Group #42128221 — subgroups

Subgroups — C₁₀₉×#10227 #42128221

Order 27904.

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

normal subgroups only

This group has 466 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	C₁₀₉×#3229	C₂	1	2	yes	maximal
2	C₁₀₉×#3162	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1960	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1518	C₂	1	2	yes	maximal
5	C₁₀₉×#1281	C₂	1	1	yes	maximal, characteristic
6	#533×C₁₀₉	C₂²	1	2	yes	—
7	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	#501×C₁₀₉	—	2	2	no	—
9	#501×C₁₀₉	—	2	2	no	—
10	#498×C₁₀₉	—	2	2	no	—
11	#487×C₁₀₉	—	2	2	no	—
12	#483×C₁₀₉	—	2	4	no	—
13	#475×C₁₀₉	—	2	4	no	—
14	#431×C₁₀₉	C₂²	1	2	yes	—
15	#427×C₁₀₉	—	2	2	no	—
16	C₂×#103×C₁₀₉	—	2	2	no	—
17	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
18	#382×C₁₀₉	C₂²	1	1	yes	characteristic
19	#328×C₁₀₉	—	2	4	no	—
20	C₂²×Q₈×C₁₀₉	—	2	1	no	—
21	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
22	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
23	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
24	C₄⋊Q₈×C₁₀₉	D₄	1	2	yes	—
25	C₄⋊Q₈×C₁₀₉	D₄	1	2	yes	—
26	C₄⋊Q₈×C₁₀₉	D₄	1	2	yes	—
27	#125×C₁₀₉	—	2	2	no	—
28	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
29	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
30	C₄×Q₈×C₁₀₉	—	2	2	no	—
31	C₄×Q₈×C₁₀₉	—	2	2	no	—
32	#117×C₁₀₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
35	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
36	#107×C₁₀₉	—	2	4	no	—
37	#106×C₁₀₉	—	2	8	no	—
38	#106×C₁₀₉	—	2	2	no	—
39	#106×C₁₀₉	—	2	2	no	—
40	C₄⋊C₈×C₁₀₉	—	2	2	no	—
41	#103×C₁₀₉	—	2	8	no	—
42	#103×C₁₀₉	—	4	4	no	—
43	C₂²⋊C₈×C₁₀₉	D₄	1	4	yes	—
44	#97×C₁₀₉	—	2	2	no	—
45	C₄×C₈×C₁₀₉	—	2	2	no	—
46	#95×C₁₀₉	—	2	2	no	—
47	C₂×Q₈×C₁₀₉	—	2	2	no	—
48	C₂×Q₈×C₁₀₉	—	2	2	no	—
49	C₂×Q₈×C₁₀₉	—	2	2	no	—
50	C₂×Q₈×C₁₀₉	—	4	1	no	—
51	C₂×Q₈×C₁₀₉	—	4	2	no	—
52	C₂×Q₈×C₁₀₉	—	4	2	no	—
53	C₂²×C₄×C₁₀₉	—	4	1	no	—
54	C₂²×C₄×C₁₀₉	—	4	1	no	—
55	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₁₀₉	—	2	2	no	—
58	C₂×C₈×C₁₀₉	—	2	2	no	—
59	C₂×C₈×C₁₀₉	—	2	4	no	—
60	C₂×C₈×C₁₀₉	SD₁₆	1	4	yes	—
61	C₄⋊C₄×C₁₀₉	—	2	4	no	—
62	C₄⋊C₄×C₁₀₉	—	2	2	no	—
63	C₄⋊C₄×C₁₀₉	—	2	2	no	—
64	C₄⋊C₄×C₁₀₉	—	2	2	no	—
65	C₄⋊C₄×C₁₀₉	—	2	2	no	—
66	C₄⋊C₄×C₁₀₉	—	4	2	no	—
67	C₄⋊C₄×C₁₀₉	—	4	2	no	—
68	C₄⋊C₄×C₁₀₉	—	4	2	no	—
69	C₄⋊C₄×C₁₀₉	—	4	2	no	—
70	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
71	C₄²×C₁₀₉	—	4	2	no	—
72	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
73	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₀₉	#41	1	1	yes	characteristic
75	C₂³×C₁₀₉	#127	1	1	yes	characteristic
76	Q₈×C₁₀₉	—	4	4	no	—
77	Q₈×C₁₀₉	—	4	4	no	—
78	Q₈×C₁₀₉	—	8	2	no	—
79	Q₈×C₁₀₉	—	8	2	no	—
80	C₂×C₄×C₁₀₉	—	4	2	no	—
81	C₂×C₄×C₁₀₉	—	4	2	no	—
82	C₂×C₄×C₁₀₉	—	4	2	no	—
83	C₂×C₄×C₁₀₉	—	4	2	no	—
84	C₂×C₄×C₁₀₉	—	8	1	no	—
85	C₂×C₄×C₁₀₉	—	8	1	no	—
86	C₂×C₄×C₁₀₉	—	8	2	no	—
87	C₂×C₄×C₁₀₉	—	8	1	no	—
88	C₂×C₄×C₁₀₉	#120	1	2	yes	—
89	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
90	C₂×C₄×C₁₀₉	#121	1	2	yes	—
91	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
93	C₂×C₄×C₁₀₉	#137	1	2	yes	—
94	C₈×C₁₀₉	—	2	4	no	—
95	C₈×C₁₀₉	—	4	4	no	—
96	C₂²×C₁₀₉	—	2	2	no	—
97	C₂²×C₁₀₉	#392	1	1	yes	characteristic
98	C₂²×C₁₀₉	#492	1	1	yes	characteristic
99	C₂²×C₁₀₉	#497	1	1	yes	characteristic
100	C₄×C₁₀₉	—	2	2	no	—
101	C₄×C₁₀₉	—	2	2	no	—
102	C₄×C₁₀₉	—	8	2	no	—
103	C₄×C₁₀₉	—	8	2	no	—
104	C₄×C₁₀₉	—	8	2	no	—
105	C₄×C₁₀₉	—	16	1	no	—
106	C₄×C₁₀₉	#448	1	2	yes	—
107	C₄×C₁₀₉	#461	1	2	yes	—
108	#10227	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×C₁₀₉	—	2	1	no	—
110	C₂×C₁₀₉	—	2	1	no	—
111	C₂×C₁₀₉	#2009	1	2	yes	—
112	C₂×C₁₀₉	#2013	1	1	yes	characteristic
113	#3229	C₂×C₁₀₉	1	2	yes	—
114	#3162	C₂×C₁₀₉	1	1	yes	characteristic
115	#1960	C₂×C₁₀₉	1	1	yes	characteristic
116	#1518	C₂×C₁₀₉	1	2	yes	—
117	#1281	C₂×C₁₀₉	1	1	yes	characteristic
118	C₁₀₉	#10227	1	1	yes	characteristic, direct product factor, Sylow
119	#533	C₂²×C₁₀₉	1	2	yes	—
120	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
121	#501	—	2	2	no	—
122	#501	—	2	2	no	—
123	#498	—	2	2	no	—
124	#487	—	2	2	no	—
125	#483	—	2	4	no	—
126	#475	—	2	4	no	—
127	#431	C₂²×C₁₀₉	1	2	yes	—
128	#427	—	2	2	no	—
129	C₂×#103	—	2	2	no	—
130	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
131	#382	C₂²×C₁₀₉	1	1	yes	characteristic
132	#328	—	2	4	no	—
133	C₂²×Q₈	—	2	1	no	—
134	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
135	C₄⋊Q₈	—	2	2	no	—
136	C₄⋊Q₈	—	2	1	no	—
137	C₄⋊Q₈	D₄×C₁₀₉	1	2	yes	—
138	C₄⋊Q₈	D₄×C₁₀₉	1	2	yes	—
139	C₄⋊Q₈	D₄×C₁₀₉	1	2	yes	—
140	#125	—	2	2	no	—
141	C₂²⋊Q₈	—	2	2	no	—
142	C₂²⋊Q₈	—	2	2	no	—
143	C₄×Q₈	—	2	2	no	—
144	C₄×Q₈	—	2	2	no	—
145	#117	—	2	2	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
149	#107	—	2	4	no	—
150	#106	—	2	2	no	—
151	#106	—	2	2	no	—
152	#106	—	2	8	no	—
153	C₄⋊C₈	—	2	2	no	—
154	#103	—	2	8	no	—
155	#103	—	4	4	no	—
156	C₂²⋊C₈	D₄×C₁₀₉	1	4	yes	—
157	#97	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	#95	—	2	2	no	—
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	2	2	no	—
163	C₂×Q₈	—	4	1	no	—
164	C₂×Q₈	—	4	2	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂²×C₄	—	4	1	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
169	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	SD₁₆×C₁₀₉	1	4	yes	—
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
186	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
187	C₄²	#41×C₁₀₉	1	1	yes	characteristic
188	C₂³	#127×C₁₀₉	1	1	yes	characteristic
189	Q₈	—	4	4	no	—
190	Q₈	—	4	4	no	—
191	Q₈	—	8	2	no	—
192	Q₈	—	8	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	8	1	no	—
198	C₂×C₄	—	8	1	no	—
199	C₂×C₄	—	8	2	no	—
200	C₂×C₄	—	8	1	no	—
201	C₂×C₄	#120×C₁₀₉	1	2	yes	—
202	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
203	C₂×C₄	#121×C₁₀₉	1	2	yes	—
204	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
205	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
206	C₂×C₄	#137×C₁₀₉	1	2	yes	—
207	C₈	—	2	4	no	—
208	C₈	—	4	4	no	—
209	C₂²	—	2	2	no	—
210	C₂²	#392×C₁₀₉	1	1	yes	characteristic
211	C₂²	#492×C₁₀₉	1	1	yes	characteristic
212	C₂²	#497×C₁₀₉	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	8	2	no	—
216	C₄	—	8	2	no	—
217	C₄	—	8	2	no	—
218	C₄	—	16	1	no	—
219	C₄	#448×C₁₀₉	1	2	yes	—
220	C₄	#461×C₁₀₉	1	2	yes	—
221	C₂	—	2	1	no	—
222	C₂	—	2	1	no	—
223	C₂	C₁₀₉×#2009	1	2	yes	—
224	C₂	C₁₀₉×#2013	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#3229

C₂

yes

maximal

C₁₀₉×#3162

C₂

yes

maximal, characteristic

C₁₀₉×#1960

C₂

yes

maximal, characteristic

C₁₀₉×#1518

C₂

yes

maximal

C₁₀₉×#1281

C₂

yes

maximal, characteristic

#533×C₁₀₉

C₂²

yes

—

C₂×C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

#501×C₁₀₉

—

#501×C₁₀₉

—

#498×C₁₀₉

—

#487×C₁₀₉

—

#483×C₁₀₉

—

#475×C₁₀₉

—

#431×C₁₀₉

C₂²

yes

—

#427×C₁₀₉

—

C₂×#103×C₁₀₉

—

C₂×#97×C₁₀₉

C₂²

yes

characteristic

#382×C₁₀₉

C₂²

yes

characteristic

#328×C₁₀₉

—

C₂²×Q₈×C₁₀₉

—

C₂²×C₈×C₁₀₉

D₄

yes

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

D₄

yes

—

C₄⋊Q₈×C₁₀₉

D₄

yes

—

C₄⋊Q₈×C₁₀₉

D₄

yes

—

#125×C₁₀₉

—

C₂²⋊Q₈×C₁₀₉

—

C₂²⋊Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

#117×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

#107×C₁₀₉

—

#106×C₁₀₉

—

#106×C₁₀₉

—

#106×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

#103×C₁₀₉

—

#103×C₁₀₉

—

C₂²⋊C₈×C₁₀₉

D₄

yes

—

#97×C₁₀₉

—

C₄×C₈×C₁₀₉

—

#95×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

SD₁₆

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×D₄

yes

—

C₄²×C₁₀₉

C₂×D₄

yes

C₄²×C₁₀₉

yes

C₂³×C₁₀₉

yes

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#120

yes

—

C₂×C₄×C₁₀₉

#120

yes

characteristic

C₂×C₄×C₁₀₉

#121

yes

—

C₂×C₄×C₁₀₉

#121

yes

characteristic

C₂×C₄×C₁₀₉

C₂×SD₁₆

yes

—

C₂×C₄×C₁₀₉

#137

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

100

C₄×C₁₀₉

—

101

C₄×C₁₀₉

—

102

C₄×C₁₀₉

—

103

C₄×C₁₀₉

—

104

C₄×C₁₀₉

—

105

C₄×C₁₀₉

—

106

C₄×C₁₀₉

#448

yes

—

107

C₄×C₁₀₉

#461

yes

—

108

#10227

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

109

C₂×C₁₀₉

—

110

C₂×C₁₀₉

—

111

C₂×C₁₀₉

#2009

yes

—

112

C₂×C₁₀₉

#2013

yes

characteristic

113

#3229

C₂×C₁₀₉

yes

—

114

#3162

C₂×C₁₀₉

yes

characteristic

115

#1960

C₂×C₁₀₉

yes

characteristic

116

#1518

C₂×C₁₀₉

yes

—

117

#1281

C₂×C₁₀₉

yes

characteristic

118

C₁₀₉

#10227

yes

characteristic, direct product factor, Sylow

119

#533

C₂²×C₁₀₉

yes

—

120

C₂×C₄⋊Q₈

C₂²×C₁₀₉

yes

characteristic

121

#501

—

122

#501

—

123

#498

—

124

#487

—

125

#483

—

126

#475

—

127

#431

C₂²×C₁₀₉

yes

—

128

#427

—

129

C₂×#103

—

130

C₂×#97

C₂²×C₁₀₉

yes

characteristic

131

#382

C₂²×C₁₀₉

yes

characteristic

132

#328

—

133

C₂²×Q₈

—

134

C₂²×C₈

D₄×C₁₀₉

yes

—

135

C₄⋊Q₈

—

136

C₄⋊Q₈

—

137

C₄⋊Q₈

D₄×C₁₀₉

yes

—

138

C₄⋊Q₈

D₄×C₁₀₉

yes

—

139

C₄⋊Q₈

D₄×C₁₀₉

yes

—

140

#125

—

141

C₂²⋊Q₈

—

142

C₂²⋊Q₈

—

143

C₄×Q₈

—

144

C₄×Q₈

—

145

#117

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄⋊C₄

—

148

C₂×C₄²

C₂³×C₁₀₉

yes

characteristic

149

#107

—

150

#106

—

151

#106

—

152

#106

—

153

C₄⋊C₈

—

154

#103

—

155

#103

—

156

C₂²⋊C₈

D₄×C₁₀₉

yes

—

157

#97

—

158

C₄×C₈

—

159

#95

—

160

C₂×Q₈

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂×Q₈

—

166

C₂²×C₄

—

167

C₂²×C₄

—

168

C₂²×C₄

C₂×D₄×C₁₀₉

yes

—

169

C₂²×C₄

C₂×D₄×C₁₀₉

yes

characteristic

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

SD₁₆×C₁₀₉

yes

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₂²⋊C₄

—

184

C₄²

—

185

C₄²

C₂×D₄×C₁₀₉

yes

—

186

C₄²

C₂×D₄×C₁₀₉

yes

characteristic

187

C₄²

#41×C₁₀₉

yes

characteristic

188

C₂³

#127×C₁₀₉

yes

characteristic

189

Q₈

—

190

Q₈

—

191

Q₈

—

192

Q₈

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

#120×C₁₀₉

yes

—

202

C₂×C₄

#120×C₁₀₉

yes

characteristic

203

C₂×C₄

#121×C₁₀₉

yes

—

204

C₂×C₄

#121×C₁₀₉

yes

characteristic

205

C₂×C₄

C₂×SD₁₆×C₁₀₉

yes

—

206

C₂×C₄

#137×C₁₀₉

yes

—

207

C₈

—

208

C₈

—

209

C₂²

—

210

C₂²

#392×C₁₀₉

yes

characteristic

211

C₂²

#492×C₁₀₉

yes

characteristic

212

C₂²

#497×C₁₀₉

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

#448×C₁₀₉

yes

—

220

C₄

#461×C₁₀₉

yes

—

221

C₂

—

222

C₂

—

223

C₂

C₁₀₉×#2009

yes

—

224

C₂

C₁₀₉×#2013

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	C₂³×C₁₀₉	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

#392

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₉

prime radical

C₂³×C₁₀₉

#127

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
108. #10227	C₁₀₉	direct (N × H)
118. C₁₀₉	#10227	direct (N × H)

H ≅ G/N

type

108. #10227

C₁₀₉

direct (N × H)

118. C₁₀₉

#10227

direct (N × H)

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

subgroup	index
C₁₀₉×#3229	2
C₁₀₉×#3162	2
C₁₀₉×#1960	2
C₁₀₉×#1518	2
C₁₀₉×#1281	2
#10227	109

subgroup

index

C₁₀₉×#3229

C₁₀₉×#3162

C₁₀₉×#1960

C₁₀₉×#1518

C₁₀₉×#1281

#10227

109

Subgroups — C₁₀₉×#10227 #42128221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#10227 #42128221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10227 #42128221