Group #28375746 — subgroups

Subgroups — #28375746

Order 13952.

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

normal subgroups only

This group has 372 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	Q₈×D₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295385	C₂	1	2	yes	maximal, complemented
3	#20295376	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295321	C₂	1	2	yes	maximal, complemented
5	#20295306	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295238	C₂	1	2	yes	maximal
7	#20295228	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295157	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295124	C₂	1	2	yes	maximal, complemented
10	C₄×#13182014	C₂	1	1	yes	maximal, characteristic
11	#556×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₀₉	—	2	1	no	—
13	C₂×#13182016	C₂²	1	2	yes	—
14	C₂×#13182014	C₂²	1	1	yes	characteristic
15	#16812699	C₂²	1	1	yes	characteristic
16	#16812699	C₂²	1	2	yes	—
17	#16812679	—	2	1	no	—
18	#16812679	—	2	2	no	—
19	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
20	C₄×D₄₃₆	C₂²	1	2	yes	—
21	#16812632	C₂²	1	2	yes	—
22	#16812632	C₂²	1	1	yes	characteristic
23	C₄⋊#11724784	C₂²	1	1	yes	characteristic
24	C₄⋊#11724784	C₂²	1	2	yes	—
25	#16812623	C₂²	1	2	yes	—
26	#16812623	C₂²	1	2	yes	—
27	#16812623	C₂²	1	2	yes	—
28	#16812620	C₂²	1	2	yes	—
29	#16812620	C₂²	1	1	yes	characteristic
30	#16812619	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	—	2	2	no	—
42	C₂×Dic₂₁₈	—	2	1	no	—
43	Q₈×D₁₀₉	—	4	4	no	—
44	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
45	C₂×C₄×D₁₀₉	—	2	2	no	—
46	C₂×C₄×D₁₀₉	—	2	1	no	—
47	#13182016	—	2	4	no	—
48	#13182014	—	2	2	no	—
49	C₂×#11724784	C₂³	1	1	yes	characteristic
50	C₂×#11724784	C₂³	1	1	yes	characteristic
51	C₂×#11724784	C₂³	1	2	yes	—
52	C₄⋊Dic₁₀₉	—	2	2	no	—
53	C₄⋊Dic₁₀₉	—	2	1	no	—
54	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
55	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
56	#13182006	—	2	2	no	—
57	#13182006	—	2	1	no	—
58	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
59	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
60	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₉	—	2	1	no	—
62	C₄⋊C₄×C₁₀₉	—	2	1	no	—
63	C₄⋊C₄×C₁₀₉	—	2	2	no	—
64	C₄⋊C₄×C₁₀₉	—	2	1	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	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
69	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
70	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
71	C₄²×C₁₀₉	—	2	2	no	—
72	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₄²×C₁₀₉	C₂³	1	2	yes	—
74	C₂³×C₁₀₉	—	2	1	no	—
75	C₂²×D₁₀₉	—	2	1	no	—
76	Q₈×C₁₀₉	—	2	2	no	—
77	Q₈×C₁₀₉	—	2	2	no	—
78	Q₈×C₁₀₉	—	2	2	no	—
79	D₄×C₁₀₉	—	2	1	no	—
80	D₄×C₁₀₉	—	2	1	no	—
81	C₂×C₄×C₁₀₉	—	2	1	no	—
82	C₂×C₄×C₁₀₉	—	2	2	no	—
83	C₂×C₄×C₁₀₉	—	4	2	no	—
84	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
89	C₂×Dic₁₀₉	—	2	1	no	—
90	C₂×Dic₁₀₉	—	2	2	no	—
91	D₄₃₆	—	2	1	no	—
92	D₄₃₆	#41	1	2	yes	—
93	C₄×D₁₀₉	—	4	4	no	—
94	C₄×D₁₀₉	—	4	2	no	—
95	Dic₂₁₈	—	4	4	no	—
96	Dic₂₁₈	—	4	2	no	—
97	#11724784	—	2	1	no	—
98	#11724784	—	2	2	no	—
99	#11724784	—	2	1	no	—
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	C₂×D₁₀₉	—	2	2	no	—
105	C₂×D₁₀₉	—	4	1	no	—
106	C₂×D₁₀₉	—	4	1	no	—
107	C₄×C₁₀₉	—	2	2	no	—
108	C₄×C₁₀₉	—	2	2	no	—
109	C₄×C₁₀₉	—	2	1	no	—
110	C₄×C₁₀₉	—	4	2	no	—
111	C₄×C₁₀₉	—	4	1	no	—
112	C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
113	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
114	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
115	Dic₁₀₉	—	4	1	no	—
116	Dic₁₀₉	—	4	2	no	—
117	C₂×C₁₀₉	—	8	1	no	—
118	C₂×C₁₀₉	#548	1	1	yes	characteristic
119	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₁₀₉	#578	1	1	yes	characteristic
121	D₁₀₉	—	4	2	no	—
122	#3367	—	109	1	no	maximal, Sylow
123	C₁₀₉	#3367	1	1	yes	characteristic, complemented, Sylow
124	#556	D₁₀₉	1	1	yes	characteristic, complemented
125	D₄×Q₈	—	109	1	no	—
126	#489	—	109	2	no	—
127	#487	—	109	1	no	—
128	#477	—	109	2	no	—
129	#474	—	109	1	no	—
130	#463	—	109	2	no	—
131	#461	—	109	1	no	—
132	#446	—	109	1	no	—
133	#440	—	109	2	no	—
134	C₄×D₈	—	109	1	no	—
135	C₂²×Q₈	—	218	1	no	—
136	C₂×SD₁₆	—	109	2	no	—
137	C₂×D₈	—	109	1	no	—
138	C₄⋊Q₈	—	109	1	no	—
139	C₄⋊Q₈	—	109	2	no	—
140	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
141	#126	—	2	2	no	—
142	#124	C₂×D₁₀₉	1	2	yes	—
143	C₂²⋊Q₈	—	2	1	no	—
144	C₂²⋊Q₈	—	218	1	no	—
145	C₂²⋊Q₈	—	218	2	no	—
146	C₄×Q₈	C₂×D₁₀₉	1	2	yes	—
147	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
148	C₄×D₄	—	109	1	no	—
149	C₄×D₄	—	109	2	no	—
150	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
151	#117	—	2	1	no	—
152	#107	—	109	1	no	—
153	#107	—	109	2	no	—
154	C₄⋊C₈	—	109	1	no	—
155	C₄⋊C₈	—	109	2	no	—
156	#103	—	109	2	no	—
157	#103	—	109	2	no	—
158	#103	—	109	2	no	—
159	#102	—	109	1	no	—
160	#102	—	109	2	no	—
161	#102	—	109	1	no	—
162	#97	—	109	2	no	—
163	C₄×C₈	—	109	1	no	—
164	C₂×Q₈	—	436	4	no	—
165	C₂×Q₈	—	218	2	no	—
166	C₂×Q₈	—	218	1	no	—
167	C₂×Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
168	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
169	C₂×D₄	—	109	1	no	—
170	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	218	2	no	—
173	C₂²×C₄	—	218	1	no	—
174	SD₁₆	—	218	4	no	—
175	D₈	—	218	2	no	—
176	C₂×C₈	—	109	1	no	—
177	C₂×C₈	—	109	1	no	—
178	C₂×C₈	—	109	2	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	109	1	no	—
183	C₄⋊C₄	—	109	2	no	—
184	C₄⋊C₄	—	218	2	no	—
185	C₄⋊C₄	—	218	1	no	—
186	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
187	C₄⋊C₄	C₂²⋊D₁₀₉	1	2	yes	—
188	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	218	2	no	—
193	C₂²⋊C₄	—	218	1	no	—
194	C₄²	—	2	2	no	—
195	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
196	C₄²	C₂²×D₁₀₉	1	2	yes	—
197	C₂³	—	2	1	no	—
198	C₂³	—	218	1	no	—
199	Q₈	—	2	2	no	—
200	Q₈	—	2	2	no	—
201	Q₈	—	2	2	no	—
202	Q₈	—	436	4	no	—
203	Q₈	—	436	2	no	—
204	D₄	—	2	1	no	—
205	D₄	—	2	1	no	—
206	D₄	—	109	2	no	—
207	D₄	—	218	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	436	4	no	—
212	C₂×C₄	—	436	2	no	—
213	C₂×C₄	—	218	2	no	—
214	C₂×C₄	—	218	1	no	—
215	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
216	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
217	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
218	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
219	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
220	C₈	—	218	1	no	—
221	C₈	—	218	2	no	—
222	C₈	—	218	1	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	436	1	no	—
227	C₂²	—	436	1	no	—
228	C₂²	—	218	2	no	—
229	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	1	no	—
235	C₄	—	436	2	no	—
236	C₄	—	436	1	no	—
237	C₄	#16812724	1	2	yes	—
238	C₄	#41×D₁₀₉	1	1	yes	characteristic
239	C₄	#16812746	1	1	yes	characteristic
240	C₂	—	8	1	no	—
241	C₂	—	436	2	no	—
242	C₂	#20295683	1	1	yes	characteristic
243	C₂	C₂×#16812724	1	1	yes	characteristic
244	C₂	#20295841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₄₃₆

C₂

yes

maximal, characteristic, complemented

#20295385

C₂

yes

maximal, complemented

#20295376

C₂

yes

maximal, characteristic, complemented

#20295321

C₂

yes

maximal, complemented

#20295306

C₂

yes

maximal, characteristic, complemented

#20295238

C₂

yes

maximal

#20295228

C₂

yes

maximal, characteristic, complemented

#20295157

C₂

yes

maximal, characteristic, complemented

#20295124

C₂

yes

maximal, complemented

C₄×#13182014

C₂

yes

maximal, characteristic

#556×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₁₀₉

—

C₂×#13182016

C₂²

yes

—

C₂×#13182014

C₂²

yes

characteristic

#16812699

C₂²

yes

characteristic

#16812699

C₂²

yes

—

#16812679

—

#16812679

—

C₄×D₄₃₆

C₂²

yes

characteristic

C₄×D₄₃₆

C₂²

yes

—

#16812632

C₂²

yes

—

#16812632

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

—

#16812623

C₂²

yes

—

#16812623

C₂²

yes

—

#16812623

C₂²

yes

—

#16812620

C₂²

yes

—

#16812620

C₂²

yes

characteristic

#16812619

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

—

C₂×Dic₂₁₈

—

Q₈×D₁₀₉

—

C₂×D₄₃₆

C₂³

yes

characteristic

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

—

#13182016

—

#13182014

—

C₂×#11724784

C₂³

yes

characteristic

C₂×#11724784

C₂³

yes

characteristic

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂³

yes

characteristic

#13182006

—

#13182006

—

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

—

C₂²×D₁₀₉

—

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

—

D₄₃₆

—

D₄₃₆

#41

yes

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724784

—

#11724784

—

#11724784

—

100

C₂²×C₁₀₉

—

101

C₂²×C₁₀₉

—

102

C₂²×C₁₀₉

—

103

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

104

C₂×D₁₀₉

—

105

C₂×D₁₀₉

—

106

C₂×D₁₀₉

—

107

C₄×C₁₀₉

—

108

C₄×C₁₀₉

—

109

C₄×C₁₀₉

—

110

C₄×C₁₀₉

—

111

C₄×C₁₀₉

—

112

C₄×C₁₀₉

AGL(1,ℤ/8)

yes

—

113

C₄×C₁₀₉

C₂×#41

yes

characteristic

114

C₄×C₁₀₉

2₋¹⁺⁴

yes

characteristic

115

Dic₁₀₉

—

116

Dic₁₀₉

—

117

C₂×C₁₀₉

—

118

C₂×C₁₀₉

#548

yes

characteristic

119

C₂×C₁₀₉

C₂×AGL(1,ℤ/8)

yes

characteristic

120

C₂×C₁₀₉

#578

yes

characteristic

121

D₁₀₉

—

122

#3367

—

109

maximal, Sylow

123

C₁₀₉

#3367

yes

characteristic, complemented, Sylow

124

#556

D₁₀₉

yes

characteristic, complemented

125

D₄×Q₈

—

109

—

126

#489

—

109

—

127

#487

—

109

—

128

#477

—

109

—

129

#474

—

109

—

130

#463

—

109

—

131

#461

—

109

—

132

#446

—

109

—

133

#440

—

109

—

134

C₄×D₈

—

109

—

135

C₂²×Q₈

—

218

—

136

C₂×SD₁₆

—

109

—

137

C₂×D₈

—

109

—

138

C₄⋊Q₈

—

109

—

139

C₄⋊Q₈

—

109

—

140

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

141

#126

—

142

#124

C₂×D₁₀₉

yes

—

143

C₂²⋊Q₈

—

144

C₂²⋊Q₈

—

218

—

145

C₂²⋊Q₈

—

218

—

146

C₄×Q₈

C₂×D₁₀₉

yes

—

147

C₄×Q₈

C₂×D₁₀₉

yes

characteristic

148

C₄×D₄

—

109

—

149

C₄×D₄

—

109

—

150

C₄×D₄

C₂×D₁₀₉

yes

characteristic

151

#117

—

152

#107

—

109

—

153

#107

—

109

—

154

C₄⋊C₈

—

109

—

155

C₄⋊C₈

—

109

—

156

#103

—

109

—

157

#103

—

109

—

158

#103

—

109

—

159

#102

—

109

—

160

#102

—

109

—

161

#102

—

109

—

162

#97

—

109

—

163

C₄×C₈

—

109

—

164

C₂×Q₈

—

436

—

165

C₂×Q₈

—

218

—

166

C₂×Q₈

—

218

—

167

C₂×Q₈

C₂²⋊D₁₀₉

yes

characteristic

168

C₂×Q₈

C₂²×D₁₀₉

yes

—

169

C₂×D₄

—

109

—

170

C₂×D₄

C₂²×D₁₀₉

yes

characteristic

171

C₂²×C₄

—

172

C₂²×C₄

—

218

—

173

C₂²×C₄

—

218

—

174

SD₁₆

—

218

—

175

D₈

—

218

—

176

C₂×C₈

—

109

—

177

C₂×C₈

—

109

—

178

C₂×C₈

—

109

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

109

—

183

C₄⋊C₄

—

109

—

184

C₄⋊C₄

—

218

—

185

C₄⋊C₄

—

218

—

186

C₄⋊C₄

C₂²⋊D₁₀₉

yes

characteristic

187

C₄⋊C₄

C₂²⋊D₁₀₉

yes

—

188

C₄⋊C₄

C₂²×D₁₀₉

yes

characteristic

189

C₂²⋊C₄

—

190

C₂²⋊C₄

—

191

C₂²⋊C₄

—

192

C₂²⋊C₄

—

218

—

193

C₂²⋊C₄

—

218

—

194

C₄²

—

195

C₄²

C₂²×D₁₀₉

yes

characteristic

196

C₄²

C₂²×D₁₀₉

yes

—

197

C₂³

—

198

C₂³

—

218

—

199

Q₈

—

200

Q₈

—

201

Q₈

—

202

Q₈

—

436

—

203

Q₈

—

436

—

204

D₄

—

205

D₄

—

206

D₄

—

109

—

207

D₄

—

218

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

436

—

212

C₂×C₄

—

436

—

213

C₂×C₄

—

218

—

214

C₂×C₄

—

218

—

215

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

216

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

217

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

218

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

219

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

220

C₈

—

218

—

221

C₈

—

218

—

222

C₈

—

218

—

223

C₂²

—

224

C₂²

—

225

C₂²

—

226

C₂²

—

436

—

227

C₂²

—

436

—

228

C₂²

—

218

—

229

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

230

C₄

—

231

C₄

—

232

C₄

—

233

C₄

—

234

C₄

—

235

C₄

—

436

—

236

C₄

—

436

—

237

C₄

#16812724

yes

—

238

C₄

#41×D₁₀₉

yes

characteristic

239

C₄

#16812746

yes

characteristic

240

C₂

—

241

C₂

—

436

—

242

C₂

#20295683

yes

characteristic

243

C₂

C₂×#16812724

yes

characteristic

244

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×#13182014	C₂²
hypercenter	#556	D₁₀₉
hypocenter	C₁₀₉	#3367
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

C₂²

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #556

N	H ≅ G/N	type
1. Q₈×D₄₃₆	C₂	semidirect
2. #20295385	C₂	semidirect
3. #20295376	C₂	semidirect
4. #20295321	C₂	semidirect
5. #20295306	C₂	semidirect
7. #20295228	C₂	semidirect
8. #20295157	C₂	semidirect
9. #20295124	C₂	semidirect
11. #556×C₁₀₉	C₂	semidirect
123. C₁₀₉	#3367	semidirect
124. #556	D₁₀₉	semidirect

H ≅ G/N

type

1. Q₈×D₄₃₆

C₂

semidirect

2. #20295385

C₂

semidirect

3. #20295376

C₂

semidirect

4. #20295321

C₂

semidirect

5. #20295306

C₂

semidirect

7. #20295228

C₂

semidirect

8. #20295157

C₂

semidirect

9. #20295124

C₂

semidirect

11. #556×C₁₀₉

C₂

semidirect

123. C₁₀₉

#3367

semidirect

124. #556

D₁₀₉

semidirect

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

subgroup	index
Q₈×D₄₃₆	2
#20295385	2
#20295376	2
#20295321	2
#20295306	2
#20295238	2
#20295228	2
#20295157	2
#20295124	2
C₄×#13182014	2
#556×C₁₀₉	2
#3367	109

subgroup

index

Q₈×D₄₃₆

#556×C₁₀₉

109

Subgroups — #28375746

Conjugacy classes of subgroups

Complements

Maximal subgroups