Subgroups — C₂×#23765534 #34442869

Order 20224.

638 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₆₃₂	C₂	1	1	yes	maximal, characteristic
2	#23765534	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19098388	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098363	C₂	1	2	yes	maximal
6	C₂×#500×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₆₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₃₁₆	—	2	1	no	—
9	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
10	C₄×D₆₃₂	—	2	2	no	—
11	C₄×D₆₃₂	C₂²	1	4	yes	—
12	C₂×#14579095	C₂²	1	1	yes	characteristic
13	C₂×#14579083	—	2	1	no	—
14	#19098388	C₂²	1	8	yes	complemented
15	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	4	yes	complemented
16	#19098363	C₂²	1	8	yes	—
17	C₂×C₄⋊Q₈×C₇₉	—	2	1	no	—
18	#500×C₇₉	—	2	2	no	—
19	#500×C₇₉	C₂²	1	4	yes	complemented
20	C₂×#107×C₇₉	—	2	1	no	—
21	C₂×#107×C₇₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂²×D₃₁₆	—	2	1	no	—
24	C₂²×C₄×D₇₉	—	4	1	no	—
25	C₂×D₆₃₂	—	2	1	no	—
26	C₂×D₆₃₂	—	2	2	no	—
27	C₂×D₆₃₂	Q₈	1	4	yes	complemented
28	C₂×D₆₃₂	C₂³	1	2	yes	—
29	C₄×D₃₁₆	—	2	4	no	—
30	C₄×D₃₁₆	—	4	2	no	—
31	C₂×C₄⋊Dic₇₉	—	2	1	no	—
32	C₂×(C₂×D₇₉)⋊C₄	—	4	1	no	—
33	C₂×C₇₉⋊C₁₆	—	2	4	no	—
34	C₂×C₇₉⋊C₁₆	C₂³	1	4	yes	—
35	#14579095	—	2	1	no	—
36	#14579095	C₂³	1	2	yes	—
37	#14579083	—	2	4	no	—
38	C₂²×Q₈×C₇₉	—	4	1	no	—
39	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₇₉	—	2	4	no	—
42	C₄⋊Q₈×C₇₉	—	4	2	no	—
43	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
45	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
46	#107×C₇₉	—	2	4	no	—
47	#107×C₇₉	—	2	2	no	—
48	#107×C₇₉	C₂³	1	4	yes	—
49	C₄×C₈×C₇₉	D₄	1	2	yes	—
50	C₄×C₈×C₇₉	C₂³	1	2	yes	—
51	C₂³×D₇₉	—	4	1	no	—
52	C₂×D₃₁₆	—	2	4	no	—
53	C₂×D₃₁₆	—	2	2	no	—
54	C₂×D₃₁₆	—	4	1	no	—
55	C₂×D₃₁₆	—	4	2	no	—
56	C₂²×Dic₇₉	—	4	1	no	—
57	C₂×C₄×D₇₉	—	4	4	no	—
58	C₂×C₄×D₇₉	—	8	4	no	—
59	D₆₃₂	—	2	4	no	—
60	D₆₃₂	—	2	2	no	—
61	D₆₃₂	C₂×Q₈	1	4	yes	complemented
62	C₄⋊Dic₇₉	—	2	2	no	—
63	C₄⋊Dic₇₉	—	4	1	no	—
64	(C₂×D₇₉)⋊C₄	—	4	4	no	—
65	C₇₉⋊C₁₆	—	2	4	no	—
66	C₂×Q₈×C₇₉	—	4	4	no	—
67	C₂×Q₈×C₇₉	—	8	4	no	—
68	C₂²×C₄×C₇₉	—	4	2	no	—
69	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₇₉	—	2	2	no	—
73	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₇₉	#41	1	2	yes	—
77	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₇₉	—	2	4	no	—
79	C₄⋊C₄×C₇₉	—	4	2	no	—
80	C₄⋊C₄×C₇₉	—	4	4	no	—
81	C₄²×C₇₉	D₈	1	2	yes	—
82	C₄²×C₇₉	C₂×D₄	1	2	yes	—
83	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
84	C₂²×D₇₉	—	4	2	no	—
85	C₂²×D₇₉	—	4	4	no	—
86	C₂²×D₇₉	—	8	2	no	—
87	C₂²×D₇₉	—	8	2	no	—
88	Q₈×C₇₉	—	8	8	no	—
89	C₂×C₄×C₇₉	—	2	2	no	—
90	C₂×C₄×C₇₉	—	4	4	no	—
91	C₂×C₄×C₇₉	—	8	4	no	—
92	C₂×C₄×C₇₉	D₁₆	1	4	yes	—
93	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
94	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₇₉	#137	1	2	yes	—
98	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
99	C₂×Dic₇₉	—	4	2	no	—
100	C₂×Dic₇₉	—	8	2	no	—
101	D₃₁₆	—	2	4	no	—
102	D₃₁₆	—	4	2	no	—
103	D₃₁₆	—	4	4	no	—
104	C₄×D₇₉	—	8	8	no	—
105	C₈×C₇₉	—	2	2	no	—
106	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
107	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
108	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
109	C₂²×C₇₉	#474	1	2	yes	—
110	C₂²×C₇₉	C₂×D₁₆	1	2	yes	—
111	C₂²×C₇₉	#510	1	2	yes	—
112	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
113	C₂×D₇₉	—	4	4	no	—
114	C₂×D₇₉	—	8	2	no	—
115	C₂×D₇₉	—	8	2	no	—
116	C₂×D₇₉	—	8	8	no	—
117	C₄×C₇₉	—	2	2	no	—
118	C₄×C₇₉	—	8	4	no	—
119	C₄×C₇₉	#474	1	2	yes	—
120	C₄×C₇₉	C₂×D₁₆	1	4	yes	—
121	C₄×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
123	Dic₇₉	—	8	2	no	—
124	C₂×#2224	—	79	1	no	maximal, Sylow
125	C₂×C₇₉	#2224	1	4	yes	complemented
126	C₂×C₇₉	C₂×#474	1	1	yes	characteristic
127	C₂×C₇₉	C₂²×D₁₆	1	1	yes	characteristic
128	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
129	D₇₉	—	8	4	no	—
130	C₂×#500	D₇₉	1	1	yes	characteristic, complemented
131	C₂×C₄×D₈	—	79	1	no	—
132	#2224	—	79	8	no	—
133	C₂×#366	—	79	2	no	—
134	C₂×C₄⋊C₁₆	—	79	1	no	—
135	C₂×#357	—	79	2	no	—
136	C₇₉	C₂×#2224	1	1	yes	characteristic, complemented, Sylow
137	C₂²×D₈	—	79	1	no	—
138	C₂×C₄⋊Q₈	—	2	1	no	—
139	C₂×C₄×D₄	—	158	1	no	—
140	C₂²×C₁₆	—	79	2	no	—
141	#500	—	2	2	no	—
142	#500	C₂×D₇₉	1	4	yes	complemented
143	C₄×D₈	—	79	4	no	—
144	C₄×D₈	—	158	2	no	—
145	C₂×#107	—	2	1	no	—
146	C₂×#107	—	79	1	no	—
147	C₂×#107	C₂×D₇₉	1	2	yes	—
148	C₂×#102	—	158	1	no	—
149	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
150	#366	—	79	8	no	—
151	C₄⋊C₁₆	—	79	4	no	—
152	#357	—	79	8	no	—
153	C₂²×Q₈	—	4	1	no	—
154	C₂²×D₄	—	158	1	no	—
155	C₂³×C₄	—	316	1	no	—
156	C₂×D₈	—	79	4	no	—
157	C₂×D₈	—	79	2	no	—
158	C₂×D₈	—	158	1	no	—
159	C₂×D₈	—	158	2	no	—
160	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
162	C₄⋊Q₈	—	2	4	no	—
163	C₄⋊Q₈	—	4	2	no	—
164	C₄×D₄	—	316	2	no	—
165	C₄×D₄	—	158	4	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	4	1	no	—
168	C₂×C₄⋊C₄	—	158	1	no	—
169	C₂×C₂²⋊C₄	—	316	1	no	—
170	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
171	C₂×C₁₆	—	79	4	no	—
172	C₂×C₁₆	—	158	4	no	—
173	#107	—	2	4	no	—
174	#107	—	2	2	no	—
175	#107	—	79	2	no	—
176	#107	—	158	1	no	—
177	#107	C₂²×D₇₉	1	4	yes	—
178	#102	—	158	4	no	—
179	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
180	C₄×C₈	C₂²×D₇₉	1	2	yes	—
181	C₂⁴	—	316	1	no	—
182	C₂×Q₈	—	4	4	no	—
183	C₂×Q₈	—	8	4	no	—
184	C₂×D₄	—	316	1	no	—
185	C₂×D₄	—	316	2	no	—
186	C₂×D₄	—	158	4	no	—
187	C₂×D₄	—	158	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	316	1	no	—
190	C₂²×C₄	—	316	4	no	—
191	C₂²×C₄	—	632	4	no	—
192	C₂²×C₄	#11929609	1	1	yes	characteristic
193	C₂²×C₄	#11929609	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
195	D₈	—	79	4	no	—
196	D₈	—	158	2	no	—
197	D₈	—	158	4	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
200	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
201	C₂×C₈	Q₈×D₇₉	1	2	yes	—
202	C₂×C₈	#11929625	1	2	yes	—
203	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	316	1	no	—
208	C₄⋊C₄	—	158	2	no	—
209	C₂²⋊C₄	—	316	4	no	—
210	C₄²	#11929609	1	2	yes	—
211	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
212	C₁₆	—	158	4	no	—
213	C₂³	—	316	2	no	—
214	C₂³	—	316	4	no	—
215	C₂³	—	632	2	no	—
216	C₂³	—	632	2	no	—
217	C₂³	C₂×#11929609	1	1	yes	characteristic
218	Q₈	—	8	8	no	—
219	D₄	—	316	4	no	—
220	D₄	—	316	2	no	—
221	D₄	—	158	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	8	4	no	—
225	C₂×C₄	—	316	2	no	—
226	C₂×C₄	—	632	2	no	—
227	C₂×C₄	—	632	8	no	—
228	C₂×C₄	#14579102	1	4	yes	—
229	C₂×C₄	#14579137	1	2	yes	—
230	C₂×C₄	C₂×#11929609	1	2	yes	—
231	C₂×C₄	C₂×#11929609	1	2	yes	—
232	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
233	C₂×C₄	#14579192	1	2	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	#14579137	1	2	yes	—
237	C₈	C₂×Q₈×D₇₉	1	1	yes	characteristic
238	C₈	C₂×#11929625	1	1	yes	characteristic
239	C₂²	—	316	4	no	—
240	C₂²	—	632	2	no	—
241	C₂²	—	632	8	no	—
242	C₂²	—	632	2	no	—
243	C₂²	#19098840	1	2	yes	—
244	C₂²	C₂×#14579102	1	2	yes	—
245	C₂²	#19099004	1	2	yes	—
246	C₂²	C₂²×#11929609	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	8	4	no	—
249	C₄	—	632	2	no	—
250	C₄	#19098840	1	2	yes	—
251	C₄	C₂×#14579102	1	4	yes	—
252	C₄	C₂×#14579137	1	1	yes	characteristic
253	C₄	C₂×#14579192	1	1	yes	characteristic
254	C₂	—	632	4	no	—
255	C₂	#23765534	1	4	yes	direct product factor
256	C₂	C₂×#19098840	1	1	yes	characteristic
257	C₂	C₂²×#14579102	1	1	yes	characteristic
258	C₂	C₂×#19099004	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#500×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂²×D₆₃₂	C₂²
hypercenter	C₂×#500	D₇₉
hypocenter	C₇₉	C₂×#2224
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #23765534	C₂	direct (N × H)
3. C₂×#19098388	C₂	semidirect
4. C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	semidirect
6. C₂×#500×C₇₉	C₂	semidirect
14. #19098388	C₂²	semidirect
15. C₄⋊(C₇₉⋊C₁₆)	C₂²	semidirect
19. #500×C₇₉	C₂²	semidirect
27. C₂×D₆₃₂	Q₈	semidirect
61. D₆₃₂	C₂×Q₈	semidirect
125. C₂×C₇₉	#2224	semidirect
130. C₂×#500	D₇₉	semidirect
136. C₇₉	C₂×#2224	semidirect
142. #500	C₂×D₇₉	semidirect
255. C₂	#23765534	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₆₃₂	2
#23765534	2
C₂×#19098388	2
C₂×C₄⋊(C₇₉⋊C₁₆)	2
C₂×#19098363	2
C₂×#500×C₇₉	2
C₂×#2224	79

Subgroups — C2×#23765534 #34442869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765534 #34442869