Subgroups — C₂×#23765597 #34443032

Order 20224.

630 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098804	C₂	1	1	yes	maximal, characteristic
2	#23765597	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19098390	C₂	1	2	yes	maximal, complemented
4	C₂×#19098366	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
6	C₂²×#107×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₁₆	C₂²	1	1	yes	characteristic
8	C₂×#14579140	—	2	1	no	—
9	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
10	#19098804	—	2	2	no	—
11	#19098804	C₂²	1	4	yes	—
12	C₂×#14579095	C₂²	1	1	yes	characteristic
13	C₂×#14579086	—	2	1	no	—
14	#19098390	C₂²	1	8	yes	complemented
15	#19098366	C₂²	1	8	yes	complemented
16	C₂²⋊(C₇₉⋊C₁₆)	C₂²	1	4	yes	—
17	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₇₉	—	2	1	no	—
19	C₂×#107×C₇₉	—	2	2	no	—
20	C₂×#107×C₇₉	—	2	1	no	—
21	C₂×#107×C₇₉	C₂²	1	4	yes	—
22	C₂×#107×C₇₉	C₂²	1	2	yes	—
23	C₂²×Dic₁₅₈	—	2	1	no	—
24	C₂×Dic₃₁₆	—	2	4	no	—
25	C₂×Dic₃₁₆	—	2	1	no	—
26	C₂×Dic₃₁₆	C₂³	1	2	yes	—
27	#14579140	—	2	4	no	—
28	#14579140	—	4	2	no	—
29	C₂×C₄⋊Dic₇₉	—	2	1	no	—
30	C₂×Dic₇₉⋊C₄	—	4	1	no	—
31	C₂×C₂²⋊Dic₇₉	—	4	1	no	—
32	C₂×C₇₉⋊C₁₆	—	2	4	no	—
33	C₂×C₇₉⋊C₁₆	C₂³	1	4	yes	—
34	#14579095	—	2	1	no	—
35	#14579095	C₂³	1	2	yes	—
36	#14579086	—	2	4	no	—
37	C₂³×C₄×C₇₉	—	4	1	no	—
38	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₉	—	2	4	no	—
40	C₂²×C₈×C₇₉	D₄	1	2	yes	—
41	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
43	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₇₉	—	2	4	no	—
45	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
47	C₂×C₄⋊C₄×C₇₉	—	4	2	no	—
48	#107×C₇₉	—	2	4	no	—
49	#107×C₇₉	—	2	2	no	—
50	#107×C₇₉	C₂³	1	4	yes	—
51	C₂×Dic₁₅₈	—	2	2	no	—
52	C₂×Dic₁₅₈	—	4	2	no	—
53	C₂×Dic₁₅₈	—	4	1	no	—
54	C₂×Dic₁₅₈	—	4	2	no	—
55	C₂²×Dic₇₉	—	4	1	no	—
56	C₂²×Dic₇₉	—	4	1	no	—
57	Dic₃₁₆	—	2	4	no	—
58	Dic₃₁₆	—	4	2	no	—
59	C₄⋊Dic₇₉	—	2	2	no	—
60	C₄⋊Dic₇₉	—	4	1	no	—
61	Dic₇₉⋊C₄	—	4	4	no	—
62	C₂²⋊Dic₇₉	—	4	4	no	—
63	C₇₉⋊C₁₆	—	2	4	no	—
64	C₂⁴×C₇₉	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	—	2	2	no	—
66	C₂²×C₄×C₇₉	—	2	2	no	—
67	C₂²×C₄×C₇₉	—	4	4	no	—
68	C₂²×C₄×C₇₉	—	4	2	no	—
69	C₂²×C₄×C₇₉	—	8	4	no	—
70	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	D₈	1	2	yes	—
72	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₇₉	—	2	4	no	—
75	C₂×C₈×C₇₉	—	2	4	no	—
76	C₂×C₈×C₇₉	—	2	2	no	—
77	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₇₉	#41	1	4	yes	—
80	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₇₉	—	2	4	no	—
82	C₄⋊C₄×C₇₉	—	4	4	no	—
83	C₄⋊C₄×C₇₉	—	4	2	no	—
84	C₂³×C₇₉	—	2	2	no	—
85	C₂³×C₇₉	—	2	2	no	—
86	C₂³×C₇₉	SD₃₂	1	4	yes	—
87	C₂³×C₇₉	C₂×D₈	1	2	yes	—
88	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₇₉	—	2	2	no	—
90	C₂×C₄×C₇₉	—	2	2	no	—
91	C₂×C₄×C₇₉	—	2	2	no	—
92	C₂×C₄×C₇₉	—	2	4	no	—
93	C₂×C₄×C₇₉	—	4	4	no	—
94	C₂×C₄×C₇₉	—	8	8	no	—
95	C₂×C₄×C₇₉	—	8	4	no	—
96	C₂×C₄×C₇₉	#123	1	2	yes	—
97	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₇₉	#137	1	2	yes	—
100	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
101	C₂×Dic₇₉	—	4	2	no	—
102	C₂×Dic₇₉	—	4	2	no	—
103	C₂×Dic₇₉	—	8	2	no	—
104	C₂×Dic₇₉	—	8	2	no	—
105	Dic₁₅₈	—	4	2	no	—
106	Dic₁₅₈	—	4	2	no	—
107	Dic₁₅₈	—	8	2	no	—
108	C₈×C₇₉	—	2	2	no	—
109	C₈×C₇₉	#123	1	2	yes	—
110	C₈×C₇₉	C₂×#41	1	2	yes	—
111	C₂²×C₇₉	—	2	2	no	—
112	C₂²×C₇₉	—	2	2	no	—
113	C₂²×C₇₉	—	2	8	no	—
114	C₂²×C₇₉	#480	1	2	yes	—
115	C₂²×C₇₉	C₂×SD₃₂	1	4	yes	—
116	C₂²×C₇₉	C₂×SD₃₂	1	2	yes	—
117	C₂²×C₇₉	#510	1	2	yes	—
118	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
119	C₄×C₇₉	—	2	2	no	—
120	C₄×C₇₉	—	8	4	no	—
121	C₄×C₇₉	#480	1	2	yes	—
122	C₄×C₇₉	C₂×#123	1	1	yes	characteristic
123	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
124	Dic₇₉	—	8	2	no	—
125	Dic₇₉	—	8	2	no	—
126	C₂×#2235	—	79	1	no	maximal, Sylow
127	C₂×C₇₉	—	2	4	no	—
128	C₂×C₇₉	#2235	1	4	yes	complemented
129	C₂×C₇₉	C₂×#480	1	1	yes	characteristic
130	C₂×C₇₉	C₂²×SD₃₂	1	1	yes	characteristic
131	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
132	C₂×#467	—	79	1	no	—
133	C₂²×#107	D₇₉	1	1	yes	characteristic
134	#2235	—	79	8	no	—
135	C₂×#367	—	79	2	no	—
136	C₂×#358	—	79	2	no	—
137	C₂×C₂²⋊C₁₆	—	79	1	no	—
138	C₇₉	C₂×#2235	1	1	yes	characteristic, complemented, Sylow
139	C₂²×Q₁₆	—	79	1	no	—
140	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
141	C₂×C₂²⋊Q₈	—	158	1	no	—
142	C₂²×C₄⋊C₄	—	2	1	no	—
143	C₂²×C₁₆	—	79	2	no	—
144	#467	—	79	4	no	—
145	#467	—	158	2	no	—
146	C₂×#107	—	2	1	no	—
147	C₂×#107	—	2	2	no	—
148	C₂×#107	—	79	1	no	—
149	C₂×#107	C₂×D₇₉	1	2	yes	—
150	C₂×#107	C₂×D₇₉	1	4	yes	—
151	C₂×#103	—	158	1	no	—
152	#367	—	79	8	no	—
153	#358	—	79	8	no	—
154	C₂²⋊C₁₆	—	79	4	no	—
155	C₂²×Q₈	—	158	1	no	—
156	C₂³×C₄	—	4	1	no	—
157	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
158	C₂×Q₁₆	—	79	2	no	—
159	C₂×Q₁₆	—	158	1	no	—
160	C₂×Q₁₆	—	158	4	no	—
161	C₂²×C₈	—	2	4	no	—
162	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
164	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
165	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
166	C₂²⋊Q₈	—	316	2	no	—
167	C₂²⋊Q₈	—	158	4	no	—
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	4	2	no	—
171	C₂×C₄⋊C₄	—	4	1	no	—
172	C₂×C₄⋊C₄	—	316	1	no	—
173	C₂×C₄⋊C₄	—	158	1	no	—
174	C₂×C₂²⋊C₄	—	316	1	no	—
175	C₂×C₁₆	—	79	4	no	—
176	C₂×C₁₆	—	158	4	no	—
177	#107	—	2	4	no	—
178	#107	—	2	2	no	—
179	#107	—	79	2	no	—
180	#107	—	158	1	no	—
181	#107	C₂²×D₇₉	1	4	yes	—
182	#103	—	158	4	no	—
183	C₂⁴	#11929609	1	1	yes	characteristic
184	C₂×Q₈	—	316	1	no	—
185	C₂×Q₈	—	316	2	no	—
186	C₂×Q₈	—	316	2	no	—
187	C₂×Q₈	—	158	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	4	4	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	8	4	no	—
193	C₂²×C₄	—	316	1	no	—
194	C₂²×C₄	—	316	1	no	—
195	C₂²×C₄	#11929609	1	2	yes	—
196	C₂²×C₄	#11929609	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
198	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
199	Q₁₆	—	316	2	no	—
200	Q₁₆	—	158	4	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
205	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
206	C₂×C₈	#11929624	1	4	yes	—
207	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	316	4	no	—
212	C₄⋊C₄	—	316	1	no	—
213	C₄⋊C₄	—	158	2	no	—
214	C₂²⋊C₄	—	316	4	no	—
215	C₁₆	—	158	4	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	#14579105	1	4	yes	—
219	C₂³	C₂×#11929609	1	2	yes	—
220	C₂³	C₂×#11929609	1	1	yes	characteristic
221	Q₈	—	316	2	no	—
222	Q₈	—	316	2	no	—
223	Q₈	—	632	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	8	8	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	—	316	2	no	—
232	C₂×C₄	—	316	2	no	—
233	C₂×C₄	—	632	2	no	—
234	C₂×C₄	—	632	2	no	—
235	C₂×C₄	#14579144	1	2	yes	—
236	C₂×C₄	C₂×#11929609	1	2	yes	—
237	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
238	C₂×C₄	#14579192	1	2	yes	—
239	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	#14579144	1	2	yes	—
242	C₈	C₂×#11929624	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	8	no	—
246	C₂²	#19098876	1	2	yes	—
247	C₂²	C₂×#14579105	1	4	yes	—
248	C₂²	C₂×#14579105	1	2	yes	—
249	C₂²	#19099004	1	2	yes	—
250	C₂²	C₂²×#11929609	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	8	4	no	—
253	C₄	—	632	2	no	—
254	C₄	—	632	2	no	—
255	C₄	#19098876	1	2	yes	—
256	C₄	C₂×#14579144	1	1	yes	characteristic
257	C₄	C₂×#14579192	1	1	yes	characteristic
258	C₂	—	2	4	no	—
259	C₂	#23765597	1	4	yes	direct product factor
260	C₂	C₂×#19098876	1	1	yes	characteristic
261	C₂	C₂²×#14579105	1	1	yes	characteristic
262	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	C₂²×#107×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂⁴×C₇₉	D₈
hypercenter	C₂²×#107	D₇₉
hypocenter	C₇₉	C₂×#2235
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
2. #23765597	C₂	direct (N × H)
3. C₂×#19098390	C₂	semidirect
4. C₂×#19098366	C₂	semidirect
14. #19098390	C₂²	semidirect
15. #19098366	C₂²	semidirect
128. C₂×C₇₉	#2235	semidirect
138. C₇₉	C₂×#2235	semidirect
259. C₂	#23765597	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×#19098804	2
#23765597	2
C₂×#19098390	2
C₂×#19098366	2
C₂×C₂²⋊(C₇₉⋊C₁₆)	2
C₂²×#107×C₇₉	2
C₂×#2235	79

Subgroups — C2×#23765597 #34443032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765597 #34443032