Subgroups — C₁₀₉×#2974 #28360228

Order 13952.

464 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#436×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	#435×C₁₀₉	C₂	1	4	yes	maximal, complemented
4	C₈×D₄×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#432×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#431×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#407×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	C₂×C₂²⋊C₈×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₂²×D₄×C₁₀₉	C₄	1	1	yes	characteristic
10	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
11	C₂×M₁₆×C₁₀₉	C₂²	1	4	yes	complemented
12	C₂²×C₈×C₁₀₉	C₂²	1	4	yes	complemented
13	C₄×D₄×C₁₀₉	C₄	1	2	yes	—
14	C₄×D₄×C₁₀₉	C₄	1	1	yes	characteristic
15	C₄×D₄×C₁₀₉	C₄	1	1	yes	characteristic
16	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
18	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₁₀₉	C₄	1	2	yes	—
21	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	complemented
23	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	complemented
24	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
26	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂²⋊C₈×C₁₀₉	C₂²	1	4	yes	complemented
28	C₂²⋊C₈×C₁₀₉	C₂²	1	4	yes	complemented
29	#97×C₁₀₉	C₂²	1	2	yes	complemented
30	C₄×C₈×C₁₀₉	C₂²	1	2	yes	complemented
31	C₂⁴×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂×D₄×C₁₀₉	—	2	4	no	—
33	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₀₉	—	4	4	no	—
37	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
45	M₁₆×C₁₀₉	—	4	4	no	—
46	C₂×C₈×C₁₀₉	—	4	4	no	—
47	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
48	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
49	C₂×C₈×C₁₀₉	C₂³	1	4	yes	—
50	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
51	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
53	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
54	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
55	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
56	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
57	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
60	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
61	C₂³×C₁₀₉	—	2	4	no	—
62	C₂³×C₁₀₉	—	2	4	no	—
63	C₂³×C₁₀₉	—	4	2	no	—
64	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
65	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
66	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
67	D₄×C₁₀₉	—	2	4	no	—
68	D₄×C₁₀₉	—	2	2	no	—
69	D₄×C₁₀₉	M₁₆	1	4	yes	complemented
70	C₂×C₄×C₁₀₉	—	2	2	no	—
71	C₂×C₄×C₁₀₉	—	2	2	no	—
72	C₂×C₄×C₁₀₉	—	2	4	no	—
73	C₂×C₄×C₁₀₉	—	2	4	no	—
74	C₂×C₄×C₁₀₉	—	2	2	no	—
75	C₂×C₄×C₁₀₉	—	4	2	no	—
76	C₂×C₄×C₁₀₉	—	4	2	no	—
77	C₂×C₄×C₁₀₉	—	4	2	no	—
78	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
86	C₈×C₁₀₉	—	2	2	no	—
87	C₈×C₁₀₉	—	2	4	no	—
88	C₈×C₁₀₉	—	2	2	no	—
89	C₂²×C₁₀₉	—	2	1	no	—
90	C₂²×C₁₀₉	—	2	2	no	—
91	C₂²×C₁₀₉	—	2	1	no	—
92	C₂²×C₁₀₉	—	2	2	no	—
93	C₂²×C₁₀₉	—	2	2	no	—
94	C₂²×C₁₀₉	—	2	2	no	—
95	C₂²×C₁₀₉	—	2	2	no	—
96	C₂²×C₁₀₉	—	4	8	no	—
97	C₂²×C₁₀₉	C₂×M₁₆	1	4	yes	—
98	C₂²×C₁₀₉	#131	1	2	yes	—
99	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
100	C₄×C₁₀₉	—	2	1	no	—
101	C₄×C₁₀₉	—	2	2	no	—
102	C₄×C₁₀₉	—	2	2	no	—
103	C₄×C₁₀₉	—	2	1	no	—
104	C₄×C₁₀₉	—	2	1	no	—
105	C₄×C₁₀₉	—	2	1	no	—
106	C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
107	C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
108	C₂×C₁₀₉	—	2	2	no	—
109	C₂×C₁₀₉	—	2	4	no	—
110	C₂×C₁₀₉	—	4	2	no	—
111	C₂×C₁₀₉	#518	1	1	yes	characteristic
112	C₂×C₁₀₉	C₂²×M₁₆	1	1	yes	characteristic
113	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
114	#2974	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₁₀₉	#2974	1	1	yes	characteristic, direct product factor, Sylow
116	C₂×C₄×D₄	C₂×C₁₀₉	1	1	yes	characteristic
117	#436	C₂×C₁₀₉	1	2	yes	complemented
118	#435	C₂×C₁₀₉	1	4	yes	complemented
119	C₈×D₄	C₂×C₁₀₉	1	2	yes	complemented
120	#432	C₂×C₁₀₉	1	1	yes	characteristic, complemented
121	#431	C₂×C₁₀₉	1	1	yes	characteristic, complemented
122	#407	C₂×C₁₀₉	1	2	yes	complemented
123	C₂×C₂²⋊C₈	C₂×C₁₀₉	1	2	yes	complemented
124	C₂²×D₄	C₄×C₁₀₉	1	1	yes	characteristic
125	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	—
126	C₂×M₁₆	C₂²×C₁₀₉	1	4	yes	complemented
127	C₂²×C₈	C₂²×C₁₀₉	1	4	yes	complemented
128	C₄×D₄	C₄×C₁₀₉	1	2	yes	—
129	C₄×D₄	C₄×C₁₀₉	1	1	yes	characteristic
130	C₄×D₄	C₄×C₁₀₉	1	1	yes	characteristic
131	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
132	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
133	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
134	C₂×C₄⋊C₄	C₄×C₁₀₉	1	1	yes	characteristic
135	C₂×C₂²⋊C₄	C₄×C₁₀₉	1	2	yes	—
136	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
137	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	complemented
138	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	complemented
139	C₂²⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
140	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
141	C₂²⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
142	C₂²⋊C₈	C₂²×C₁₀₉	1	4	yes	complemented
143	C₂²⋊C₈	C₂²×C₁₀₉	1	4	yes	complemented
144	#97	C₂²×C₁₀₉	1	2	yes	complemented
145	C₄×C₈	C₂²×C₁₀₉	1	2	yes	complemented
146	C₂⁴	C₂×C₄×C₁₀₉	1	2	yes	—
147	C₂×D₄	—	2	4	no	—
148	C₂×D₄	C₂×C₄×C₁₀₉	1	2	yes	—
149	C₂×D₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
150	C₂×D₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
151	C₂²×C₄	—	4	4	no	—
152	C₂²×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₄×C₁₀₉	1	2	yes	—
155	C₂²×C₄	C₂×C₄×C₁₀₉	1	2	yes	—
156	C₂²×C₄	C₂×C₄×C₁₀₉	1	2	yes	—
157	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
159	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
160	M₁₆	—	4	4	no	—
161	C₂×C₈	—	4	4	no	—
162	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
163	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
164	C₂×C₈	C₂³×C₁₀₉	1	4	yes	—
165	C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
166	C₄⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
167	C₄⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
168	C₂²⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
169	C₂²⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
170	C₂²⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
171	C₂²⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
172	C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
173	C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
174	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
175	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
176	C₂³	—	2	4	no	—
177	C₂³	—	2	4	no	—
178	C₂³	—	4	2	no	—
179	C₂³	C₂²×C₄×C₁₀₉	1	2	yes	—
180	C₂³	C₂²×C₄×C₁₀₉	1	2	yes	—
181	C₂³	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
182	D₄	—	2	4	no	—
183	D₄	—	2	2	no	—
184	D₄	M₁₆×C₁₀₉	1	4	yes	complemented
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
194	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
196	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
197	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
198	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
199	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
200	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
201	C₈	—	2	2	no	—
202	C₈	—	2	4	no	—
203	C₈	—	2	2	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	4	8	no	—
212	C₂²	C₂×M₁₆×C₁₀₉	1	4	yes	—
213	C₂²	#131×C₁₀₉	1	2	yes	—
214	C₂²	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
222	C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	2	4	no	—
225	C₂	—	4	2	no	—
226	C₂	#518×C₁₀₉	1	1	yes	characteristic
227	C₂	C₂²×M₁₆×C₁₀₉	1	1	yes	characteristic
228	C₂	C₂×#131×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
socle	C₂²×C₁₀₉	C₂³×C₄
prime radical	C₂²×D₄×C₁₀₉	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #436×C₁₀₉	C₂	semidirect
3. #435×C₁₀₉	C₂	semidirect
4. C₈×D₄×C₁₀₉	C₂	semidirect
5. #432×C₁₀₉	C₂	semidirect
6. #431×C₁₀₉	C₂	semidirect
7. #407×C₁₀₉	C₂	semidirect
8. C₂×C₂²⋊C₈×C₁₀₉	C₂	semidirect
11. C₂×M₁₆×C₁₀₉	C₂²	semidirect
12. C₂²×C₈×C₁₀₉	C₂²	semidirect
22. C₄⋊C₈×C₁₀₉	C₂²	semidirect
23. C₄⋊C₈×C₁₀₉	C₂²	semidirect
27. C₂²⋊C₈×C₁₀₉	C₂²	semidirect
28. C₂²⋊C₈×C₁₀₉	C₂²	semidirect
29. #97×C₁₀₉	C₂²	semidirect
30. C₄×C₈×C₁₀₉	C₂²	semidirect
69. D₄×C₁₀₉	M₁₆	semidirect
114. #2974	C₁₀₉	direct (N × H)
115. C₁₀₉	#2974	direct (N × H)
117. #436	C₂×C₁₀₉	semidirect
118. #435	C₂×C₁₀₉	semidirect
119. C₈×D₄	C₂×C₁₀₉	semidirect
120. #432	C₂×C₁₀₉	semidirect
121. #431	C₂×C₁₀₉	semidirect
122. #407	C₂×C₁₀₉	semidirect
123. C₂×C₂²⋊C₈	C₂×C₁₀₉	semidirect
126. C₂×M₁₆	C₂²×C₁₀₉	semidirect
127. C₂²×C₈	C₂²×C₁₀₉	semidirect
137. C₄⋊C₈	C₂²×C₁₀₉	semidirect
138. C₄⋊C₈	C₂²×C₁₀₉	semidirect
142. C₂²⋊C₈	C₂²×C₁₀₉	semidirect
143. C₂²⋊C₈	C₂²×C₁₀₉	semidirect
144. #97	C₂²×C₁₀₉	semidirect
145. C₄×C₈	C₂²×C₁₀₉	semidirect
184. D₄	M₁₆×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₁₀₉	2
#436×C₁₀₉	2
#435×C₁₀₉	2
C₈×D₄×C₁₀₉	2
#432×C₁₀₉	2
#431×C₁₀₉	2
#407×C₁₀₉	2
C₂×C₂²⋊C₈×C₁₀₉	2
#2974	109

Subgroups — C109×#2974 #28360228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2974 #28360228