Subgroups — #34178170

Order 20224.

598 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772033	C₂	1	2	yes	maximal
2	#23761270	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
4	#19098620	C₄	1	4	yes	—
5	#19098620	C₂²	1	4	yes	—
6	#19098616	—	2	2	no	—
7	#19098616	—	2	2	no	—
8	#19098587	—	2	2	no	—
9	#19098527	—	2	2	no	—
10	C₄×#11929608	C₄	1	4	yes	—
11	C₂³×C₈×C₇₉	C₄	1	2	yes	—
12	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₇₉	—	2	2	no	—
14	C₂×C₄⋊C₈×C₇₉	—	2	2	no	—
15	C₂×C₄⋊C₈×C₇₉	C₄	1	2	yes	—
16	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
17	C₂×#11929608	—	2	4	no	—
18	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
19	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
20	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
21	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
22	#14579072	C₂×C₄	1	8	yes	—
23	C₄×C₇₉⋊C₈	—	2	4	no	—
24	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₇₉	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₇₉	—	2	2	no	—
28	C₂²×C₈×C₇₉	—	2	2	no	—
29	C₂²×C₈×C₇₉	—	4	4	no	—
30	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₉	D₄	1	2	yes	—
35	C₂×C₄²×C₇₉	D₄	1	2	yes	—
36	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₇₉	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₇₉	—	2	4	no	—
40	C₄⋊C₈×C₇₉	—	2	4	no	—
41	#11929608	—	4	8	no	—
42	C₂×C₇₉⋊C₈	—	2	8	no	—
43	C₂⁴×C₇₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₉	—	2	2	no	—
45	C₂²×C₄×C₇₉	—	2	2	no	—
46	C₂²×C₄×C₇₉	—	2	4	no	—
47	C₂²×C₄×C₇₉	—	4	2	no	—
48	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₇₉	—	2	4	no	—
63	C₂×C₈×C₇₉	—	4	4	no	—
64	C₂×C₈×C₇₉	—	4	4	no	—
65	C₂×C₈×C₇₉	—	4	4	no	—
66	C₄²×C₇₉	C₄²	1	2	yes	—
67	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
71	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
72	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
73	C₄²×C₇₉	C₂×D₄	1	2	yes	—
74	C₂³×C₇₉	—	2	2	no	—
75	C₂³×C₇₉	—	2	2	no	—
76	C₂³×C₇₉	—	2	2	no	—
77	C₂³×C₇₉	#95	1	1	yes	characteristic
78	C₂³×C₇₉	#95	1	1	yes	characteristic
79	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₇₉	—	2	4	no	—
81	C₂×C₄×C₇₉	—	2	2	no	—
82	C₂×C₄×C₇₉	—	2	2	no	—
83	C₂×C₄×C₇₉	—	2	4	no	—
84	C₂×C₄×C₇₉	—	2	4	no	—
85	C₂×C₄×C₇₉	—	2	2	no	—
86	C₂×C₄×C₇₉	—	2	2	no	—
87	C₂×C₄×C₇₉	—	2	1	no	—
88	C₂×C₄×C₇₉	—	2	1	no	—
89	C₂×C₄×C₇₉	—	2	1	no	—
90	C₂×C₄×C₇₉	—	2	1	no	—
91	C₂×C₄×C₇₉	—	4	2	no	—
92	C₂×C₄×C₇₉	—	4	1	no	—
93	C₂×C₄×C₇₉	—	4	1	no	—
94	C₂×C₄×C₇₉	#95	1	2	yes	—
95	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
96	C₂×C₄×C₇₉	#95	1	2	yes	—
97	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
98	C₂×C₄×C₇₉	#100	1	2	yes	—
99	C₂×C₄×C₇₉	#101	1	2	yes	—
100	C₂×C₄×C₇₉	#108	1	4	yes	—
101	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
105	C₈×C₇₉	—	4	4	no	—
106	C₇₉⋊C₈	—	4	8	no	—
107	C₂²×C₇₉	—	2	8	no	—
108	C₂²×C₇₉	—	2	1	no	—
109	C₂²×C₇₉	—	2	1	no	—
110	C₂²×C₇₉	—	2	1	no	—
111	C₂²×C₇₉	—	2	1	no	—
112	C₂²×C₇₉	—	2	1	no	—
113	C₂²×C₇₉	—	2	1	no	—
114	C₂²×C₇₉	H(3,ℤ/4)	1	2	yes	—
115	C₂²×C₇₉	#343	1	2	yes	—
116	C₂²×C₇₉	C₂×#95	1	1	yes	characteristic
117	C₂²×C₇₉	C₂×#108	1	2	yes	—
118	C₄×C₇₉	—	2	2	no	—
119	C₄×C₇₉	—	2	2	no	—
120	C₄×C₇₉	—	2	2	no	—
121	C₄×C₇₉	—	2	1	no	—
122	C₄×C₇₉	—	2	1	no	—
123	C₄×C₇₉	#341	1	4	yes	—
124	C₄×C₇₉	#343	1	2	yes	—
125	C₄×C₇₉	C₂×#100	1	1	yes	characteristic
126	C₄×C₇₉	C₂×#101	1	1	yes	characteristic
127	#7988	—	79	1	no	maximal, Sylow
128	C₂×C₇₉	—	2	2	no	—
129	C₂×C₇₉	—	2	2	no	—
130	C₂×C₇₉	—	2	2	no	—
131	C₂×C₇₉	C₂×H(3,ℤ/4)	1	1	yes	characteristic
132	C₂×C₇₉	C₂×#341	1	1	yes	characteristic
133	C₂×C₇₉	C₂×#343	1	1	yes	characteristic
134	#2920	—	79	2	no	—
135	C₂²×C₄⋊C₈	D₇₉	1	1	yes	characteristic
136	#1290	—	79	4	no	—
137	C₇₉	#7988	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₈	Dic₇₉	1	2	yes	—
139	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
140	#432	—	79	4	no	—
141	#432	—	79	4	no	—
142	#431	—	158	2	no	—
143	#431	—	158	2	no	—
144	#423	—	158	2	no	—
145	C₂×C₄⋊C₈	—	2	2	no	—
146	C₂×C₄⋊C₈	—	2	2	no	—
147	C₂×C₄⋊C₈	Dic₇₉	1	2	yes	—
148	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
149	#407	—	158	2	no	—
150	C₄×M₁₆	—	79	4	no	—
151	C₂³×C₄	Dic₁₅₈	1	1	yes	characteristic
152	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
153	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
154	C₂×M₁₆	—	158	4	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	4	4	no	—
158	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
159	C₂×C₄²	Dic₁₅₈	1	1	yes	characteristic
160	C₂×C₄²	C₄×D₇₉	1	2	yes	—
161	C₂×C₄²	C₄×D₇₉	1	2	yes	—
162	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
163	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
165	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
166	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
167	C₄⋊C₈	—	2	4	no	—
168	C₄⋊C₈	—	2	4	no	—
169	C₄⋊C₈	—	158	4	no	—
170	C₄⋊C₈	—	158	4	no	—
171	C₂²⋊C₈	—	158	4	no	—
172	C₂²⋊C₈	—	158	4	no	—
173	#97	—	79	8	no	—
174	C₄×C₈	—	158	4	no	—
175	C₂⁴	C₄⋊Dic₇₉	1	1	yes	characteristic
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
181	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
182	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
183	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
184	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
185	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
186	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
187	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
188	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
189	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
191	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
193	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
194	M₁₆	—	316	8	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	4	4	no	—
197	C₂×C₈	—	4	4	no	—
198	C₂×C₈	—	4	4	no	—
199	C₂×C₈	—	158	8	no	—
200	C₄²	C₄×Dic₇₉	1	2	yes	—
201	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
202	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
203	C₄²	C₂²⋊Dic₇₉	1	2	yes	—
204	C₄²	Dic₇₉⋊C₄	1	2	yes	—
205	C₄²	Dic₇₉⋊C₄	1	2	yes	—
206	C₄²	C₂×C₄×D₇₉	1	2	yes	—
207	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	C₇₉⋊#95	1	1	yes	characteristic
212	C₂³	C₇₉⋊#95	1	1	yes	characteristic
213	C₂³	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	C₇₉⋊#95	1	2	yes	—
229	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
230	C₂×C₄	C₇₉⋊#95	1	2	yes	—
231	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
232	C₂×C₄	#14579078	1	2	yes	—
233	C₂×C₄	#14579080	1	2	yes	—
234	C₂×C₄	#14579097	1	4	yes	—
235	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
236	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
238	C₂×C₄	C₂×Dic₇₉⋊C₄	1	2	yes	—
239	C₈	—	4	4	no	—
240	C₈	—	316	8	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	#19098320	1	2	yes	—
249	C₂²	#19098332	1	2	yes	—
250	C₂²	C₂×C₇₉⋊#95	1	1	yes	characteristic
251	C₂²	C₂×#14579097	1	2	yes	—
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	#19098328	1	4	yes	—
258	C₄	#19098332	1	2	yes	—
259	C₄	C₂×#14579078	1	1	yes	characteristic
260	C₄	C₂×#14579080	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	2	2	no	—
263	C₂	—	2	2	no	—
264	C₂	C₂×#19098320	1	1	yes	characteristic
265	C₂	C₂×#19098328	1	1	yes	characteristic
266	C₂	C₂×#19098332	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₇₉⋊#95
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂²×C₄⋊C₈×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#95
prime radical	C₂⁴×C₇₉	C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂²×C₄⋊C₈	D₇₉
hypocenter	C₇₉	#7988
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #23761270	C₂	semidirect
137. C₇₉	#7988	semidirect

Maximal subgroups

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

subgroup	index
#23772033	2
#23761270	2
C₂²×C₄⋊C₈×C₇₉	2
#7988	79