Group #42128571 — subgroups

Subgroups — C₁₀₉×#10577 #42128571

Order 27904.

406 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 146 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×#2571	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1947	C₂	1	2	yes	maximal
3	C₁₀₉×#1844	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1296	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1279	C₂	1	2	yes	maximal
6	C₂×#107×C₁₀₉	—	2	2	no	—
7	C₂×#106×C₁₀₉	—	2	2	no	—
8	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	#400×C₁₀₉	—	2	2	no	—
11	#389×C₁₀₉	—	2	2	no	—
12	#384×C₁₀₉	C₂²	1	1	yes	characteristic
13	#384×C₁₀₉	C₂²	1	1	yes	characteristic
14	#382×C₁₀₉	—	2	2	no	—
15	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
16	#340×C₁₀₉	—	2	2	no	—
17	C₂²×C₈×C₁₀₉	—	2	1	no	—
18	C₂²×C₈×C₁₀₉	—	2	2	no	—
19	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
22	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
24	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
26	C₂×C₄²×C₁₀₉	—	2	2	no	—
27	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
28	#107×C₁₀₉	—	4	4	no	—
29	#106×C₁₀₉	—	4	4	no	—
30	C₄⋊C₈×C₁₀₉	—	2	2	no	—
31	C₄⋊C₈×C₁₀₉	Q₈	1	4	yes	—
32	C₄×C₈×C₁₀₉	—	2	1	no	—
33	C₄×C₈×C₁₀₉	Q₈	1	2	yes	—
34	#95×C₁₀₉	—	2	2	no	—
35	#95×C₁₀₉	—	2	4	no	—
36	#95×C₁₀₉	—	2	2	no	—
37	C₂²×C₄×C₁₀₉	—	2	2	no	—
38	C₂²×C₄×C₁₀₉	—	2	2	no	—
39	C₂²×C₄×C₁₀₉	—	2	2	no	—
40	C₂²×C₄×C₁₀₉	—	4	1	no	—
41	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
43	C₂×C₈×C₁₀₉	—	2	2	no	—
44	C₂×C₈×C₁₀₉	—	2	2	no	—
45	C₂×C₈×C₁₀₉	—	2	2	no	—
46	C₂×C₈×C₁₀₉	—	2	4	no	—
47	C₂×C₈×C₁₀₉	—	4	2	no	—
48	C₂×C₈×C₁₀₉	—	4	2	no	—
49	C₄⋊C₄×C₁₀₉	—	2	2	no	—
50	C₄⋊C₄×C₁₀₉	—	2	2	no	—
51	C₄⋊C₄×C₁₀₉	—	2	2	no	—
52	C₄⋊C₄×C₁₀₉	—	2	2	no	—
53	C₄⋊C₄×C₁₀₉	—	4	2	no	—
54	C₄⋊C₄×C₁₀₉	—	4	2	no	—
55	C₄⋊C₄×C₁₀₉	—	4	2	no	—
56	C₄⋊C₄×C₁₀₉	—	4	2	no	—
57	C₄⋊C₄×C₁₀₉	—	4	2	no	—
58	C₄⋊C₄×C₁₀₉	—	4	2	no	—
59	C₄⋊C₄×C₁₀₉	—	8	2	no	—
60	C₄²×C₁₀₉	—	4	2	no	—
61	C₄²×C₁₀₉	—	4	2	no	—
62	C₄²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₁₀₉	C₂×Q₈	1	2	yes	—
64	C₄²×C₁₀₉	#41	1	1	yes	characteristic
65	C₂³×C₁₀₉	#120	1	1	yes	characteristic
66	C₂×C₄×C₁₀₉	—	2	1	no	—
67	C₂×C₄×C₁₀₉	—	2	1	no	—
68	C₂×C₄×C₁₀₉	—	4	2	no	—
69	C₂×C₄×C₁₀₉	—	4	2	no	—
70	C₂×C₄×C₁₀₉	—	4	2	no	—
71	C₂×C₄×C₁₀₉	—	4	2	no	—
72	C₂×C₄×C₁₀₉	—	4	4	no	—
73	C₂×C₄×C₁₀₉	—	4	2	no	—
74	C₂×C₄×C₁₀₉	—	4	2	no	—
75	C₂×C₄×C₁₀₉	—	4	2	no	—
76	C₂×C₄×C₁₀₉	—	8	2	no	—
77	C₂×C₄×C₁₀₉	—	8	1	no	—
78	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
80	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
82	C₂×C₄×C₁₀₉	#135	1	2	yes	—
83	C₂×C₄×C₁₀₉	#135	1	2	yes	—
84	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
85	C₂×C₄×C₁₀₉	#137	1	2	yes	—
86	C₈×C₁₀₉	—	2	4	no	—
87	C₈×C₁₀₉	—	4	4	no	—
88	C₂²×C₁₀₉	#395	1	1	yes	characteristic
89	C₂²×C₁₀₉	#449	1	2	yes	—
90	C₂²×C₁₀₉	#452	1	2	yes	—
91	C₂²×C₁₀₉	#453	1	1	yes	characteristic
92	C₂²×C₁₀₉	#456	1	1	yes	characteristic
93	C₄×C₁₀₉	—	2	1	no	—
94	C₄×C₁₀₉	—	2	1	no	—
95	C₄×C₁₀₉	—	8	2	no	—
96	C₄×C₁₀₉	—	8	2	no	—
97	C₄×C₁₀₉	—	8	2	no	—
98	C₄×C₁₀₉	—	16	1	no	—
99	C₄×C₁₀₉	#478	1	2	yes	—
100	C₄×C₁₀₉	#479	1	2	yes	—
101	C₄×C₁₀₉	#499	1	2	yes	—
102	C₄×C₁₀₉	#501	1	2	yes	—
103	#10577	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂×C₁₀₉	#1624	1	1	yes	characteristic
105	C₂×C₁₀₉	#1630	1	1	yes	characteristic
106	C₂×C₁₀₉	#1652	1	2	yes	—
107	C₂×C₁₀₉	#2059	1	2	yes	—
108	C₂×C₁₀₉	#2060	1	1	yes	characteristic
109	#2571	C₂×C₁₀₉	1	1	yes	characteristic
110	#1947	C₂×C₁₀₉	1	2	yes	—
111	#1844	C₂×C₁₀₉	1	1	yes	characteristic
112	#1296	C₂×C₁₀₉	1	1	yes	characteristic
113	#1279	C₂×C₁₀₉	1	2	yes	—
114	C₁₀₉	#10577	1	1	yes	characteristic, direct product factor, Sylow
115	C₂×#107	—	2	2	no	—
116	C₂×#106	—	2	2	no	—
117	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
119	#400	—	2	2	no	—
120	#389	—	2	2	no	—
121	#384	C₂²×C₁₀₉	1	1	yes	characteristic
122	#384	C₂²×C₁₀₉	1	1	yes	characteristic
123	#382	—	2	2	no	—
124	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
125	#340	—	2	2	no	—
126	C₂²×C₈	—	2	1	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	1	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
133	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
134	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
137	#107	—	4	4	no	—
138	#106	—	4	4	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	Q₈×C₁₀₉	1	4	yes	—
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	Q₈×C₁₀₉	1	2	yes	—
143	#95	—	2	2	no	—
144	#95	—	2	4	no	—
145	#95	—	2	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	4	1	no	—
150	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	8	2	no	—
169	C₄²	—	4	2	no	—
170	C₄²	—	4	2	no	—
171	C₄²	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
172	C₄²	C₂×Q₈×C₁₀₉	1	2	yes	—
173	C₄²	#41×C₁₀₉	1	1	yes	characteristic
174	C₂³	#120×C₁₀₉	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	8	2	no	—
186	C₂×C₄	—	8	1	no	—
187	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
188	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
189	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
190	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	2	yes	—
191	C₂×C₄	#135×C₁₀₉	1	2	yes	—
192	C₂×C₄	#135×C₁₀₉	1	2	yes	—
193	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
194	C₂×C₄	#137×C₁₀₉	1	2	yes	—
195	C₈	—	2	4	no	—
196	C₈	—	4	4	no	—
197	C₂²	#395×C₁₀₉	1	1	yes	characteristic
198	C₂²	#449×C₁₀₉	1	2	yes	—
199	C₂²	#452×C₁₀₉	1	2	yes	—
200	C₂²	#453×C₁₀₉	1	1	yes	characteristic
201	C₂²	#456×C₁₀₉	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	8	2	no	—
205	C₄	—	8	2	no	—
206	C₄	—	8	2	no	—
207	C₄	—	16	1	no	—
208	C₄	#478×C₁₀₉	1	2	yes	—
209	C₄	#479×C₁₀₉	1	2	yes	—
210	C₄	#499×C₁₀₉	1	2	yes	—
211	C₄	#501×C₁₀₉	1	2	yes	—
212	C₂	C₁₀₉×#1624	1	1	yes	characteristic
213	C₂	C₁₀₉×#1630	1	1	yes	characteristic
214	C₂	C₁₀₉×#1652	1	2	yes	—
215	C₂	C₁₀₉×#2059	1	2	yes	—
216	C₂	C₁₀₉×#2060	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2571

C₂

yes

maximal, characteristic

C₁₀₉×#1947

C₂

yes

maximal

C₁₀₉×#1844

C₂

yes

maximal, characteristic

C₁₀₉×#1296

C₂

yes

maximal, characteristic

C₁₀₉×#1279

C₂

yes

maximal

C₂×#107×C₁₀₉

—

C₂×#106×C₁₀₉

—

C₂×C₄⋊C₈×C₁₀₉

C₂²

yes

—

C₂×C₄×C₈×C₁₀₉

C₂²

yes

characteristic

#400×C₁₀₉

—

#389×C₁₀₉

—

#384×C₁₀₉

C₂²

yes

characteristic

#384×C₁₀₉

C₂²

yes

characteristic

#382×C₁₀₉

—

C₄×C₄⋊C₄×C₁₀₉

C₂²

yes

—

#340×C₁₀₉

—

C₂²×C₈×C₁₀₉

—

C₂²×C₈×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

D₄

yes

—

C₂×C₄⋊C₄×C₁₀₉

D₄

yes

—

C₂×C₄⋊C₄×C₁₀₉

D₄

yes

—

C₂×C₄²×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

#107×C₁₀₉

—

#106×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

Q₈

yes

—

C₄×C₈×C₁₀₉

—

C₄×C₈×C₁₀₉

Q₈

yes

—

#95×C₁₀₉

—

#95×C₁₀₉

—

#95×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×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

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×Q₈

yes

characteristic

C₄²×C₁₀₉

C₂×Q₈

yes

—

C₄²×C₁₀₉

yes

C₂³×C₁₀₉

yes

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

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊Q₈

yes

—

C₂×C₄×C₁₀₉

#135

yes

—

C₂×C₄×C₁₀₉

#135

yes

—

C₂×C₄×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₀₉

#137

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

#395

yes

characteristic

C₂²×C₁₀₉

#449

yes

—

C₂²×C₁₀₉

#452

yes

—

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#478

yes

—

100

C₄×C₁₀₉

#479

yes

—

101

C₄×C₁₀₉

#499

yes

—

102

C₄×C₁₀₉

#501

yes

—

103

#10577

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

104

C₂×C₁₀₉

#1624

yes

characteristic

105

C₂×C₁₀₉

#1630

yes

characteristic

106

C₂×C₁₀₉

#1652

yes

—

107

C₂×C₁₀₉

#2059

yes

—

108

C₂×C₁₀₉

#2060

yes

characteristic

109

#2571

C₂×C₁₀₉

yes

characteristic

110

#1947

C₂×C₁₀₉

yes

—

111

#1844

C₂×C₁₀₉

yes

characteristic

112

#1296

C₂×C₁₀₉

yes

characteristic

113

#1279

C₂×C₁₀₉

yes

—

114

C₁₀₉

#10577

yes

characteristic, direct product factor, Sylow

115

C₂×#107

—

116

C₂×#106

—

117

C₂×C₄⋊C₈

C₂²×C₁₀₉

yes

—

118

C₂×C₄×C₈

C₂²×C₁₀₉

yes

characteristic

119

#400

—

120

#389

—

121

#384

C₂²×C₁₀₉

yes

characteristic

122

#384

C₂²×C₁₀₉

yes

characteristic

123

#382

—

124

C₄×C₄⋊C₄

C₂²×C₁₀₉

yes

—

125

#340

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

D₄×C₁₀₉

yes

—

133

C₂×C₄⋊C₄

D₄×C₁₀₉

yes

—

134

C₂×C₄⋊C₄

D₄×C₁₀₉

yes

—

135

C₂×C₄²

—

136

C₂×C₄²

C₂³×C₁₀₉

yes

characteristic

137

#107

—

138

#106

—

139

C₄⋊C₈

—

140

C₄⋊C₈

Q₈×C₁₀₉

yes

—

141

C₄×C₈

—

142

C₄×C₈

Q₈×C₁₀₉

yes

—

143

#95

—

144

#95

—

145

#95

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂×D₄×C₁₀₉

yes

characteristic

151

C₂²×C₄

C₂×D₄×C₁₀₉

yes

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄²

—

170

C₄²

—

171

C₄²

C₂×Q₈×C₁₀₉

yes

characteristic

172

C₄²

C₂×Q₈×C₁₀₉

yes

—

173

C₄²

#41×C₁₀₉

yes

characteristic

174

C₂³

#120×C₁₀₉

yes

characteristic

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

C₂²⋊Q₈×C₁₀₉

yes

characteristic

188

C₂×C₄

C₂²⋊Q₈×C₁₀₉

yes

—

189

C₂×C₄

C₄⋊Q₈×C₁₀₉

yes

characteristic

190

C₂×C₄

C₄⋊Q₈×C₁₀₉

yes

—

191

C₂×C₄

#135×C₁₀₉

yes

—

192

C₂×C₄

#135×C₁₀₉

yes

—

193

C₂×C₄

AGL(1,ℤ/8)×C₁₀₉

yes

—

194

C₂×C₄

#137×C₁₀₉

yes

—

195

C₈

—

196

C₈

—

197

C₂²

#395×C₁₀₉

yes

characteristic

198

C₂²

#449×C₁₀₉

yes

—

199

C₂²

#452×C₁₀₉

yes

—

200

C₂²

#453×C₁₀₉

yes

characteristic

201

C₂²

#456×C₁₀₉

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

#478×C₁₀₉

yes

—

209

C₄

#479×C₁₀₉

yes

—

210

C₄

#499×C₁₀₉

yes

—

211

C₄

#501×C₁₀₉

yes

—

212

C₂

C₁₀₉×#1624

yes

characteristic

213

C₂

C₁₀₉×#1630

yes

characteristic

214

C₂

C₁₀₉×#1652

yes

—

215

C₂

C₁₀₉×#2059

yes

—

216

C₂

C₁₀₉×#2060

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₀₉

#120

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₉

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
103. #10577	C₁₀₉	direct (N × H)
114. C₁₀₉	#10577	direct (N × H)

H ≅ G/N

type

103. #10577

C₁₀₉

direct (N × H)

114. C₁₀₉

#10577

direct (N × H)

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

subgroup	index
C₁₀₉×#2571	2
C₁₀₉×#1947	2
C₁₀₉×#1844	2
C₁₀₉×#1296	2
C₁₀₉×#1279	2
#10577	109

subgroup

index

C₁₀₉×#2571

C₁₀₉×#1947

C₁₀₉×#1844

C₁₀₉×#1296

C₁₀₉×#1279

#10577

109

Subgroups — C₁₀₉×#10577 #42128571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#10577 #42128571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10577 #42128571