Subgroups — #34305072

Order 20224.

1086 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099292	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#23764323	C₂	1	3	yes	maximal
4	#23764201	C₂	1	3	yes	maximal
5	#23763402	C₂	1	3	yes	maximal
6	C₄×#14579084	C₂	1	3	yes	maximal
7	Q₈×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
8	#19099292	—	2	4	no	—
9	#19099292	C₄	1	8	yes	complemented
10	C₂×#14579161	C₂²	1	3	yes	—
11	C₂×#14579152	—	2	3	no	—
12	C₂×Q₈×Dic₇₉	—	2	1	no	—
13	C₂×C₄×Dic₁₅₈	C₂²	1	3	yes	—
14	Q₈×C₇₉⋊C₈	—	2	4	no	—
15	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	3	yes	—
16	C₂×#14579084	C₂²	1	3	yes	—
17	C₂×#14579084	C₂²	1	3	yes	—
18	C₂×#14579084	C₂²	1	6	yes	—
19	C₂×C₄×C₇₉⋊C₈	C₂²	1	3	yes	—
20	#19098325	C₂²	1	1	yes	characteristic
21	#19098325	C₂²	1	3	yes	—
22	C₂×C₄×Q₈×C₇₉	—	2	1	no	—
23	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
24	#389×C₇₉	—	2	3	no	—
25	#384×C₇₉	C₂²	1	3	yes	—
26	C₄×C₄⋊C₄×C₇₉	C₂²	1	3	yes	—
27	C₂²×Dic₁₅₈	C₂³	1	3	yes	—
28	C₂²×C₇₉⋊C₈	C₂³	1	3	yes	—
29	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
30	#14579161	—	2	6	no	—
31	#14579161	C₂×C₄	1	12	yes	—
32	#14579152	—	2	24	no	—
33	Q₈×Dic₇₉	—	2	8	no	—
34	C₄×Dic₁₅₈	—	2	6	no	—
35	C₄×Dic₁₅₈	C₂×C₄	1	12	yes	—
36	C₂×C₄⋊Dic₇₉	—	2	3	no	—
37	C₂×C₄⋊Dic₇₉	—	2	3	no	—
38	C₂×C₄⋊Dic₇₉	C₂³	1	1	yes	characteristic
39	C₂×Dic₇₉⋊C₄	—	2	3	no	—
40	C₂×C₄×Dic₇₉	—	2	3	no	—
41	C₄⋊(C₇₉⋊C₈)	—	2	6	no	—
42	#14579084	—	2	6	no	—
43	#14579084	—	2	6	no	—
44	#14579084	—	4	6	no	—
45	C₄×C₇₉⋊C₈	—	2	6	no	—
46	C₂²×Q₈×C₇₉	D₄	1	1	yes	characteristic
47	C₄×Q₈×C₇₉	—	4	4	no	—
48	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
49	C₄×Q₈×C₇₉	D₄	1	4	yes	—
50	C₂×C₄⋊C₄×C₇₉	—	2	3	no	—
51	C₂×C₄⋊C₄×C₇₉	—	2	3	no	—
52	C₂×C₄⋊C₄×C₇₉	D₄	1	3	yes	—
53	C₂×C₄⋊C₄×C₇₉	C₂³	1	3	yes	—
54	C₂×C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₇₉	—	2	3	no	—
56	C₂×C₄²×C₇₉	C₂³	1	3	yes	—
57	#95×C₇₉	—	2	3	no	—
58	C₂×Dic₁₅₈	—	2	3	no	—
59	C₂×Dic₁₅₈	—	2	3	no	—
60	C₂×Dic₁₅₈	—	2	3	no	—
61	C₂×Dic₁₅₈	—	2	3	no	—
62	C₂×Dic₁₅₈	—	2	3	no	—
63	C₂×Dic₁₅₈	C₂²×C₄	1	6	yes	—
64	C₂²×Dic₇₉	—	2	3	no	—
65	C₂²×Dic₇₉	—	2	1	no	—
66	C₂×C₇₉⋊C₈	—	2	1	no	—
67	C₂×C₇₉⋊C₈	—	2	3	no	—
68	C₂×C₇₉⋊C₈	—	2	3	no	—
69	C₂×C₇₉⋊C₈	—	2	3	no	—
70	C₂×C₇₉⋊C₈	—	2	1	no	—
71	C₂×C₇₉⋊C₈	#41	1	2	yes	—
72	C₄⋊Dic₇₉	—	2	12	no	—
73	C₄⋊Dic₇₉	—	2	12	no	—
74	C₄⋊Dic₇₉	—	2	1	no	—
75	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
76	Dic₇₉⋊C₄	—	2	12	no	—
77	C₄×Dic₇₉	—	2	12	no	—
78	C₂×Q₈×C₇₉	C₂²⋊C₄	1	4	yes	—
79	C₂×Q₈×C₇₉	SD₁₆	1	4	yes	—
80	C₂×Q₈×C₇₉	Q₁₆	1	4	yes	—
81	C₂²×C₄×C₇₉	—	2	1	no	—
82	C₂²×C₄×C₇₉	—	2	3	no	—
83	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
84	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
85	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₇₉	—	2	1	no	—
87	C₄⋊C₄×C₇₉	—	2	3	no	—
88	C₄⋊C₄×C₇₉	—	2	3	no	—
89	C₄⋊C₄×C₇₉	—	4	6	no	—
90	C₄⋊C₄×C₇₉	—	8	3	no	—
91	C₄⋊C₄×C₇₉	C₂²⋊C₄	1	12	yes	—
92	C₄⋊C₄×C₇₉	#41	1	2	yes	—
93	C₄²×C₇₉	—	4	6	no	—
94	C₄²×C₇₉	C₂²×C₄	1	6	yes	—
95	C₄²×C₇₉	C₂×D₄	1	6	yes	—
96	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₇₉	#103	1	16	yes	complemented
98	C₂×C₄×C₇₉	—	2	3	no	—
99	C₂×C₄×C₇₉	—	2	3	no	—
100	C₂×C₄×C₇₉	—	2	2	no	—
101	C₂×C₄×C₇₉	—	4	1	no	—
102	C₂×C₄×C₇₉	—	4	1	no	—
103	C₂×C₄×C₇₉	—	4	3	no	—
104	C₂×C₄×C₇₉	—	4	3	no	—
105	C₂×C₄×C₇₉	—	4	3	no	—
106	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	6	yes	—
107	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	6	yes	—
108	C₂×C₄×C₇₉	C₂×SD₁₆	1	6	yes	—
109	C₂×C₄×C₇₉	C₂×Q₁₆	1	6	yes	—
110	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×Dic₇₉	—	2	2	no	—
117	C₂×Dic₇₉	—	2	6	no	—
118	C₂×Dic₇₉	—	4	3	no	—
119	C₂×Dic₇₉	—	4	3	no	—
120	C₂×Dic₇₉	—	4	1	no	—
121	C₂×Dic₇₉	—	4	1	no	—
122	Dic₁₅₈	—	2	6	no	—
123	Dic₁₅₈	—	2	6	no	—
124	Dic₁₅₈	—	4	3	no	—
125	Dic₁₅₈	—	4	3	no	—
126	C₇₉⋊C₈	—	2	2	no	—
127	C₇₉⋊C₈	—	4	3	no	—
128	C₂²×C₇₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₇₉	#548	1	1	yes	characteristic
130	C₂²×C₇₉	#550	1	1	yes	characteristic
131	C₂²×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
132	C₂²×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
133	C₂²×C₇₉	#577	1	1	yes	characteristic
134	C₂²×C₇₉	#578	1	1	yes	characteristic
135	C₄×C₇₉	—	2	6	no	—
136	C₄×C₇₉	—	4	2	no	—
137	C₄×C₇₉	—	8	3	no	—
138	C₄×C₇₉	C₂×#103	1	12	yes	—
139	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
140	C₄×C₇₉	#520	1	1	yes	characteristic
141	C₄×C₇₉	#548	1	1	yes	characteristic
142	C₄×C₇₉	#550	1	1	yes	characteristic
143	Dic₇₉	—	4	6	no	—
144	Dic₇₉	—	4	2	no	—
145	#22542	—	79	1	no	maximal, Sylow
146	C₂×C₇₉	#2341	1	1	yes	characteristic
147	C₂×C₇₉	C₂²×#103	1	1	yes	characteristic
148	C₂×C₇₉	#2898	1	1	yes	characteristic
149	C₂×C₇₉	#3359	1	1	yes	characteristic
150	C₂×C₇₉	#3363	1	1	yes	characteristic
151	C₂×C₇₉	#3392	1	1	yes	characteristic
152	C₂×C₇₉	#3395	1	1	yes	characteristic
153	C₂×#557	—	79	1	no	—
154	C₂×C₈×Q₈	—	79	1	no	—
155	Q₈×C₄⋊C₄	D₇₉	1	1	yes	characteristic
156	#1981	—	79	3	no	—
157	#1956	—	79	3	no	—
158	#1803	—	79	3	no	—
159	C₄×#103	—	79	3	no	—
160	C₇₉	#22542	1	1	yes	characteristic, complemented, Sylow
161	#557	—	79	8	no	—
162	#557	—	158	4	no	—
163	C₂×C₄⋊Q₈	—	79	3	no	—
164	C₂×#125	—	158	3	no	—
165	C₂×C₄×Q₈	—	2	1	no	—
166	C₂×C₄×Q₈	—	79	3	no	—
167	C₂×C₄×Q₈	—	158	1	no	—
168	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
169	C₈×Q₈	—	158	4	no	—
170	C₂×C₄⋊C₈	—	79	3	no	—
171	C₂×#103	—	79	6	no	—
172	C₂×#103	—	79	3	no	—
173	C₂×#103	—	79	3	no	—
174	C₂×C₄×C₈	—	79	3	no	—
175	#389	—	2	3	no	—
176	#384	C₂×D₇₉	1	3	yes	—
177	C₄×C₄⋊C₄	C₂×D₇₉	1	3	yes	—
178	#340	—	79	3	no	—
179	#340	—	79	1	no	—
180	C₂²×Q₈	—	79	3	no	—
181	C₂²×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
182	C₂²×C₈	—	79	1	no	—
183	C₂²×C₈	—	79	3	no	—
184	C₄⋊Q₈	—	79	12	no	—
185	C₄⋊Q₈	—	158	6	no	—
186	#125	—	158	24	no	—
187	C₄×Q₈	—	4	4	no	—
188	C₄×Q₈	—	79	12	no	—
189	C₄×Q₈	—	158	8	no	—
190	C₄×Q₈	—	158	6	no	—
191	C₄×Q₈	C₄×D₇₉	1	4	yes	—
192	C₄×Q₈	D₃₁₆	1	4	yes	—
193	C₂×C₄⋊C₄	—	2	3	no	—
194	C₂×C₄⋊C₄	—	2	3	no	—
195	C₂×C₄⋊C₄	—	79	1	no	—
196	C₂×C₄⋊C₄	—	158	3	no	—
197	C₂×C₄⋊C₄	—	158	3	no	—
198	C₂×C₄⋊C₄	—	158	3	no	—
199	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	3	yes	—
200	C₂×C₄⋊C₄	C₂²×D₇₉	1	3	yes	—
201	C₂×C₄⋊C₄	C₂²×D₇₉	1	1	yes	characteristic
202	C₂×C₄²	—	2	3	no	—
203	C₂×C₄²	—	158	3	no	—
204	C₂×C₄²	C₂²×D₇₉	1	3	yes	—
205	C₄⋊C₈	—	158	6	no	—
206	#103	—	316	6	no	—
207	#103	—	158	6	no	—
208	#103	—	158	6	no	—
209	C₄×C₈	—	158	6	no	—
210	#95	—	2	3	no	—
211	C₂×Q₈	—	79	6	no	—
212	C₂×Q₈	—	158	3	no	—
213	C₂×Q₈	—	158	3	no	—
214	C₂×Q₈	—	158	3	no	—
215	C₂×Q₈	—	158	3	no	—
216	C₂×Q₈	—	158	3	no	—
217	C₂×Q₈	(C₂×D₇₉)⋊C₄	1	4	yes	—
218	C₂×Q₈	#11929612	1	4	yes	—
219	C₂×Q₈	#11929614	1	4	yes	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	3	no	—
222	C₂²×C₄	—	158	1	no	—
223	C₂²×C₄	—	158	3	no	—
224	C₂²×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
225	C₂²×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
226	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
227	C₂×C₈	—	79	2	no	—
228	C₂×C₈	—	158	3	no	—
229	C₂×C₈	—	158	3	no	—
230	C₂×C₈	—	158	3	no	—
231	C₂×C₈	—	158	1	no	—
232	C₂×C₈	—	158	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	3	no	—
235	C₄⋊C₄	—	2	3	no	—
236	C₄⋊C₄	—	4	6	no	—
237	C₄⋊C₄	—	8	3	no	—
238	C₄⋊C₄	—	79	2	no	—
239	C₄⋊C₄	—	158	12	no	—
240	C₄⋊C₄	—	158	12	no	—
241	C₄⋊C₄	—	158	12	no	—
242	C₄⋊C₄	—	158	1	no	—
243	C₄⋊C₄	(C₂×D₇₉)⋊C₄	1	12	yes	—
244	C₄⋊C₄	#11929625	1	2	yes	—
245	C₄²	—	4	6	no	—
246	C₄²	—	158	12	no	—
247	C₄²	C₂×C₄×D₇₉	1	6	yes	—
248	C₄²	C₂×D₃₁₆	1	6	yes	—
249	C₂³	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
250	Q₈	—	316	3	no	—
251	Q₈	—	316	3	no	—
252	Q₈	—	158	6	no	—
253	Q₈	—	158	6	no	—
254	Q₈	#14579084	1	16	yes	complemented
255	C₂×C₄	—	2	3	no	—
256	C₂×C₄	—	2	3	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	3	no	—
261	C₂×C₄	—	4	3	no	—
262	C₂×C₄	—	4	3	no	—
263	C₂×C₄	—	316	3	no	—
264	C₂×C₄	—	316	3	no	—
265	C₂×C₄	—	316	1	no	—
266	C₂×C₄	—	316	1	no	—
267	C₂×C₄	—	158	2	no	—
268	C₂×C₄	—	158	6	no	—
269	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	6	yes	—
270	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	6	yes	—
271	C₂×C₄	C₂×#11929612	1	6	yes	—
272	C₂×C₄	C₂×#11929614	1	6	yes	—
273	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
275	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
276	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
277	C₂×C₄	#14579207	1	1	yes	characteristic
278	C₂×C₄	#14579209	1	1	yes	characteristic
279	C₈	—	316	3	no	—
280	C₈	—	158	2	no	—
281	C₂²	C₂²×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
282	C₂²	#19099224	1	1	yes	characteristic
283	C₂²	#19099232	1	1	yes	characteristic
284	C₂²	C₂²×#11929612	1	1	yes	characteristic
285	C₂²	C₂²×#11929614	1	1	yes	characteristic
286	C₂²	#19099374	1	1	yes	characteristic
287	C₂²	#19099378	1	1	yes	characteristic
288	C₄	—	2	6	no	—
289	C₄	—	4	2	no	—
290	C₄	—	8	3	no	—
291	C₄	—	316	2	no	—
292	C₄	—	316	6	no	—
293	C₄	C₂×#14579084	1	12	yes	—
294	C₄	C₄×#11929625	1	1	yes	characteristic
295	C₄	#19099045	1	1	yes	characteristic
296	C₄	#19099225	1	1	yes	characteristic
297	C₄	#19099235	1	1	yes	characteristic
298	C₂	#23766238	1	1	yes	characteristic
299	C₂	C₂²×#14579084	1	1	yes	characteristic
300	C₂	#23771930	1	1	yes	characteristic
301	C₂	#23776042	1	1	yes	characteristic
302	C₂	#23776078	1	1	yes	characteristic
303	C₂	#23776381	1	1	yes	characteristic
304	C₂	#23776403	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	Q₈×C₄⋊C₄×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	Q₈×C₄⋊C₄	D₇₉
hypocenter	C₇₉	#22542
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ Q₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. #19099292	C₄	semidirect
97. Q₈×C₇₉	#103	semidirect
160. C₇₉	#22542	semidirect
254. Q₈	#14579084	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099292	2
C₂×Q₈×C₇₉⋊C₈	2
#23764323	2
#23764201	2
#23763402	2
C₄×#14579084	2
Q₈×C₄⋊C₄×C₇₉	2
#22542	79