Subgroups — #34188359

Order 20224.

484 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23771778	C₂	1	1	yes	maximal, characteristic
2	C₄×#14579072	C₂	1	1	yes	maximal, characteristic
3	#23761282	C₂	1	4	yes	maximal
4	C₇₉×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579161	C₄	1	1	yes	characteristic
6	C₂×#14579161	C₄	1	1	yes	characteristic
7	C₂×#14579072	—	2	1	no	—
8	C₂×#14579072	C₂²	1	2	yes	—
9	C₂×C₄×C₇₉⋊C₈	—	2	1	no	—
10	#19098469	—	2	1	no	—
11	#19098469	—	2	1	no	—
12	#19098469	—	2	1	no	—
13	#19098469	—	2	1	no	—
14	#19098429	C₄	1	2	yes	—
15	#19098429	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₇₉	—	2	1	no	—
17	C₂×#97×C₇₉	C₂²	1	2	yes	—
18	C₄³×C₇₉	C₂²	1	1	yes	characteristic
19	#336×C₇₉	—	2	1	no	—
20	C₂²×Dic₁₅₈	—	2	1	no	—
21	C₂²×Dic₁₅₈	—	2	1	no	—
22	C₂²×C₇₉⋊C₈	—	2	2	no	—
23	#14579161	—	4	2	no	—
24	#14579161	—	4	2	no	—
25	C₂×C₄⋊Dic₇₉	—	2	1	no	—
26	C₂×C₄⋊Dic₇₉	—	2	1	no	—
27	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
29	#14579072	—	2	4	no	—
30	#14579072	—	2	2	no	—
31	#14579072	Q₈	1	4	yes	complemented
32	C₄×C₇₉⋊C₈	—	2	4	no	—
33	C₇₉⋊#95	—	2	2	no	—
34	C₇₉⋊#95	—	2	2	no	—
35	C₇₉⋊#95	—	2	2	no	—
36	C₇₉⋊#95	—	2	2	no	—
37	C₂²×C₈×C₇₉	—	2	2	no	—
38	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₇₉	—	2	4	no	—
46	#97×C₇₉	—	2	4	no	—
47	C₂×Dic₁₅₈	—	4	2	no	—
48	C₂×Dic₁₅₈	—	4	2	no	—
49	C₂×Dic₁₅₈	—	4	2	no	—
50	C₂×Dic₁₅₈	—	4	2	no	—
51	C₂×Dic₁₅₈	—	4	2	no	—
52	C₂×Dic₁₅₈	—	4	2	no	—
53	C₂²×Dic₇₉	—	2	2	no	—
54	C₂²×Dic₇₉	—	2	2	no	—
55	C₂×C₇₉⋊C₈	—	2	4	no	—
56	C₂×C₇₉⋊C₈	—	2	4	no	—
57	C₂×C₇₉⋊C₈	—	2	4	no	—
58	C₄⋊Dic₇₉	—	4	2	no	—
59	C₄⋊Dic₇₉	—	4	2	no	—
60	C₄⋊Dic₇₉	—	4	2	no	—
61	C₄⋊Dic₇₉	—	4	2	no	—
62	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₇₉	—	2	4	no	—
70	C₂×C₈×C₇₉	—	4	2	no	—
71	C₂×C₈×C₇₉	—	4	2	no	—
72	C₄²×C₇₉	—	2	2	no	—
73	C₄²×C₇₉	—	2	2	no	—
74	C₄²×C₇₉	—	2	2	no	—
75	C₄²×C₇₉	—	2	2	no	—
76	C₄²×C₇₉	—	2	2	no	—
77	C₄²×C₇₉	—	4	1	no	—
78	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
79	C₄²×C₇₉	#41	1	2	yes	—
80	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₉	—	2	1	no	—
82	C₂×C₄×C₇₉	—	2	1	no	—
83	C₂×C₄×C₇₉	—	2	1	no	—
84	C₂×C₄×C₇₉	—	2	1	no	—
85	C₂×C₄×C₇₉	—	2	1	no	—
86	C₂×C₄×C₇₉	—	2	1	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	—	2	1	no	—
92	C₂×C₄×C₇₉	—	2	1	no	—
93	C₂×C₄×C₇₉	—	2	1	no	—
94	C₂×C₄×C₇₉	#100	1	2	yes	—
95	C₂×C₄×C₇₉	#101	1	2	yes	—
96	C₂×C₄×C₇₉	#104	1	2	yes	—
97	C₂×C₄×C₇₉	#104	1	2	yes	—
98	C₂×C₄×C₇₉	#104	1	2	yes	—
99	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
101	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
103	C₂×Dic₇₉	—	4	2	no	—
104	C₂×Dic₇₉	—	4	2	no	—
105	C₂×Dic₇₉	—	4	2	no	—
106	C₂×Dic₇₉	—	4	2	no	—
107	C₂×Dic₇₉	—	4	2	no	—
108	C₂×Dic₇₉	—	4	2	no	—
109	Dic₁₅₈	—	8	4	no	—
110	Dic₁₅₈	—	8	4	no	—
111	C₈×C₇₉	—	4	4	no	—
112	C₇₉⋊C₈	—	2	8	no	—
113	C₂²×C₇₉	#391	1	1	yes	characteristic
114	C₂²×C₇₉	C₂×#100	1	1	yes	characteristic
115	C₂²×C₇₉	C₂×#101	1	1	yes	characteristic
116	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
117	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
118	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
119	C₂²×C₇₉	#421	1	1	yes	characteristic
120	C₄×C₇₉	—	2	1	no	—
121	C₄×C₇₉	—	2	1	no	—
122	C₄×C₇₉	—	2	2	no	—
123	C₄×C₇₉	—	2	2	no	—
124	C₄×C₇₉	—	2	2	no	—
125	C₄×C₇₉	—	2	2	no	—
126	C₄×C₇₉	—	4	1	no	—
127	C₄×C₇₉	#329	1	4	yes	—
128	Dic₇₉	—	8	2	no	—
129	Dic₇₉	—	8	2	no	—
130	#9890	—	79	1	no	maximal, Sylow
131	C₂×C₇₉	C₂×#329	1	1	yes	characteristic
132	C₂×C₇₉	#1526	1	1	yes	characteristic
133	C₂×C₇₉	#1527	1	1	yes	characteristic
134	C₂×C₇₉	#1531	1	1	yes	characteristic
135	C₂×C₇₉	#1986	1	1	yes	characteristic
136	C₂×C₇₉	#1994	1	1	yes	characteristic
137	C₂×C₇₉	#1997	1	1	yes	characteristic
138	#2862	—	79	1	no	—
139	#1767	D₇₉	1	1	yes	characteristic
140	C₄×#97	—	79	1	no	—
141	#1295	—	79	4	no	—
142	C₇₉	#9890	1	1	yes	characteristic, complemented, Sylow
143	C₂×C₄⋊Q₈	—	79	1	no	—
144	C₂×C₄⋊Q₈	—	79	1	no	—
145	C₂×C₄⋊C₈	—	2	1	no	—
146	C₂×#97	—	79	2	no	—
147	C₂×#97	—	158	1	no	—
148	C₂×#97	C₂×D₇₉	1	2	yes	—
149	C₂×C₄×C₈	—	158	1	no	—
150	#391	—	158	1	no	—
151	#391	—	158	1	no	—
152	#391	—	158	1	no	—
153	#391	—	158	1	no	—
154	#382	—	79	2	no	—
155	#382	—	79	2	no	—
156	C₄³	C₂×D₇₉	1	1	yes	characteristic
157	#336	—	2	1	no	—
158	C₂²×Q₈	—	158	1	no	—
159	C₂²×Q₈	—	158	1	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	158	2	no	—
162	C₄⋊Q₈	—	316	2	no	—
163	C₄⋊Q₈	—	316	2	no	—
164	C₂×C₄⋊C₄	—	79	2	no	—
165	C₂×C₄⋊C₄	—	79	2	no	—
166	C₂×C₄⋊C₄	—	158	1	no	—
167	C₂×C₄⋊C₄	—	158	1	no	—
168	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
169	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
170	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
171	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
175	C₄⋊C₈	—	2	4	no	—
176	#97	—	2	4	no	—
177	#97	—	79	4	no	—
178	#97	—	158	4	no	—
179	#97	—	158	2	no	—
180	C₄×C₈	—	158	4	no	—
181	#95	—	158	2	no	—
182	#95	—	158	2	no	—
183	#95	—	158	2	no	—
184	#95	—	158	2	no	—
185	C₂×Q₈	—	316	2	no	—
186	C₂×Q₈	—	316	2	no	—
187	C₂×Q₈	—	316	2	no	—
188	C₂×Q₈	—	316	2	no	—
189	C₂×Q₈	—	316	2	no	—
190	C₂×Q₈	—	316	2	no	—
191	C₂²×C₄	—	158	2	no	—
192	C₂²×C₄	—	158	2	no	—
193	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
194	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
195	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
196	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	158	4	no	—
204	C₂×C₈	—	158	4	no	—
205	C₂×C₈	—	158	4	no	—
206	C₄⋊C₄	—	316	2	no	—
207	C₄⋊C₄	—	316	2	no	—
208	C₄⋊C₄	—	316	2	no	—
209	C₄⋊C₄	—	316	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	4	1	no	—
216	C₄²	Q₈×D₇₉	1	2	yes	—
217	C₄²	#11929623	1	2	yes	—
218	C₂³	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
219	Q₈	—	632	4	no	—
220	Q₈	—	632	4	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	316	2	no	—
235	C₂×C₄	—	316	2	no	—
236	C₂×C₄	—	316	2	no	—
237	C₂×C₄	—	316	2	no	—
238	C₂×C₄	—	316	2	no	—
239	C₂×C₄	—	316	2	no	—
240	C₂×C₄	#14579077	1	2	yes	—
241	C₂×C₄	#14579079	1	2	yes	—
242	C₂×C₄	#14579087	1	2	yes	—
243	C₂×C₄	#14579087	1	2	yes	—
244	C₂×C₄	#14579089	1	2	yes	—
245	C₂×C₄	#14579126	1	2	yes	—
246	C₂×C₄	#14579141	1	2	yes	—
247	C₂×C₄	#14579148	1	1	yes	characteristic
248	C₂×C₄	#14579159	1	1	yes	characteristic
249	C₈	—	4	4	no	—
250	C₈	—	158	8	no	—
251	C₂²	#19098468	1	1	yes	characteristic
252	C₂²	C₂×#14579077	1	1	yes	characteristic
253	C₂²	C₂×#14579079	1	1	yes	characteristic
254	C₂²	C₂×#14579087	1	1	yes	characteristic
255	C₂²	C₂×#14579087	1	1	yes	characteristic
256	C₂²	C₂×#14579089	1	1	yes	characteristic
257	C₂²	#19098581	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	632	2	no	—
266	C₄	—	632	2	no	—
267	C₄	#19098302	1	4	yes	—
268	C₂	C₂×#19098302	1	1	yes	characteristic
269	C₂	#23761994	1	1	yes	characteristic
270	C₂	#23761999	1	1	yes	characteristic
271	C₂	#23762016	1	1	yes	characteristic
272	C₂	#23764343	1	1	yes	characteristic
273	C₂	#23764376	1	1	yes	characteristic
274	C₂	#23764388	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1767	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1767	D₇₉
hypocenter	C₇₉	#9890
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
31. #14579072	Q₈	semidirect
142. C₇₉	#9890	semidirect

Maximal subgroups

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

subgroup	index
#23771778	2
C₄×#14579072	2
#23761282	2
C₇₉×#1767	2
#9890	79