Subgroups — #34445007

Order 20224.

696 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	C₂×#19098995	C₂	1	2	yes	maximal, complemented
2	#23775160	C₂	1	2	yes	maximal, complemented
3	#23765650	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765486	C₂	1	2	yes	maximal, complemented
5	#23765441	C₂	1	2	yes	maximal, complemented
6	#23765325	C₂	1	4	yes	maximal, complemented
7	C₄×Dic₆₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14579183	C₂²	1	4	yes	—
10	#19099234	—	2	2	no	—
11	#19098995	—	2	8	no	—
12	C₂×Dic₆₃₂	C₂²	1	1	yes	characteristic
13	C₂×#14579106	C₂²	1	4	yes	—
14	C₂×D₁₂₆₄	C₂²	1	2	yes	—
15	C₂×D₁₂₆₄	C₂²	1	2	yes	—
16	#19098940	C₂²	1	2	yes	—
17	#19098799	C₂²	1	4	yes	—
18	#19098764	—	2	2	no	—
19	#19098710	—	2	4	no	—
20	C₄×Dic₃₁₆	C₂²	1	2	yes	—
21	#19098389	C₂²	1	1	yes	characteristic
22	#19098368	C₂²	1	2	yes	—
23	#19098365	C₂²	1	4	yes	—
24	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
25	C₈×D₄×C₇₉	C₂²	1	1	yes	characteristic
26	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
27	C₂²⋊C₁₆×C₇₉	C₂²	1	2	yes	—
28	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
29	C₂×#11929624	—	2	4	no	—
30	#14579183	—	2	8	no	—
31	#14579183	—	4	4	no	—
32	C₂×Dic₃₁₆	C₂³	1	2	yes	—
33	C₂×#11929613	—	2	4	no	—
34	C₂×D₆₃₂	—	2	2	no	—
35	C₂×D₆₃₂	C₂³	1	4	yes	—
36	#14579158	—	2	2	no	—
37	#14579157	—	4	2	no	—
38	#14579135	—	2	4	no	—
39	#14579134	—	4	4	no	—
40	C₄×Dic₁₅₈	—	2	2	no	—
41	#14579123	—	4	2	no	—
42	Dic₆₃₂	D₄	1	4	yes	complemented
43	#14579106	—	2	8	no	—
44	D₁₂₆₄	—	2	4	no	—
45	D₁₂₆₄	—	4	2	no	—
46	#14579095	C₂³	1	2	yes	—
47	#14579086	—	2	2	no	—
48	#14579083	—	2	4	no	—
49	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
50	C₄×D₄×C₇₉	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₇₉	—	2	4	no	—
52	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
53	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
54	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
56	C₂²⋊C₈×C₇₉	D₄	1	2	yes	—
57	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
58	#11929624	—	4	8	no	—
59	#11929624	—	8	4	no	—
60	C₂×Dic₁₅₈	—	2	2	no	—
61	C₂×D₃₁₆	—	2	4	no	—
62	C₂×D₃₁₆	—	4	2	no	—
63	C₂×D₃₁₆	—	4	2	no	—
64	C₂×C₂²⋊D₇₉	—	4	4	no	—
65	C₂×C₂²⋊D₇₉	—	4	4	no	—
66	C₂×C₄×D₇₉	—	4	4	no	—
67	Dic₃₁₆	—	2	2	no	—
68	Dic₃₁₆	C₂×D₄	1	4	yes	—
69	#11929613	—	4	8	no	—
70	D₆₃₂	—	2	4	no	—
71	D₆₃₂	—	2	4	no	—
72	D₆₃₂	—	8	2	no	—
73	C₄⋊Dic₇₉	—	2	2	no	—
74	Dic₇₉⋊C₄	—	4	2	no	—
75	(C₂×D₇₉)⋊C₄	—	4	4	no	—
76	C₄×Dic₇₉	—	4	2	no	—
77	C₂×D₄×C₇₉	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₇₉	—	2	2	no	—
80	C₂×C₈×C₇₉	—	2	2	no	—
81	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
85	C₂²⋊C₄×C₇₉	D₈	1	2	yes	—
86	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
87	C₁₆×C₇₉	—	2	2	no	—
88	C₁₆×C₇₉	—	2	1	no	—
89	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
90	C₂³×C₇₉	C₂×D₈	1	2	yes	—
91	C₂²×D₇₉	—	4	4	no	—
92	D₄×C₇₉	—	2	2	no	—
93	C₂×C₄×C₇₉	—	2	2	no	—
94	C₂×C₄×C₇₉	—	2	2	no	—
95	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
99	C₂²⋊D₇₉	—	8	8	no	—
100	C₂²⋊D₇₉	—	16	4	no	—
101	C₂×Dic₇₉	—	4	2	no	—
102	C₂×Dic₇₉	—	4	2	no	—
103	D₃₁₆	—	4	4	no	—
104	D₃₁₆	—	4	4	no	—
105	D₃₁₆	—	8	4	no	—
106	D₃₁₆	—	16	2	no	—
107	C₄×D₇₉	—	8	8	no	—
108	Dic₁₅₈	—	2	4	no	—
109	Dic₁₅₈	—	4	2	no	—
110	C₈×C₇₉	—	2	2	no	—
111	C₈×C₇₉	—	2	1	no	—
112	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
113	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂²×C₇₉	—	2	2	no	—
115	C₂²×C₇₉	—	2	2	no	—
116	C₂²×C₇₉	—	2	2	no	—
117	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
118	C₂×D₇₉	—	8	4	no	—
119	C₂×D₇₉	—	8	4	no	—
120	C₂×D₇₉	—	8	4	no	—
121	C₄×C₇₉	—	2	1	no	—
122	C₄×C₇₉	—	2	2	no	—
123	C₄×C₇₉	#508	1	2	yes	complemented
124	C₄×C₇₉	D₄²	1	1	yes	characteristic
125	C₄×C₇₉	#576	1	1	yes	characteristic
126	Dic₇₉	—	4	4	no	—
127	Dic₇₉	—	8	2	no	—
128	#33853	—	79	1	no	maximal, Sylow
129	C₂×C₇₉	—	4	2	no	—
130	C₂×C₇₉	D₄×D₈	1	1	yes	characteristic
131	C₂×C₇₉	C₂×#508	1	1	yes	characteristic
132	C₂×C₇₉	#3415	1	1	yes	characteristic
133	D₇₉	—	16	4	no	—
134	C₂×#508	—	79	2	no	—
135	#3287	—	79	2	no	—
136	#2246	—	79	1	no	—
137	#2215	—	79	2	no	—
138	#2207	—	79	2	no	—
139	#2186	—	79	4	no	—
140	C₄×Q₃₂	—	79	1	no	—
141	D₄×C₁₆	D₇₉	1	1	yes	characteristic, complemented
142	C₇₉	#33853	1	1	yes	characteristic, complemented, Sylow
143	C₂×#135	—	79	4	no	—
144	#550	—	158	2	no	—
145	#508	—	158	8	no	—
146	C₂×Q₃₂	—	79	1	no	—
147	C₂×SD₃₂	—	79	4	no	—
148	C₂×D₁₆	—	79	2	no	—
149	C₂×D₁₆	—	79	2	no	—
150	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
151	#493	—	79	2	no	—
152	#466	—	79	4	no	—
153	#460	—	158	2	no	—
154	#449	—	158	4	no	—
155	C₄×Q₁₆	—	79	2	no	—
156	C₈×D₄	C₂×D₇₉	1	1	yes	characteristic
157	#366	—	79	1	no	—
158	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
159	#358	—	79	2	no	—
160	#357	—	79	4	no	—
161	C₂²⋊C₁₆	C₂×D₇₉	1	2	yes	—
162	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
163	C₂×#41	—	158	4	no	—
164	#135	—	316	4	no	—
165	#135	—	158	8	no	—
166	C₂×Q₁₆	—	79	2	no	—
167	C₂×SD₁₆	—	158	4	no	—
168	C₂×D₈	—	79	4	no	—
169	C₂×D₈	—	158	2	no	—
170	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
171	#127	—	316	2	no	—
172	#127	—	158	2	no	—
173	#121	—	316	4	no	—
174	#121	—	158	4	no	—
175	C₄×Q₈	—	158	2	no	—
176	C₄×D₄	—	316	2	no	—
177	C₄×D₄	D₃₁₆	1	1	yes	characteristic
178	Q₃₂	—	79	4	no	—
179	SD₃₂	—	158	8	no	—
180	D₁₆	—	316	2	no	—
181	D₁₆	—	158	4	no	—
182	C₂×C₁₆	—	2	4	no	—
183	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
184	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
185	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
186	#107	—	79	2	no	—
187	C₄⋊C₈	D₃₁₆	1	1	yes	characteristic
188	#103	—	158	2	no	—
189	#102	—	158	4	no	—
190	C₂²⋊C₈	D₃₁₆	1	2	yes	—
191	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
192	#41	—	316	8	no	—
193	#41	—	632	4	no	—
194	C₂×Q₈	—	158	2	no	—
195	C₂×D₄	—	316	4	no	—
196	C₂×D₄	—	316	4	no	—
197	C₂×D₄	—	316	2	no	—
198	C₂×D₄	—	316	2	no	—
199	C₂×D₄	—	158	4	no	—
200	C₂×D₄	D₆₃₂	1	1	yes	characteristic
201	C₂²×C₄	—	316	4	no	—
202	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
203	Q₁₆	—	79	4	no	—
204	Q₁₆	—	158	2	no	—
205	SD₁₆	—	316	8	no	—
206	D₈	—	632	2	no	—
207	D₈	—	158	4	no	—
208	D₈	—	158	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
212	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
213	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
214	C₄⋊C₄	—	316	2	no	—
215	C₄⋊C₄	—	158	2	no	—
216	C₄⋊C₄	D₆₃₂	1	1	yes	characteristic
217	C₂²⋊C₄	—	316	4	no	—
218	C₂²⋊C₄	D₆₃₂	1	2	yes	—
219	C₄²	—	316	2	no	—
220	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
221	C₁₆	—	2	1	no	—
222	C₁₆	—	2	2	no	—
223	C₁₆	D₄×D₇₉	1	2	yes	—
224	C₂³	—	316	4	no	—
225	C₂³	C₂×D₆₃₂	1	2	yes	—
226	Q₈	—	316	2	no	—
227	Q₈	—	158	4	no	—
228	D₄	—	2	2	no	—
229	D₄	—	316	4	no	—
230	D₄	—	316	4	no	—
231	D₄	—	632	8	no	—
232	D₄	—	632	4	no	—
233	D₄	—	1264	2	no	—
234	D₄	—	1264	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	316	2	no	—
238	C₂×C₄	—	316	2	no	—
239	C₂×C₄	—	632	8	no	—
240	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
241	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
242	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
247	C₈	#14579207	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	632	4	no	—
252	C₂²	—	632	4	no	—
253	C₂²	—	632	4	no	—
254	C₂²	C₂²×D₆₃₂	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	2	no	—
257	C₄	—	316	4	no	—
258	C₄	—	632	2	no	—
259	C₄	#19098995	1	2	yes	complemented
260	C₄	D₄×D₃₁₆	1	1	yes	characteristic
261	C₄	#19099368	1	1	yes	characteristic
262	C₂	—	4	2	no	—
263	C₂	—	1264	4	no	—
264	C₂	D₄×D₆₃₂	1	1	yes	characteristic
265	C₂	C₂×#19098995	1	1	yes	characteristic
266	C₂	#23776541	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₃₂
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	D₄×C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
hypercenter	D₄×C₁₆	D₇₉
hypocenter	C₇₉	#33853
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#19098995	C₂	semidirect
2. #23775160	C₂	semidirect
3. #23765650	C₂	semidirect
4. #23765486	C₂	semidirect
5. #23765441	C₂	semidirect
6. #23765325	C₂	semidirect
7. C₄×Dic₆₃₂	C₂	semidirect
8. D₄×C₁₆×C₇₉	C₂	semidirect
42. Dic₆₃₂	D₄	semidirect
123. C₄×C₇₉	#508	semidirect
141. D₄×C₁₆	D₇₉	semidirect
142. C₇₉	#33853	semidirect
259. C₄	#19098995	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098995	2
#23775160	2
#23765650	2
#23765486	2
#23765441	2
#23765325	2
C₄×Dic₆₃₂	2
D₄×C₁₆×C₇₉	2
#33853	79