Group #14307435 — subgroups

Subgroups — #14307435

Order 1920.

568 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 122 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	#11752498	C₂	1	1	yes	maximal, characteristic
2	C₄²×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₃×#65682	C₂	1	1	yes	maximal, characteristic
4	#11746524	C₂	1	4	yes	maximal
5	#10604587	—	3	1	no	maximal, Hall
6	C₂×C₄×ΓL(1,𝔽₁₆)	—	2	2	no	—
7	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
8	C₂×C₃×#700	C₂²	1	2	yes	—
9	#91759	C₄	1	1	yes	characteristic, complemented
10	C₅×#870	C₄	1	1	yes	characteristic, complemented
11	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
12	#91468	C₄	1	2	yes	complemented
13	#91358	C₂²	1	2	yes	—
14	#91349	—	2	2	no	—
15	#91323	—	2	2	no	—
16	C₃×#4004	—	2	2	no	—
17	#70514	—	5	1	no	maximal, Hall
18	#66022	—	3	1	no	—
19	C₄²×AGL(1,𝔽₅)	—	3	1	no	—
20	#65682	D₃	1	1	yes	characteristic
21	#64875	—	3	4	no	—
22	C₂²×ΓL(1,𝔽₁₆)	—	2	2	no	—
23	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
24	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
25	C₂×#1204	—	2	2	no	—
26	C₄×ΓL(1,𝔽₁₆)	—	2	8	no	—
27	C₄×ΓL(1,𝔽₁₆)	—	2	2	no	—
28	C₄×ΓL(1,𝔽₁₆)	Q₈	1	4	yes	complemented
29	C₂×C₃×#170	—	2	2	no	—
30	C₃×#700	—	2	4	no	—
31	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
33	#6712	—	2	2	no	—
34	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
35	C₅×Dic₃⋊C₄	—	2	2	no	—
36	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄×Dic₅	D₄	1	2	yes	—
38	#6668	—	2	2	no	—
39	#6665	—	2	2	no	—
40	Dic₃×Dic₅	—	2	2	no	—
41	#5470	—	5	1	no	—
42	C₃×#423	—	5	1	no	—
43	C₄²×Dic₃	—	5	1	no	—
44	#4555	—	5	4	no	—
45	C₂×C₄×AGL(1,𝔽₅)	—	3	2	no	—
46	C₂×C₄×AGL(1,𝔽₅)	—	6	2	no	—
47	C₂×#700	C₂×D₃	1	2	yes	—
48	C₅×#125	—	3	1	no	—
49	#4042	—	6	2	no	—
50	#4038	—	3	2	no	—
51	#4034	—	3	2	no	—
52	#4033	—	6	2	no	—
53	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
54	#4016	—	3	1	no	—
55	#4004	—	2	2	no	—
56	#1994	—	15	1	no	Sylow
57	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
58	C₂×ΓL(1,𝔽₁₆)	—	2	8	no	—
59	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
60	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×Dic₁₅	—	2	2	no	—
63	C₂×C₅×Dic₃	—	2	2	no	—
64	C₂×C₃×Dic₅	C₂²⋊C₄	1	2	yes	—
65	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
66	C₃×C₄×D₅	—	2	4	no	—
67	C₃×C₄×D₅	C₂×Q₈	1	2	yes	—
68	C₃×C₄×D₅	#41	1	2	yes	—
69	#1204	—	4	4	no	—
70	C₃×#170	—	4	2	no	—
71	C₂×C₃×M₁₆	—	5	2	no	—
72	C₂×C₃×C₄²	—	5	1	no	—
73	#926	—	5	2	no	—
74	#924	—	10	2	no	—
75	C₂×C₄×Dic₃	—	5	2	no	—
76	C₂×C₄×Dic₃	—	10	2	no	—
77	#870	—	5	1	no	—
78	#870	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
79	#869	—	5	2	no	—
80	C₄×Dic₆	—	10	2	no	—
81	C₃×C₄⋊C₈	—	10	2	no	—
82	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
83	C₂×C₄×D₅	D₁₂	1	2	yes	—
84	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
85	C₂×Dic₁₀	—	6	2	no	—
86	C₂×#170	—	2	2	no	—
87	C₄×AGL(1,𝔽₅)	—	3	4	no	—
88	C₄×AGL(1,𝔽₅)	—	6	8	no	—
89	C₄×AGL(1,𝔽₅)	—	6	2	no	—
90	#700	—	2	4	no	—
91	C₅×C₄⋊C₄	—	3	2	no	—
92	C₅×C₄⋊C₄	—	6	2	no	—
93	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
94	#685	—	6	2	no	—
95	C₄⋊Dic₅	—	3	2	no	—
96	#683	—	6	2	no	—
97	#683	—	6	2	no	—
98	C₄×Dic₅	—	6	2	no	—
99	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
100	C₄×Dic₅	#67	1	2	yes	—
101	#533	—	15	1	no	—
102	#423	—	5	1	no	—
103	C₄³	—	15	1	no	—
104	#339	—	15	4	no	—
105	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₃×D₅	—	2	2	no	—
107	C₂×C₃×D₅	C₂²⋊Q₈	1	2	yes	—
108	ΓL(1,𝔽₁₆)	—	2	8	no	—
109	C₃×C₄×C₅	—	2	2	no	—
110	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
111	Dic₁₅	—	4	2	no	—
112	C₃×Dic₅	#104	1	4	yes	—
113	C₃×Dic₅	#124	1	1	yes	characteristic
114	C₃×Dic₅	C₄⋊Q₈	1	1	yes	characteristic
115	C₅×Dic₃	—	4	2	no	—
116	C₂²×C₃×C₄	—	5	1	no	—
117	C₂²×C₃×C₄	—	5	2	no	—
118	C₂²×Dic₃	—	10	2	no	—
119	C₂×Dic₆	—	10	2	no	—
120	C₃×M₁₆	—	10	4	no	—
121	C₂×C₃×C₈	—	10	2	no	—
122	C₃×C₄²	—	5	2	no	—
123	C₃×C₄²	—	5	1	no	—
124	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
125	#217	—	10	2	no	—
126	C₄⋊Dic₃	—	5	2	no	—
127	C₄⋊Dic₃	—	10	2	no	—
128	C₄⋊Dic₃	C₂×AGL(1,𝔽₅)	1	2	yes	—
129	Dic₃⋊C₄	—	2	2	no	—
130	Dic₃⋊C₄	—	10	2	no	—
131	C₄×Dic₃	—	5	4	no	—
132	C₄×Dic₃	—	10	2	no	—
133	C₄×Dic₃	—	10	2	no	—
134	C₄×Dic₃	—	10	8	no	—
135	C₂²×D₅	C₂×D₁₂	1	1	yes	characteristic
136	C₂×AGL(1,𝔽₅)	—	6	8	no	—
137	C₂×AGL(1,𝔽₅)	—	6	4	no	—
138	C₂×C₄×C₅	—	6	2	no	—
139	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
140	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
141	C₂×Dic₅	—	6	2	no	—
142	C₂×Dic₅	(C₂×D₃)⋊C₄	1	2	yes	—
143	C₂×Dic₅	C₂×#67	1	1	yes	characteristic
144	C₄×D₅	—	2	4	no	—
145	C₄×D₅	#237	1	2	yes	—
146	C₄×D₅	D₃×Q₈	1	2	yes	—
147	Dic₁₀	—	12	4	no	—
148	#170	—	4	2	no	—
149	C₂×M₁₆	—	5	2	no	—
150	C₄⋊Q₈	—	15	2	no	—
151	#125	—	3	1	no	—
152	#125	—	15	1	no	—
153	C₂²⋊Q₈	—	30	2	no	—
154	C₄×Q₈	—	30	2	no	—
155	#117	—	15	2	no	—
156	C₂×C₄²	—	5	1	no	—
157	C₂×C₄²	—	15	2	no	—
158	C₂×C₄²	—	30	2	no	—
159	C₄⋊C₈	—	10	2	no	—
160	C₂×C₃×C₅	#391	1	1	yes	characteristic
161	C₂×C₃×C₅	C₂×#104	1	2	yes	—
162	C₃×D₅	—	2	2	no	—
163	C₂³×C₃	—	5	1	no	—
164	C₂×C₃×C₄	—	5	2	no	—
165	C₂×C₃×C₄	—	5	1	no	—
166	C₂×C₃×C₄	—	5	2	no	—
167	C₂×C₃×C₄	—	5	2	no	—
168	C₂×C₃×C₄	—	10	4	no	—
169	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
170	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₂×Dic₃	—	2	2	no	—
172	C₂×Dic₃	—	10	4	no	—
173	C₂×Dic₃	—	10	8	no	—
174	C₂×Dic₃	—	10	2	no	—
175	Dic₆	—	20	4	no	—
176	C₃×C₈	—	20	2	no	—
177	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
178	C₂×D₅	—	2	2	no	—
179	C₂×D₅	#897	1	2	yes	—
180	AGL(1,𝔽₅)	—	6	8	no	—
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	12	2	no	—
183	C₄×C₅	#887	1	2	yes	—
184	Dic₅	—	12	2	no	—
185	Dic₅	#825	1	4	yes	—
186	Dic₅	#936	1	1	yes	characteristic
187	Dic₅	#944	1	1	yes	characteristic
188	C₂×Q₈	—	30	2	no	—
189	C₂²×C₄	—	5	1	no	—
190	C₂²×C₄	—	5	2	no	—
191	C₂²×C₄	—	30	2	no	—
192	M₁₆	—	10	4	no	—
193	C₂×C₈	—	10	2	no	—
194	C₄⋊C₄	—	3	2	no	—
195	C₄⋊C₄	—	6	2	no	—
196	C₄⋊C₄	—	15	2	no	—
197	C₄⋊C₄	—	30	2	no	—
198	C₄⋊C₄	—	30	2	no	—
199	C₂²⋊C₄	—	30	2	no	—
200	C₄²	—	5	2	no	—
201	C₄²	—	5	1	no	—
202	C₄²	—	15	4	no	—
203	C₄²	—	30	2	no	—
204	C₄²	—	30	2	no	—
205	C₄²	—	30	8	no	—
206	C₄²	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
207	C₃×C₅	#1994	1	1	yes	characteristic, complemented, Hall
208	C₂²×C₃	—	5	2	no	—
209	C₂²×C₃	—	10	2	no	—
210	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	5	4	no	—
213	C₃×C₄	—	5	1	no	—
214	C₃×C₄	—	5	1	no	—
215	C₃×C₄	Q₈×AGL(1,𝔽₅)	1	2	yes	—
216	Dic₃	—	4	2	no	—
217	Dic₃	—	10	8	no	—
218	Dic₃	—	20	2	no	—
219	C₂×C₅	#4973	1	1	yes	characteristic
220	C₂×C₅	C₂×#825	1	2	yes	—
221	D₅	—	2	2	no	—
222	C₂³	—	5	1	no	—
223	Q₈	—	60	4	no	—
224	C₂×C₄	—	5	1	no	—
225	C₂×C₄	—	5	2	no	—
226	C₂×C₄	—	5	2	no	—
227	C₂×C₄	—	5	2	no	—
228	C₂×C₄	—	6	2	no	—
229	C₂×C₄	—	10	4	no	—
230	C₂×C₄	—	30	4	no	—
231	C₂×C₄	—	30	8	no	—
232	C₂×C₄	—	30	2	no	—
233	C₂×C₄	(C₂×D₃)⋊AGL(1,𝔽₅)	1	2	yes	—
234	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
235	C₈	—	20	2	no	—
236	C₂×C₃	—	10	2	no	—
237	C₂×C₃	(C₂×AGL(1,𝔽₅))⋊Q₈	1	1	yes	characteristic
238	C₂×C₃	D₅⋊#104	1	2	yes	—
239	C₅	#70514	1	1	yes	characteristic, complemented, Sylow
240	C₂²	—	5	2	no	—
241	C₂²	—	10	2	no	—
242	C₂²	C₂×(C₂×D₃)⋊AGL(1,𝔽₅)	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	5	4	no	—
245	C₄	—	5	1	no	—
246	C₄	—	5	1	no	—
247	C₄	—	12	2	no	—
248	C₄	—	30	8	no	—
249	C₄	—	60	2	no	—
250	C₄	ΓL(1,𝔽₁₆)⋊Q₈	1	2	yes	—
251	C₃	#10604587	1	1	yes	characteristic, complemented, Sylow
252	C₂	—	10	2	no	—
253	C₂	D₅⋊#825	1	2	yes	—
254	C₂	#11751752	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11752498

C₂

yes

maximal, characteristic

C₄²×ΓL(1,𝔽₁₆)

C₂

yes

maximal, characteristic

C₃×#65682

C₂

yes

maximal, characteristic

#11746524

C₂

yes

maximal

#10604587

—

maximal, Hall

C₂×C₄×ΓL(1,𝔽₁₆)

—

C₂×C₄×ΓL(1,𝔽₁₆)

C₂²

yes

—

C₂×C₃×#700

C₂²

yes

—

#91759

C₄

yes

characteristic, complemented

C₅×#870

C₄

yes

characteristic, complemented

C₃×C₄²×D₅

C₂²

yes

characteristic

#91468

C₄

yes

complemented

#91358

C₂²

yes

—

—

—

—

—

—

C₄²×AGL(1,𝔽₅)

—

#65682

D₃

yes

characteristic

#64875

—

C₂²×ΓL(1,𝔽₁₆)

—

C₂×C₃×C₄×D₅

D₄

yes

—

C₂×C₃×C₄×D₅

C₂³

yes

characteristic

C₂×#1204

—

C₄×ΓL(1,𝔽₁₆)

—

C₄×ΓL(1,𝔽₁₆)

—

C₄×ΓL(1,𝔽₁₆)

Q₈

yes

complemented

C₂×C₃×#170

—

C₃×#700

—

C₃×C₄²×C₅

C₂×C₄

yes

characteristic

C₄⋊Dic₁₅

C₂×C₄

yes

—

#6712

—

C₅×C₄⋊Dic₃

C₂×C₄

yes

—

C₅×Dic₃⋊C₄

—

C₃×C₄×Dic₅

C₂×C₄

yes

characteristic

C₃×C₄×Dic₅

D₄

yes

—

#6668

—

#6665

—

Dic₃×Dic₅

—

#5470

—

C₃×#423

—

C₄²×Dic₃

—

#4555

—

C₂×C₄×AGL(1,𝔽₅)

—

C₂×C₄×AGL(1,𝔽₅)

—

C₂×#700

C₂×D₃

yes

—

—

—

—

—

—

C₄²×D₅

C₂×D₃

yes

—

—

—

C₂²×C₃×D₅

C₂×D₄

yes

characteristic

C₂×ΓL(1,𝔽₁₆)

—

C₂×ΓL(1,𝔽₁₆)

—

C₂×C₃×C₄×C₅

C₂²⋊C₄

yes

—

C₂×C₃×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₁₅

—

C₂×C₅×Dic₃

—

C₂×C₃×Dic₅

C₂²⋊C₄

yes

—

C₂×C₃×Dic₅

C₂×D₄

yes

characteristic

C₃×C₄×D₅

—

C₃×C₄×D₅

C₂×Q₈

yes

—

C₃×C₄×D₅

#41

yes

—

#1204

—

C₃×#170

—

C₂×C₃×M₁₆

—

C₂×C₃×C₄²

—

#926

—

#924

—

C₂×C₄×Dic₃

—

C₂×C₄×Dic₃

—

#870

—

#870

AGL(1,𝔽₅)

yes

characteristic, complemented

#869

—

C₄×Dic₆

—

C₃×C₄⋊C₈

—

C₂²×AGL(1,𝔽₅)

—

C₂×C₄×D₅

D₁₂

yes

—

C₂×C₄×D₅

C₂²×D₃

yes

characteristic

C₂×Dic₁₀

—

—

—

—

—

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₄×D₃

yes

characteristic

#685

—

C₄⋊Dic₅

—

#683

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

C₄×D₃

yes

characteristic

100

C₄×Dic₅

#67

yes

—

101

#533

—

102

#423

—

103

C₄³

—

104

#339

—

105

C₂²×C₃×C₅

C₂×C₂²⋊C₄

yes

characteristic

106

C₂×C₃×D₅

—

107

C₂×C₃×D₅

C₂²⋊Q₈

yes

—

108

ΓL(1,𝔽₁₆)

—

109

C₃×C₄×C₅

—

110

C₃×C₄×C₅

C₄×Q₈

yes

—

111

Dic₁₅

—

112

C₃×Dic₅

#104

yes

—

113

C₃×Dic₅

#124

yes

characteristic

114

C₃×Dic₅

C₄⋊Q₈

yes

characteristic

115

C₅×Dic₃

—

116

C₂²×C₃×C₄

—

117

C₂²×C₃×C₄

—

118

C₂²×Dic₃

—

119

C₂×Dic₆

—

120

C₃×M₁₆

—

121

C₂×C₃×C₈

—

122

C₃×C₄²

—

123

C₃×C₄²

—

124

C₃×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

125

#217

—

126

C₄⋊Dic₃

—

127

C₄⋊Dic₃

—

128

C₄⋊Dic₃

C₂×AGL(1,𝔽₅)

yes

—

129

Dic₃⋊C₄

—

130

Dic₃⋊C₄

—

131

C₄×Dic₃

—

132

C₄×Dic₃

—

133

C₄×Dic₃

—

134

C₄×Dic₃

—

135

C₂²×D₅

C₂×D₁₂

yes

characteristic

136

C₂×AGL(1,𝔽₅)

—

137

C₂×AGL(1,𝔽₅)

—

138

C₂×C₄×C₅

—

139

C₂×C₄×C₅

(C₂×D₃)⋊C₄

yes

—

140

C₂×C₄×C₅

C₂×C₄×D₃

yes

characteristic

141

C₂×Dic₅

—

142

C₂×Dic₅

(C₂×D₃)⋊C₄

yes

—

143

C₂×Dic₅

C₂×#67

yes

characteristic

144

C₄×D₅

—

145

C₄×D₅

#237

yes

—

146

C₄×D₅

D₃×Q₈

yes

—

147

Dic₁₀

—

148

#170

—

149

C₂×M₁₆

—

150

C₄⋊Q₈

—

151

#125

—

152

#125

—

153

C₂²⋊Q₈

—

154

C₄×Q₈

—

155

#117

—

156

C₂×C₄²

—

157

C₂×C₄²

—

158

C₂×C₄²

—

159

C₄⋊C₈

—

160

C₂×C₃×C₅

#391

yes

characteristic

161

C₂×C₃×C₅

C₂×#104

yes

—

162

C₃×D₅

—

163

C₂³×C₃

—

164

C₂×C₃×C₄

—

165

C₂×C₃×C₄

—

166

C₂×C₃×C₄

—

167

C₂×C₃×C₄

—

168

C₂×C₃×C₄

—

169

C₂×C₃×C₄

C₂²⋊AGL(1,𝔽₅)

yes

—

170

C₂×C₃×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

171

C₂×Dic₃

—

172

C₂×Dic₃

—

173

C₂×Dic₃

—

174

C₂×Dic₃

—

175

Dic₆

—

176

C₃×C₈

—

177

C₂²×C₅

C₂×(C₂×D₃)⋊C₄

yes

characteristic

178

C₂×D₅

—

179

C₂×D₅

#897

yes

—

180

AGL(1,𝔽₅)

—

181

C₄×C₅

—

182

C₄×C₅

—

183

C₄×C₅

#887

yes

—

184

Dic₅

—

185

Dic₅

#825

yes

—

186

Dic₅

#936

yes

characteristic

187

Dic₅

#944

yes

characteristic

188

C₂×Q₈

—

189

C₂²×C₄

—

190

C₂²×C₄

—

191

C₂²×C₄

—

192

M₁₆

—

193

C₂×C₈

—

194

C₄⋊C₄

—

195

C₄⋊C₄

—

196

C₄⋊C₄

—

197

C₄⋊C₄

—

198

C₄⋊C₄

—

199

C₂²⋊C₄

—

200

C₄²

—

201

C₄²

—

202

C₄²

—

203

C₄²

—

204

C₄²

—

205

C₄²

—

206

C₄²

D₃×AGL(1,𝔽₅)

yes

characteristic

207

C₃×C₅

#1994

yes

characteristic, complemented, Hall

208

C₂²×C₃

—

209

C₂²×C₃

—

210

C₂²×C₃

C₂×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

211

C₃×C₄

—

212

C₃×C₄

—

213

C₃×C₄

—

214

C₃×C₄

—

215

C₃×C₄

Q₈×AGL(1,𝔽₅)

yes

—

216

Dic₃

—

217

Dic₃

—

218

Dic₃

—

219

C₂×C₅

#4973

yes

characteristic

220

C₂×C₅

C₂×#825

yes

—

221

D₅

—

222

C₂³

—

223

Q₈

—

224

C₂×C₄

—

225

C₂×C₄

—

226

C₂×C₄

—

227

C₂×C₄

—

228

C₂×C₄

—

229

C₂×C₄

—

230

C₂×C₄

—

231

C₂×C₄

—

232

C₂×C₄

—

233

C₂×C₄

(C₂×D₃)⋊AGL(1,𝔽₅)

yes

—

234

C₂×C₄

C₂×D₃×AGL(1,𝔽₅)

yes

characteristic

235

C₈

—

236

C₂×C₃

—

237

C₂×C₃

(C₂×AGL(1,𝔽₅))⋊Q₈

yes

characteristic

238

C₂×C₃

D₅⋊#104

yes

—

239

C₅

#70514

yes

characteristic, complemented, Sylow

240

C₂²

—

241

C₂²

—

242

C₂²

C₂×(C₂×D₃)⋊AGL(1,𝔽₅)

yes

characteristic

243

C₄

—

244

C₄

—

245

C₄

—

246

C₄

—

247

C₄

—

248

C₄

—

249

C₄

—

250

C₄

ΓL(1,𝔽₁₆)⋊Q₈

yes

—

251

C₃

#10604587

yes

characteristic, complemented, Sylow

252

C₂

—

253

C₂

D₅⋊#825

yes

—

254

C₂

#11751752

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₃)⋊AGL(1,𝔽₅)
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
Fitting	C₃×C₄²×C₅	C₂×C₄
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×D₅	C₂×D₄
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	C₄²	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#1994
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

C₂×(C₂×D₃)⋊AGL(1,𝔽₅)

commutator

C₂×C₃×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂×D₃×AGL(1,𝔽₅)

Fitting

C₃×C₄²×C₅

C₂×C₄

socle

C₂²×C₃×C₅

C₂×C₂²⋊C₄

prime radical

C₂²×C₃×D₅

C₂×D₄

cosocle

C₂×C₃×C₄×D₅

C₂³

C₄²

C₃×C₅

G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
9. #91759	C₄	semidirect
10. C₅×#870	C₄	semidirect
12. #91468	C₄	semidirect
28. C₄×ΓL(1,𝔽₁₆)	Q₈	semidirect
78. #870	AGL(1,𝔽₅)	semidirect
207. C₃×C₅	#1994	semidirect
239. C₅	#70514	semidirect
251. C₃	#10604587	semidirect

H ≅ G/N

type

9. #91759

C₄

semidirect

10. C₅×#870

C₄

semidirect

12. #91468

C₄

semidirect

28. C₄×ΓL(1,𝔽₁₆)

Q₈

semidirect

78. #870

AGL(1,𝔽₅)

semidirect

207. C₃×C₅

#1994

semidirect

239. C₅

#70514

semidirect

251. C₃

#10604587

semidirect

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

subgroup	index
#11752498	2
C₄²×ΓL(1,𝔽₁₆)	2
C₃×#65682	2
#11746524	2
#10604587	3
#70514	5

subgroup

index

C₄²×ΓL(1,𝔽₁₆)

Subgroups — #14307435

Conjugacy classes of subgroups

Complements

Maximal subgroups