Group #28376063 — subgroups

Subgroups — #28376063

Order 13952.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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	#20295421	C₂	1	2	yes	maximal
3	#20295405	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295245	C₂	1	2	yes	maximal
5	#20295236	C₂	1	2	yes	maximal, complemented
6	#20295233	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295231	C₂	1	1	yes	maximal, characteristic
8	#20295157	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295131	C₂	1	2	yes	maximal, complemented
10	C₄×#13182017	C₂	1	1	yes	maximal, characteristic
11	#556×C₁₀₉	C₂	1	1	yes	maximal, characteristic
12	C₂×#13182019	C₂²	1	2	yes	—
13	C₂×#13182019	C₂²	1	2	yes	—
14	C₂×#13182019	C₂²	1	2	yes	—
15	C₂×#13182017	C₂²	1	1	yes	characteristic
16	C₂×#13182017	C₂²	1	2	yes	—
17	#16812699	C₂²	1	1	yes	characteristic
18	#16812699	C₂²	1	2	yes	—
19	#16812690	—	2	1	no	—
20	#16812690	—	2	2	no	—
21	Q₈×Dic₁₀₉	—	2	1	no	—
22	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
23	C₄×Dic₂₁₈	C₂²	1	2	yes	—
24	#16812630	C₂²	1	1	yes	characteristic
25	C₄⋊#11724784	C₂²	1	1	yes	characteristic
26	C₄⋊#11724784	C₂²	1	2	yes	—
27	#16812623	C₂²	1	2	yes	—
28	#16812622	C₂²	1	2	yes	—
29	#16812622	C₂²	1	1	yes	characteristic
30	#16812620	C₂²	1	1	yes	characteristic
31	#16812610	C₂²	1	2	yes	—
32	C₄×#11724784	C₂²	1	1	yes	characteristic
33	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
34	#126×C₁₀₉	—	2	2	no	—
35	#124×C₁₀₉	C₂²	1	2	yes	—
36	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
37	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
38	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
40	#117×C₁₀₉	—	2	1	no	—
41	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
42	C₂×Dic₂₁₈	C₂³	1	2	yes	—
43	#13182019	—	2	4	no	—
44	#13182019	—	4	2	no	—
45	#13182019	—	4	2	no	—
46	#13182017	—	2	2	no	—
47	#13182017	—	4	2	no	—
48	C₂×#11724784	C₂³	1	1	yes	characteristic
49	C₂×#11724784	C₂³	1	2	yes	—
50	C₂×#11724784	C₂³	1	1	yes	characteristic
51	C₄⋊Dic₁₀₉	—	2	2	no	—
52	C₄⋊Dic₁₀₉	—	2	1	no	—
53	C₄⋊Dic₁₀₉	—	2	2	no	—
54	C₄⋊Dic₁₀₉	—	2	1	no	—
55	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
56	#13182008	—	2	2	no	—
57	#13182008	—	2	1	no	—
58	C₄×Dic₁₀₉	—	2	2	no	—
59	C₄×Dic₁₀₉	—	2	1	no	—
60	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
61	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
62	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₉	—	2	1	no	—
64	C₄⋊C₄×C₁₀₉	—	2	1	no	—
65	C₄⋊C₄×C₁₀₉	—	2	2	no	—
66	C₄⋊C₄×C₁₀₉	—	2	1	no	—
67	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
69	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
70	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
71	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
72	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
73	C₄²×C₁₀₉	—	2	2	no	—
74	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
75	C₄²×C₁₀₉	C₂³	1	2	yes	—
76	C₂³×C₁₀₉	—	2	1	no	—
77	Q₈×C₁₀₉	—	2	2	no	—
78	Q₈×C₁₀₉	—	2	2	no	—
79	Q₈×C₁₀₉	—	2	2	no	—
80	D₄×C₁₀₉	—	2	1	no	—
81	D₄×C₁₀₉	—	2	1	no	—
82	C₂×C₄×C₁₀₉	—	2	1	no	—
83	C₂×C₄×C₁₀₉	—	2	2	no	—
84	C₂×C₄×C₁₀₉	—	4	2	no	—
85	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
90	C₂×Dic₁₀₉	—	2	1	no	—
91	C₂×Dic₁₀₉	—	2	1	no	—
92	C₂×Dic₁₀₉	—	2	2	no	—
93	Dic₂₁₈	—	2	1	no	—
94	Dic₂₁₈	—	2	2	no	—
95	Dic₂₁₈	—	2	2	no	—
96	Dic₂₁₈	—	2	1	no	—
97	#11724784	—	2	2	no	—
98	#11724784	—	2	1	no	—
99	#11724784	#41	1	2	yes	—
100	C₂²×C₁₀₉	—	4	1	no	—
101	C₂²×C₁₀₉	—	4	1	no	—
102	C₂²×C₁₀₉	—	4	1	no	—
103	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
104	C₄×C₁₀₉	—	2	2	no	—
105	C₄×C₁₀₉	—	2	2	no	—
106	C₄×C₁₀₉	—	2	1	no	—
107	C₄×C₁₀₉	—	4	2	no	—
108	C₄×C₁₀₉	—	4	1	no	—
109	C₄×C₁₀₉	#137	1	2	yes	—
110	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
111	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
112	Dic₁₀₉	—	4	1	no	—
113	Dic₁₀₉	—	4	2	no	—
114	Dic₁₀₉	—	4	1	no	—
115	C₂×C₁₀₉	—	8	1	no	—
116	C₂×C₁₀₉	#550	1	1	yes	characteristic
117	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
118	C₂×C₁₀₉	#578	1	1	yes	characteristic
119	#3398	—	109	1	no	maximal, Sylow
120	C₁₀₉	#3398	1	1	yes	characteristic, complemented, Sylow
121	#557	—	109	1	no	—
122	#556	D₁₀₉	1	1	yes	characteristic
123	#497	—	109	2	no	—
124	#494	—	109	1	no	—
125	#464	—	109	2	no	—
126	#462	—	109	1	no	—
127	#462	—	109	2	no	—
128	#461	—	109	1	no	—
129	#446	—	109	1	no	—
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	2	no	—
139	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
140	#126	—	2	2	no	—
141	#125	—	218	1	no	—
142	#125	—	218	2	no	—
143	#124	C₂×D₁₀₉	1	2	yes	—
144	C₂²⋊Q₈	—	2	1	no	—
145	C₄×Q₈	—	109	1	no	—
146	C₄×Q₈	—	109	2	no	—
147	C₄×Q₈	—	218	1	no	—
148	C₄×Q₈	C₂×D₁₀₉	1	2	yes	—
149	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
150	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
151	#117	—	2	1	no	—
152	#106	—	109	1	no	—
153	C₄⋊C₈	—	109	1	no	—
154	C₄⋊C₈	—	109	2	no	—
155	#103	—	109	2	no	—
156	#103	—	109	1	no	—
157	#103	—	109	2	no	—
158	#102	—	109	1	no	—
159	#97	—	109	2	no	—
160	C₄×C₈	—	109	1	no	—
161	C₂×Q₈	—	109	1	no	—
162	C₂×Q₈	—	109	2	no	—
163	C₂×Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
164	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
165	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
166	C₂²×C₄	—	2	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	—	109	1	no	—
173	C₂×C₈	—	109	1	no	—
174	C₂×C₈	—	109	2	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	109	1	no	—
179	C₄⋊C₄	—	218	2	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	C₂²⋊D₁₀₉	1	1	yes	characteristic
186	C₄⋊C₄	C₂²⋊D₁₀₉	1	2	yes	—
187	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₄²	—	2	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	—	2	1	no	—
197	Q₈	—	2	2	no	—
198	Q₈	—	2	2	no	—
199	Q₈	—	2	2	no	—
200	Q₈	—	218	1	no	—
201	Q₈	—	218	2	no	—
202	Q₈	—	218	2	no	—
203	Q₈	—	218	1	no	—
204	D₄	—	2	1	no	—
205	D₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	218	1	no	—
210	C₂×C₄	—	218	1	no	—
211	C₂×C₄	—	218	2	no	—
212	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
213	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
214	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
215	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
217	C₈	—	109	2	no	—
218	C₈	—	218	2	no	—
219	C₈	—	218	1	no	—
220	C₂²	—	4	1	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	1	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	1	no	—
229	C₄	—	436	1	no	—
230	C₄	—	436	2	no	—
231	C₄	—	436	1	no	—
232	C₄	#16812729	1	2	yes	—
233	C₄	#41×D₁₀₉	1	1	yes	characteristic
234	C₄	#16812745	1	1	yes	characteristic
235	C₂	—	8	1	no	—
236	C₂	#20295695	1	1	yes	characteristic
237	C₂	C₂×#16812729	1	1	yes	characteristic
238	C₂	#20295841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295755

C₂

yes

maximal, characteristic, complemented

#20295421

C₂

yes

maximal

#20295405

C₂

yes

maximal, characteristic, complemented

#20295245

C₂

yes

maximal

#20295236

C₂

yes

maximal, complemented

#20295233

C₂

yes

maximal, characteristic, complemented

#20295231

C₂

yes

maximal, characteristic

#20295157

C₂

yes

maximal, characteristic, complemented

#20295131

C₂

yes

maximal, complemented

C₄×#13182017

C₂

yes

maximal, characteristic

#556×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182019

C₂²

yes

—

C₂×#13182019

C₂²

yes

—

C₂×#13182019

C₂²

yes

—

C₂×#13182017

C₂²

yes

characteristic

C₂×#13182017

C₂²

yes

—

#16812699

C₂²

yes

characteristic

#16812699

C₂²

yes

—

#16812690

—

#16812690

—

Q₈×Dic₁₀₉

—

C₄×Dic₂₁₈

C₂²

yes

characteristic

C₄×Dic₂₁₈

C₂²

yes

—

#16812630

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

—

#16812623

C₂²

yes

—

#16812622

C₂²

yes

—

#16812622

C₂²

yes

characteristic

#16812620

C₂²

yes

characteristic

#16812610

C₂²

yes

—

C₄×#11724784

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

#126×C₁₀₉

—

#124×C₁₀₉

C₂²

yes

—

C₂²⋊Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

C₂²

yes

—

C₄×Q₈×C₁₀₉

C₂²

yes

characteristic

C₄×D₄×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

—

C₂×Dic₂₁₈

C₂³

yes

characteristic

C₂×Dic₂₁₈

C₂³

yes

—

—

—

—

—

—

C₂³

yes

characteristic

C₂×#11724784

C₂³

yes

—

C₂×#11724784

C₂³

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

characteristic

#13182008

—

#13182008

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

characteristic

C₂×Q₈×C₁₀₉

C₂³

yes

—

C₂×D₄×C₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

D₄

yes

characteristic

C₄⋊C₄×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×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

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

—

—

yes

—

100

C₂²×C₁₀₉

—

101

C₂²×C₁₀₉

—

102

C₂²×C₁₀₉

—

103

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

104

C₄×C₁₀₉

—

105

C₄×C₁₀₉

—

106

C₄×C₁₀₉

—

107

C₄×C₁₀₉

—

108

C₄×C₁₀₉

—

109

C₄×C₁₀₉

#137

yes

—

110

C₄×C₁₀₉

C₂×#41

yes

characteristic

111

C₄×C₁₀₉

H(5,𝔽₂)

yes

characteristic

112

Dic₁₀₉

—

113

Dic₁₀₉

—

114

Dic₁₀₉

—

115

C₂×C₁₀₉

—

116

C₂×C₁₀₉

#550

yes

characteristic

117

C₂×C₁₀₉

C₂×#137

yes

characteristic

118

C₂×C₁₀₉

#578

yes

characteristic

119

#3398

—

109

maximal, Sylow

120

C₁₀₉

#3398

yes

characteristic, complemented, Sylow

121

#557

—

109

—

122

#556

D₁₀₉

yes

characteristic

123

#497

—

109

—

124

#494

—

109

—

125

#464

—

109

—

126

#462

—

109

—

127

#462

—

109

—

128

#461

—

109

—

129

#446

—

109

—

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

C₂×D₁₀₉

yes

characteristic

140

#126

—

141

#125

—

218

—

142

#125

—

218

—

143

#124

C₂×D₁₀₉

yes

—

144

C₂²⋊Q₈

—

145

C₄×Q₈

—

109

—

146

C₄×Q₈

—

109

—

147

C₄×Q₈

—

218

—

148

C₄×Q₈

C₂×D₁₀₉

yes

—

149

C₄×Q₈

C₂×D₁₀₉

yes

characteristic

150

C₄×D₄

C₂×D₁₀₉

yes

characteristic

151

#117

—

152

#106

—

109

—

153

C₄⋊C₈

—

109

—

154

C₄⋊C₈

—

109

—

155

#103

—

109

—

156

#103

—

109

—

157

#103

—

109

—

158

#102

—

109

—

159

#97

—

109

—

160

C₄×C₈

—

109

—

161

C₂×Q₈

—

109

—

162

C₂×Q₈

—

109

—

163

C₂×Q₈

C₂²⋊D₁₀₉

yes

characteristic

164

C₂×Q₈

C₂²×D₁₀₉

yes

—

165

C₂×D₄

C₂²×D₁₀₉

yes

characteristic

166

C₂²×C₄

—

167

Q₁₆

—

436

—

168

Q₁₆

—

436

—

169

Q₁₆

—

218

—

170

SD₁₆

—

436

—

171

SD₁₆

—

218

—

172

C₂×C₈

—

109

—

173

C₂×C₈

—

109

—

174

C₂×C₈

—

109

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

109

—

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

C₂²⋊D₁₀₉

yes

characteristic

186

C₄⋊C₄

C₂²⋊D₁₀₉

yes

—

187

C₄⋊C₄

C₂²×D₁₀₉

yes

characteristic

188

C₂²⋊C₄

—

189

C₂²⋊C₄

—

190

C₂²⋊C₄

—

191

C₄²

—

192

C₄²

—

218

—

193

C₄²

—

218

—

194

C₄²

C₂²×D₁₀₉

yes

characteristic

195

C₄²

C₂²×D₁₀₉

yes

—

196

C₂³

—

197

Q₈

—

198

Q₈

—

199

Q₈

—

200

Q₈

—

218

—

201

Q₈

—

218

—

202

Q₈

—

218

—

203

Q₈

—

218

—

204

D₄

—

205

D₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

218

—

210

C₂×C₄

—

218

—

211

C₂×C₄

—

218

—

212

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

213

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

214

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

215

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

216

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

217

C₈

—

109

—

218

C₈

—

218

—

219

C₈

—

218

—

220

C₂²

—

221

C₂²

—

222

C₂²

—

223

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

228

C₄

—

229

C₄

—

436

—

230

C₄

—

436

—

231

C₄

—

436

—

232

C₄

#16812729

yes

—

233

C₄

#41×D₁₀₉

yes

characteristic

234

C₄

#16812745

yes

characteristic

235

C₂

—

236

C₂

#20295695

yes

characteristic

237

C₂

C₂×#16812729

yes

characteristic

238

C₂

#20295841

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#556×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×D₄×C₁₀₉	C₂³
hypercenter	#556	D₁₀₉
hypocenter	C₁₀₉	#3398
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #556

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#556×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

prime radical

C₂×D₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #556

N	H ≅ G/N	type
1. #20295755	C₂	semidirect
3. #20295405	C₂	semidirect
5. #20295236	C₂	semidirect
6. #20295233	C₂	semidirect
8. #20295157	C₂	semidirect
9. #20295131	C₂	semidirect
120. C₁₀₉	#3398	semidirect

H ≅ G/N

type

1. #20295755

C₂

semidirect

3. #20295405

C₂

semidirect

5. #20295236

C₂

semidirect

6. #20295233

C₂

semidirect

8. #20295157

C₂

semidirect

9. #20295131

C₂

semidirect

120. C₁₀₉

#3398

semidirect

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

subgroup	index
#20295755	2
#20295421	2
#20295405	2
#20295245	2
#20295236	2
#20295233	2
#20295231	2
#20295157	2
#20295131	2
C₄×#13182017	2
#556×C₁₀₉	2
#3398	109

subgroup

index

#556×C₁₀₉

109

Subgroups — #28376063

Conjugacy classes of subgroups

Complements

Maximal subgroups