—
222	Q₈	—	316	48	no	—
223	Q₈	—	79	48	no	—
224	Q₈	—	79	16	no	—
225	C₂×C₄	—	4	12	no	—
226	C₂×C₄	—	316	12	no	—
227	C₂×C₄	—	79	8	no	—
228	C₂×C₄	—	79	48	no	—
229	C₂×C₄	—	79	24	no	—
230	C₂×C₄	C₂×Q₈×D₇₉	1	12	yes	—
231	C₂×C₄	C₂×Q₈×D₇₉	1	12	yes	—
232	C₂×C₄	C₂⁴×D₇₉	1	24	yes	—
233	C₂×C₄	C₂⁴×D₇₉	1	3	yes	—
234	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
235	C₂²	—	2	6	no	—
236	C₂²	—	4	4	no	—
237	C₂²	C₂²×Q₈×D₇₉	1	6	yes	—
238	C₂²	C₂⁵×D₇₉	1	1	yes	characteristic
239	C₄	—	2	24	no	—
240	C₄	—	79	16	no	—
241	C₄	—	158	24	no	—
242	C₄	C₂²×Q₈×D₇₉	1	6	yes	—
243	C₄	C₂²×Q₈×D₇₉	1	2	yes	—
244	C₂	—	2	6	no	—
245	C₂	C₂³×Q₈×D₇₉	1	1	yes	characteristic
246	C₂	#23778213	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₇₉
commutator, socle, cosocle	C₂²×C₇₉	C₂⁶
Fitting	C₇₉×#3458	C₂
prime radical	C₂⁴×C₇₉	C₂⁴
hypercenter	#3458	D₇₉
hypocenter	C₇₉	#62849
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ #3458

Complements

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

N	H ≅ G/N	type
1. #23777208	C₂	semidirect
2. #23777205	C₂	semidirect
8. Q₈×Dic₁₅₈	C₂²	semidirect
9. #19099293	C₂²	semidirect
10. #19099291	C₂²	semidirect
11. #19099290	C₂²	semidirect
12. #19099288	C₂²	semidirect
30. C₂²×Dic₁₅₈	Q₈	semidirect
130. C₇₉	#62849	semidirect

Maximal subgroups

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

subgroup	index
#23777208	2
#23777205	2
#23776805	2
#23776802	2
C₂×#19099040	2
C₇₉×#3458	2
#62849	79