Group #28375681 — subgroups

Subgroups — #28375681

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	#20295749	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295383	C₂	1	2	yes	maximal, complemented
3	#20295378	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295330	C₂	1	2	yes	maximal, complemented
5	#20295313	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295241	C₂	1	2	yes	maximal
7	#20295231	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13182017	C₂	1	2	yes	maximal, complemented
10	C₄×#13182014	C₂	1	1	yes	maximal, characteristic
11	#554×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182017	C₂²	1	2	yes	—
13	C₂×#13182014	C₂²	1	1	yes	characteristic
14	#16812699	C₂²	1	1	yes	characteristic
15	#16812694	—	2	2	no	—
16	#16812687	C₂²	1	2	yes	—
17	#16812675	—	2	1	no	—
18	C₄×Dic₂₁₈	C₂²	1	2	yes	—
19	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
20	#16812657	—	2	1	no	—
21	#16812632	C₂²	1	1	yes	characteristic
22	#16812630	C₂²	1	2	yes	—
23	C₄⋊#11724784	C₂²	1	1	yes	characteristic
24	C₄⋊#11724784	C₂²	1	2	yes	—
25	#16812622	C₂²	1	2	yes	—
26	#16812622	C₂²	1	2	yes	—
27	#16812622	C₂²	1	2	yes	—
28	#16812620	C₂²	1	1	yes	characteristic
29	#16812619	C₂²	1	2	yes	—
30	#16812619	C₂²	1	1	yes	characteristic
31	C₄×#11724784	C₂²	1	1	yes	characteristic
32	C₄×#11724784	C₂²	1	2	yes	—
33	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
34	#125×C₁₀₉	C₂²	1	2	yes	—
35	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
36	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
37	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
41	C₂×Dic₂₁₈	C₂³	1	2	yes	—
42	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
43	C₂×C₄×D₁₀₉	—	2	1	no	—
44	#13182017	—	2	4	no	—
45	#13182014	—	2	2	no	—
46	C₂×#11724784	C₂³	1	1	yes	characteristic
47	C₂×#11724784	C₂³	1	2	yes	—
48	C₂×#11724784	C₂³	1	1	yes	characteristic
49	C₄⋊Dic₁₀₉	—	2	1	no	—
50	C₄⋊Dic₁₀₉	—	2	1	no	—
51	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
52	#13182008	—	2	2	no	—
53	#13182006	—	2	2	no	—
54	#13182006	—	2	1	no	—
55	#13182006	—	2	2	no	—
56	C₄×Dic₁₀₉	—	2	2	no	—
57	C₂×Q₈×C₁₀₉	—	2	1	no	—
58	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
59	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	—	2	1	no	—
61	C₂²×C₄×C₁₀₉	—	2	2	no	—
62	C₄⋊C₄×C₁₀₉	—	2	2	no	—
63	C₄⋊C₄×C₁₀₉	—	2	1	no	—
64	C₄⋊C₄×C₁₀₉	—	2	2	no	—
65	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
67	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
69	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
70	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
71	C₄²×C₁₀₉	C₂³	1	2	yes	—
72	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₂³×C₁₀₉	—	2	1	no	—
74	C₂²×D₁₀₉	—	2	1	no	—
75	Q₈×C₁₀₉	—	2	2	no	—
76	Q₈×C₁₀₉	—	8	1	no	—
77	D₄×C₁₀₉	—	2	1	no	—
78	D₄×C₁₀₉	#41	1	2	yes	—
79	C₂×C₄×C₁₀₉	—	2	1	no	—
80	C₂×C₄×C₁₀₉	—	2	2	no	—
81	C₂×C₄×C₁₀₉	—	4	2	no	—
82	C₂×C₄×C₁₀₉	—	8	2	no	—
83	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
88	C₂×Dic₁₀₉	—	2	1	no	—
89	C₂×Dic₁₀₉	—	2	2	no	—
90	D₄₃₆	—	2	1	no	—
91	D₄₃₆	—	2	1	no	—
92	C₄×D₁₀₉	—	4	2	no	—
93	Dic₂₁₈	—	2	2	no	—
94	Dic₂₁₈	—	2	2	no	—
95	#11724784	—	2	1	no	—
96	#11724784	—	2	2	no	—
97	#11724784	—	2	1	no	—
98	C₂²×C₁₀₉	—	2	2	no	—
99	C₂²×C₁₀₉	—	4	1	no	—
100	C₂²×C₁₀₉	—	4	1	no	—
101	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
102	C₂×D₁₀₉	—	4	1	no	—
103	C₂×D₁₀₉	—	4	1	no	—
104	C₂×D₁₀₉	—	4	1	no	—
105	C₄×C₁₀₉	—	2	2	no	—
106	C₄×C₁₀₉	—	2	2	no	—
107	C₄×C₁₀₉	—	2	1	no	—
108	C₄×C₁₀₉	—	4	1	no	—
109	C₄×C₁₀₉	—	4	2	no	—
110	C₄×C₁₀₉	#135	1	2	yes	—
111	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
112	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
113	Dic₁₀₉	—	4	2	no	—
114	Dic₁₀₉	—	4	1	no	—
115	C₂×C₁₀₉	—	4	2	no	—
116	C₂×C₁₀₉	#548	1	1	yes	characteristic
117	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
118	C₂×C₁₀₉	#577	1	1	yes	characteristic
119	D₁₀₉	—	8	1	no	—
120	#3360	—	109	1	no	maximal, Sylow
121	C₁₀₉	#3360	1	1	yes	characteristic, complemented, Sylow
122	#556	—	109	1	no	—
123	#554	D₁₀₉	1	1	yes	characteristic, complemented
124	#488	—	109	2	no	—
125	#487	—	109	1	no	—
126	#479	—	109	2	no	—
127	#476	—	109	1	no	—
128	#463	—	109	2	no	—
129	#461	—	109	1	no	—
130	C₈×Q₈	—	109	1	no	—
131	C₄×SD₁₆	—	109	2	no	—
132	C₄×D₈	—	109	1	no	—
133	C₂×SD₁₆	—	109	2	no	—
134	C₂×D₈	—	109	1	no	—
135	C₄⋊Q₈	—	109	1	no	—
136	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
137	#126	—	218	2	no	—
138	#125	C₂×D₁₀₉	1	2	yes	—
139	#124	—	109	2	no	—
140	C₂²⋊Q₈	—	2	1	no	—
141	C₂²⋊Q₈	—	2	2	no	—
142	C₂²⋊Q₈	—	218	1	no	—
143	C₄×Q₈	—	109	2	no	—
144	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
145	C₄×D₄	—	109	1	no	—
146	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
147	C₄×D₄	C₂×D₁₀₉	1	2	yes	—
148	#117	—	218	1	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	#107	—	109	1	no	—
151	#106	—	109	2	no	—
152	C₄⋊C₈	—	109	1	no	—
153	C₄⋊C₈	—	109	2	no	—
154	#103	—	109	2	no	—
155	#103	—	109	2	no	—
156	#103	—	109	2	no	—
157	#102	—	109	1	no	—
158	#102	—	109	1	no	—
159	#102	—	109	2	no	—
160	C₄×C₈	—	109	1	no	—
161	C₄×C₈	—	109	2	no	—
162	C₂×Q₈	—	2	1	no	—
163	C₂×Q₈	—	109	2	no	—
164	C₂×Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
165	C₂×D₄	—	109	1	no	—
166	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	218	1	no	—
170	SD₁₆	—	218	4	no	—
171	D₈	—	218	2	no	—
172	C₂×C₈	—	109	1	no	—
173	C₂×C₈	—	109	2	no	—
174	C₂×C₈	—	109	1	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	109	1	no	—
179	C₄⋊C₄	—	218	2	no	—
180	C₄⋊C₄	—	218	1	no	—
181	C₄⋊C₄	—	218	1	no	—
182	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
183	C₄⋊C₄	C₂²⋊D₁₀₉	1	2	yes	—
184	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
185	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
186	C₂²⋊C₄	—	2	1	no	—
187	C₂²⋊C₄	—	2	2	no	—
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	C₂²×D₁₀₉	1	1	yes	characteristic
193	C₄²	C₂²×D₁₀₉	1	2	yes	—
194	C₂³	—	2	1	no	—
195	C₂³	—	218	1	no	—
196	Q₈	—	2	2	no	—
197	Q₈	—	8	1	no	—
198	Q₈	—	218	2	no	—
199	Q₈	—	218	2	no	—
200	D₄	—	2	1	no	—
201	D₄	—	218	1	no	—
202	D₄	—	218	1	no	—
203	D₄	#13182030	1	2	yes	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	8	2	no	—
208	C₂×C₄	—	436	2	no	—
209	C₂×C₄	—	218	1	no	—
210	C₂×C₄	—	218	2	no	—
211	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
212	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
213	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
214	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
215	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
216	C₈	—	218	1	no	—
217	C₈	—	218	2	no	—
218	C₈	—	218	1	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	1	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	436	1	no	—
223	C₂²	—	436	1	no	—
224	C₂²	—	436	1	no	—
225	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	4	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	436	2	no	—
232	C₄	—	436	1	no	—
233	C₄	#16812719	1	2	yes	—
234	C₄	C₂×#13182030	1	1	yes	characteristic
235	C₄	#16812747	1	1	yes	characteristic
236	C₂	—	4	2	no	—
237	C₂	—	872	1	no	—
238	C₂	#20295687	1	1	yes	characteristic
239	C₂	C₂×#16812719	1	1	yes	characteristic
240	C₂	#20295837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295749

C₂

yes

maximal, characteristic, complemented

#20295383

C₂

yes

maximal, complemented

#20295378

C₂

yes

maximal, characteristic, complemented

#20295330

C₂

yes

maximal, complemented

#20295313

C₂

yes

maximal, characteristic, complemented

#20295241

C₂

yes

maximal

#20295231

C₂

yes

maximal, characteristic, complemented

Q₈×#11724784

C₂

yes

maximal, characteristic, complemented

C₄×#13182017

C₂

yes

maximal, complemented

C₄×#13182014

C₂

yes

maximal, characteristic

#554×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182017

C₂²

yes

—

C₂×#13182014

C₂²

yes

characteristic

#16812699

C₂²

yes

characteristic

#16812694

—

#16812687

C₂²

yes

—

#16812675

—

C₄×Dic₂₁₈

C₂²

yes

—

C₄×D₄₃₆

C₂²

yes

characteristic

#16812657

—

#16812632

C₂²

yes

characteristic

#16812630

C₂²

yes

—

C₄⋊#11724784

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

—

#16812622

C₂²

yes

—

#16812622

C₂²

yes

—

#16812622

C₂²

yes

—

#16812620

C₂²

yes

characteristic

#16812619

C₂²

yes

—

#16812619

C₂²

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₄×#11724784

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

#125×C₁₀₉

C₂²

yes

—

C₂²⋊Q₈×C₁₀₉

—

C₂²⋊Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

C₂²

yes

characteristic

C₄×D₄×C₁₀₉

C₂²

yes

characteristic

C₄×D₄×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×Dic₂₁₈

C₂³

yes

—

C₂×D₄₃₆

C₂³

yes

characteristic

C₂×C₄×D₁₀₉

—

#13182017

—

#13182014

—

C₂×#11724784

C₂³

yes

characteristic

C₂×#11724784

C₂³

yes

—

C₂×#11724784

C₂³

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

—

—

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

characteristic

C₂×D₄×C₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

—

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

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂²⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

—

C₂²×D₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

#41

yes

—

C₂×C₄×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

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724784

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

100

C₂²×C₁₀₉

—

101

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

102

C₂×D₁₀₉

—

103

C₂×D₁₀₉

—

104

C₂×D₁₀₉

—

105

C₄×C₁₀₉

—

106

C₄×C₁₀₉

—

107

C₄×C₁₀₉

—

108

C₄×C₁₀₉

—

109

C₄×C₁₀₉

—

110

C₄×C₁₀₉

#135

yes

—

111

C₄×C₁₀₉

C₂×#41

yes

characteristic

112

C₄×C₁₀₉

2₋¹⁺⁴

yes

characteristic

113

Dic₁₀₉

—

114

Dic₁₀₉

—

115

C₂×C₁₀₉

—

116

C₂×C₁₀₉

#548

yes

characteristic

117

C₂×C₁₀₉

C₂×#135

yes

characteristic

118

C₂×C₁₀₉

#577

yes

characteristic

119

D₁₀₉

—

120

#3360

—

109

maximal, Sylow

121

C₁₀₉

#3360

yes

characteristic, complemented, Sylow

122

#556

—

109

—

123

#554

D₁₀₉

yes

characteristic, complemented

124

#488

—

109

—

125

#487

—

109

—

126

#479

—

109

—

127

#476

—

109

—

128

#463

—

109

—

129

#461

—

109

—

130

C₈×Q₈

—

109

—

131

C₄×SD₁₆

—

109

—

132

C₄×D₈

—

109

—

133

C₂×SD₁₆

—

109

—

134

C₂×D₈

—

109

—

135

C₄⋊Q₈

—

109

—

136

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

137

#126

—

218

—

138

#125

C₂×D₁₀₉

yes

—

139

#124

—

109

—

140

C₂²⋊Q₈

—

141

C₂²⋊Q₈

—

142

C₂²⋊Q₈

—

218

—

143

C₄×Q₈

—

109

—

144

C₄×Q₈

C₂×D₁₀₉

yes

characteristic

145

C₄×D₄

—

109

—

146

C₄×D₄

C₂×D₁₀₉

yes

characteristic

147

C₄×D₄

C₂×D₁₀₉

yes

—

148

#117

—

218

—

149

C₂×C₄⋊C₄

—

150

#107

—

109

—

151

#106

—

109

—

152

C₄⋊C₈

—

109

—

153

C₄⋊C₈

—

109

—

154

#103

—

109

—

155

#103

—

109

—

156

#103

—

109

—

157

#102

—

109

—

158

#102

—

109

—

159

#102

—

109

—

160

C₄×C₈

—

109

—

161

C₄×C₈

—

109

—

162

C₂×Q₈

—

163

C₂×Q₈

—

109

—

164

C₂×Q₈

C₂²⋊D₁₀₉

yes

characteristic

165

C₂×D₄

—

109

—

166

C₂×D₄

C₂²×D₁₀₉

yes

characteristic

167

C₂²×C₄

—

168

C₂²×C₄

—

169

C₂²×C₄

—

218

—

170

SD₁₆

—

218

—

171

D₈

—

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

C₂²⋊D₁₀₉

yes

characteristic

183

C₄⋊C₄

C₂²⋊D₁₀₉

yes

—

184

C₄⋊C₄

C₂²×D₁₀₉

yes

—

185

C₄⋊C₄

C₂²×D₁₀₉

yes

characteristic

186

C₂²⋊C₄

—

187

C₂²⋊C₄

—

188

C₂²⋊C₄

—

218

—

189

C₂²⋊C₄

—

218

—

190

C₂²⋊C₄

—

218

—

191

C₄²

—

218

—

192

C₄²

C₂²×D₁₀₉

yes

characteristic

193

C₄²

C₂²×D₁₀₉

yes

—

194

C₂³

—

195

C₂³

—

218

—

196

Q₈

—

197

Q₈

—

198

Q₈

—

218

—

199

Q₈

—

218

—

200

D₄

—

201

D₄

—

218

—

202

D₄

—

218

—

203

D₄

#13182030

yes

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

436

—

209

C₂×C₄

—

218

—

210

C₂×C₄

—

218

—

211

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

212

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

213

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

214

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

215

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

216

C₈

—

218

—

217

C₈

—

218

—

218

C₈

—

218

—

219

C₂²

—

220

C₂²

—

221

C₂²

—

222

C₂²

—

436

—

223

C₂²

—

436

—

224

C₂²

—

436

—

225

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

226

C₄

—

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

—

231

C₄

—

436

—

232

C₄

—

436

—

233

C₄

#16812719

yes

—

234

C₄

C₂×#13182030

yes

characteristic

235

C₄

#16812747

yes

characteristic

236

C₂

—

237

C₂

—

872

—

238

C₂

#20295687

yes

characteristic

239

C₂

C₂×#16812719

yes

characteristic

240

C₂

#20295837

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#554×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#13182014	C₂²
hypercenter	#554	D₁₀₉
hypocenter	C₁₀₉	#3360
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #554

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#554×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

C₂²

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #554

N	H ≅ G/N	type
1. #20295749	C₂	semidirect
2. #20295383	C₂	semidirect
3. #20295378	C₂	semidirect
4. #20295330	C₂	semidirect
5. #20295313	C₂	semidirect
7. #20295231	C₂	semidirect
8. Q₈×#11724784	C₂	semidirect
9. C₄×#13182017	C₂	semidirect
11. #554×C₁₀₉	C₂	semidirect
121. C₁₀₉	#3360	semidirect
123. #554	D₁₀₉	semidirect

H ≅ G/N

type

1. #20295749

C₂

semidirect

2. #20295383

C₂

semidirect

3. #20295378

C₂

semidirect

4. #20295330

C₂

semidirect

5. #20295313

C₂

semidirect

7. #20295231

C₂

semidirect

8. Q₈×#11724784

C₂

semidirect

9. C₄×#13182017

C₂

semidirect

11. #554×C₁₀₉

C₂

semidirect

121. C₁₀₉

#3360

semidirect

123. #554

D₁₀₉

semidirect

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

subgroup	index
#20295749	2
#20295383	2
#20295378	2
#20295330	2
#20295313	2
#20295241	2
#20295231	2
Q₈×#11724784	2
C₄×#13182017	2
C₄×#13182014	2
#554×C₁₀₉	2
#3360	109

subgroup

index

#554×C₁₀₉

109

Subgroups — #28375681

Conjugacy classes of subgroups

Complements

Maximal subgroups