Subgroups — C₁₀₉×#3402 #28360656

Order 13952.

356 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#556×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	#497×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	#495×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#464×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#463×C₁₀₉	C₂	1	2	yes	maximal
6	#462×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#440×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₄×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
13	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
15	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
17	#126×C₁₀₉	—	2	2	no	—
18	#126×C₁₀₉	—	2	2	no	—
19	#124×C₁₀₉	C₂²	1	2	yes	—
20	#124×C₁₀₉	C₂²	1	2	yes	—
21	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
22	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
23	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
24	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
26	#117×C₁₀₉	—	2	2	no	—
27	#107×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
29	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
30	#103×C₁₀₉	C₂²	1	2	yes	—
31	#103×C₁₀₉	C₂²	1	2	yes	—
32	#102×C₁₀₉	C₂²	1	2	yes	—
33	#97×C₁₀₉	C₂²	1	2	yes	—
34	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
36	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
37	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
38	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₀₉	—	2	2	no	—
40	Q₁₆×C₁₀₉	—	2	2	no	—
41	Q₁₆×C₁₀₉	—	4	1	no	—
42	Q₁₆×C₁₀₉	—	4	2	no	—
43	SD₁₆×C₁₀₉	—	2	4	no	—
44	SD₁₆×C₁₀₉	—	4	2	no	—
45	D₈×C₁₀₉	—	4	1	no	—
46	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
48	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
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	D₄	1	2	yes	—
54	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
55	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
56	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
57	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
58	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
59	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
60	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
61	C₄²×C₁₀₉	—	2	2	no	—
62	C₄²×C₁₀₉	—	2	2	no	—
63	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
64	C₄²×C₁₀₉	C₂³	1	2	yes	—
65	C₂³×C₁₀₉	—	2	2	no	—
66	Q₈×C₁₀₉	—	2	2	no	—
67	Q₈×C₁₀₉	—	2	2	no	—
68	Q₈×C₁₀₉	—	2	2	no	—
69	Q₈×C₁₀₉	—	2	2	no	—
70	Q₈×C₁₀₉	—	4	1	no	—
71	D₄×C₁₀₉	—	2	2	no	—
72	D₄×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	4	no	—
77	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
82	C₈×C₁₀₉	—	2	1	no	—
83	C₈×C₁₀₉	—	2	2	no	—
84	C₈×C₁₀₉	#41	1	2	yes	—
85	C₂²×C₁₀₉	—	4	2	no	—
86	C₂²×C₁₀₉	—	4	2	no	—
87	C₂²×C₁₀₉	—	4	2	no	—
88	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₁₀₉	—	2	2	no	—
90	C₄×C₁₀₉	—	2	1	no	—
91	C₄×C₁₀₉	—	2	1	no	—
92	C₄×C₁₀₉	—	2	2	no	—
93	C₄×C₁₀₉	—	4	2	no	—
94	C₄×C₁₀₉	—	4	2	no	—
95	C₄×C₁₀₉	—	4	2	no	—
96	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
97	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
98	C₂×C₁₀₉	—	8	2	no	—
99	C₂×C₁₀₉	#550	1	1	yes	characteristic
100	C₂×C₁₀₉	#575	1	1	yes	characteristic
101	C₂×C₁₀₉	#578	1	1	yes	characteristic
102	#3402	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₁₀₉	#3402	1	1	yes	characteristic, direct product factor, Sylow
104	#556	C₂×C₁₀₉	1	2	yes	complemented
105	#497	C₂×C₁₀₉	1	2	yes	complemented
106	#495	C₂×C₁₀₉	1	1	yes	characteristic
107	#464	C₂×C₁₀₉	1	2	yes	complemented
108	#463	C₂×C₁₀₉	1	2	yes	—
109	#462	C₂×C₁₀₉	1	2	yes	complemented
110	#446	C₂×C₁₀₉	1	1	yes	characteristic, complemented
111	#440	C₂×C₁₀₉	1	2	yes	complemented
112	C₄×Q₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
113	C₂×Q₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
114	C₂×Q₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
115	C₂×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
116	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
117	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
118	C₂×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
119	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
120	#126	—	2	2	no	—
121	#126	—	2	2	no	—
122	#124	C₂²×C₁₀₉	1	2	yes	—
123	#124	C₂²×C₁₀₉	1	2	yes	—
124	C₂²⋊Q₈	—	2	2	no	—
125	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
126	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
127	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
128	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
129	#117	—	2	2	no	—
130	#107	C₂²×C₁₀₉	1	1	yes	characteristic
131	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
132	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
133	#103	C₂²×C₁₀₉	1	2	yes	—
134	#103	C₂²×C₁₀₉	1	2	yes	—
135	#102	C₂²×C₁₀₉	1	2	yes	—
136	#97	C₂²×C₁₀₉	1	2	yes	—
137	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
138	C₂×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
139	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
140	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
141	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
142	C₂²×C₄	—	2	2	no	—
143	Q₁₆	—	2	2	no	—
144	Q₁₆	—	4	1	no	—
145	Q₁₆	—	4	2	no	—
146	SD₁₆	—	2	4	no	—
147	SD₁₆	—	4	2	no	—
148	D₈	—	4	1	no	—
149	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
150	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
151	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
157	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
158	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	2	2	no	—
166	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
167	C₄²	C₂³×C₁₀₉	1	2	yes	—
168	C₂³	—	2	2	no	—
169	Q₈	—	2	2	no	—
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	Q₈	—	2	2	no	—
173	Q₈	—	4	1	no	—
174	D₄	—	2	2	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
181	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
182	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
183	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
184	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	2	no	—
187	C₈	#41×C₁₀₉	1	2	yes	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	2	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	4	2	no	—
199	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
200	C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
201	C₂	—	8	2	no	—
202	C₂	#550×C₁₀₉	1	1	yes	characteristic
203	C₂	#575×C₁₀₉	1	1	yes	characteristic
204	C₂	#578×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	C₂×D₈×C₁₀₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄×Q₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #556×C₁₀₉	C₂	semidirect
2. #497×C₁₀₉	C₂	semidirect
4. #464×C₁₀₉	C₂	semidirect
6. #462×C₁₀₉	C₂	semidirect
7. #446×C₁₀₉	C₂	semidirect
8. #440×C₁₀₉	C₂	semidirect
9. C₄×Q₁₆×C₁₀₉	C₂	semidirect
102. #3402	C₁₀₉	direct (N × H)
103. C₁₀₉	#3402	direct (N × H)
104. #556	C₂×C₁₀₉	semidirect
105. #497	C₂×C₁₀₉	semidirect
107. #464	C₂×C₁₀₉	semidirect
109. #462	C₂×C₁₀₉	semidirect
110. #446	C₂×C₁₀₉	semidirect
111. #440	C₂×C₁₀₉	semidirect
112. C₄×Q₁₆	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#556×C₁₀₉	2
#497×C₁₀₉	2
#495×C₁₀₉	2
#464×C₁₀₉	2
#463×C₁₀₉	2
#462×C₁₀₉	2
#446×C₁₀₉	2
#440×C₁₀₉	2
C₄×Q₁₆×C₁₀₉	2
#3402	109

Subgroups — C109×#3402 #28360656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3402 #28360656