Group #46810864 — subgroups

Subgroups — #46810864

Order 34432.

400 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×D₂₆₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#22575382	C₂	1	2	yes	maximal, complemented
3	C₂×#22575365	C₂	1	2	yes	maximal
4	#31770360	C₂	1	1	yes	maximal, characteristic, complemented
5	#336×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₂₆₉	C₄	1	1	yes	characteristic
7	C₂²×C₄×D₂₆₉	C₄	1	1	yes	characteristic
8	C₂²×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
9	C₂²×#14524130	C₂²	1	2	yes	—
10	C₂×C₄×Dic₂₆₉	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₂₆₉	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
13	C₄²×D₂₆₉	—	2	4	no	—
14	#22575382	—	2	4	no	—
15	#22575365	—	2	4	no	—
16	C₂²×C₈×C₂₆₉	C₂²	1	2	yes	—
17	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
18	C₂³×D₂₆₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₂₆₉	—	2	2	no	—
23	C₂×C₄×D₂₆₉	—	2	2	no	—
24	C₂×C₄×D₂₆₉	—	2	2	no	—
25	C₂×C₄×D₂₆₉	—	2	2	no	—
26	C₂×C₄×D₂₆₉	—	2	2	no	—
27	C₂×C₄×D₂₆₉	—	2	2	no	—
28	C₂×C₄×D₂₆₉	—	2	2	no	—
29	C₂×C₄×D₂₆₉	C₈	1	4	yes	complemented
30	C₂×C₄×D₂₆₉	D₄	1	2	yes	—
31	C₂×C₄×D₂₆₉	Q₈	1	2	yes	—
32	C₂×#14524130	—	2	2	no	—
33	C₂×#14524130	—	2	2	no	—
34	C₂×#14524130	—	2	2	no	—
35	C₄×Dic₂₆₉	—	2	2	no	—
36	C₄×Dic₂₆₉	—	2	2	no	—
37	C₄×Dic₂₆₉	D₄	1	4	yes	—
38	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₂₆₉	—	2	2	no	—
42	C₂×C₈×C₂₆₉	—	2	2	no	—
43	C₂×C₈×C₂₆₉	—	2	2	no	—
44	C₄²×C₂₆₉	—	2	2	no	—
45	C₂²×D₂₆₉	—	2	1	no	—
46	C₂²×D₂₆₉	—	2	1	no	—
47	C₂²×D₂₆₉	—	2	1	no	—
48	C₂²×D₂₆₉	—	2	1	no	—
49	C₂²×D₂₆₉	C₄⋊C₄	1	2	yes	—
50	C₂²×D₂₆₉	C₂×C₈	1	2	yes	—
51	C₂²×D₂₆₉	M₁₆	1	2	yes	—
52	C₂×Dic₂₆₉	—	2	1	no	—
53	C₂×Dic₂₆₉	—	2	1	no	—
54	C₂×Dic₂₆₉	—	2	1	no	—
55	C₂×Dic₂₆₉	—	2	1	no	—
56	C₂×Dic₂₆₉	C₂²⋊C₄	1	2	yes	—
57	C₂×Dic₂₆₉	C₂²⋊C₄	1	2	yes	—
58	C₂×Dic₂₆₉	C₄⋊C₄	1	2	yes	—
59	C₂×Dic₂₆₉	C₂×C₈	1	2	yes	—
60	C₂×Dic₂₆₉	M₁₆	1	2	yes	—
61	C₄×D₂₆₉	—	2	8	no	—
62	C₄×D₂₆₉	—	2	4	no	—
63	C₄×D₂₆₉	—	2	4	no	—
64	C₄×D₂₆₉	—	2	8	no	—
65	#14524130	—	4	2	no	—
66	C₂³×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₆₉	—	2	1	no	—
68	C₂×C₄×C₂₆₉	—	2	1	no	—
69	C₂×C₄×C₂₆₉	—	2	1	no	—
70	C₂×C₄×C₂₆₉	—	2	1	no	—
71	C₂×C₄×C₂₆₉	C₂×C₈	1	2	yes	—
72	C₂×C₄×C₂₆₉	M₁₆	1	2	yes	—
73	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₂₆₉	#41	1	2	yes	—
77	C₈×C₂₆₉	—	4	2	no	—
78	C₂²×C₂₆₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₂₆₉	#117	1	1	yes	characteristic
81	C₂²×C₂₆₉	C₂²×C₈	1	1	yes	characteristic
82	C₂²×C₂₆₉	C₂×M₁₆	1	1	yes	characteristic
83	C₂²×C₂₆₉	C₂×M₁₆	1	1	yes	characteristic
84	C₂²×C₂₆₉	C₂×M₁₆	1	1	yes	characteristic
85	C₂×D₂₆₉	—	2	2	no	—
86	C₂×D₂₆₉	—	2	2	no	—
87	C₂×D₂₆₉	—	2	2	no	—
88	C₂×D₂₆₉	—	2	2	no	—
89	C₂×D₂₆₉	—	2	2	no	—
90	C₂×D₂₆₉	—	2	2	no	—
91	C₂×D₂₆₉	C₄⋊C₈	1	4	yes	complemented
92	C₄×C₂₆₉	—	2	2	no	—
93	C₄×C₂₆₉	—	2	2	no	—
94	C₄×C₂₆₉	#121	1	2	yes	—
95	C₄×C₂₆₉	C₂²⋊Q₈	1	2	yes	—
96	Dic₂₆₉	—	2	2	no	—
97	Dic₂₆₉	—	2	2	no	—
98	Dic₂₆₉	C₂²⋊C₈	1	4	yes	complemented
99	C₂×C₂₆₉	#380	1	1	yes	characteristic
100	C₂×C₂₆₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
101	C₂×C₂₆₉	#407	1	1	yes	characteristic
102	C₂×C₂₆₉	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₂×C₂₆₉	#423	1	1	yes	characteristic
104	C₂×C₂₆₉	#431	1	1	yes	characteristic
105	C₂×C₂₆₉	#432	1	1	yes	characteristic
106	D₂₆₉	—	2	4	no	—
107	C₂₆₉	#1797	1	1	yes	characteristic, complemented, Sylow
108	#1797	—	269	1	no	maximal, Sylow
109	C₂²×C₄²	—	269	1	no	—
110	C₂×C₄⋊C₈	—	269	2	no	—
111	C₂×C₂²⋊C₈	—	269	2	no	—
112	#336	—	269	1	no	—
113	#336	D₂₆₉	1	1	yes	characteristic, complemented
114	C₂³×C₄	—	269	1	no	—
115	C₂³×C₄	—	269	1	no	—
116	C₂³×C₄	—	269	1	no	—
117	C₂²×C₈	—	269	2	no	—
118	C₂²×C₈	C₂×D₂₆₉	1	2	yes	—
119	C₂×C₄²	—	269	1	no	—
120	C₂×C₄²	—	269	1	no	—
121	C₂×C₄²	—	269	1	no	—
122	C₂×C₄²	—	538	4	no	—
123	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
124	C₄⋊C₈	—	538	4	no	—
125	C₂²⋊C₈	—	538	4	no	—
126	C₂⁴	—	269	1	no	—
127	C₂²×C₄	—	269	1	no	—
128	C₂²×C₄	—	269	1	no	—
129	C₂²×C₄	—	269	1	no	—
130	C₂²×C₄	—	269	4	no	—
131	C₂²×C₄	—	269	2	no	—
132	C₂²×C₄	—	269	2	no	—
133	C₂²×C₄	—	538	2	no	—
134	C₂²×C₄	—	538	2	no	—
135	C₂²×C₄	—	538	2	no	—
136	C₂²×C₄	—	538	2	no	—
137	C₂²×C₄	—	538	2	no	—
138	C₂²×C₄	—	538	2	no	—
139	C₂²×C₄	—	538	2	no	—
140	C₂²×C₄	C₄×D₂₆₉	1	1	yes	characteristic
141	C₂²×C₄	C₄×D₂₆₉	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	538	2	no	—
147	C₂×C₈	—	538	2	no	—
148	C₂×C₈	—	538	2	no	—
149	C₄²	—	2	2	no	—
150	C₄²	—	269	4	no	—
151	C₄²	—	538	2	no	—
152	C₄²	—	538	2	no	—
153	C₂³	—	269	2	no	—
154	C₂³	—	269	2	no	—
155	C₂³	—	269	2	no	—
156	C₂³	—	538	1	no	—
157	C₂³	—	538	1	no	—
158	C₂³	—	538	1	no	—
159	C₂³	—	538	1	no	—
160	C₂³	C₂×C₄×D₂₆₉	1	1	yes	characteristic
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	269	2	no	—
166	C₂×C₄	—	269	2	no	—
167	C₂×C₄	—	269	2	no	—
168	C₂×C₄	—	269	2	no	—
169	C₂×C₄	—	269	2	no	—
170	C₂×C₄	—	538	8	no	—
171	C₂×C₄	—	538	8	no	—
172	C₂×C₄	—	538	1	no	—
173	C₂×C₄	—	538	1	no	—
174	C₂×C₄	—	538	1	no	—
175	C₂×C₄	—	538	1	no	—
176	C₂×C₄	—	538	4	no	—
177	C₂×C₄	—	538	4	no	—
178	C₂×C₄	C₈×D₂₆₉	1	2	yes	—
179	C₂×C₄	#16864385	1	2	yes	—
180	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
181	C₂×C₄	D₄×D₂₆₉	1	2	yes	—
182	C₂×C₄	Q₈×D₂₆₉	1	1	yes	characteristic
183	C₂×C₄	#16864402	1	2	yes	—
184	C₈	—	4	2	no	—
185	C₈	—	1076	2	no	—
186	C₂²	—	269	4	no	—
187	C₂²	—	538	2	no	—
188	C₂²	—	538	2	no	—
189	C₂²	—	538	2	no	—
190	C₂²	—	538	2	no	—
191	C₂²	—	538	2	no	—
192	C₂²	—	538	2	no	—
193	C₂²	C₂²⋊C₄×D₂₆₉	1	1	yes	characteristic
194	C₂²	C₄⋊C₄×D₂₆₉	1	1	yes	characteristic
195	C₂²	#22575410	1	1	yes	characteristic
196	C₂²	C₂×C₈×D₂₆₉	1	1	yes	characteristic
197	C₂²	M₁₆×D₂₆₉	1	1	yes	characteristic
198	C₂²	M₁₆×D₂₆₉	1	1	yes	characteristic
199	C₂²	C₂×#16864385	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	269	4	no	—
203	C₄	—	538	2	no	—
204	C₄	—	538	2	no	—
205	C₄	#22575426	1	2	yes	—
206	C₄	#22575431	1	2	yes	—
207	C₂	—	538	4	no	—
208	C₂	#31770464	1	1	yes	characteristic
209	C₂	C₂²⋊C₈×D₂₆₉	1	1	yes	characteristic
210	C₂	#31770567	1	1	yes	characteristic
211	C₂	C₄⋊C₈×D₂₆₉	1	1	yes	characteristic
212	C₂	#31770627	1	1	yes	characteristic
213	C₂	#31770656	1	1	yes	characteristic
214	C₂	#31770660	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄²×D₂₆₉

C₂

yes

maximal, characteristic

C₂×#22575382

C₂

yes

maximal, complemented

C₂×#22575365

C₂

yes

maximal

#31770360

C₂

yes

maximal, characteristic, complemented

#336×C₂₆₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₂₆₉

C₄

yes

characteristic

C₂²×C₄×D₂₆₉

C₄

yes

characteristic

C₂²×C₄×D₂₆₉

C₂²

yes

characteristic

C₂²×#14524130

C₂²

yes

—

C₂×C₄×Dic₂₆₉

C₄

yes

characteristic

C₂×C₄×Dic₂₆₉

C₄

yes

characteristic

C₂×C₄×Dic₂₆₉

C₂²

yes

characteristic

C₄²×D₂₆₉

—

#22575382

—

#22575365

—

C₂²×C₈×C₂₆₉

C₂²

yes

—

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

C₂³×D₂₆₉

C₂×C₄

yes

characteristic

C₂²×Dic₂₆₉

C₂×C₄

yes

characteristic

C₂²×Dic₂₆₉

C₂×C₄

yes

characteristic

C₂²×Dic₂₆₉

C₂×C₄

yes

characteristic

C₂×C₄×D₂₆₉

—

C₂×C₄×D₂₆₉

—

C₂×C₄×D₂₆₉

—

C₂×C₄×D₂₆₉

—

C₂×C₄×D₂₆₉

—

C₂×C₄×D₂₆₉

—

C₂×C₄×D₂₆₉

—

C₂×C₄×D₂₆₉

C₈

yes

complemented

C₂×C₄×D₂₆₉

D₄

yes

—

C₂×C₄×D₂₆₉

Q₈

yes

—

C₂×#14524130

—

C₂×#14524130

—

C₂×#14524130

—

C₄×Dic₂₆₉

—

C₄×Dic₂₆₉

—

C₄×Dic₂₆₉

D₄

yes

—

C₂²×C₄×C₂₆₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₆₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

—

C₄²×C₂₆₉

—

C₂²×D₂₆₉

—

C₂²×D₂₆₉

—

C₂²×D₂₆₉

—

C₂²×D₂₆₉

—

C₂²×D₂₆₉

C₄⋊C₄

yes

—

C₂²×D₂₆₉

C₂×C₈

yes

—

C₂²×D₂₆₉

M₁₆

yes

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

C₂²⋊C₄

yes

—

C₂×Dic₂₆₉

C₂²⋊C₄

yes

—

C₂×Dic₂₆₉

C₄⋊C₄

yes

—

C₂×Dic₂₆₉

C₂×C₈

yes

—

C₂×Dic₂₆₉

M₁₆

yes

—

C₄×D₂₆₉

—

C₄×D₂₆₉

—

C₄×D₂₆₉

—

C₄×D₂₆₉

—

#14524130

—

C₂³×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂×C₈

yes

—

C₂×C₄×C₂₆₉

M₁₆

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₆₉

#41

yes

—

C₈×C₂₆₉

—

C₂²×C₂₆₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₆₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₆₉

#117

yes

characteristic

C₂²×C₂₆₉

C₂²×C₈

yes

characteristic

C₂²×C₂₆₉

C₂×M₁₆

yes

characteristic

C₂²×C₂₆₉

C₂×M₁₆

yes

characteristic

C₂²×C₂₆₉

C₂×M₁₆

yes

characteristic

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₂×D₂₆₉

C₄⋊C₈

yes

complemented

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

#121

yes

—

C₄×C₂₆₉

C₂²⋊Q₈

yes

—

Dic₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

C₂²⋊C₈

yes

complemented

C₂×C₂₆₉

#380

yes

characteristic

100

C₂×C₂₆₉

C₂×C₂²⋊C₈

yes

characteristic

101

C₂×C₂₆₉

#407

yes

characteristic

102

C₂×C₂₆₉

C₂×C₄⋊C₈

yes

characteristic

103

C₂×C₂₆₉

#423

yes

characteristic

104

C₂×C₂₆₉

#431

yes

characteristic

105

C₂×C₂₆₉

#432

yes

characteristic

106

D₂₆₉

—

107

C₂₆₉

#1797

yes

characteristic, complemented, Sylow

108

#1797

—

269

maximal, Sylow

109

C₂²×C₄²

—

269

—

110

C₂×C₄⋊C₈

—

269

—

111

C₂×C₂²⋊C₈

—

269

—

112

#336

—

269

—

113

#336

D₂₆₉

yes

characteristic, complemented

114

C₂³×C₄

—

269

—

115

C₂³×C₄

—

269

—

116

C₂³×C₄

—

269

—

117

C₂²×C₈

—

269

—

118

C₂²×C₈

C₂×D₂₆₉

yes

—

119

C₂×C₄²

—

269

—

120

C₂×C₄²

—

269

—

121

C₂×C₄²

—

269

—

122

C₂×C₄²

—

538

—

123

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

124

C₄⋊C₈

—

538

—

125

C₂²⋊C₈

—

538

—

126

C₂⁴

—

269

—

127

C₂²×C₄

—

269

—

128

C₂²×C₄

—

269

—

129

C₂²×C₄

—

269

—

130

C₂²×C₄

—

269

—

131

C₂²×C₄

—

269

—

132

C₂²×C₄

—

269

—

133

C₂²×C₄

—

538

—

134

C₂²×C₄

—

538

—

135

C₂²×C₄

—

538

—

136

C₂²×C₄

—

538

—

137

C₂²×C₄

—

538

—

138

C₂²×C₄

—

538

—

139

C₂²×C₄

—

538

—

140

C₂²×C₄

C₄×D₂₆₉

yes

characteristic

141

C₂²×C₄

C₄×D₂₆₉

yes

characteristic

142

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

538

—

147

C₂×C₈

—

538

—

148

C₂×C₈

—

538

—

149

C₄²

—

150

C₄²

—

269

—

151

C₄²

—

538

—

152

C₄²

—

538

—

153

C₂³

—

269

—

154

C₂³

—

269

—

155

C₂³

—

269

—

156

C₂³

—

538

—

157

C₂³

—

538

—

158

C₂³

—

538

—

159

C₂³

—

538

—

160

C₂³

C₂×C₄×D₂₆₉

yes

characteristic

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

269

—

166

C₂×C₄

—

269

—

167

C₂×C₄

—

269

—

168

C₂×C₄

—

269

—

169

C₂×C₄

—

269

—

170

C₂×C₄

—

538

—

171

C₂×C₄

—

538

—

172

C₂×C₄

—

538

—

173

C₂×C₄

—

538

—

174

C₂×C₄

—

538

—

175

C₂×C₄

—

538

—

176

C₂×C₄

—

538

—

177

C₂×C₄

—

538

—

178

C₂×C₄

C₈×D₂₆₉

yes

—

179

C₂×C₄

#16864385

yes

—

180

C₂×C₄

D₄×D₂₆₉

yes

characteristic

181

C₂×C₄

D₄×D₂₆₉

yes

—

182

C₂×C₄

Q₈×D₂₆₉

yes

characteristic

183

C₂×C₄

#16864402

yes

—

184

C₈

—

185

C₈

—

1076

—

186

C₂²

—

269

—

187

C₂²

—

538

—

188

C₂²

—

538

—

189

C₂²

—

538

—

190

C₂²

—

538

—

191

C₂²

—

538

—

192

C₂²

—

538

—

193

C₂²

C₂²⋊C₄×D₂₆₉

yes

characteristic

194

C₂²

C₄⋊C₄×D₂₆₉

yes

characteristic

195

C₂²

#22575410

yes

characteristic

196

C₂²

C₂×C₈×D₂₆₉

yes

characteristic

197

C₂²

M₁₆×D₂₆₉

yes

characteristic

198

C₂²

M₁₆×D₂₆₉

yes

characteristic

199

C₂²

C₂×#16864385

yes

characteristic

200

C₄

—

201

C₄

—

202

C₄

—

269

—

203

C₄

—

538

—

204

C₄

—

538

—

205

C₄

#22575426

yes

—

206

C₄

#22575431

yes

—

207

C₂

—

538

—

208

C₂

#31770464

yes

characteristic

209

C₂

C₂²⋊C₈×D₂₆₉

yes

characteristic

210

C₂

#31770567

yes

characteristic

211

C₂

C₄⋊C₈×D₂₆₉

yes

characteristic

212

C₂

#31770627

yes

characteristic

213

C₂

#31770656

yes

characteristic

214

C₂

#31770660

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₆₉
commutator	C₂²×C₂₆₉	C₂²×C₈
Fitting	#336×C₂₆₉	C₂
socle	C₂³×C₂₆₉	C₂²×C₄
prime radical	C₂³×D₂₆₉	C₂×C₄
cosocle	C₂²×C₄×C₂₆₉	C₂³
hypercenter	#336	D₂₆₉
hypocenter	C₂₆₉	#1797
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₆₉

commutator

C₂²×C₂₆₉

C₂²×C₈

Fitting

#336×C₂₆₉

C₂

socle

C₂³×C₂₆₉

C₂²×C₄

prime radical

C₂³×D₂₆₉

C₂×C₄

cosocle

C₂²×C₄×C₂₆₉

C₂³

D₂₆₉

C₂₆₉

G ▷ C₂²×C₂₆₉ ▷ C₂₆₉

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
2. C₂×#22575382	C₂	semidirect
4. #31770360	C₂	semidirect
5. #336×C₂₆₉	C₂	semidirect
29. C₂×C₄×D₂₆₉	C₈	semidirect
91. C₂×D₂₆₉	C₄⋊C₈	semidirect
98. Dic₂₆₉	C₂²⋊C₈	semidirect
107. C₂₆₉	#1797	semidirect
113. #336	D₂₆₉	semidirect

H ≅ G/N

type

2. C₂×#22575382

C₂

semidirect

4. #31770360

C₂

semidirect

5. #336×C₂₆₉

C₂

semidirect

29. C₂×C₄×D₂₆₉

C₈

semidirect

91. C₂×D₂₆₉

C₄⋊C₈

semidirect

98. Dic₂₆₉

C₂²⋊C₈

semidirect

107. C₂₆₉

#1797

semidirect

113. #336

D₂₆₉

semidirect

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

subgroup	index
C₂×C₄²×D₂₆₉	2
C₂×#22575382	2
C₂×#22575365	2
#31770360	2
#336×C₂₆₉	2
#1797	269

subgroup

index

C₂×C₄²×D₂₆₉

#336×C₂₆₉

269

Subgroups — #46810864

Conjugacy classes of subgroups

Complements

Maximal subgroups