Group #42128281 — subgroups

Subgroups — C₁₀₉×#10287 #42128281

Order 27904.

566 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 130 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×#3206	C₂	1	2	yes	maximal
2	C₁₀₉×#2862	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#1962	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1548	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#1276	C₂	1	1	yes	maximal, characteristic, complemented
6	#532×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
8	#492×C₁₀₉	—	2	2	no	—
9	#487×C₁₀₉	—	2	2	no	—
10	#461×C₁₀₉	—	2	4	no	—
11	#448×C₁₀₉	—	2	4	no	—
12	#431×C₁₀₉	C₂²	1	2	yes	—
13	#423×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×#104×C₁₀₉	—	2	2	no	—
15	#391×C₁₀₉	—	2	4	no	—
16	#382×C₁₀₉	—	2	2	no	—
17	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
18	#333×C₁₀₉	—	2	4	no	—
19	C₂×#41×C₁₀₉	—	2	1	no	—
20	C₂²×Q₈×C₁₀₉	D₄	1	4	yes	—
21	C₂×SD₁₆×C₁₀₉	—	2	8	no	—
22	C₂×M₁₆×C₁₀₉	—	2	1	no	—
23	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
24	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
25	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
26	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
27	C₄⋊Q₈×C₁₀₉	D₄	1	1	yes	characteristic
28	#127×C₁₀₉	D₄	1	1	yes	characteristic
29	#124×C₁₀₉	—	2	1	no	—
30	#121×C₁₀₉	—	2	2	no	—
31	C₄×D₄×C₁₀₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
34	C₂×C₄²×C₁₀₉	—	2	2	no	—
35	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
36	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₁₀₉	—	2	2	no	—
38	C₄⋊C₈×C₁₀₉	D₄	1	4	yes	—
39	#104×C₁₀₉	—	4	4	no	—
40	#103×C₁₀₉	—	2	8	no	—
41	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
42	C₄×C₈×C₁₀₉	—	2	2	no	—
43	#95×C₁₀₉	—	2	8	no	—
44	#41×C₁₀₉	—	4	2	no	—
45	#41×C₁₀₉	—	4	2	no	—
46	C₂×Q₈×C₁₀₉	—	2	4	no	—
47	C₂×Q₈×C₁₀₉	—	2	1	no	—
48	C₂×Q₈×C₁₀₉	—	2	4	no	—
49	C₂×Q₈×C₁₀₉	—	4	8	no	—
50	C₂×D₄×C₁₀₉	—	2	2	no	—
51	C₂×D₄×C₁₀₉	—	2	1	no	—
52	C₂×D₄×C₁₀₉	—	4	1	no	—
53	C₂²×C₄×C₁₀₉	—	2	2	no	—
54	C₂²×C₄×C₁₀₉	—	2	4	no	—
55	C₂²×C₄×C₁₀₉	—	4	1	no	—
56	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
58	SD₁₆×C₁₀₉	—	8	8	no	—
59	M₁₆×C₁₀₉	—	4	2	no	—
60	C₂×C₈×C₁₀₉	—	2	4	no	—
61	C₂×C₈×C₁₀₉	—	4	1	no	—
62	C₄⋊C₄×C₁₀₉	—	2	2	no	—
63	C₄⋊C₄×C₁₀₉	—	4	2	no	—
64	C₄⋊C₄×C₁₀₉	—	4	2	no	—
65	C₄⋊C₄×C₁₀₉	—	4	2	no	—
66	C₄⋊C₄×C₁₀₉	—	4	2	no	—
67	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
68	C₄²×C₁₀₉	—	2	2	no	—
69	C₄²×C₁₀₉	—	4	4	no	—
70	C₄²×C₁₀₉	SD₁₆	1	4	yes	complemented
71	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
73	C₄²×C₁₀₉	#41	1	1	yes	characteristic
74	C₂³×C₁₀₉	—	4	1	no	—
75	C₂³×C₁₀₉	#120	1	1	yes	characteristic
76	Q₈×C₁₀₉	—	4	8	no	—
77	Q₈×C₁₀₉	—	4	2	no	—
78	Q₈×C₁₀₉	—	8	4	no	—
79	D₄×C₁₀₉	—	4	4	no	—
80	D₄×C₁₀₉	—	4	2	no	—
81	D₄×C₁₀₉	—	8	2	no	—
82	C₂×C₄×C₁₀₉	—	2	4	no	—
83	C₂×C₄×C₁₀₉	—	2	4	no	—
84	C₂×C₄×C₁₀₉	—	4	4	no	—
85	C₂×C₄×C₁₀₉	—	4	8	no	—
86	C₂×C₄×C₁₀₉	—	4	1	no	—
87	C₂×C₄×C₁₀₉	—	4	4	no	—
88	C₂×C₄×C₁₀₉	—	8	1	no	—
89	C₂×C₄×C₁₀₉	—	8	1	no	—
90	C₂×C₄×C₁₀₉	#120	1	2	yes	—
91	C₂×C₄×C₁₀₉	#121	1	2	yes	—
92	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
94	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
95	C₂×C₄×C₁₀₉	#137	1	2	yes	—
96	C₈×C₁₀₉	—	4	4	no	—
97	C₈×C₁₀₉	—	8	1	no	—
98	C₂²×C₁₀₉	—	2	2	no	—
99	C₂²×C₁₀₉	—	8	2	no	—
100	C₂²×C₁₀₉	—	8	1	no	—
101	C₂²×C₁₀₉	#392	1	1	yes	characteristic
102	C₂²×C₁₀₉	#448	1	2	yes	—
103	C₄×C₁₀₉	—	2	2	no	—
104	C₄×C₁₀₉	—	4	4	no	—
105	C₄×C₁₀₉	—	8	4	no	—
106	C₄×C₁₀₉	—	8	1	no	—
107	C₄×C₁₀₉	#455	1	4	yes	—
108	C₄×C₁₀₉	#461	1	2	yes	—
109	C₄×C₁₀₉	#492	1	1	yes	characteristic
110	C₄×C₁₀₉	#497	1	1	yes	characteristic
111	#10287	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₁₀₉	—	2	2	no	—
113	C₂×C₁₀₉	—	16	1	no	—
114	C₂×C₁₀₉	#2004	1	2	yes	—
115	C₂×C₁₀₉	#2013	1	1	yes	characteristic
116	#3206	C₂×C₁₀₉	1	2	yes	—
117	#2862	C₂×C₁₀₉	1	1	yes	characteristic, complemented
118	#1962	C₂×C₁₀₉	1	1	yes	characteristic
119	#1548	C₂×C₁₀₉	1	2	yes	complemented
120	#1276	C₂×C₁₀₉	1	1	yes	characteristic, complemented
121	C₁₀₉	#10287	1	1	yes	characteristic, direct product factor, Sylow
122	#532	C₂²×C₁₀₉	1	1	yes	characteristic
123	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
124	#492	—	2	2	no	—
125	#487	—	2	2	no	—
126	#461	—	2	4	no	—
127	#448	—	2	4	no	—
128	#431	C₂²×C₁₀₉	1	2	yes	—
129	#423	C₂²×C₁₀₉	1	1	yes	characteristic
130	C₂×#104	—	2	2	no	—
131	#391	—	2	4	no	—
132	#382	—	2	2	no	—
133	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
134	#333	—	2	4	no	—
135	C₂×#41	—	2	1	no	—
136	C₂²×Q₈	D₄×C₁₀₉	1	4	yes	—
137	C₂×SD₁₆	—	2	8	no	—
138	C₂×M₁₆	—	2	1	no	—
139	C₄⋊Q₈	—	2	2	no	—
140	C₄⋊Q₈	—	2	2	no	—
141	C₄⋊Q₈	—	2	2	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	D₄×C₁₀₉	1	1	yes	characteristic
144	#127	D₄×C₁₀₉	1	1	yes	characteristic
145	#124	—	2	1	no	—
146	#121	—	2	2	no	—
147	C₄×D₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄²	—	2	2	no	—
151	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
152	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	D₄×C₁₀₉	1	4	yes	—
155	#104	—	4	4	no	—
156	#103	—	2	8	no	—
157	C₂²⋊C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	#95	—	2	8	no	—
160	#41	—	4	2	no	—
161	#41	—	4	2	no	—
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	4	8	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	—	4	1	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	4	1	no	—
172	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
173	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
174	SD₁₆	—	8	8	no	—
175	M₁₆	—	4	2	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	4	1	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	—	2	2	no	—
185	C₄²	—	4	4	no	—
186	C₄²	SD₁₆×C₁₀₉	1	4	yes	complemented
187	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
188	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
189	C₄²	#41×C₁₀₉	1	1	yes	characteristic
190	C₂³	—	4	1	no	—
191	C₂³	#120×C₁₀₉	1	1	yes	characteristic
192	Q₈	—	4	8	no	—
193	Q₈	—	4	2	no	—
194	Q₈	—	8	4	no	—
195	D₄	—	4	4	no	—
196	D₄	—	4	2	no	—
197	D₄	—	8	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	8	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	8	1	no	—
205	C₂×C₄	—	8	1	no	—
206	C₂×C₄	#120×C₁₀₉	1	2	yes	—
207	C₂×C₄	#121×C₁₀₉	1	2	yes	—
208	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
209	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
210	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
211	C₂×C₄	#137×C₁₀₉	1	2	yes	—
212	C₈	—	4	4	no	—
213	C₈	—	8	1	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	8	2	no	—
216	C₂²	—	8	1	no	—
217	C₂²	#392×C₁₀₉	1	1	yes	characteristic
218	C₂²	#448×C₁₀₉	1	2	yes	—
219	C₄	—	2	2	no	—
220	C₄	—	4	4	no	—
221	C₄	—	8	4	no	—
222	C₄	—	8	1	no	—
223	C₄	#455×C₁₀₉	1	4	yes	—
224	C₄	#461×C₁₀₉	1	2	yes	—
225	C₄	#492×C₁₀₉	1	1	yes	characteristic
226	C₄	#497×C₁₀₉	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	—	16	1	no	—
229	C₂	C₁₀₉×#2004	1	2	yes	—
230	C₂	C₁₀₉×#2013	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#3206

C₂

yes

maximal

C₁₀₉×#2862

C₂

yes

maximal, characteristic, complemented

C₁₀₉×#1962

C₂

yes

maximal, characteristic

C₁₀₉×#1548

C₂

yes

maximal, complemented

C₁₀₉×#1276

C₂

yes

maximal, characteristic, complemented

#532×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₁₀₉

C₂²

yes

—

#492×C₁₀₉

—

#487×C₁₀₉

—

#461×C₁₀₉

—

#448×C₁₀₉

—

#431×C₁₀₉

C₂²

yes

—

#423×C₁₀₉

C₂²

yes

characteristic

C₂×#104×C₁₀₉

—

#391×C₁₀₉

—

#382×C₁₀₉

—

C₄³×C₁₀₉

C₂²

yes

characteristic

#333×C₁₀₉

—

C₂×#41×C₁₀₉

—

C₂²×Q₈×C₁₀₉

D₄

yes

—

C₂×SD₁₆×C₁₀₉

—

C₂×M₁₆×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

D₄

yes

characteristic

#127×C₁₀₉

D₄

yes

characteristic

#124×C₁₀₉

—

#121×C₁₀₉

—

C₄×D₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄²×C₁₀₉

—

C₂×C₄²×C₁₀₉

D₄

yes

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

D₄

yes

—

#104×C₁₀₉

—

#103×C₁₀₉

—

C₂²⋊C₈×C₁₀₉

—

C₄×C₈×C₁₀₉

—

#95×C₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

characteristic

SD₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

SD₁₆

yes

complemented

C₄²×C₁₀₉

C₂×D₄

yes

characteristic

C₄²×C₁₀₉

C₂×D₄

yes

—

C₄²×C₁₀₉

#41

yes

characteristic

C₂³×C₁₀₉

—

C₂³×C₁₀₉

#120

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×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

#121

yes

—

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

C₂×SD₁₆

yes

—

C₂×C₄×C₁₀₉

#137

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

100

C₂²×C₁₀₉

—

101

C₂²×C₁₀₉

#392

yes

characteristic

102

C₂²×C₁₀₉

#448

yes

—

103

C₄×C₁₀₉

—

104

C₄×C₁₀₉

—

105

C₄×C₁₀₉

—

106

C₄×C₁₀₉

—

107

C₄×C₁₀₉

#455

yes

—

108

C₄×C₁₀₉

#461

yes

—

109

C₄×C₁₀₉

#492

yes

characteristic

110

C₄×C₁₀₉

#497

yes

characteristic

111

#10287

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

112

C₂×C₁₀₉

—

113

C₂×C₁₀₉

—

114

C₂×C₁₀₉

#2004

yes

—

115

C₂×C₁₀₉

#2013

yes

characteristic

116

#3206

C₂×C₁₀₉

yes

—

117

#2862

C₂×C₁₀₉

yes

characteristic, complemented

118

#1962

C₂×C₁₀₉

yes

characteristic

119

#1548

C₂×C₁₀₉

yes

complemented

120

#1276

C₂×C₁₀₉

yes

characteristic, complemented

121

C₁₀₉

#10287

yes

characteristic, direct product factor, Sylow

122

#532

C₂²×C₁₀₉

yes

characteristic

123

C₂×C₄⋊Q₈

C₂²×C₁₀₉

yes

—

124

#492

—

125

#487

—

126

#461

—

127

#448

—

128

#431

C₂²×C₁₀₉

yes

—

129

#423

C₂²×C₁₀₉

yes

characteristic

130

C₂×#104

—

131

#391

—

132

#382

—

133

C₄³

C₂²×C₁₀₉

yes

characteristic

134

#333

—

135

C₂×#41

—

136

C₂²×Q₈

D₄×C₁₀₉

yes

—

137

C₂×SD₁₆

—

138

C₂×M₁₆

—

139

C₄⋊Q₈

—

140

C₄⋊Q₈

—

141

C₄⋊Q₈

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

D₄×C₁₀₉

yes

characteristic

144

#127

D₄×C₁₀₉

yes

characteristic

145

#124

—

146

#121

—

147

C₄×D₄

—

148

C₂×C₄⋊C₄

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄²

—

151

C₂×C₄²

D₄×C₁₀₉

yes

—

152

C₂×C₄²

C₂³×C₁₀₉

yes

characteristic

153

C₄⋊C₈

—

154

C₄⋊C₈

D₄×C₁₀₉

yes

—

155

#104

—

156

#103

—

157

C₂²⋊C₈

—

158

C₄×C₈

—

159

#95

—

160

#41

—

161

#41

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂×Q₈

—

166

C₂×D₄

—

167

C₂×D₄

—

168

C₂×D₄

—

169

C₂²×C₄

—

170

C₂²×C₄

—

171

C₂²×C₄

—

172

C₂²×C₄

C₂×D₄×C₁₀₉

yes

—

173

C₂²×C₄

C₂×D₄×C₁₀₉

yes

characteristic

174

SD₁₆

—

175

M₁₆

—

176

C₂×C₈

—

177

C₂×C₈

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₂²⋊C₄

—

184

C₄²

—

185

C₄²

—

186

C₄²

SD₁₆×C₁₀₉

yes

complemented

187

C₄²

C₂×D₄×C₁₀₉

yes

characteristic

188

C₄²

C₂×D₄×C₁₀₉

yes

—

189

C₄²

#41×C₁₀₉

yes

characteristic

190

C₂³

—

191

C₂³

#120×C₁₀₉

yes

characteristic

192

Q₈

—

193

Q₈

—

194

Q₈

—

195

D₄

—

196

D₄

—

197

D₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

#120×C₁₀₉

yes

—

207

C₂×C₄

#121×C₁₀₉

yes

—

208

C₂×C₄

#121×C₁₀₉

yes

characteristic

209

C₂×C₄

#127×C₁₀₉

yes

characteristic

210

C₂×C₄

C₂×SD₁₆×C₁₀₉

yes

—

211

C₂×C₄

#137×C₁₀₉

yes

—

212

C₈

—

213

C₈

—

214

C₂²

—

215

C₂²

—

216

C₂²

—

217

C₂²

#392×C₁₀₉

yes

characteristic

218

C₂²

#448×C₁₀₉

yes

—

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

223

C₄

#455×C₁₀₉

yes

—

224

C₄

#461×C₁₀₉

yes

—

225

C₄

#492×C₁₀₉

yes

characteristic

226

C₄

#497×C₁₀₉

yes

characteristic

227

C₂

—

228

C₂

—

229

C₂

C₁₀₉×#2004

yes

—

230

C₂

C₁₀₉×#2013

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	#532×C₁₀₉	C₂²
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

#532×C₁₀₉

C₂²

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
2. C₁₀₉×#2862	C₂	semidirect
4. C₁₀₉×#1548	C₂	semidirect
5. C₁₀₉×#1276	C₂	semidirect
70. C₄²×C₁₀₉	SD₁₆	semidirect
111. #10287	C₁₀₉	direct (N × H)
117. #2862	C₂×C₁₀₉	semidirect
119. #1548	C₂×C₁₀₉	semidirect
120. #1276	C₂×C₁₀₉	semidirect
121. C₁₀₉	#10287	direct (N × H)
186. C₄²	SD₁₆×C₁₀₉	semidirect

H ≅ G/N

type

2. C₁₀₉×#2862

C₂

semidirect

4. C₁₀₉×#1548

C₂

semidirect

5. C₁₀₉×#1276

C₂

semidirect

70. C₄²×C₁₀₉

SD₁₆

semidirect

111. #10287

C₁₀₉

direct (N × H)

117. #2862

C₂×C₁₀₉

semidirect

119. #1548

C₂×C₁₀₉

semidirect

120. #1276

C₂×C₁₀₉

semidirect

121. C₁₀₉

#10287

direct (N × H)

186. C₄²

SD₁₆×C₁₀₉

semidirect

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

subgroup	index
C₁₀₉×#3206	2
C₁₀₉×#2862	2
C₁₀₉×#1962	2
C₁₀₉×#1548	2
C₁₀₉×#1276	2
#10287	109

subgroup

index

C₁₀₉×#3206

C₁₀₉×#2862

C₁₀₉×#1962

C₁₀₉×#1548

C₁₀₉×#1276

#10287

109

Subgroups — C₁₀₉×#10287 #42128281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#10287 #42128281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10287 #42128281