Subgroups — C₂×#23773924 #35157885

Order 20224.

1902 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099120	C₂	1	2	yes	maximal
2	#23773924	C₂	1	16	yes	maximal, direct product factor
3	C₂×#19098972	C₂	1	8	yes	maximal, complemented
4	C₂×#19098601	C₂	1	4	yes	maximal
5	C₂²×#97×C₇₉	C₂	1	1	yes	maximal, characteristic
6	#19099120	—	2	8	no	—
7	#19099120	C₂²	1	16	yes	—
8	C₂×#14579161	C₂²	1	4	yes	—
9	C₂×#14579153	C₂²	1	4	yes	—
10	C₂×#14579140	—	2	4	no	—
11	C₂×C₄×Dic₁₅₈	—	2	4	no	—
12	C₂×#14579120	C₂²	1	4	yes	—
13	#19098972	C₂²	1	64	yes	complemented
14	#19098601	C₂²	1	32	yes	—
15	C₂×#14579095	C₂²	1	8	yes	—
16	C₂×#14579093	C₂²	1	8	yes	—
17	C₂³×C₈×C₇₉	C₂²	1	2	yes	—
18	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×#97×C₇₉	C₂²	1	4	yes	—
20	C₂×#97×C₇₉	C₂²	1	8	yes	—
21	C₂²×Dic₁₅₈	—	2	4	no	—
22	#14579161	—	2	8	no	—
23	#14579161	C₂³	1	16	yes	—
24	#14579153	—	2	8	no	—
25	#14579153	C₂³	1	16	yes	—
26	#14579140	—	2	32	no	—
27	C₄×Dic₁₅₈	—	2	32	no	—
28	#14579120	—	2	8	no	—
29	#14579120	C₂³	1	16	yes	—
30	C₂×C₄⋊Dic₇₉	—	2	4	no	—
31	C₂×C₄⋊Dic₇₉	C₂³	1	8	yes	—
32	C₂×Dic₇₉⋊C₄	—	2	8	no	—
33	C₂×C₂²⋊Dic₇₉	—	2	4	no	—
34	C₂×C₄×Dic₇₉	—	2	4	no	—
35	#14579095	C₂³	1	32	yes	—
36	#14579093	C₂³	1	32	yes	—
37	C₂³×C₄×C₇₉	D₄	1	2	yes	—
38	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₇₉	Q₈	1	16	yes	complemented
40	C₂²×C₈×C₇₉	C₂³	1	4	yes	—
41	C₂²×C₈×C₇₉	C₂³	1	8	yes	—
42	C₂×C₄²×C₇₉	D₄	1	2	yes	—
43	C₂×C₄²×C₇₉	D₄	1	4	yes	—
44	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
45	C₂×C₄²×C₇₉	C₂³	1	4	yes	—
46	#97×C₇₉	C₂³	1	16	yes	—
47	C₂×Dic₁₅₈	—	2	16	no	—
48	C₂×Dic₁₅₈	—	4	16	no	—
49	C₂²×Dic₇₉	—	2	8	no	—
50	C₄⋊Dic₇₉	—	2	16	no	—
51	C₄⋊Dic₇₉	—	2	8	no	—
52	C₄⋊Dic₇₉	C₂⁴	1	16	yes	—
53	Dic₇₉⋊C₄	—	2	32	no	—
54	C₂²⋊Dic₇₉	—	2	16	no	—
55	C₄×Dic₇₉	—	2	16	no	—
56	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₉	—	4	4	no	—
58	C₂²×C₄×C₇₉	C₂×D₄	1	4	yes	—
59	C₂²×C₄×C₇₉	C₂×D₄	1	8	yes	—
60	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₉	C₂×Q₈	1	8	yes	—
63	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
64	C₂²×C₄×C₇₉	C₂⁴	1	2	yes	—
65	C₂×C₈×C₇₉	C₂×Q₈	1	16	yes	—
66	C₂×C₈×C₇₉	C₂×Q₈	1	32	yes	complemented
67	C₂×C₈×C₇₉	C₂⁴	1	8	yes	—
68	C₄²×C₇₉	C₂×D₄	1	8	yes	—
69	C₄²×C₇₉	C₂×D₄	1	4	yes	—
70	C₄²×C₇₉	C₂⁴	1	4	yes	—
71	C₂³×C₇₉	—	2	4	no	—
72	C₂³×C₇₉	C₄⋊Q₈	1	4	yes	—
73	C₂³×C₇₉	C₂²×D₄	1	2	yes	—
74	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₉	—	2	8	no	—
76	C₂×C₄×C₇₉	—	4	8	no	—
77	C₂×C₄×C₇₉	C₄⋊Q₈	1	8	yes	—
78	C₂×C₄×C₇₉	C₄⋊Q₈	1	4	yes	—
79	C₂×C₄×C₇₉	C₂²×D₄	1	8	yes	—
80	C₂×C₄×C₇₉	C₂²×D₄	1	2	yes	—
81	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₇₉	C₂²×Q₈	1	4	yes	—
83	C₂×C₄×C₇₉	C₂²×Q₈	1	8	yes	—
84	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
85	C₂×Dic₇₉	—	2	16	no	—
86	C₂×Dic₇₉	—	4	16	no	—
87	Dic₁₅₈	—	4	32	no	—
88	C₈×C₇₉	C₂²×Q₈	1	16	yes	—
89	C₂²×C₇₉	—	2	8	no	—
90	C₂²×C₇₉	—	2	4	no	—
91	C₂²×C₇₉	C₂×C₄⋊Q₈	1	4	yes	—
92	C₂²×C₇₉	C₂×C₄⋊Q₈	1	2	yes	—
93	C₂²×C₇₉	#575	1	4	yes	—
94	C₂²×C₇₉	C₂³×D₄	1	1	yes	characteristic
95	C₄×C₇₉	—	2	8	no	—
96	C₄×C₇₉	C₂×C₄⋊Q₈	1	4	yes	—
97	C₄×C₇₉	C₂×C₄⋊Q₈	1	2	yes	—
98	C₄×C₇₉	C₂³×Q₈	1	2	yes	—
99	Dic₇₉	—	4	16	no	—
100	C₂×#3162	—	79	1	no	maximal, Sylow
101	C₂×C₇₉	—	2	4	no	—
102	C₂×C₇₉	#3162	1	4	yes	complemented
103	C₂×C₇₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₇₉	C₂×#575	1	2	yes	—
105	C₂×#533	—	79	2	no	—
106	#3162	—	79	16	no	—
107	C₂×#501	—	79	8	no	—
108	C₂×#427	—	79	4	no	—
109	C₂²×#97	D₇₉	1	1	yes	characteristic
110	C₇₉	C₂×#3162	1	1	yes	characteristic, complemented, Sylow
111	C₂³×C₈	C₂×D₇₉	1	2	yes	—
112	#533	—	79	16	no	—
113	#533	—	158	8	no	—
114	C₂×C₄⋊Q₈	—	79	4	no	—
115	C₂×#125	—	79	4	no	—
116	C₂×C₂²⋊Q₈	—	158	4	no	—
117	C₂×C₄×Q₈	—	158	4	no	—
118	C₂×#117	—	79	4	no	—
119	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
120	#501	—	79	64	no	—
121	#427	—	79	32	no	—
122	C₂×#107	—	79	8	no	—
123	C₂×#106	—	79	8	no	—
124	C₂×#97	C₂×D₇₉	1	4	yes	—
125	C₂×#97	C₂×D₇₉	1	8	yes	—
126	C₂²×Q₈	—	158	4	no	—
127	C₂³×C₄	D₃₁₆	1	2	yes	—
128	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
129	C₂²×C₈	Dic₁₅₈	1	16	yes	complemented
130	C₂²×C₈	C₂²×D₇₉	1	4	yes	—
131	C₂²×C₈	C₂²×D₇₉	1	8	yes	—
132	C₄⋊Q₈	—	79	16	no	—
133	C₄⋊Q₈	—	158	8	no	—
134	#125	—	79	16	no	—
135	#125	—	158	8	no	—
136	C₂²⋊Q₈	—	158	32	no	—
137	C₄×Q₈	—	158	32	no	—
138	#117	—	79	16	no	—
139	#117	—	158	8	no	—
140	C₂×C₄⋊C₄	—	79	8	no	—
141	C₂×C₄⋊C₄	—	158	8	no	—
142	C₂×C₄⋊C₄	—	158	4	no	—
143	C₂×C₂²⋊C₄	—	158	4	no	—
144	C₂×C₄²	—	158	4	no	—
145	C₂×C₄²	D₃₁₆	1	4	yes	—
146	C₂×C₄²	D₃₁₆	1	2	yes	—
147	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
148	C₂×C₄²	C₂²×D₇₉	1	4	yes	—
149	#107	—	79	32	no	—
150	#106	—	79	32	no	—
151	#97	C₂²×D₇₉	1	16	yes	—
152	C₂⁴	C₂×D₃₁₆	1	1	yes	characteristic
153	C₂×Q₈	—	316	16	no	—
154	C₂×Q₈	—	158	16	no	—
155	C₂²×C₄	—	4	4	no	—
156	C₂²×C₄	—	158	8	no	—
157	C₂²×C₄	C₂×D₃₁₆	1	4	yes	—
158	C₂²×C₄	C₂×D₃₁₆	1	8	yes	—
159	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
160	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
161	C₂²×C₄	C₂×Dic₁₅₈	1	8	yes	—
162	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
163	C₂²×C₄	C₂³×D₇₉	1	2	yes	—
164	C₂×C₈	C₂×Dic₁₅₈	1	16	yes	—
165	C₂×C₈	C₂×Dic₁₅₈	1	32	yes	complemented
166	C₂×C₈	C₂³×D₇₉	1	8	yes	—
167	C₄⋊C₄	—	79	16	no	—
168	C₄⋊C₄	—	158	32	no	—
169	C₄⋊C₄	—	158	8	no	—
170	C₄⋊C₄	—	158	16	no	—
171	C₂²⋊C₄	—	158	16	no	—
172	C₄²	—	158	16	no	—
173	C₄²	C₂×D₃₁₆	1	8	yes	—
174	C₄²	C₂×D₃₁₆	1	4	yes	—
175	C₄²	C₂³×D₇₉	1	4	yes	—
176	C₂³	—	2	4	no	—
177	C₂³	#14579161	1	4	yes	—
178	C₂³	C₂²×D₃₁₆	1	2	yes	—
179	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
180	Q₈	—	316	32	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	316	16	no	—
184	C₂×C₄	—	158	16	no	—
185	C₂×C₄	#14579161	1	8	yes	—
186	C₂×C₄	#14579161	1	4	yes	—
187	C₂×C₄	C₂²×D₃₁₆	1	8	yes	—
188	C₂×C₄	C₂²×D₃₁₆	1	2	yes	—
189	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₁₅₈	1	4	yes	—
191	C₂×C₄	C₂²×Dic₁₅₈	1	8	yes	—
192	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
193	C₈	C₂²×Dic₁₅₈	1	16	yes	—
194	C₂²	—	2	4	no	—
195	C₂²	—	2	8	no	—
196	C₂²	C₂×#14579161	1	4	yes	—
197	C₂²	C₂×#14579161	1	2	yes	—
198	C₂²	#19099363	1	4	yes	—
199	C₂²	C₂³×D₃₁₆	1	1	yes	characteristic
200	C₄	—	2	8	no	—
201	C₄	—	316	16	no	—
202	C₄	C₂×#14579161	1	2	yes	—
203	C₄	C₂×#14579161	1	4	yes	—
204	C₄	C₂³×Dic₁₅₈	1	2	yes	—
205	C₂	—	2	4	no	—
206	C₂	#23773924	1	4	yes	direct product factor
207	C₂	C₂²×#14579161	1	1	yes	characteristic
208	C₂	C₂×#19099363	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₉
Fitting	C₂²×#97×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×D₄
prime radical	C₂⁴×C₇₉	C₂×D₄
hypercenter	C₂²×#97	D₇₉
hypocenter	C₇₉	C₂×#3162
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #23773924	C₂	direct (N × H)
3. C₂×#19098972	C₂	semidirect
13. #19098972	C₂²	semidirect
39. C₂²×C₈×C₇₉	Q₈	semidirect
66. C₂×C₈×C₇₉	C₂×Q₈	semidirect
102. C₂×C₇₉	#3162	semidirect
110. C₇₉	C₂×#3162	semidirect
129. C₂²×C₈	Dic₁₅₈	semidirect
165. C₂×C₈	C₂×Dic₁₅₈	semidirect
206. C₂	#23773924	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×#19099120	2
#23773924	2
C₂×#19098972	2
C₂×#19098601	2
C₂²×#97×C₇₉	2
C₂×#3162	79

Subgroups — C2×#23773924 #35157885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23773924 #35157885