Group #42128279 — subgroups

Subgroups — C₁₀₉×#10285 #42128279

Order 27904.

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

normal subgroups only

This group has 542 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×#3227	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×#1549	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	#498×C₁₀₉	—	2	2	no	—
9	#486×C₁₀₉	—	2	2	no	—
10	#481×C₁₀₉	—	2	4	no	—
11	#476×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	#332×C₁₀₉	—	2	4	no	—
19	C₂×#41×C₁₀₉	—	2	1	no	—
20	C₂²×Q₈×C₁₀₉	—	2	2	no	—
21	C₂×M₁₆×C₁₀₉	—	2	1	no	—
22	C₄⋊Q₈×C₁₀₉	—	2	2	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	D₄	1	1	yes	characteristic
27	#127×C₁₀₉	D₄	1	1	yes	characteristic
28	#124×C₁₀₉	—	2	1	no	—
29	#121×C₁₀₉	—	2	2	no	—
30	C₄×D₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₀₉	D₄	1	4	yes	—
33	C₂×C₄²×C₁₀₉	—	2	2	no	—
34	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
35	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
36	#106×C₁₀₉	—	2	8	no	—
37	C₄⋊C₈×C₁₀₉	—	2	2	no	—
38	C₄⋊C₈×C₁₀₉	D₄	1	4	yes	—
39	#104×C₁₀₉	—	4	4	no	—
40	#102×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	1	no	—
47	C₂×Q₈×C₁₀₉	—	4	8	no	—
48	C₂×Q₈×C₁₀₉	—	4	2	no	—
49	C₂×Q₈×C₁₀₉	—	4	2	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	M₁₆×C₁₀₉	—	4	2	no	—
59	C₂×C₈×C₁₀₉	—	2	4	no	—
60	C₂×C₈×C₁₀₉	—	4	1	no	—
61	C₄⋊C₄×C₁₀₉	—	2	4	no	—
62	C₄⋊C₄×C₁₀₉	—	2	2	no	—
63	C₄⋊C₄×C₁₀₉	—	2	4	no	—
64	C₄⋊C₄×C₁₀₉	—	4	2	no	—
65	C₄⋊C₄×C₁₀₉	—	4	2	no	—
66	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
67	C₄²×C₁₀₉	—	2	2	no	—
68	C₄²×C₁₀₉	—	4	4	no	—
69	C₄²×C₁₀₉	SD₁₆	1	4	yes	—
70	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
72	C₄²×C₁₀₉	#41	1	1	yes	characteristic
73	C₂³×C₁₀₉	—	4	1	no	—
74	C₂³×C₁₀₉	#120	1	1	yes	characteristic
75	Q₈×C₁₀₉	—	4	2	no	—
76	Q₈×C₁₀₉	—	8	4	no	—
77	Q₈×C₁₀₉	—	8	4	no	—
78	D₄×C₁₀₉	—	4	4	no	—
79	D₄×C₁₀₉	—	4	2	no	—
80	D₄×C₁₀₉	—	8	2	no	—
81	C₂×C₄×C₁₀₉	—	2	4	no	—
82	C₂×C₄×C₁₀₉	—	2	4	no	—
83	C₂×C₄×C₁₀₉	—	4	4	no	—
84	C₂×C₄×C₁₀₉	—	4	8	no	—
85	C₂×C₄×C₁₀₉	—	4	1	no	—
86	C₂×C₄×C₁₀₉	—	4	4	no	—
87	C₂×C₄×C₁₀₉	—	8	1	no	—
88	C₂×C₄×C₁₀₉	—	8	1	no	—
89	C₂×C₄×C₁₀₉	#120	1	2	yes	—
90	C₂×C₄×C₁₀₉	#121	1	2	yes	—
91	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
94	C₂×C₄×C₁₀₉	#137	1	2	yes	—
95	C₈×C₁₀₉	—	4	4	no	—
96	C₈×C₁₀₉	—	8	1	no	—
97	C₂²×C₁₀₉	—	2	2	no	—
98	C₂²×C₁₀₉	—	8	2	no	—
99	C₂²×C₁₀₉	—	8	1	no	—
100	C₂²×C₁₀₉	#392	1	1	yes	characteristic
101	C₂²×C₁₀₉	#448	1	2	yes	—
102	C₄×C₁₀₉	—	2	2	no	—
103	C₄×C₁₀₉	—	4	4	no	—
104	C₄×C₁₀₉	—	8	4	no	—
105	C₄×C₁₀₉	—	8	1	no	—
106	C₄×C₁₀₉	#454	1	4	yes	—
107	C₄×C₁₀₉	#461	1	2	yes	—
108	C₄×C₁₀₉	#492	1	1	yes	characteristic
109	C₄×C₁₀₉	#497	1	1	yes	characteristic
110	#10285	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂×C₁₀₉	—	2	2	no	—
112	C₂×C₁₀₉	—	16	1	no	—
113	C₂×C₁₀₉	#2003	1	2	yes	—
114	C₂×C₁₀₉	#2013	1	1	yes	characteristic
115	#3227	C₂×C₁₀₉	1	2	yes	—
116	#2862	C₂×C₁₀₉	1	1	yes	characteristic, complemented
117	#1962	C₂×C₁₀₉	1	1	yes	characteristic
118	#1549	C₂×C₁₀₉	1	2	yes	complemented
119	#1276	C₂×C₁₀₉	1	1	yes	characteristic, complemented
120	C₁₀₉	#10285	1	1	yes	characteristic, direct product factor, Sylow
121	#532	C₂²×C₁₀₉	1	1	yes	characteristic
122	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
123	#498	—	2	2	no	—
124	#486	—	2	2	no	—
125	#481	—	2	4	no	—
126	#476	—	2	4	no	—
127	#431	C₂²×C₁₀₉	1	2	yes	—
128	#423	C₂²×C₁₀₉	1	1	yes	characteristic
129	C₂×#104	—	2	2	no	—
130	#391	—	2	4	no	—
131	#382	—	2	2	no	—
132	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
133	#332	—	2	4	no	—
134	C₂×#41	—	2	1	no	—
135	C₂²×Q₈	—	2	2	no	—
136	C₂×M₁₆	—	2	1	no	—
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	—	2	2	no	—
139	C₄⋊Q₈	—	2	2	no	—
140	C₄⋊Q₈	—	2	2	no	—
141	C₄⋊Q₈	D₄×C₁₀₉	1	1	yes	characteristic
142	#127	D₄×C₁₀₉	1	1	yes	characteristic
143	#124	—	2	1	no	—
144	#121	—	2	2	no	—
145	C₄×D₄	—	2	2	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	D₄×C₁₀₉	1	4	yes	—
148	C₂×C₄²	—	2	2	no	—
149	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
150	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
151	#106	—	2	8	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	D₄×C₁₀₉	1	4	yes	—
154	#104	—	4	4	no	—
155	#102	—	2	8	no	—
156	C₂²⋊C₈	—	2	2	no	—
157	C₄×C₈	—	2	2	no	—
158	#95	—	2	8	no	—
159	#41	—	4	2	no	—
160	#41	—	4	2	no	—
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	—	4	8	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	2	no	—
165	C₂×D₄	—	2	2	no	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	—	4	1	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	4	1	no	—
171	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
172	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
173	M₁₆	—	4	2	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	4	1	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	—	4	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	4	4	no	—
184	C₄²	SD₁₆×C₁₀₉	1	4	yes	—
185	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
186	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
187	C₄²	#41×C₁₀₉	1	1	yes	characteristic
188	C₂³	—	4	1	no	—
189	C₂³	#120×C₁₀₉	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	8	4	no	—
192	Q₈	—	8	4	no	—
193	D₄	—	4	4	no	—
194	D₄	—	4	2	no	—
195	D₄	—	8	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	8	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	8	1	no	—
203	C₂×C₄	—	8	1	no	—
204	C₂×C₄	#120×C₁₀₉	1	2	yes	—
205	C₂×C₄	#121×C₁₀₉	1	2	yes	—
206	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
207	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
208	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
209	C₂×C₄	#137×C₁₀₉	1	2	yes	—
210	C₈	—	4	4	no	—
211	C₈	—	8	1	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	8	2	no	—
214	C₂²	—	8	1	no	—
215	C₂²	#392×C₁₀₉	1	1	yes	characteristic
216	C₂²	#448×C₁₀₉	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	4	4	no	—
219	C₄	—	8	4	no	—
220	C₄	—	8	1	no	—
221	C₄	#454×C₁₀₉	1	4	yes	—
222	C₄	#461×C₁₀₉	1	2	yes	—
223	C₄	#492×C₁₀₉	1	1	yes	characteristic
224	C₄	#497×C₁₀₉	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	—	16	1	no	—
227	C₂	C₁₀₉×#2003	1	2	yes	—
228	C₂	C₁₀₉×#2013	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#3227

C₂

yes

maximal

C₁₀₉×#2862

C₂

yes

maximal, characteristic, complemented

C₁₀₉×#1962

C₂

yes

maximal, characteristic

C₁₀₉×#1549

C₂

yes

maximal, complemented

C₁₀₉×#1276

C₂

yes

maximal, characteristic, complemented

#532×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₁₀₉

C₂²

yes

—

#498×C₁₀₉

—

#486×C₁₀₉

—

#481×C₁₀₉

—

#476×C₁₀₉

—

#431×C₁₀₉

C₂²

yes

—

#423×C₁₀₉

C₂²

yes

characteristic

C₂×#104×C₁₀₉

—

#391×C₁₀₉

—

#382×C₁₀₉

—

C₄³×C₁₀₉

C₂²

yes

characteristic

#332×C₁₀₉

—

C₂×#41×C₁₀₉

—

C₂²×Q₈×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

D₄

yes

—

C₂×C₄²×C₁₀₉

—

C₂×C₄²×C₁₀₉

D₄

yes

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

#106×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

D₄

yes

—

#104×C₁₀₉

—

#102×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

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

—

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

—

C₂²×C₁₀₉

—

100

C₂²×C₁₀₉

#392

yes

characteristic

101

C₂²×C₁₀₉

#448

yes

—

102

C₄×C₁₀₉

—

103

C₄×C₁₀₉

—

104

C₄×C₁₀₉

—

105

C₄×C₁₀₉

—

106

C₄×C₁₀₉

#454

yes

—

107

C₄×C₁₀₉

#461

yes

—

108

C₄×C₁₀₉

#492

yes

characteristic

109

C₄×C₁₀₉

#497

yes

characteristic

110

#10285

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

111

C₂×C₁₀₉

—

112

C₂×C₁₀₉

—

113

C₂×C₁₀₉

#2003

yes

—

114

C₂×C₁₀₉

#2013

yes

characteristic

115

#3227

C₂×C₁₀₉

yes

—

116

#2862

C₂×C₁₀₉

yes

characteristic, complemented

117

#1962

C₂×C₁₀₉

yes

characteristic

118

#1549

C₂×C₁₀₉

yes

complemented

119

#1276

C₂×C₁₀₉

yes

characteristic, complemented

120

C₁₀₉

#10285

yes

characteristic, direct product factor, Sylow

121

#532

C₂²×C₁₀₉

yes

characteristic

122

C₂×C₄⋊Q₈

C₂²×C₁₀₉

yes

—

123

#498

—

124

#486

—

125

#481

—

126

#476

—

127

#431

C₂²×C₁₀₉

yes

—

128

#423

C₂²×C₁₀₉

yes

characteristic

129

C₂×#104

—

130

#391

—

131

#382

—

132

C₄³

C₂²×C₁₀₉

yes

characteristic

133

#332

—

134

C₂×#41

—

135

C₂²×Q₈

—

136

C₂×M₁₆

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

—

139

C₄⋊Q₈

—

140

C₄⋊Q₈

—

141

C₄⋊Q₈

D₄×C₁₀₉

yes

characteristic

142

#127

D₄×C₁₀₉

yes

characteristic

143

#124

—

144

#121

—

145

C₄×D₄

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄⋊C₄

D₄×C₁₀₉

yes

—

148

C₂×C₄²

—

149

C₂×C₄²

D₄×C₁₀₉

yes

—

150

C₂×C₄²

C₂³×C₁₀₉

yes

characteristic

151

#106

—

152

C₄⋊C₈

—

153

C₄⋊C₈

D₄×C₁₀₉

yes

—

154

#104

—

155

#102

—

156

C₂²⋊C₈

—

157

C₄×C₈

—

158

#95

—

159

#41

—

160

#41

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂×D₄

—

166

C₂×D₄

—

167

C₂×D₄

—

168

C₂²×C₄

—

169

C₂²×C₄

—

170

C₂²×C₄

—

171

C₂²×C₄

C₂×D₄×C₁₀₉

yes

—

172

C₂²×C₄

C₂×D₄×C₁₀₉

yes

characteristic

173

M₁₆

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₂²⋊C₄

—

182

C₄²

—

183

C₄²

—

184

C₄²

SD₁₆×C₁₀₉

yes

—

185

C₄²

C₂×D₄×C₁₀₉

yes

characteristic

186

C₄²

C₂×D₄×C₁₀₉

yes

—

187

C₄²

#41×C₁₀₉

yes

characteristic

188

C₂³

—

189

C₂³

#120×C₁₀₉

yes

characteristic

190

Q₈

—

191

Q₈

—

192

Q₈

—

193

D₄

—

194

D₄

—

195

D₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

#120×C₁₀₉

yes

—

205

C₂×C₄

#121×C₁₀₉

yes

—

206

C₂×C₄

#121×C₁₀₉

yes

characteristic

207

C₂×C₄

#127×C₁₀₉

yes

characteristic

208

C₂×C₄

C₂×SD₁₆×C₁₀₉

yes

—

209

C₂×C₄

#137×C₁₀₉

yes

—

210

C₈

—

211

C₈

—

212

C₂²

—

213

C₂²

—

214

C₂²

—

215

C₂²

#392×C₁₀₉

yes

characteristic

216

C₂²

#448×C₁₀₉

yes

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

221

C₄

#454×C₁₀₉

yes

—

222

C₄

#461×C₁₀₉

yes

—

223

C₄

#492×C₁₀₉

yes

characteristic

224

C₄

#497×C₁₀₉

yes

characteristic

225

C₂

—

226

C₂

—

227

C₂

C₁₀₉×#2003

yes

—

228

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×#1549	C₂	semidirect
5. C₁₀₉×#1276	C₂	semidirect
110. #10285	C₁₀₉	direct (N × H)
116. #2862	C₂×C₁₀₉	semidirect
118. #1549	C₂×C₁₀₉	semidirect
119. #1276	C₂×C₁₀₉	semidirect
120. C₁₀₉	#10285	direct (N × H)

H ≅ G/N

type

2. C₁₀₉×#2862

C₂

semidirect

4. C₁₀₉×#1549

C₂

semidirect

5. C₁₀₉×#1276

C₂

semidirect

110. #10285

C₁₀₉

direct (N × H)

116. #2862

C₂×C₁₀₉

semidirect

118. #1549

C₂×C₁₀₉

semidirect

119. #1276

C₂×C₁₀₉

semidirect

120. C₁₀₉

#10285

direct (N × H)

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

subgroup	index
C₁₀₉×#3227	2
C₁₀₉×#2862	2
C₁₀₉×#1962	2
C₁₀₉×#1549	2
C₁₀₉×#1276	2
#10285	109

subgroup

index

C₁₀₉×#3227

C₁₀₉×#2862

C₁₀₉×#1962

C₁₀₉×#1549

C₁₀₉×#1276

#10285

109

Subgroups — C₁₀₉×#10285 #42128279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#10285 #42128279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10285 #42128279