Group #42131409 — subgroups

Subgroups — C₁₀₉×#13415 #42131409

Order 27904.

436 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 114 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	C₂×#494×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#362×C₁₀₉	C₂	1	2	yes	maximal
3	C₁₀₉×#2141	C₂	1	2	yes	maximal
4	C₁₀₉×#1945	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₁₀₉	—	2	1	no	—
7	C₂²×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈×C₁₀₉	—	2	1	no	—
9	C₂×M₃₂×C₁₀₉	C₂²	1	2	yes	—
10	#494×C₁₀₉	—	2	4	no	—
11	C₂×#108×C₁₀₉	C₂²	1	1	yes	characteristic
12	#427×C₁₀₉	C₂²	1	1	yes	characteristic
13	#424×C₁₀₉	—	2	1	no	—
14	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	#362×C₁₀₉	—	2	4	no	—
16	#358×C₁₀₉	—	2	4	no	—
17	#339×C₁₀₉	—	2	1	no	—
18	C₂²×Q₈×C₁₀₉	—	2	2	no	—
19	C₂×Q₁₆×C₁₀₉	—	2	4	no	—
20	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
21	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
22	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
23	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
24	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
25	C₂×Q₁₆×C₁₀₉	—	4	2	no	—
26	C₂×Q₁₆×C₁₀₉	—	4	2	no	—
27	C₂×Q₁₆×C₁₀₉	D₄	1	4	yes	—
28	C₂×M₁₆×C₁₀₉	—	2	1	no	—
29	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₁₀₉	—	2	4	no	—
32	C₄⋊Q₈×C₁₀₉	—	4	2	no	—
33	#117×C₁₀₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₀₉	—	4	2	no	—
35	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
36	M₃₂×C₁₀₉	—	2	2	no	—
37	M₃₂×C₁₀₉	—	2	2	no	—
38	C₂×C₁₆×C₁₀₉	—	2	2	no	—
39	#108×C₁₀₉	—	2	2	no	—
40	#107×C₁₀₉	D₄	1	2	yes	—
41	#106×C₁₀₉	—	2	1	no	—
42	C₄⋊C₈×C₁₀₉	—	4	2	no	—
43	C₄×C₈×C₁₀₉	—	2	2	no	—
44	C₂×Q₈×C₁₀₉	—	2	4	no	—
45	C₂×Q₈×C₁₀₉	—	4	2	no	—
46	C₂×Q₈×C₁₀₉	—	4	2	no	—
47	C₂×Q₈×C₁₀₉	—	4	2	no	—
48	C₂×Q₈×C₁₀₉	—	4	2	no	—
49	C₂×Q₈×C₁₀₉	—	4	2	no	—
50	C₂²×C₄×C₁₀₉	—	2	1	no	—
51	C₂²×C₄×C₁₀₉	—	4	2	no	—
52	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
53	Q₁₆×C₁₀₉	—	2	4	no	—
54	Q₁₆×C₁₀₉	—	2	4	no	—
55	Q₁₆×C₁₀₉	—	4	8	no	—
56	Q₁₆×C₁₀₉	—	4	2	no	—
57	Q₁₆×C₁₀₉	—	4	2	no	—
58	M₁₆×C₁₀₉	—	4	1	no	—
59	M₁₆×C₁₀₉	—	4	1	no	—
60	C₂×C₈×C₁₀₉	—	2	1	no	—
61	C₂×C₈×C₁₀₉	—	2	1	no	—
62	C₂×C₈×C₁₀₉	—	4	1	no	—
63	C₂×C₈×C₁₀₉	D₈	1	2	yes	—
64	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₁₀₉	#41	1	2	yes	—
67	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
68	C₄⋊C₄×C₁₀₉	—	2	2	no	—
69	C₄⋊C₄×C₁₀₉	—	4	8	no	—
70	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
71	C₄²×C₁₀₉	—	2	1	no	—
72	C₄²×C₁₀₉	—	4	1	no	—
73	C₄²×C₁₀₉	SD₁₆	1	2	yes	—
74	C₁₆×C₁₀₉	—	4	2	no	—
75	C₂³×C₁₀₉	AGL(1,ℤ/8)	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	—	2	4	no	—
81	C₂×C₄×C₁₀₉	—	2	2	no	—
82	C₂×C₄×C₁₀₉	—	4	4	no	—
83	C₂×C₄×C₁₀₉	—	4	2	no	—
84	C₂×C₄×C₁₀₉	—	8	2	no	—
85	C₂×C₄×C₁₀₉	—	8	2	no	—
86	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
87	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
88	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
92	C₈×C₁₀₉	—	2	2	no	—
93	C₈×C₁₀₉	—	8	1	no	—
94	C₈×C₁₀₉	#121	1	2	yes	—
95	C₈×C₁₀₉	#123	1	2	yes	—
96	C₂²×C₁₀₉	—	2	2	no	—
97	C₂²×C₁₀₉	#450	1	1	yes	characteristic
98	C₂²×C₁₀₉	#459	1	1	yes	characteristic
99	C₂²×C₁₀₉	#490	1	1	yes	characteristic
100	C₄×C₁₀₉	—	2	4	no	—
101	C₄×C₁₀₉	—	8	2	no	—
102	C₄×C₁₀₉	—	8	4	no	—
103	C₄×C₁₀₉	#394	1	1	yes	characteristic
104	C₄×C₁₀₉	#447	1	1	yes	characteristic
105	C₄×C₁₀₉	#460	1	1	yes	characteristic
106	C₄×C₁₀₉	#486	1	1	yes	characteristic
107	#13415	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂×C₁₀₉	—	2	1	no	—
109	C₂×C₁₀₉	—	2	1	no	—
110	C₂×C₁₀₉	#2041	1	1	yes	characteristic
111	C₂×C₁₀₉	#2214	1	2	yes	—
112	C₂×#494	C₂×C₁₀₉	1	1	yes	characteristic
113	C₂×#362	C₂×C₁₀₉	1	2	yes	—
114	#2141	C₂×C₁₀₉	1	2	yes	—
115	#1945	C₂×C₁₀₉	1	1	yes	characteristic
116	#1376	C₂×C₁₀₉	1	1	yes	characteristic
117	C₁₀₉	#13415	1	1	yes	characteristic, direct product factor, Sylow
118	C₂²×Q₁₆	—	2	1	no	—
119	C₂²×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
120	C₂×C₄⋊Q₈	—	2	1	no	—
121	C₂×M₃₂	C₂²×C₁₀₉	1	2	yes	—
122	#494	—	2	4	no	—
123	C₂×#108	C₂²×C₁₀₉	1	1	yes	characteristic
124	#427	C₂²×C₁₀₉	1	1	yes	characteristic
125	#424	—	2	1	no	—
126	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
127	#362	—	2	4	no	—
128	#358	—	2	4	no	—
129	#339	—	2	1	no	—
130	C₂²×Q₈	—	2	2	no	—
131	C₂×Q₁₆	—	2	4	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	4	2	no	—
138	C₂×Q₁₆	—	4	2	no	—
139	C₂×Q₁₆	D₄×C₁₀₉	1	4	yes	—
140	C₂×M₁₆	—	2	1	no	—
141	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
142	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	4	no	—
144	C₄⋊Q₈	—	4	2	no	—
145	#117	—	2	1	no	—
146	C₂×C₄⋊C₄	—	4	2	no	—
147	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
148	M₃₂	—	2	2	no	—
149	M₃₂	—	2	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	#108	—	2	2	no	—
152	#107	D₄×C₁₀₉	1	2	yes	—
153	#106	—	2	1	no	—
154	C₄⋊C₈	—	4	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
165	Q₁₆	—	2	4	no	—
166	Q₁₆	—	2	4	no	—
167	Q₁₆	—	4	8	no	—
168	Q₁₆	—	4	2	no	—
169	Q₁₆	—	4	2	no	—
170	M₁₆	—	4	1	no	—
171	M₁₆	—	4	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	D₈×C₁₀₉	1	2	yes	—
176	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
177	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
178	C₂×C₈	#41×C₁₀₉	1	2	yes	—
179	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	4	8	no	—
182	C₂²⋊C₄	—	4	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	4	1	no	—
185	C₄²	SD₁₆×C₁₀₉	1	2	yes	—
186	C₁₆	—	4	2	no	—
187	C₂³	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
188	Q₈	—	4	4	no	—
189	Q₈	—	4	4	no	—
190	Q₈	—	8	2	no	—
191	Q₈	—	8	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
199	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
200	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
201	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
202	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
203	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	8	1	no	—
206	C₈	#121×C₁₀₉	1	2	yes	—
207	C₈	#123×C₁₀₉	1	2	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	#450×C₁₀₉	1	1	yes	characteristic
210	C₂²	#459×C₁₀₉	1	1	yes	characteristic
211	C₂²	#490×C₁₀₉	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	8	2	no	—
214	C₄	—	8	4	no	—
215	C₄	#394×C₁₀₉	1	1	yes	characteristic
216	C₄	#447×C₁₀₉	1	1	yes	characteristic
217	C₄	#460×C₁₀₉	1	1	yes	characteristic
218	C₄	#486×C₁₀₉	1	1	yes	characteristic
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	C₁₀₉×#2041	1	1	yes	characteristic
222	C₂	C₁₀₉×#2214	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#494×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#362×C₁₀₉

C₂

yes

maximal

C₁₀₉×#2141

C₂

yes

maximal

C₁₀₉×#1945

C₂

yes

maximal, characteristic

C₁₀₉×#1376

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₁₀₉

—

C₂²×Q₁₆×C₁₀₉

C₂²

yes

—

C₂×C₄⋊Q₈×C₁₀₉

—

C₂×M₃₂×C₁₀₉

C₂²

yes

—

#494×C₁₀₉

—

C₂×#108×C₁₀₉

C₂²

yes

characteristic

#427×C₁₀₉

C₂²

yes

characteristic

#424×C₁₀₉

—

C₂×C₄×C₈×C₁₀₉

C₂²

yes

characteristic

#362×C₁₀₉

—

#358×C₁₀₉

—

#339×C₁₀₉

—

C₂²×Q₈×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

D₄

yes

—

C₂×M₁₆×C₁₀₉

—

C₂²×C₈×C₁₀₉

D₄

yes

characteristic

C₂²×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

#117×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄²×C₁₀₉

D₄

yes

characteristic

M₃₂×C₁₀₉

—

M₃₂×C₁₀₉

—

C₂×C₁₆×C₁₀₉

—

#108×C₁₀₉

—

#107×C₁₀₉

D₄

yes

—

#106×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

C₄×C₈×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

characteristic

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₈

yes

—

C₂×C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×D₄

yes

—

C₂×C₈×C₁₀₉

#41

yes

—

C₂×C₈×C₁₀₉

#41

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

SD₁₆

yes

—

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

AGL(1,ℤ/8)

yes

characteristic

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

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

C₂×D₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₀₉

AGL(1,ℤ/8)

yes

characteristic

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

#121

yes

—

C₈×C₁₀₉

#123

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

100

C₄×C₁₀₉

—

101

C₄×C₁₀₉

—

102

C₄×C₁₀₉

—

103

C₄×C₁₀₉

#394

yes

characteristic

104

C₄×C₁₀₉

#447

yes

characteristic

105

C₄×C₁₀₉

#460

yes

characteristic

106

C₄×C₁₀₉

#486

yes

characteristic

107

#13415

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

108

C₂×C₁₀₉

—

109

C₂×C₁₀₉

—

110

C₂×C₁₀₉

#2041

yes

characteristic

111

C₂×C₁₀₉

#2214

yes

—

112

C₂×#494

C₂×C₁₀₉

yes

characteristic

113

C₂×#362

C₂×C₁₀₉

yes

—

114

#2141

C₂×C₁₀₉

yes

—

115

#1945

C₂×C₁₀₉

yes

characteristic

116

#1376

C₂×C₁₀₉

yes

characteristic

117

C₁₀₉

#13415

yes

characteristic, direct product factor, Sylow

118

C₂²×Q₁₆

—

119

C₂²×Q₁₆

C₂²×C₁₀₉

yes

—

120

C₂×C₄⋊Q₈

—

121

C₂×M₃₂

C₂²×C₁₀₉

yes

—

122

#494

—

123

C₂×#108

C₂²×C₁₀₉

yes

characteristic

124

#427

C₂²×C₁₀₉

yes

characteristic

125

#424

—

126

C₂×C₄×C₈

C₂²×C₁₀₉

yes

characteristic

127

#362

—

128

#358

—

129

#339

—

130

C₂²×Q₈

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

133

C₂×Q₁₆

—

134

C₂×Q₁₆

—

135

C₂×Q₁₆

—

136

C₂×Q₁₆

—

137

C₂×Q₁₆

—

138

C₂×Q₁₆

—

139

C₂×Q₁₆

D₄×C₁₀₉

yes

—

140

C₂×M₁₆

—

141

C₂²×C₈

D₄×C₁₀₉

yes

characteristic

142

C₂²×C₈

C₂³×C₁₀₉

yes

characteristic

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

#117

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄²

D₄×C₁₀₉

yes

characteristic

148

M₃₂

—

149

M₃₂

—

150

C₂×C₁₆

—

151

#108

—

152

#107

D₄×C₁₀₉

yes

—

153

#106

—

154

C₄⋊C₈

—

155

C₄×C₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂×Q₈

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

C₂²×C₄

C₂×D₄×C₁₀₉

yes

characteristic

165

Q₁₆

—

166

Q₁₆

—

167

Q₁₆

—

168

Q₁₆

—

169

Q₁₆

—

170

M₁₆

—

171

M₁₆

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

D₈×C₁₀₉

yes

—

176

C₂×C₈

C₂×D₄×C₁₀₉

yes

characteristic

177

C₂×C₈

C₂×D₄×C₁₀₉

yes

—

178

C₂×C₈

#41×C₁₀₉

yes

—

179

C₂×C₈

#41×C₁₀₉

yes

characteristic

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₂²⋊C₄

—

183

C₄²

—

184

C₄²

—

185

C₄²

SD₁₆×C₁₀₉

yes

—

186

C₁₆

—

187

C₂³

AGL(1,ℤ/8)×C₁₀₉

yes

characteristic

188

Q₈

—

189

Q₈

—

190

Q₈

—

191

Q₈

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

#120×C₁₀₉

yes

characteristic

199

C₂×C₄

#121×C₁₀₉

yes

characteristic

200

C₂×C₄

#124×C₁₀₉

yes

characteristic

201

C₂×C₄

C₂×D₈×C₁₀₉

yes

characteristic

202

C₂×C₄

C₂×SD₁₆×C₁₀₉

yes

characteristic

203

C₂×C₄

AGL(1,ℤ/8)×C₁₀₉

yes

characteristic

204

C₈

—

205

C₈

—

206

C₈

#121×C₁₀₉

yes

—

207

C₈

#123×C₁₀₉

yes

—

208

C₂²

—

209

C₂²

#450×C₁₀₉

yes

characteristic

210

C₂²

#459×C₁₀₉

yes

characteristic

211

C₂²

#490×C₁₀₉

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

#394×C₁₀₉

yes

characteristic

216

C₄

#447×C₁₀₉

yes

characteristic

217

C₄

#460×C₁₀₉

yes

characteristic

218

C₄

#486×C₁₀₉

yes

characteristic

219

C₂

—

220

C₂

—

221

C₂

C₁₀₉×#2041

yes

characteristic

222

C₂

C₁₀₉×#2214

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#459
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
prime radical	C₂³×C₁₀₉	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

#459

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₁₀₉

prime radical

C₂³×C₁₀₉

AGL(1,ℤ/8)

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

N	H ≅ G/N	type
107. #13415	C₁₀₉	direct (N × H)
117. C₁₀₉	#13415	direct (N × H)

H ≅ G/N

type

107. #13415

C₁₀₉

direct (N × H)

117. C₁₀₉

#13415

direct (N × H)

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

subgroup	index
C₂×#494×C₁₀₉	2
C₂×#362×C₁₀₉	2
C₁₀₉×#2141	2
C₁₀₉×#1945	2
C₁₀₉×#1376	2
#13415	109

subgroup

index

C₂×#494×C₁₀₉

C₂×#362×C₁₀₉

C₁₀₉×#2141

C₁₀₉×#1945

C₁₀₉×#1376

#13415

109

Subgroups — C₁₀₉×#13415 #42131409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#13415 #42131409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#13415 #42131409