Subgroups — C₂×#31770369 #46810608

Order 34432.

392 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16864386	C₂	1	1	yes	maximal, characteristic
2	C₂³×#14524130	C₂	1	1	yes	maximal, characteristic
3	#31770369	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×C₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#16864386	—	2	1	no	—
6	C₂×#16864386	—	2	2	no	—
7	C₂×#16864386	C₄	1	4	yes	—
8	C₂×#16864386	C₂²	1	2	yes	—
9	C₂²×#14524130	—	2	2	no	—
10	C₂²×#14524130	—	2	2	no	—
11	C₂²×#14524130	—	2	1	no	—
12	C₂²×#14524130	C₄	1	2	yes	—
13	C₂²×#14524130	C₄	1	2	yes	—
14	C₂²×#14524130	C₂²	1	2	yes	—
15	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₂₆₉	—	2	2	no	—
17	C₂×M₁₆×C₂₆₉	—	2	1	no	—
18	C₂×M₁₆×C₂₆₉	C₄	1	4	yes	—
19	C₂×M₁₆×C₂₆₉	C₂²	1	2	yes	—
20	C₂²×C₈×C₂₆₉	—	2	1	no	—
21	#16864386	—	2	4	no	—
22	#16864386	—	2	2	no	—
23	#16864386	C₂×C₄	1	4	yes	—
24	C₂×#14524130	—	2	4	no	—
25	C₂×#14524130	—	2	4	no	—
26	C₂×#14524130	—	2	4	no	—
27	C₂×#14524130	—	2	2	no	—
28	C₂×#14524130	—	2	2	no	—
29	C₂×#14524130	—	2	2	no	—
30	C₂×#14524130	C₂×C₄	1	4	yes	—
31	C₂⁴×C₂₆₉	Q₈	1	1	yes	characteristic
32	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₂₆₉	D₄	1	2	yes	—
37	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₆₉	D₄	1	2	yes	—
40	C₂²×C₄×C₂₆₉	Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
42	M₁₆×C₂₆₉	—	2	4	no	—
43	M₁₆×C₂₆₉	—	2	2	no	—
44	M₁₆×C₂₆₉	C₂×C₄	1	4	yes	—
45	C₂×C₈×C₂₆₉	—	2	4	no	—
46	C₂×C₈×C₂₆₉	—	2	2	no	—
47	#14524130	—	2	4	no	—
48	#14524130	—	2	4	no	—
49	C₂³×C₂₆₉	—	2	4	no	—
50	C₂³×C₂₆₉	C₄⋊C₄	1	2	yes	—
51	C₂³×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
52	C₂³×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂³×C₂₆₉	C₄⋊C₄	1	2	yes	—
54	C₂³×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₂₆₉	C₄²	1	2	yes	—
56	C₂×C₄×C₂₆₉	C₄²	1	2	yes	—
57	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₂₆₉	C₄⋊C₄	1	2	yes	—
64	C₂×C₄×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₆₉	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
73	C₈×C₂₆₉	—	2	4	no	—
74	C₂²×C₂₆₉	—	2	4	no	—
75	C₂²×C₂₆₉	—	2	4	no	—
76	C₂²×C₂₆₉	—	2	2	no	—
77	C₂²×C₂₆₉	—	2	2	no	—
78	C₂²×C₂₆₉	#95	1	2	yes	—
79	C₂²×C₂₆₉	#95	1	2	yes	—
80	C₂²×C₂₆₉	#108	1	4	yes	—
81	C₂²×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₂₆₉	#95	1	2	yes	—
85	C₄×C₂₆₉	#95	1	2	yes	—
86	C₄×C₂₆₉	C₂×C₄²	1	1	yes	characteristic
87	C₄×C₂₆₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₄×C₂₆₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₄×C₂₆₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₂₆₉	—	2	2	no	—
91	C₂×C₂₆₉	—	2	1	no	—
92	C₂×C₂₆₉	—	2	1	no	—
93	C₂×C₂₆₉	#341	1	4	yes	complemented
94	C₂×C₂₆₉	C₂×#95	1	1	yes	characteristic
95	C₂×C₂₆₉	C₂×#108	1	2	yes	—
96	C₂₆₉	C₂×#341	1	1	yes	characteristic, complemented, Sylow
97	C₂×#341	—	269	1	no	maximal, Sylow
98	C₂²×M₁₆	—	269	1	no	—
99	C₂²×M₁₆	D₂₆₉	1	1	yes	characteristic
100	C₂³×C₈	—	269	1	no	—
101	#341	—	269	4	no	—
102	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
103	C₂×M₁₆	—	2	2	no	—
104	C₂×M₁₆	—	2	1	no	—
105	C₂×M₁₆	—	269	4	no	—
106	C₂×M₁₆	—	269	2	no	—
107	C₂×M₁₆	—	538	1	no	—
108	C₂×M₁₆	—	538	2	no	—
109	C₂×M₁₆	Dic₂₆₉	1	4	yes	—
110	C₂×M₁₆	C₂×D₂₆₉	1	2	yes	—
111	C₂²×C₈	—	2	1	no	—
112	C₂²×C₈	—	269	2	no	—
113	C₂²×C₈	—	269	2	no	—
114	C₂²×C₈	—	269	2	no	—
115	C₂²×C₈	—	538	2	no	—
116	C₂²×C₈	—	538	1	no	—
117	C₂²×C₈	—	538	2	no	—
118	C₂⁴	Dic₅₃₈	1	1	yes	characteristic
119	C₂²×C₄	C₄×D₂₆₉	1	1	yes	characteristic
120	C₂²×C₄	C₄×D₂₆₉	1	1	yes	characteristic
121	C₂²×C₄	C₄×D₂₆₉	1	2	yes	—
122	C₂²×C₄	C₂×Dic₂₆₉	1	2	yes	—
123	C₂²×C₄	C₂²⋊D₂₆₉	1	2	yes	—
124	C₂²×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
125	C₂²×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
126	C₂²×C₄	D₁₀₇₆	1	2	yes	—
127	C₂²×C₄	Dic₅₃₈	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
129	M₁₆	—	2	4	no	—
130	M₁₆	—	2	2	no	—
131	M₁₆	—	269	4	no	—
132	M₁₆	—	538	4	no	—
133	M₁₆	—	538	2	no	—
134	M₁₆	C₂×Dic₂₆₉	1	4	yes	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	269	4	no	—
138	C₂×C₈	—	538	4	no	—
139	C₂×C₈	—	538	4	no	—
140	C₂×C₈	—	538	4	no	—
141	C₂×C₈	—	538	2	no	—
142	C₂×C₈	—	538	2	no	—
143	C₂×C₈	—	538	2	no	—
144	C₂³	—	2	4	no	—
145	C₂³	#16864381	1	2	yes	—
146	C₂³	#16864381	1	1	yes	characteristic
147	C₂³	#16864381	1	1	yes	characteristic
148	C₂³	C₄⋊Dic₂₆₉	1	2	yes	—
149	C₂³	C₂×Dic₅₃₈	1	1	yes	characteristic
150	C₂×C₄	C₄×Dic₂₆₉	1	2	yes	—
151	C₂×C₄	C₄×Dic₂₆₉	1	2	yes	—
152	C₂×C₄	#16864379	1	2	yes	—
153	C₂×C₄	#16864379	1	2	yes	—
154	C₂×C₄	#16864379	1	2	yes	—
155	C₂×C₄	#16864379	1	2	yes	—
156	C₂×C₄	C₂²⋊Dic₂₆₉	1	2	yes	—
157	C₂×C₄	C₂²⋊Dic₂₆₉	1	2	yes	—
158	C₂×C₄	#16864381	1	2	yes	—
159	C₂×C₄	#16864381	1	1	yes	characteristic
160	C₂×C₄	#16864381	1	1	yes	characteristic
161	C₂×C₄	C₄⋊Dic₂₆₉	1	2	yes	—
162	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
163	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
164	C₂×C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
165	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
166	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
167	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
168	C₈	—	2	4	no	—
169	C₈	—	538	4	no	—
170	C₈	—	538	4	no	—
171	C₂²	—	2	4	no	—
172	C₂²	—	2	4	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	2	no	—
175	C₂²	#22575360	1	2	yes	—
176	C₂²	#22575360	1	2	yes	—
177	C₂²	#22575388	1	4	yes	—
178	C₂²	C₂×#16864381	1	1	yes	characteristic
179	C₂²	C₂×#16864381	1	1	yes	characteristic
180	C₂²	C₂×C₄⋊Dic₂₆₉	1	1	yes	characteristic
181	C₄	#22575360	1	2	yes	—
182	C₄	#22575360	1	2	yes	—
183	C₄	C₂×C₄×Dic₂₆₉	1	1	yes	characteristic
184	C₄	C₂×#16864379	1	1	yes	characteristic
185	C₄	C₂×#16864379	1	1	yes	characteristic
186	C₄	C₂×C₂²⋊Dic₂₆₉	1	1	yes	characteristic
187	C₂	—	2	2	no	—
188	C₂	—	2	1	no	—
189	C₂	—	2	1	no	—
190	C₂	#31770369	1	4	yes	direct product factor
191	C₂	C₂×#22575360	1	1	yes	characteristic
192	C₂	C₂×#22575388	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₂₆₉
commutator	C₄×C₂₆₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₆₉
Fitting	C₂²×M₁₆×C₂₆₉	C₂
socle	C₂³×C₂₆₉	C₄⋊C₄
prime radical	C₂⁴×C₂₆₉	Q₈
cosocle	C₂²×C₄×C₂₆₉	C₂³
hypercenter	C₂²×M₁₆	D₂₆₉
hypocenter	C₂₆₉	C₂×#341
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #31770369	C₂	direct (N × H)
93. C₂×C₂₆₉	#341	semidirect
96. C₂₆₉	C₂×#341	semidirect
190. C₂	#31770369	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#16864386	2
C₂³×#14524130	2
#31770369	2
C₂²×M₁₆×C₂₆₉	2
C₂×#341	269

Subgroups — C2×#31770369 #46810608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31770369 #46810608