Group #28376065 — subgroups

Subgroups — #28376065

Order 13952.

350 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 125 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	#20295755	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295749	C₂	1	1	yes	maximal, characteristic
3	#20295420	C₂	1	2	yes	maximal
4	#20295406	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295249	C₂	1	2	yes	maximal
6	#20295237	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295237	C₂	1	2	yes	maximal, complemented
8	#20295232	C₂	1	1	yes	maximal, characteristic
9	#20295134	C₂	1	2	yes	maximal, complemented
10	C₄×#13182018	C₂	1	1	yes	maximal, characteristic
11	#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₄₃₆	C₂²	1	2	yes	—
13	C₂×Dic₄₃₆	C₂²	1	2	yes	—
14	C₂×Dic₄₃₆	C₂²	1	2	yes	—
15	C₂×#13182018	C₂²	1	1	yes	characteristic
16	C₂×#13182018	C₂²	1	2	yes	—
17	#16812699	C₂²	1	1	yes	characteristic
18	#16812699	C₂²	1	1	yes	characteristic
19	#16812699	C₂²	1	2	yes	—
20	#16812694	—	2	2	no	—
21	#16812690	—	2	1	no	—
22	#16812690	—	2	2	no	—
23	#16812687	C₂²	1	2	yes	—
24	#16812675	—	2	1	no	—
25	Q₈×Dic₁₀₉	—	2	1	no	—
26	C₄×Dic₂₁₈	C₂²	1	2	yes	—
27	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₁₈	C₂²	1	2	yes	—
29	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
30	#16812657	—	2	1	no	—
31	#16812631	C₂²	1	1	yes	characteristic
32	#16812624	C₂²	1	2	yes	—
33	#16812624	C₂²	1	1	yes	characteristic
34	#16812624	C₂²	1	2	yes	—
35	#16812621	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
39	#97×C₁₀₉	C₂²	1	2	yes	—
40	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₁₈	C₂³	1	2	yes	—
42	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
43	C₂×Dic₂₁₈	C₂³	1	2	yes	—
44	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
45	C₂×C₄×D₁₀₉	—	2	1	no	—
46	Dic₄₃₆	—	2	4	no	—
47	Dic₄₃₆	—	4	2	no	—
48	Dic₄₃₆	—	4	2	no	—
49	#13182018	—	2	2	no	—
50	#13182018	—	4	2	no	—
51	C₄⋊Dic₁₀₉	—	2	1	no	—
52	C₄⋊Dic₁₀₉	—	2	2	no	—
53	C₄⋊Dic₁₀₉	—	2	1	no	—
54	C₄⋊Dic₁₀₉	—	2	2	no	—
55	C₄⋊Dic₁₀₉	—	2	1	no	—
56	C₄⋊Dic₁₀₉	—	2	1	no	—
57	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
59	#13182008	—	2	2	no	—
60	#13182008	—	2	2	no	—
61	#13182008	—	2	1	no	—
62	#13182006	—	2	2	no	—
63	#13182006	—	2	1	no	—
64	#13182006	—	2	2	no	—
65	C₄×Dic₁₀₉	—	2	2	no	—
66	C₄×Dic₁₀₉	—	2	2	no	—
67	C₄×Dic₁₀₉	—	2	1	no	—
68	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
71	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
74	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
75	C₄²×C₁₀₉	C₂³	1	2	yes	—
76	C₂²×D₁₀₉	—	2	1	no	—
77	Q₈×C₁₀₉	—	2	2	no	—
78	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
83	C₂×Dic₁₀₉	—	2	1	no	—
84	C₂×Dic₁₀₉	—	2	1	no	—
85	C₂×Dic₁₀₉	—	2	2	no	—
86	C₂×Dic₁₀₉	—	2	1	no	—
87	C₂×Dic₁₀₉	—	2	2	no	—
88	D₄₃₆	—	2	1	no	—
89	D₄₃₆	—	2	1	no	—
90	C₄×D₁₀₉	—	4	2	no	—
91	Dic₂₁₈	—	2	1	no	—
92	Dic₂₁₈	—	2	2	no	—
93	Dic₂₁₈	—	2	2	no	—
94	Dic₂₁₈	—	2	2	no	—
95	Dic₂₁₈	—	2	2	no	—
96	Dic₂₁₈	—	2	1	no	—
97	C₈×C₁₀₉	—	2	2	no	—
98	C₈×C₁₀₉	—	2	1	no	—
99	C₈×C₁₀₉	#41	1	2	yes	—
100	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
101	C₂×D₁₀₉	—	4	1	no	—
102	C₂×D₁₀₉	—	4	1	no	—
103	C₂×D₁₀₉	—	4	1	no	—
104	C₄×C₁₀₉	—	2	2	no	—
105	C₄×C₁₀₉	—	2	2	no	—
106	C₄×C₁₀₉	—	2	1	no	—
107	C₄×C₁₀₉	#137	1	2	yes	—
108	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
109	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
110	Dic₁₀₉	—	4	1	no	—
111	Dic₁₀₉	—	4	2	no	—
112	Dic₁₀₉	—	4	2	no	—
113	Dic₁₀₉	—	4	1	no	—
114	Dic₁₀₉	—	4	1	no	—
115	C₂×C₁₀₉	#550	1	1	yes	characteristic
116	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
117	C₂×C₁₀₉	#578	1	1	yes	characteristic
118	D₁₀₉	—	8	1	no	—
119	#3398	—	109	1	no	maximal, Sylow
120	C₁₀₉	#3398	1	1	yes	characteristic, complemented, Sylow
121	#557	—	109	1	no	—
122	#556	—	109	1	no	—
123	#497	—	109	2	no	—
124	#494	—	109	1	no	—
125	#464	—	109	2	no	—
126	#462	—	109	2	no	—
127	#462	—	109	1	no	—
128	#461	—	109	1	no	—
129	#446	D₁₀₉	1	1	yes	characteristic, complemented
130	#441	—	109	2	no	—
131	C₄×SD₁₆	—	109	1	no	—
132	C₂×Q₁₆	—	109	2	no	—
133	C₂×Q₁₆	—	109	2	no	—
134	C₂×Q₁₆	—	109	2	no	—
135	C₂×SD₁₆	—	109	2	no	—
136	C₂×SD₁₆	—	109	1	no	—
137	C₄⋊Q₈	—	109	1	no	—
138	C₄⋊Q₈	—	109	1	no	—
139	C₄⋊Q₈	—	109	2	no	—
140	#126	—	218	2	no	—
141	#125	—	218	1	no	—
142	#125	—	218	2	no	—
143	#124	—	109	2	no	—
144	C₂²⋊Q₈	—	218	1	no	—
145	C₄×Q₈	—	109	2	no	—
146	C₄×Q₈	—	109	1	no	—
147	C₄×Q₈	—	109	2	no	—
148	C₄×Q₈	—	218	1	no	—
149	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
150	C₄×D₄	—	109	1	no	—
151	#117	—	218	1	no	—
152	#106	—	109	1	no	—
153	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
154	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
155	#103	—	109	1	no	—
156	#103	—	109	2	no	—
157	#103	—	109	2	no	—
158	#102	—	109	1	no	—
159	#97	C₂×D₁₀₉	1	2	yes	—
160	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
161	C₂×Q₈	—	109	2	no	—
162	C₂×Q₈	—	109	1	no	—
163	C₂×Q₈	—	109	2	no	—
164	C₂×Q₈	D₄₃₆	1	1	yes	characteristic
165	C₂×D₄	—	109	1	no	—
166	C₂²×C₄	—	218	1	no	—
167	Q₁₆	—	436	2	no	—
168	Q₁₆	—	436	2	no	—
169	Q₁₆	—	218	4	no	—
170	SD₁₆	—	436	2	no	—
171	SD₁₆	—	218	2	no	—
172	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
174	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
175	C₄⋊C₄	—	109	1	no	—
176	C₄⋊C₄	—	109	1	no	—
177	C₄⋊C₄	—	218	2	no	—
178	C₄⋊C₄	—	218	2	no	—
179	C₄⋊C₄	—	218	1	no	—
180	C₄⋊C₄	—	218	1	no	—
181	C₄⋊C₄	—	218	2	no	—
182	C₄⋊C₄	—	218	1	no	—
183	C₄⋊C₄	—	218	2	no	—
184	C₄⋊C₄	—	218	1	no	—
185	C₄⋊C₄	—	218	1	no	—
186	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
187	C₄⋊C₄	D₄₃₆	1	2	yes	—
188	C₂²⋊C₄	—	218	2	no	—
189	C₂²⋊C₄	—	218	1	no	—
190	C₂²⋊C₄	—	218	2	no	—
191	C₄²	—	218	2	no	—
192	C₄²	—	218	2	no	—
193	C₄²	—	218	1	no	—
194	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
195	C₄²	C₂²×D₁₀₉	1	2	yes	—
196	C₂³	—	218	1	no	—
197	Q₈	—	2	2	no	—
198	Q₈	—	218	1	no	—
199	Q₈	—	218	2	no	—
200	Q₈	—	218	2	no	—
201	Q₈	—	218	2	no	—
202	Q₈	—	218	2	no	—
203	Q₈	—	218	1	no	—
204	D₄	—	218	1	no	—
205	D₄	—	218	1	no	—
206	C₂×C₄	—	436	2	no	—
207	C₂×C₄	—	218	1	no	—
208	C₂×C₄	—	218	1	no	—
209	C₂×C₄	—	218	2	no	—
210	C₂×C₄	—	218	1	no	—
211	C₂×C₄	—	218	2	no	—
212	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
213	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
214	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
215	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	2	1	no	—
219	C₈	#13182030	1	2	yes	—
220	C₂²	—	436	1	no	—
221	C₂²	—	436	1	no	—
222	C₂²	—	436	1	no	—
223	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	1	no	—
227	C₄	—	436	2	no	—
228	C₄	—	436	1	no	—
229	C₄	—	436	2	no	—
230	C₄	—	436	1	no	—
231	C₄	—	436	1	no	—
232	C₄	#16812732	1	2	yes	—
233	C₄	C₂×#13182030	1	1	yes	characteristic
234	C₄	#16812745	1	1	yes	characteristic
235	C₂	—	872	1	no	—
236	C₂	#20295698	1	1	yes	characteristic
237	C₂	C₂×#16812732	1	1	yes	characteristic
238	C₂	#20295843	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295755

C₂

yes

maximal, characteristic, complemented

#20295749

C₂

yes

maximal, characteristic

#20295420

C₂

yes

maximal

#20295406

C₂

yes

maximal, characteristic, complemented

#20295249

C₂

yes

maximal

#20295237

C₂

yes

maximal, characteristic, complemented

#20295237

C₂

yes

maximal, complemented

#20295232

C₂

yes

maximal, characteristic

#20295134

C₂

yes

maximal, complemented

C₄×#13182018

C₂

yes

maximal, characteristic

#446×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₄₃₆

C₂²

yes

—

C₂×Dic₄₃₆

C₂²

yes

—

C₂×Dic₄₃₆

C₂²

yes

—

C₂×#13182018

C₂²

yes

characteristic

C₂×#13182018

C₂²

yes

—

#16812699

C₂²

yes

characteristic

#16812699

C₂²

yes

characteristic

#16812699

C₂²

yes

—

—

—

—

C₂²

yes

—

#16812675

—

Q₈×Dic₁₀₉

—

C₄×Dic₂₁₈

C₂²

yes

—

C₄×Dic₂₁₈

C₂²

yes

characteristic

C₄×Dic₂₁₈

C₂²

yes

—

C₄×D₄₃₆

C₂²

yes

characteristic

#16812657

—

#16812631

C₂²

yes

characteristic

#16812624

C₂²

yes

—

#16812624

C₂²

yes

characteristic

#16812624

C₂²

yes

—

#16812621

C₂²

yes

characteristic

C₄×Q₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

C₂²

yes

—

#97×C₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

C₂³

yes

—

C₂×Dic₂₁₈

C₂³

yes

characteristic

C₂×Dic₂₁₈

C₂³

yes

—

C₂×D₄₃₆

C₂³

yes

characteristic

C₂×C₄×D₁₀₉

—

Dic₄₃₆

—

Dic₄₃₆

—

Dic₄₃₆

—

#13182018

—

#13182018

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

characteristic

C₄⋊Dic₁₀₉

C₂³

yes

—

—

—

—

—

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

D₄

yes

characteristic

C₄⋊C₄×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

—

C₂²×D₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

#41

yes

—

100

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

101

C₂×D₁₀₉

—

102

C₂×D₁₀₉

—

103

C₂×D₁₀₉

—

104

C₄×C₁₀₉

—

105

C₄×C₁₀₉

—

106

C₄×C₁₀₉

—

107

C₄×C₁₀₉

#137

yes

—

108

C₄×C₁₀₉

C₂×#41

yes

characteristic

109

C₄×C₁₀₉

H(5,𝔽₂)

yes

characteristic

110

Dic₁₀₉

—

111

Dic₁₀₉

—

112

Dic₁₀₉

—

113

Dic₁₀₉

—

114

Dic₁₀₉

—

115

C₂×C₁₀₉

#550

yes

characteristic

116

C₂×C₁₀₉

C₂×#137

yes

characteristic

117

C₂×C₁₀₉

#578

yes

characteristic

118

D₁₀₉

—

119

#3398

—

109

maximal, Sylow

120

C₁₀₉

#3398

yes

characteristic, complemented, Sylow

121

#557

—

109

—

122

#556

—

109

—

123

#497

—

109

—

124

#494

—

109

—

125

#464

—

109

—

126

#462

—

109

—

127

#462

—

109

—

128

#461

—

109

—

129

#446

D₁₀₉

yes

characteristic, complemented

130

#441

—

109

—

131

C₄×SD₁₆

—

109

—

132

C₂×Q₁₆

—

109

—

133

C₂×Q₁₆

—

109

—

134

C₂×Q₁₆

—

109

—

135

C₂×SD₁₆

—

109

—

136

C₂×SD₁₆

—

109

—

137

C₄⋊Q₈

—

109

—

138

C₄⋊Q₈

—

109

—

139

C₄⋊Q₈

—

109

—

140

#126

—

218

—

141

#125

—

218

—

142

#125

—

218

—

143

#124

—

109

—

144

C₂²⋊Q₈

—

218

—

145

C₄×Q₈

—

109

—

146

C₄×Q₈

—

109

—

147

C₄×Q₈

—

109

—

148

C₄×Q₈

—

218

—

149

C₄×Q₈

C₂×D₁₀₉

yes

characteristic

150

C₄×D₄

—

109

—

151

#117

—

218

—

152

#106

—

109

—

153

C₄⋊C₈

C₂×D₁₀₉

yes

characteristic

154

C₄⋊C₈

C₂×D₁₀₉

yes

—

155

#103

—

109

—

156

#103

—

109

—

157

#103

—

109

—

158

#102

—

109

—

159

#97

C₂×D₁₀₉

yes

—

160

C₄×C₈

C₂×D₁₀₉

yes

characteristic

161

C₂×Q₈

—

109

—

162

C₂×Q₈

—

109

—

163

C₂×Q₈

—

109

—

164

C₂×Q₈

D₄₃₆

yes

characteristic

165

C₂×D₄

—

109

—

166

C₂²×C₄

—

218

—

167

Q₁₆

—

436

—

168

Q₁₆

—

436

—

169

Q₁₆

—

218

—

170

SD₁₆

—

436

—

171

SD₁₆

—

218

—

172

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

173

C₂×C₈

C₂²×D₁₀₉

yes

—

174

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

175

C₄⋊C₄

—

109

—

176

C₄⋊C₄

—

109

—

177

C₄⋊C₄

—

218

—

178

C₄⋊C₄

—

218

—

179

C₄⋊C₄

—

218

—

180

C₄⋊C₄

—

218

—

181

C₄⋊C₄

—

218

—

182

C₄⋊C₄

—

218

—

183

C₄⋊C₄

—

218

—

184

C₄⋊C₄

—

218

—

185

C₄⋊C₄

—

218

—

186

C₄⋊C₄

D₄₃₆

yes

characteristic

187

C₄⋊C₄

D₄₃₆

yes

—

188

C₂²⋊C₄

—

218

—

189

C₂²⋊C₄

—

218

—

190

C₂²⋊C₄

—

218

—

191

C₄²

—

218

—

192

C₄²

—

218

—

193

C₄²

—

218

—

194

C₄²

C₂²×D₁₀₉

yes

characteristic

195

C₄²

C₂²×D₁₀₉

yes

—

196

C₂³

—

218

—

197

Q₈

—

198

Q₈

—

218

—

199

Q₈

—

218

—

200

Q₈

—

218

—

201

Q₈

—

218

—

202

Q₈

—

218

—

203

Q₈

—

218

—

204

D₄

—

218

—

205

D₄

—

218

—

206

C₂×C₄

—

436

—

207

C₂×C₄

—

218

—

208

C₂×C₄

—

218

—

209

C₂×C₄

—

218

—

210

C₂×C₄

—

218

—

211

C₂×C₄

—

218

—

212

C₂×C₄

C₂×D₄₃₆

yes

—

213

C₂×C₄

C₂×D₄₃₆

yes

—

214

C₂×C₄

C₂×D₄₃₆

yes

characteristic

215

C₂×C₄

C₂×D₄₃₆

yes

characteristic

216

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

217

C₈

—

218

C₈

—

219

C₈

#13182030

yes

—

220

C₂²

—

436

—

221

C₂²

—

436

—

222

C₂²

—

436

—

223

C₂²

C₂²×D₄₃₆

yes

characteristic

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

436

—

228

C₄

—

436

—

229

C₄

—

436

—

230

C₄

—

436

—

231

C₄

—

436

—

232

C₄

#16812732

yes

—

233

C₄

C₂×#13182030

yes

characteristic

234

C₄

#16812745

yes

characteristic

235

C₂

—

872

—

236

C₂

#20295698

yes

characteristic

237

C₂

C₂×#16812732

yes

characteristic

238

C₂

#20295843

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₃₆
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#446×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×D₄₃₆	C₂³
hypercenter	#446	D₁₀₉
hypocenter	C₁₀₉	#3398
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

subgroup

G/H

center

C₂²

C₂²×D₄₃₆

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#446×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

prime radical

C₂×D₄₃₆

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

N	H ≅ G/N	type
1. #20295755	C₂	semidirect
4. #20295406	C₂	semidirect
6. #20295237	C₂	semidirect
7. #20295237	C₂	semidirect
9. #20295134	C₂	semidirect
11. #446×C₁₀₉	C₂	semidirect
120. C₁₀₉	#3398	semidirect
129. #446	D₁₀₉	semidirect

H ≅ G/N

type

1. #20295755

C₂

semidirect

4. #20295406

C₂

semidirect

6. #20295237

C₂

semidirect

7. #20295237

C₂

semidirect

9. #20295134

C₂

semidirect

11. #446×C₁₀₉

C₂

semidirect

120. C₁₀₉

#3398

semidirect

129. #446

D₁₀₉

semidirect

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

subgroup	index
#20295755	2
#20295749	2
#20295420	2
#20295406	2
#20295249	2
#20295237	2
#20295232	2
#20295134	2
C₄×#13182018	2
#446×C₁₀₉	2
#3398	109

subgroup

index

#446×C₁₀₉

109

Subgroups — #28376065

Conjugacy classes of subgroups

Complements

Maximal subgroups