Subgroups — C₁₀₉×#3205 #28360459

Order 13952.

532 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#532×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#127×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#493×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#486×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#459×C₁₀₉	C₂	1	4	yes	maximal, complemented
6	#447×C₁₀₉	C₂	1	4	yes	maximal, complemented
7	#431×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₁₀₉	—	2	1	no	—
9	C₂²×D₄×C₁₀₉	—	2	1	no	—
10	C₂²×D₄×C₁₀₉	—	2	1	no	—
11	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	—
12	C₂×D₈×C₁₀₉	C₂²	1	8	yes	complemented
13	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
14	#127×C₁₀₉	—	2	1	no	—
15	#127×C₁₀₉	—	2	1	no	—
16	#127×C₁₀₉	C₂²	1	2	yes	—
17	#127×C₁₀₉	C₂²	1	2	yes	—
18	#127×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
19	#124×C₁₀₉	—	2	1	no	—
20	#121×C₁₀₉	—	2	1	no	—
21	#121×C₁₀₉	C₂²	1	2	yes	—
22	C₄×D₄×C₁₀₉	—	2	1	no	—
23	C₄×D₄×C₁₀₉	C₂²	1	2	yes	complemented
24	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	complemented
26	#102×C₁₀₉	C₂²	1	8	yes	complemented
27	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
28	C₄×C₈×C₁₀₉	C₂²	1	2	yes	complemented
29	C₂⁴×C₁₀₉	—	2	2	no	—
30	#41×C₁₀₉	—	4	4	no	—
31	C₂×Q₈×C₁₀₉	—	2	1	no	—
32	C₂×D₄×C₁₀₉	—	2	2	no	—
33	C₂×D₄×C₁₀₉	—	2	1	no	—
34	C₂×D₄×C₁₀₉	—	2	1	no	—
35	C₂×D₄×C₁₀₉	—	2	2	no	—
36	C₂×D₄×C₁₀₉	—	2	2	no	—
37	C₂×D₄×C₁₀₉	—	2	2	no	—
38	C₂×D₄×C₁₀₉	—	2	2	no	—
39	C₂×D₄×C₁₀₉	—	2	2	no	—
40	C₂×D₄×C₁₀₉	—	4	4	no	—
41	C₂×D₄×C₁₀₉	—	4	4	no	—
42	C₂×D₄×C₁₀₉	—	4	4	no	—
43	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
44	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₀₉	—	2	2	no	—
46	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
49	D₈×C₁₀₉	—	4	8	no	—
50	C₂×C₈×C₁₀₉	C₂³	1	4	yes	—
51	C₄⋊C₄×C₁₀₉	—	2	1	no	—
52	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
53	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
54	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
55	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
56	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
57	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
58	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
59	C₂³×C₁₀₉	—	2	2	no	—
60	C₂³×C₁₀₉	—	2	4	no	—
61	C₂³×C₁₀₉	—	4	4	no	—
62	C₂³×C₁₀₉	—	4	4	no	—
63	C₂³×C₁₀₉	—	4	4	no	—
64	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
65	Q₈×C₁₀₉	—	4	2	no	—
66	D₄×C₁₀₉	—	2	4	no	—
67	D₄×C₁₀₉	—	2	8	no	—
68	D₄×C₁₀₉	—	4	4	no	—
69	D₄×C₁₀₉	—	4	4	no	—
70	D₄×C₁₀₉	—	4	4	no	—
71	D₄×C₁₀₉	—	4	4	no	—
72	D₄×C₁₀₉	—	4	4	no	—
73	D₄×C₁₀₉	—	4	2	no	—
74	D₄×C₁₀₉	—	4	2	no	—
75	D₄×C₁₀₉	—	4	4	no	—
76	C₂×C₄×C₁₀₉	—	2	2	no	—
77	C₂×C₄×C₁₀₉	—	2	2	no	—
78	C₂×C₄×C₁₀₉	—	2	2	no	—
79	C₂×C₄×C₁₀₉	—	4	4	no	—
80	C₂×C₄×C₁₀₉	D₈	1	4	yes	complemented
81	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
87	C₈×C₁₀₉	—	2	4	no	—
88	C₂²×C₁₀₉	—	2	1	no	—
89	C₂²×C₁₀₉	—	2	1	no	—
90	C₂²×C₁₀₉	—	4	4	no	—
91	C₂²×C₁₀₉	—	4	2	no	—
92	C₂²×C₁₀₉	—	4	2	no	—
93	C₂²×C₁₀₉	—	4	4	no	—
94	C₂²×C₁₀₉	—	4	4	no	—
95	C₂²×C₁₀₉	—	4	2	no	—
96	C₂²×C₁₀₉	—	8	8	no	—
97	C₂²×C₁₀₉	C₂×D₈	1	2	yes	—
98	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
99	C₄×C₁₀₉	—	2	1	no	—
100	C₄×C₁₀₉	—	2	1	no	—
101	C₄×C₁₀₉	—	4	2	no	—
102	C₄×C₁₀₉	C₂×D₈	1	2	yes	—
103	C₄×C₁₀₉	C₂×D₈	1	2	yes	—
104	C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
105	C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
106	C₂×C₁₀₉	—	2	2	no	—
107	C₂×C₁₀₉	—	8	2	no	—
108	C₂×C₁₀₉	—	8	4	no	—
109	C₂×C₁₀₉	#535	1	1	yes	characteristic
110	C₂×C₁₀₉	C₂²×D₈	1	1	yes	characteristic
111	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
112	#3205	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₁₀₉	#3205	1	1	yes	characteristic, direct product factor, Sylow
114	#532	C₂×C₁₀₉	1	1	yes	characteristic, complemented
115	C₂×#127	C₂×C₁₀₉	1	1	yes	characteristic, complemented
116	#493	C₂×C₁₀₉	1	2	yes	complemented
117	#486	C₂×C₁₀₉	1	2	yes	complemented
118	#459	C₂×C₁₀₉	1	4	yes	complemented
119	#447	C₂×C₁₀₉	1	4	yes	complemented
120	#431	C₂×C₁₀₉	1	1	yes	characteristic, complemented
121	C₂×#41	—	2	1	no	—
122	C₂²×D₄	—	2	1	no	—
123	C₂²×D₄	—	2	1	no	—
124	C₂²×D₄	C₂²×C₁₀₉	1	2	yes	—
125	C₂×D₈	C₂²×C₁₀₉	1	8	yes	complemented
126	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
127	#127	—	2	1	no	—
128	#127	—	2	1	no	—
129	#127	C₂²×C₁₀₉	1	2	yes	—
130	#127	C₂²×C₁₀₉	1	2	yes	—
131	#127	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
132	#124	—	2	1	no	—
133	#121	—	2	1	no	—
134	#121	C₂²×C₁₀₉	1	2	yes	—
135	C₄×D₄	—	2	1	no	—
136	C₄×D₄	C₂²×C₁₀₉	1	2	yes	complemented
137	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
138	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	complemented
139	#102	C₂²×C₁₀₉	1	8	yes	complemented
140	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
141	C₄×C₈	C₂²×C₁₀₉	1	2	yes	complemented
142	C₂⁴	—	2	2	no	—
143	#41	—	4	4	no	—
144	C₂×Q₈	—	2	1	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	2	1	no	—
147	C₂×D₄	—	2	1	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	—	2	2	no	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	—	4	4	no	—
154	C₂×D₄	—	4	4	no	—
155	C₂×D₄	—	4	4	no	—
156	C₂×D₄	C₂³×C₁₀₉	1	4	yes	—
157	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
160	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
161	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
162	D₈	—	4	8	no	—
163	C₂×C₈	C₂³×C₁₀₉	1	4	yes	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	—	2	2	no	—
168	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
169	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
170	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
171	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
172	C₂³	—	2	2	no	—
173	C₂³	—	2	4	no	—
174	C₂³	—	4	4	no	—
175	C₂³	—	4	4	no	—
176	C₂³	—	4	4	no	—
177	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
178	Q₈	—	4	2	no	—
179	D₄	—	2	4	no	—
180	D₄	—	2	8	no	—
181	D₄	—	4	4	no	—
182	D₄	—	4	4	no	—
183	D₄	—	4	4	no	—
184	D₄	—	4	4	no	—
185	D₄	—	4	4	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	4	no	—
188	D₄	—	4	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	D₈×C₁₀₉	1	4	yes	complemented
194	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
195	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
196	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
197	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
198	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
199	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
200	C₈	—	2	4	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	4	4	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	—	4	4	no	—
207	C₂²	—	4	4	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	8	8	no	—
210	C₂²	C₂×D₈×C₁₀₉	1	2	yes	—
211	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	2	no	—
215	C₄	C₂×D₈×C₁₀₉	1	2	yes	—
216	C₄	C₂×D₈×C₁₀₉	1	2	yes	—
217	C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
218	C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
219	C₂	—	2	2	no	—
220	C₂	—	8	4	no	—
221	C₂	—	8	2	no	—
222	C₂	#535×C₁₀₉	1	1	yes	characteristic
223	C₂	C₂²×D₈×C₁₀₉	1	1	yes	characteristic
224	C₂	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #532×C₁₀₉	C₂	semidirect
2. C₂×#127×C₁₀₉	C₂	semidirect
3. #493×C₁₀₉	C₂	semidirect
4. #486×C₁₀₉	C₂	semidirect
5. #459×C₁₀₉	C₂	semidirect
6. #447×C₁₀₉	C₂	semidirect
7. #431×C₁₀₉	C₂	semidirect
12. C₂×D₈×C₁₀₉	C₂²	semidirect
13. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
18. #127×C₁₀₉	C₂²	semidirect
23. C₄×D₄×C₁₀₉	C₂²	semidirect
25. C₄⋊C₈×C₁₀₉	C₂²	semidirect
26. #102×C₁₀₉	C₂²	semidirect
28. C₄×C₈×C₁₀₉	C₂²	semidirect
80. C₂×C₄×C₁₀₉	D₈	semidirect
112. #3205	C₁₀₉	direct (N × H)
113. C₁₀₉	#3205	direct (N × H)
114. #532	C₂×C₁₀₉	semidirect
115. C₂×#127	C₂×C₁₀₉	semidirect
116. #493	C₂×C₁₀₉	semidirect
117. #486	C₂×C₁₀₉	semidirect
118. #459	C₂×C₁₀₉	semidirect
119. #447	C₂×C₁₀₉	semidirect
120. #431	C₂×C₁₀₉	semidirect
125. C₂×D₈	C₂²×C₁₀₉	semidirect
126. C₄⋊Q₈	C₂²×C₁₀₉	semidirect
131. #127	C₂²×C₁₀₉	semidirect
136. C₄×D₄	C₂²×C₁₀₉	semidirect
138. C₄⋊C₈	C₂²×C₁₀₉	semidirect
139. #102	C₂²×C₁₀₉	semidirect
141. C₄×C₈	C₂²×C₁₀₉	semidirect
193. C₂×C₄	D₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#532×C₁₀₉	2
C₂×#127×C₁₀₉	2
#493×C₁₀₉	2
#486×C₁₀₉	2
#459×C₁₀₉	2
#447×C₁₀₉	2
#431×C₁₀₉	2
#3205	109

Subgroups — C109×#3205 #28360459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3205 #28360459