Subgroups — C₂²×#392×C₁₀₉ #42152525

Order 27904.

13646 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₁₀₉	C₂	1	3	yes	maximal, complemented
2	C₂³×C₂²⋊C₄×C₁₀₉	C₂	1	3	yes	maximal, complemented
3	C₂×#392×C₁₀₉	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#95×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁶×C₁₀₉	C₂²	1	3	yes	—
6	C₂³×D₄×C₁₀₉	—	2	48	no	—
7	C₂³×D₄×C₁₀₉	—	2	24	no	—
8	C₂³×D₄×C₁₀₉	C₂²	1	36	yes	complemented
9	C₂⁴×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₁₀₉	C₂²	1	3	yes	—
11	C₂²×C₂²⋊C₄×C₁₀₉	—	2	24	no	—
12	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	36	yes	complemented
13	#392×C₁₀₉	C₂²	1	64	yes	direct product factor
14	C₂×#95×C₁₀₉	C₂²	1	12	yes	complemented
15	C₂⁵×C₁₀₉	—	2	12	no	—
16	C₂⁵×C₁₀₉	—	2	24	no	—
17	C₂⁵×C₁₀₉	—	2	24	no	—
18	C₂⁵×C₁₀₉	—	2	12	no	—
19	C₂⁵×C₁₀₉	D₄	1	24	yes	complemented
20	C₂⁵×C₁₀₉	C₂³	1	18	yes	—
21	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₂²×D₄×C₁₀₉	—	2	384	no	—
23	C₂²×D₄×C₁₀₉	—	2	288	no	—
24	C₂²×D₄×C₁₀₉	—	2	144	no	—
25	C₂²×D₄×C₁₀₉	C₂³	1	48	yes	complemented
26	C₂³×C₄×C₁₀₉	—	2	24	no	—
27	C₂³×C₄×C₁₀₉	—	2	12	no	—
28	C₂³×C₄×C₁₀₉	D₄	1	24	yes	complemented
29	C₂³×C₄×C₁₀₉	C₂³	1	6	yes	—
30	C₂³×C₄×C₁₀₉	C₂³	1	18	yes	—
31	C₂×C₂²⋊C₄×C₁₀₉	—	2	144	no	—
32	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	48	yes	complemented
33	#95×C₁₀₉	C₂³	1	16	yes	complemented
34	C₂⁴×C₁₀₉	—	2	192	no	—
35	C₂⁴×C₁₀₉	—	2	96	no	—
36	C₂⁴×C₁₀₉	—	2	36	no	—
37	C₂⁴×C₁₀₉	—	2	72	no	—
38	C₂⁴×C₁₀₉	—	2	72	no	—
39	C₂⁴×C₁₀₉	—	2	36	no	—
40	C₂⁴×C₁₀₉	—	4	48	no	—
41	C₂⁴×C₁₀₉	—	4	96	no	—
42	C₂⁴×C₁₀₉	—	4	48	no	—
43	C₂⁴×C₁₀₉	C₂×D₄	1	12	yes	—
44	C₂⁴×C₁₀₉	C₂×D₄	1	12	yes	—
45	C₂⁴×C₁₀₉	C₂×D₄	1	72	yes	complemented
46	C₂⁴×C₁₀₉	#41	1	4	yes	—
47	C₂⁴×C₁₀₉	C₂⁴	1	12	yes	—
48	C₂⁴×C₁₀₉	C₂⁴	1	3	yes	—
49	C₂×D₄×C₁₀₉	—	2	1152	no	—
50	C₂×D₄×C₁₀₉	—	2	192	no	—
51	C₂×D₄×C₁₀₉	—	2	96	no	—
52	C₂²×C₄×C₁₀₉	—	2	96	no	—
53	C₂²×C₄×C₁₀₉	—	2	72	no	—
54	C₂²×C₄×C₁₀₉	—	2	36	no	—
55	C₂²×C₄×C₁₀₉	—	4	16	no	—
56	C₂²×C₄×C₁₀₉	C₂×D₄	1	72	yes	complemented
57	C₂²×C₄×C₁₀₉	C₂⁴	1	4	yes	—
58	C₂²×C₄×C₁₀₉	C₂⁴	1	12	yes	—
59	C₂²⋊C₄×C₁₀₉	—	2	96	no	—
60	C₂³×C₁₀₉	—	2	288	no	—
61	C₂³×C₁₀₉	—	2	144	no	—
62	C₂³×C₁₀₉	—	2	12	no	—
63	C₂³×C₁₀₉	—	2	24	no	—
64	C₂³×C₁₀₉	—	2	24	no	—
65	C₂³×C₁₀₉	—	2	12	no	—
66	C₂³×C₁₀₉	—	4	72	no	—
67	C₂³×C₁₀₉	—	4	144	no	—
68	C₂³×C₁₀₉	—	4	72	no	—
69	C₂³×C₁₀₉	—	4	384	no	—
70	C₂³×C₁₀₉	#120	1	48	yes	—
71	C₂³×C₁₀₉	#121	1	48	yes	—
72	C₂³×C₁₀₉	#127	1	16	yes	—
73	C₂³×C₁₀₉	C₂²×D₄	1	18	yes	—
74	C₂³×C₁₀₉	C₂²×D₄	1	18	yes	—
75	C₂³×C₁₀₉	C₂²×D₄	1	24	yes	complemented
76	C₂³×C₁₀₉	C₂×#41	1	6	yes	—
77	C₂³×C₁₀₉	C₂⁵	1	1	yes	characteristic
78	D₄×C₁₀₉	—	2	384	no	—
79	C₂×C₄×C₁₀₉	—	2	144	no	—
80	C₂×C₄×C₁₀₉	—	2	24	no	—
81	C₂×C₄×C₁₀₉	—	2	12	no	—
82	C₂×C₄×C₁₀₉	—	4	24	no	—
83	C₂×C₄×C₁₀₉	C₂²×D₄	1	24	yes	complemented
84	C₂²×C₁₀₉	—	2	48	no	—
85	C₂²×C₁₀₉	—	2	24	no	—
86	C₂²×C₁₀₉	—	4	288	no	—
87	C₂²×C₁₀₉	—	4	12	no	—
88	C₂²×C₁₀₉	—	4	24	no	—
89	C₂²×C₁₀₉	—	4	12	no	—
90	C₂²×C₁₀₉	#392	1	64	yes	direct product factor
91	C₂²×C₁₀₉	C₂×#120	1	36	yes	—
92	C₂²×C₁₀₉	C₂×#121	1	36	yes	—
93	C₂²×C₁₀₉	C₂×#127	1	12	yes	—
94	C₂²×C₁₀₉	C₂³×D₄	1	3	yes	—
95	C₂²×C₁₀₉	C₂³×D₄	1	3	yes	—
96	C₂²×C₁₀₉	C₂²×#41	1	1	yes	characteristic
97	C₄×C₁₀₉	—	2	24	no	—
98	C₄×C₁₀₉	—	4	4	no	—
99	C₂²×#392	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂×C₁₀₉	—	4	24	no	—
101	C₂×C₁₀₉	C₂×#392	1	24	yes	direct product factor
102	C₂×C₁₀₉	C₂²×#120	1	3	yes	—
103	C₂×C₁₀₉	C₂²×#121	1	3	yes	—
104	C₂×C₁₀₉	C₂²×#127	1	1	yes	characteristic
105	C₂⁴×D₄	C₂×C₁₀₉	1	3	yes	complemented
106	C₂³×C₂²⋊C₄	C₂×C₁₀₉	1	3	yes	complemented
107	C₂×#392	C₂×C₁₀₉	1	24	yes	direct product factor
108	C₂²×#95	C₂×C₁₀₉	1	1	yes	characteristic, complemented
109	C₁₀₉	C₂²×#392	1	1	yes	characteristic, direct product factor, Sylow
110	C₂⁶	C₂²×C₁₀₉	1	3	yes	—
111	C₂³×D₄	—	2	48	no	—
112	C₂³×D₄	—	2	24	no	—
113	C₂³×D₄	C₂²×C₁₀₉	1	36	yes	complemented
114	C₂⁴×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
115	C₂⁴×C₄	C₂²×C₁₀₉	1	3	yes	—
116	C₂²×C₂²⋊C₄	—	2	24	no	—
117	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	36	yes	complemented
118	#392	C₂²×C₁₀₉	1	64	yes	direct product factor
119	C₂×#95	C₂²×C₁₀₉	1	12	yes	complemented
120	C₂⁵	—	2	12	no	—
121	C₂⁵	—	2	24	no	—
122	C₂⁵	—	2	24	no	—
123	C₂⁵	—	2	12	no	—
124	C₂⁵	D₄×C₁₀₉	1	24	yes	complemented
125	C₂⁵	C₂³×C₁₀₉	1	18	yes	—
126	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
127	C₂²×D₄	—	2	384	no	—
128	C₂²×D₄	—	2	288	no	—
129	C₂²×D₄	—	2	144	no	—
130	C₂²×D₄	C₂³×C₁₀₉	1	48	yes	complemented
131	C₂³×C₄	—	2	24	no	—
132	C₂³×C₄	—	2	12	no	—
133	C₂³×C₄	D₄×C₁₀₉	1	24	yes	complemented
134	C₂³×C₄	C₂³×C₁₀₉	1	6	yes	—
135	C₂³×C₄	C₂³×C₁₀₉	1	18	yes	—
136	C₂×C₂²⋊C₄	—	2	144	no	—
137	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	48	yes	complemented
138	#95	C₂³×C₁₀₉	1	16	yes	complemented
139	C₂⁴	—	2	192	no	—
140	C₂⁴	—	2	96	no	—
141	C₂⁴	—	2	36	no	—
142	C₂⁴	—	2	72	no	—
143	C₂⁴	—	2	72	no	—
144	C₂⁴	—	2	36	no	—
145	C₂⁴	—	4	48	no	—
146	C₂⁴	—	4	96	no	—
147	C₂⁴	—	4	48	no	—
148	C₂⁴	C₂×D₄×C₁₀₉	1	12	yes	—
149	C₂⁴	C₂×D₄×C₁₀₉	1	12	yes	—
150	C₂⁴	C₂×D₄×C₁₀₉	1	72	yes	complemented
151	C₂⁴	#41×C₁₀₉	1	4	yes	—
152	C₂⁴	C₂⁴×C₁₀₉	1	12	yes	—
153	C₂⁴	C₂⁴×C₁₀₉	1	3	yes	—
154	C₂×D₄	—	2	1152	no	—
155	C₂×D₄	—	2	192	no	—
156	C₂×D₄	—	2	96	no	—
157	C₂²×C₄	—	2	96	no	—
158	C₂²×C₄	—	2	72	no	—
159	C₂²×C₄	—	2	36	no	—
160	C₂²×C₄	—	4	16	no	—
161	C₂²×C₄	C₂×D₄×C₁₀₉	1	72	yes	complemented
162	C₂²×C₄	C₂⁴×C₁₀₉	1	4	yes	—
163	C₂²×C₄	C₂⁴×C₁₀₉	1	12	yes	—
164	C₂²⋊C₄	—	2	96	no	—
165	C₂³	—	2	288	no	—
166	C₂³	—	2	144	no	—
167	C₂³	—	2	12	no	—
168	C₂³	—	2	24	no	—
169	C₂³	—	2	24	no	—
170	C₂³	—	2	12	no	—
171	C₂³	—	4	72	no	—
172	C₂³	—	4	144	no	—
173	C₂³	—	4	72	no	—
174	C₂³	—	4	384	no	—
175	C₂³	#120×C₁₀₉	1	48	yes	—
176	C₂³	#121×C₁₀₉	1	48	yes	—
177	C₂³	#127×C₁₀₉	1	16	yes	—
178	C₂³	C₂²×D₄×C₁₀₉	1	18	yes	—
179	C₂³	C₂²×D₄×C₁₀₉	1	18	yes	—
180	C₂³	C₂²×D₄×C₁₀₉	1	24	yes	complemented
181	C₂³	C₂×#41×C₁₀₉	1	6	yes	—
182	C₂³	C₂⁵×C₁₀₉	1	1	yes	characteristic
183	D₄	—	2	384	no	—
184	C₂×C₄	—	2	144	no	—
185	C₂×C₄	—	2	24	no	—
186	C₂×C₄	—	2	12	no	—
187	C₂×C₄	—	4	24	no	—
188	C₂×C₄	C₂²×D₄×C₁₀₉	1	24	yes	complemented
189	C₂²	—	2	48	no	—
190	C₂²	—	2	24	no	—
191	C₂²	—	4	288	no	—
192	C₂²	—	4	12	no	—
193	C₂²	—	4	24	no	—
194	C₂²	—	4	12	no	—
195	C₂²	#392×C₁₀₉	1	64	yes	direct product factor
196	C₂²	C₂×#120×C₁₀₉	1	36	yes	—
197	C₂²	C₂×#121×C₁₀₉	1	36	yes	—
198	C₂²	C₂×#127×C₁₀₉	1	12	yes	—
199	C₂²	C₂³×D₄×C₁₀₉	1	3	yes	—
200	C₂²	C₂³×D₄×C₁₀₉	1	3	yes	—
201	C₂²	C₂²×#41×C₁₀₉	1	1	yes	characteristic
202	C₄	—	2	24	no	—
203	C₄	—	4	4	no	—
204	C₂	—	4	24	no	—
205	C₂	C₂×#392×C₁₀₉	1	24	yes	direct product factor
206	C₂	C₂²×#120×C₁₀₉	1	3	yes	—
207	C₂	C₂²×#121×C₁₀₉	1	3	yes	—
208	C₂	C₂²×#127×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₁₀₉	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₀₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₁₀₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂⁴×D₄×C₁₀₉	C₂	semidirect
2. C₂³×C₂²⋊C₄×C₁₀₉	C₂	semidirect
3. C₂×#392×C₁₀₉	C₂	direct (N × H)
4. C₂²×#95×C₁₀₉	C₂	semidirect
8. C₂³×D₄×C₁₀₉	C₂²	semidirect
12. C₂²×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
13. #392×C₁₀₉	C₂²	direct (N × H)
14. C₂×#95×C₁₀₉	C₂²	semidirect
19. C₂⁵×C₁₀₉	D₄	semidirect
25. C₂²×D₄×C₁₀₉	C₂³	semidirect
28. C₂³×C₄×C₁₀₉	D₄	semidirect
32. C₂×C₂²⋊C₄×C₁₀₉	C₂³	semidirect
33. #95×C₁₀₉	C₂³	semidirect
45. C₂⁴×C₁₀₉	C₂×D₄	semidirect
56. C₂²×C₄×C₁₀₉	C₂×D₄	semidirect
75. C₂³×C₁₀₉	C₂²×D₄	semidirect
83. C₂×C₄×C₁₀₉	C₂²×D₄	semidirect
90. C₂²×C₁₀₉	#392	direct (N × H)
99. C₂²×#392	C₁₀₉	direct (N × H)
101. C₂×C₁₀₉	C₂×#392	direct (N × H)
105. C₂⁴×D₄	C₂×C₁₀₉	semidirect
106. C₂³×C₂²⋊C₄	C₂×C₁₀₉	semidirect
107. C₂×#392	C₂×C₁₀₉	direct (N × H)
108. C₂²×#95	C₂×C₁₀₉	semidirect
109. C₁₀₉	C₂²×#392	direct (N × H)
113. C₂³×D₄	C₂²×C₁₀₉	semidirect
117. C₂²×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
118. #392	C₂²×C₁₀₉	direct (N × H)
119. C₂×#95	C₂²×C₁₀₉	semidirect
124. C₂⁵	D₄×C₁₀₉	semidirect
130. C₂²×D₄	C₂³×C₁₀₉	semidirect
133. C₂³×C₄	D₄×C₁₀₉	semidirect
137. C₂×C₂²⋊C₄	C₂³×C₁₀₉	semidirect
138. #95	C₂³×C₁₀₉	semidirect
150. C₂⁴	C₂×D₄×C₁₀₉	semidirect
161. C₂²×C₄	C₂×D₄×C₁₀₉	semidirect
180. C₂³	C₂²×D₄×C₁₀₉	semidirect
188. C₂×C₄	C₂²×D₄×C₁₀₉	semidirect
195. C₂²	#392×C₁₀₉	direct (N × H)
205. C₂	C₂×#392×C₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₁₀₉	2
C₂³×C₂²⋊C₄×C₁₀₉	2
C₂×#392×C₁₀₉	2
C₂²×#95×C₁₀₉	2
C₂²×#392	109

Subgroups — C22×#392×C109 #42152525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#392×C₁₀₉ #42152525