Subgroups — C₂×#23761438 #34199475

Order 20224.

390 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098943	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098384	C₂	1	1	yes	maximal, characteristic
3	#23761438	C₂	1	4	yes	maximal, direct product factor
4	C₂×#364×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₁₆	—	2	1	no	—
6	C₂²×Dic₃₁₆	—	2	1	no	—
7	C₂×#14579161	C₄	1	2	yes	—
8	C₂×#14579101	—	2	1	no	—
9	#19098943	—	2	1	no	—
10	#19098943	—	2	1	no	—
11	#19098943	C₄	1	2	yes	—
12	#19098943	C₂²	1	2	yes	—
13	#19098384	—	2	1	no	—
14	#19098384	C₂²	1	2	yes	—
15	C₂×M₃₂×C₇₉	—	2	1	no	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	#364×C₇₉	—	2	1	no	—
18	#364×C₇₉	C₂²	1	2	yes	—
19	C₂²×Dic₁₅₈	—	2	2	no	—
20	C₂×Dic₃₁₆	—	2	2	no	—
21	C₂×Dic₃₁₆	—	2	2	no	—
22	C₂×Dic₃₁₆	—	2	2	no	—
23	C₂×Dic₃₁₆	—	2	2	no	—
24	C₂×Dic₃₁₆	—	4	1	no	—
25	C₂×Dic₃₁₆	—	4	1	no	—
26	C₂×Dic₃₁₆	—	4	1	no	—
27	C₂×Dic₃₁₆	—	4	1	no	—
28	C₂×Dic₃₁₆	—	4	1	no	—
29	C₂×Dic₃₁₆	—	4	1	no	—
30	C₂×Dic₃₁₆	—	4	1	no	—
31	C₂×Dic₃₁₆	—	4	1	no	—
32	#14579161	—	2	2	no	—
33	#14579161	—	4	2	no	—
34	#14579161	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊Dic₇₉	—	4	2	no	—
36	#14579101	—	2	2	no	—
37	#14579101	—	4	1	no	—
38	C₂×C₇₉⋊C₁₆	—	4	1	no	—
39	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
42	M₃₂×C₇₉	—	2	2	no	—
43	M₃₂×C₇₉	—	4	1	no	—
44	C₂×C₁₆×C₇₉	—	4	1	no	—
45	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
48	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
49	C₂×Dic₁₅₈	—	2	4	no	—
50	C₂×Dic₁₅₈	—	4	2	no	—
51	C₂×Dic₁₅₈	—	4	2	no	—
52	C₂×Dic₁₅₈	—	4	2	no	—
53	C₂×Dic₁₅₈	—	4	2	no	—
54	C₂×Dic₁₅₈	—	4	2	no	—
55	C₂²×Dic₇₉	—	4	2	no	—
56	Dic₃₁₆	—	4	2	no	—
57	Dic₃₁₆	—	4	2	no	—
58	Dic₃₁₆	—	4	2	no	—
59	Dic₃₁₆	—	4	2	no	—
60	Dic₃₁₆	—	4	2	no	—
61	Dic₃₁₆	—	4	2	no	—
62	Dic₃₁₆	—	4	2	no	—
63	Dic₃₁₆	—	4	2	no	—
64	C₄⋊Dic₇₉	—	4	4	no	—
65	C₄⋊Dic₇₉	—	4	4	no	—
66	C₇₉⋊C₁₆	—	4	2	no	—
67	C₂²×C₄×C₇₉	—	2	1	no	—
68	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
72	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
73	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
74	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
75	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
76	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
77	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
78	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
79	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₇₉	—	2	1	no	—
82	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
84	C₁₆×C₇₉	—	4	2	no	—
85	C₂³×C₇₉	#100	1	1	yes	characteristic
86	C₂×C₄×C₇₉	—	2	4	no	—
87	C₂×C₄×C₇₉	—	2	1	no	—
88	C₂×C₄×C₇₉	—	2	1	no	—
89	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
90	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
91	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
92	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
93	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
94	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×Dic₇₉	—	4	4	no	—
96	C₂×Dic₇₉	—	8	2	no	—
97	C₂×Dic₇₉	—	8	2	no	—
98	Dic₁₅₈	—	4	4	no	—
99	Dic₁₅₈	—	4	4	no	—
100	Dic₁₅₈	—	8	2	no	—
101	Dic₁₅₈	—	8	2	no	—
102	C₈×C₇₉	#102	1	1	yes	characteristic
103	C₈×C₇₉	#102	1	1	yes	characteristic
104	C₈×C₇₉	#102	1	1	yes	characteristic
105	C₈×C₇₉	#102	1	1	yes	characteristic
106	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
107	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
108	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
109	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
110	C₂²×C₇₉	—	2	2	no	—
111	C₂²×C₇₉	#331	1	1	yes	characteristic
112	C₂²×C₇₉	#331	1	1	yes	characteristic
113	C₂²×C₇₉	C₂×#100	1	1	yes	characteristic
114	C₄×C₇₉	—	2	2	no	—
115	C₄×C₇₉	—	2	2	no	—
116	C₄×C₇₉	#331	1	1	yes	characteristic
117	C₄×C₇₉	#331	1	1	yes	characteristic
118	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
119	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
120	Dic₇₉	—	8	4	no	—
121	C₂×#1358	—	79	1	no	maximal, Sylow
122	C₂×C₇₉	—	2	1	no	—
123	C₂×C₇₉	—	2	1	no	—
124	C₂×C₇₉	#1358	1	2	yes	complemented
125	C₂×C₇₉	C₂×#331	1	1	yes	characteristic
126	C₂×#494	—	79	1	no	—
127	C₂×#364	—	79	1	no	—
128	C₂×#364	D₇₉	1	1	yes	characteristic
129	#1358	—	79	4	no	—
130	C₇₉	C₂×#1358	1	1	yes	characteristic, complemented, Sylow
131	C₂²×Q₁₆	—	158	1	no	—
132	C₂²×Q₁₆	—	158	1	no	—
133	C₂×C₄⋊Q₈	—	79	2	no	—
134	C₂×M₃₂	—	2	1	no	—
135	C₂×M₃₂	—	158	1	no	—
136	#494	—	79	2	no	—
137	#494	—	79	2	no	—
138	#494	—	158	1	no	—
139	#494	—	158	1	no	—
140	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
141	#364	—	2	1	no	—
142	#364	—	79	2	no	—
143	#364	—	158	1	no	—
144	#364	C₂×D₇₉	1	2	yes	—
145	C₂²×Q₈	—	158	2	no	—
146	C₂×Q₁₆	—	316	1	no	—
147	C₂×Q₁₆	—	316	1	no	—
148	C₂×Q₁₆	—	316	1	no	—
149	C₂×Q₁₆	—	316	1	no	—
150	C₂×Q₁₆	—	316	1	no	—
151	C₂×Q₁₆	—	316	1	no	—
152	C₂×Q₁₆	—	316	1	no	—
153	C₂×Q₁₆	—	316	1	no	—
154	C₂×Q₁₆	—	158	2	no	—
155	C₂×Q₁₆	—	158	2	no	—
156	C₂×Q₁₆	—	158	2	no	—
157	C₂×Q₁₆	—	158	2	no	—
158	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
160	C₄⋊Q₈	—	316	2	no	—
161	C₄⋊Q₈	—	79	4	no	—
162	C₄⋊Q₈	—	158	2	no	—
163	C₂×C₄⋊C₄	—	316	2	no	—
164	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
165	M₃₂	—	2	2	no	—
166	M₃₂	—	4	1	no	—
167	M₃₂	—	316	1	no	—
168	M₃₂	—	158	2	no	—
169	C₂×C₁₆	—	4	1	no	—
170	C₂×C₁₆	—	316	1	no	—
171	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
172	C₄×C₈	D₃₁₆	1	1	yes	characteristic
173	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
174	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
175	C₂×Q₈	—	316	2	no	—
176	C₂×Q₈	—	316	2	no	—
177	C₂×Q₈	—	316	2	no	—
178	C₂×Q₈	—	316	2	no	—
179	C₂×Q₈	—	316	2	no	—
180	C₂×Q₈	—	158	4	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	316	2	no	—
183	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
184	Q₁₆	—	316	2	no	—
185	Q₁₆	—	316	2	no	—
186	Q₁₆	—	316	2	no	—
187	Q₁₆	—	316	2	no	—
188	Q₁₆	—	316	2	no	—
189	Q₁₆	—	316	2	no	—
190	Q₁₆	—	316	2	no	—
191	Q₁₆	—	316	2	no	—
192	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
193	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
194	C₂×C₈	#11929609	1	1	yes	characteristic
195	C₂×C₈	#11929609	1	1	yes	characteristic
196	C₂×C₈	D₆₃₂	1	1	yes	characteristic
197	C₂×C₈	D₆₃₂	1	1	yes	characteristic
198	C₂×C₈	#11929611	1	1	yes	characteristic
199	C₂×C₈	#11929611	1	1	yes	characteristic
200	C₂×C₈	#11929613	1	1	yes	characteristic
201	C₂×C₈	#11929613	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
203	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
204	C₄⋊C₄	—	316	4	no	—
205	C₄⋊C₄	—	316	4	no	—
206	C₄²	—	2	1	no	—
207	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
208	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
209	C₁₆	—	4	2	no	—
210	C₁₆	—	316	2	no	—
211	C₂³	#14579077	1	1	yes	characteristic
212	Q₈	—	316	4	no	—
213	Q₈	—	316	4	no	—
214	Q₈	—	632	2	no	—
215	Q₈	—	632	2	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	316	4	no	—
220	C₂×C₄	—	632	2	no	—
221	C₂×C₄	—	632	2	no	—
222	C₂×C₄	#14579077	1	1	yes	characteristic
223	C₂×C₄	#14579081	1	1	yes	characteristic
224	C₂×C₄	#14579081	1	1	yes	characteristic
225	C₂×C₄	#14579083	1	1	yes	characteristic
226	C₂×C₄	#14579083	1	1	yes	characteristic
227	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
228	C₈	#14579081	1	1	yes	characteristic
229	C₈	#14579081	1	1	yes	characteristic
230	C₈	#14579083	1	1	yes	characteristic
231	C₈	#14579083	1	1	yes	characteristic
232	C₈	C₂×#11929609	1	1	yes	characteristic
233	C₈	C₂×D₆₃₂	1	1	yes	characteristic
234	C₈	C₂×#11929611	1	1	yes	characteristic
235	C₈	C₂×#11929613	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	#19098306	1	1	yes	characteristic
238	C₂²	#19098306	1	1	yes	characteristic
239	C₂²	C₂×#14579077	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	632	4	no	—
243	C₄	#19098306	1	1	yes	characteristic
244	C₄	#19098306	1	1	yes	characteristic
245	C₄	C₂×#14579081	1	1	yes	characteristic
246	C₄	C₂×#14579083	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	2	1	no	—
249	C₂	#23761438	1	2	yes	direct product factor
250	C₂	C₂×#19098306	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098306
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₂×#364×C₇₉	C₂
socle	C₂²×C₇₉	#331
prime radical	C₂³×C₇₉	#100
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×#364	D₇₉
hypocenter	C₇₉	C₂×#1358
LCS	G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. #23761438	C₂	direct (N × H)
124. C₂×C₇₉	#1358	semidirect
130. C₇₉	C₂×#1358	semidirect
249. C₂	#23761438	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×#19098943	2
C₂×#19098384	2
#23761438	2
C₂×#364×C₇₉	2
C₂×#1358	79

Subgroups — C2×#23761438 #34199475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23761438 #34199475