Subgroups — C₁₀₉×#2968 #28360222

Order 13952.

372 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#529×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#446×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	#435×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#435×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	C₈×D₄×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#433×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#431×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#405×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₂×#97×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
13	#126×C₁₀₉	C₄	1	2	yes	—
14	#125×C₁₀₉	C₄	1	1	yes	characteristic
15	#124×C₁₀₉	C₄	1	1	yes	characteristic
16	#123×C₁₀₉	C₄	1	2	yes	—
17	C₂²⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
18	#121×C₁₀₉	C₄	1	1	yes	characteristic
19	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
21	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
22	#117×C₁₀₉	C₂²	1	2	yes	—
23	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
25	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
26	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
27	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
30	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
31	#97×C₁₀₉	C₂²	1	1	yes	characteristic
32	#97×C₁₀₉	C₂²	1	1	yes	characteristic
33	#97×C₁₀₉	C₂²	1	2	yes	—
34	#97×C₁₀₉	C₂²	1	2	yes	—
35	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
36	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
37	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
44	M₁₆×C₁₀₉	—	2	4	no	—
45	C₂×C₈×C₁₀₉	—	2	4	no	—
46	C₂×C₈×C₁₀₉	—	2	4	no	—
47	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
48	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
49	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
50	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
51	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
52	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
53	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
55	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
57	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
58	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
59	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
60	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
61	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
62	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
63	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₄²×C₁₀₉	C₂³	1	2	yes	—
66	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
68	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
69	Q₈×C₁₀₉	—	4	1	no	—
70	D₄×C₁₀₉	—	2	4	no	—
71	D₄×C₁₀₉	—	4	1	no	—
72	C₂×C₄×C₁₀₉	—	2	2	no	—
73	C₂×C₄×C₁₀₉	—	2	2	no	—
74	C₂×C₄×C₁₀₉	—	2	2	no	—
75	C₂×C₄×C₁₀₉	—	2	2	no	—
76	C₂×C₄×C₁₀₉	—	4	1	no	—
77	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
85	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
86	C₈×C₁₀₉	—	2	2	no	—
87	C₈×C₁₀₉	—	2	2	no	—
88	C₈×C₁₀₉	—	2	2	no	—
89	C₈×C₁₀₉	#41	1	4	yes	—
90	C₂²×C₁₀₉	—	2	1	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	#131	1	2	yes	—
96	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
97	C₄×C₁₀₉	—	2	1	no	—
98	C₄×C₁₀₉	—	2	2	no	—
99	C₄×C₁₀₉	—	2	1	no	—
100	C₄×C₁₀₉	—	2	1	no	—
101	C₄×C₁₀₉	—	2	2	no	—
102	C₄×C₁₀₉	—	2	1	no	—
103	C₄×C₁₀₉	—	2	1	no	—
104	C₄×C₁₀₉	—	2	2	no	—
105	C₄×C₁₀₉	C₂×#41	1	2	yes	—
106	C₂×C₁₀₉	—	2	2	no	—
107	C₂×C₁₀₉	—	4	2	no	—
108	C₂×C₁₀₉	C₄×#41	1	1	yes	characteristic
109	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
110	C₂×C₁₀₉	#568	1	1	yes	characteristic
111	#2968	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₁₀₉	#2968	1	1	yes	characteristic, direct product factor, Sylow
113	#529	C₂×C₁₀₉	1	1	yes	characteristic
114	#446	C₂×C₁₀₉	1	2	yes	complemented
115	#435	C₂×C₁₀₉	1	2	yes	complemented
116	#435	C₂×C₁₀₉	1	2	yes	complemented
117	C₈×D₄	C₂×C₁₀₉	1	2	yes	complemented
118	#433	C₂×C₁₀₉	1	2	yes	complemented
119	#431	C₂×C₁₀₉	1	1	yes	characteristic, complemented
120	#405	C₂×C₁₀₉	1	2	yes	complemented
121	C₂×#97	C₂×C₁₀₉	1	1	yes	characteristic, complemented
122	C₂×M₁₆	C₂²×C₁₀₉	1	2	yes	—
123	C₂²×C₈	C₂²×C₁₀₉	1	2	yes	—
124	C₂²×C₈	C₂²×C₁₀₉	1	2	yes	—
125	#126	C₄×C₁₀₉	1	2	yes	—
126	#125	C₄×C₁₀₉	1	1	yes	characteristic
127	#124	C₄×C₁₀₉	1	1	yes	characteristic
128	#123	C₄×C₁₀₉	1	2	yes	—
129	C₂²⋊Q₈	C₄×C₁₀₉	1	1	yes	characteristic
130	#121	C₄×C₁₀₉	1	1	yes	characteristic
131	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
132	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
133	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
134	#117	C₂²×C₁₀₉	1	2	yes	—
135	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
136	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
137	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
138	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
139	C₂²⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
140	C₂²⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
141	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
142	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
143	#97	C₂²×C₁₀₉	1	1	yes	characteristic
144	#97	C₂²×C₁₀₉	1	1	yes	characteristic
145	#97	C₂²×C₁₀₉	1	2	yes	—
146	#97	C₂²×C₁₀₉	1	2	yes	—
147	C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
148	C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
149	C₂×Q₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
150	C₂×D₄	C₂×C₄×C₁₀₉	1	2	yes	—
151	C₂×D₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
154	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
155	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
156	M₁₆	—	2	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
160	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
161	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
162	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
163	C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
164	C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
165	C₄⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
166	C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
167	C₄⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
168	C₄⋊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	C₂²⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
173	C₂²⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
174	C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
175	C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
176	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
177	C₄²	C₂³×C₁₀₉	1	2	yes	—
178	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
179	C₂³	C₂²×C₄×C₁₀₉	1	2	yes	—
180	C₂³	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
181	Q₈	—	4	1	no	—
182	D₄	—	2	4	no	—
183	D₄	—	4	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
190	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
191	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
192	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
193	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
194	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
195	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
196	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
197	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	2	2	no	—
200	C₈	—	2	2	no	—
201	C₈	#41×C₁₀₉	1	4	yes	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	#131×C₁₀₉	1	2	yes	—
208	C₂²	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	2	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	C₂×#41×C₁₀₉	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	4	2	no	—
220	C₂	C₄×#41×C₁₀₉	1	1	yes	characteristic
221	C₂	C₂×#131×C₁₀₉	1	1	yes	characteristic
222	C₂	#568×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	#121×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. #446×C₁₀₉	C₂	semidirect
3. #435×C₁₀₉	C₂	semidirect
4. #435×C₁₀₉	C₂	semidirect
5. C₈×D₄×C₁₀₉	C₂	semidirect
6. #433×C₁₀₉	C₂	semidirect
7. #431×C₁₀₉	C₂	semidirect
8. #405×C₁₀₉	C₂	semidirect
9. C₂×#97×C₁₀₉	C₂	semidirect
111. #2968	C₁₀₉	direct (N × H)
112. C₁₀₉	#2968	direct (N × H)
114. #446	C₂×C₁₀₉	semidirect
115. #435	C₂×C₁₀₉	semidirect
116. #435	C₂×C₁₀₉	semidirect
117. C₈×D₄	C₂×C₁₀₉	semidirect
118. #433	C₂×C₁₀₉	semidirect
119. #431	C₂×C₁₀₉	semidirect
120. #405	C₂×C₁₀₉	semidirect
121. C₂×#97	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#529×C₁₀₉	2
#446×C₁₀₉	2
#435×C₁₀₉	2
C₈×D₄×C₁₀₉	2
#433×C₁₀₉	2
#431×C₁₀₉	2
#405×C₁₀₉	2
C₂×#97×C₁₀₉	2
#2968	109

Subgroups — C109×#2968 #28360222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2968 #28360222