Subgroups — C₁₀₉×#2536 #28359790

Order 13952.

604 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Q₈×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#398×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#394×C₁₀₉	C₂	1	2	yes	maximal
6	#388×C₁₀₉	C₂	1	4	yes	maximal, complemented
7	#382×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#380×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
12	C₂²⋊Q₈×C₁₀₉	—	2	8	no	—
13	C₄×Q₈×C₁₀₉	—	2	4	no	—
14	C₄×D₄×C₁₀₉	—	2	4	no	—
15	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
22	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
23	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
25	#95×C₁₀₉	C₂²	1	4	yes	—
26	#95×C₁₀₉	C₂²	1	2	yes	—
27	#95×C₁₀₉	C₂²	1	2	yes	—
28	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
29	C₂×Q₈×C₁₀₉	—	2	2	no	—
30	C₂×Q₈×C₁₀₉	—	2	2	no	—
31	C₂×Q₈×C₁₀₉	D₄	1	4	yes	complemented
32	C₂×D₄×C₁₀₉	—	2	4	no	—
33	C₂×D₄×C₁₀₉	—	2	2	no	—
34	C₂²×C₄×C₁₀₉	—	2	4	no	—
35	C₂²×C₄×C₁₀₉	—	2	4	no	—
36	C₂²×C₄×C₁₀₉	—	2	4	no	—
37	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
38	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₁₀₉	—	2	2	no	—
45	C₄⋊C₄×C₁₀₉	—	2	4	no	—
46	C₄⋊C₄×C₁₀₉	—	2	2	no	—
47	C₄⋊C₄×C₁₀₉	—	2	4	no	—
48	C₄⋊C₄×C₁₀₉	—	2	8	no	—
49	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
50	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
51	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
52	C₄²×C₁₀₉	—	2	4	no	—
53	C₄²×C₁₀₉	D₄	1	4	yes	—
54	C₂³×C₁₀₉	—	2	2	no	—
55	C₂³×C₁₀₉	—	2	2	no	—
56	C₂³×C₁₀₉	—	2	2	no	—
57	C₂³×C₁₀₉	—	2	2	no	—
58	C₂³×C₁₀₉	—	2	2	no	—
59	C₂³×C₁₀₉	—	2	2	no	—
60	C₂³×C₁₀₉	#41	1	4	yes	—
61	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
62	Q₈×C₁₀₉	—	2	8	no	—
63	D₄×C₁₀₉	—	2	8	no	—
64	C₂×C₄×C₁₀₉	—	2	2	no	—
65	C₂×C₄×C₁₀₉	—	2	2	no	—
66	C₂×C₄×C₁₀₉	—	2	2	no	—
67	C₂×C₄×C₁₀₉	—	2	2	no	—
68	C₂×C₄×C₁₀₉	—	2	2	no	—
69	C₂×C₄×C₁₀₉	—	2	4	no	—
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	—	2	4	no	—
76	C₂×C₄×C₁₀₉	—	4	8	no	—
77	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
80	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₁₀₉	#41	1	2	yes	—
83	C₂²×C₁₀₉	—	2	4	no	—
84	C₂²×C₁₀₉	—	2	4	no	—
85	C₂²×C₁₀₉	—	2	4	no	—
86	C₂²×C₁₀₉	—	4	2	no	—
87	C₂²×C₁₀₉	—	4	2	no	—
88	C₂²×C₁₀₉	—	4	2	no	—
89	C₂²×C₁₀₉	—	4	2	no	—
90	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
91	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
92	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
93	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
94	C₂²×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
95	C₄×C₁₀₉	—	2	4	no	—
96	C₄×C₁₀₉	—	2	2	no	—
97	C₄×C₁₀₉	—	2	2	no	—
98	C₄×C₁₀₉	—	4	2	no	—
99	C₄×C₁₀₉	—	4	2	no	—
100	C₄×C₁₀₉	—	4	4	no	—
101	C₄×C₁₀₉	#121	1	4	yes	—
102	C₂×C₁₀₉	—	4	4	no	—
103	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
104	C₂×C₁₀₉	#525	1	1	yes	characteristic
105	C₂×C₁₀₉	#536	1	1	yes	characteristic
106	C₂×C₁₀₉	#548	1	2	yes	—
107	C₂×C₁₀₉	#552	1	2	yes	—
108	#2536	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₁₀₉	#2536	1	1	yes	characteristic, direct product factor, Sylow
110	C₂×C₂²⋊Q₈	C₂×C₁₀₉	1	2	yes	complemented
111	C₂×C₄×Q₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
112	C₂×C₄×D₄	C₂×C₁₀₉	1	1	yes	characteristic
113	#398	C₂×C₁₀₉	1	2	yes	complemented
114	#394	C₂×C₁₀₉	1	2	yes	—
115	#388	C₂×C₁₀₉	1	4	yes	complemented
116	#382	C₂×C₁₀₉	1	1	yes	characteristic, complemented
117	#380	C₂×C₁₀₉	1	2	yes	complemented
118	C₂²×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
119	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
120	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	—
121	C₂²⋊Q₈	—	2	8	no	—
122	C₄×Q₈	—	2	4	no	—
123	C₄×D₄	—	2	4	no	—
124	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
125	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
127	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
128	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
129	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
130	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
131	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
132	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
133	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
134	#95	C₂²×C₁₀₉	1	4	yes	—
135	#95	C₂²×C₁₀₉	1	2	yes	—
136	#95	C₂²×C₁₀₉	1	2	yes	—
137	C₂⁴	C₂³×C₁₀₉	1	2	yes	—
138	C₂×Q₈	—	2	2	no	—
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	D₄×C₁₀₉	1	4	yes	complemented
141	C₂×D₄	—	2	4	no	—
142	C₂×D₄	—	2	2	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
147	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
148	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
149	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
150	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
151	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
152	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	2	8	no	—
158	C₂²⋊C₄	—	2	8	no	—
159	C₂²⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	2	8	no	—
161	C₄²	—	2	4	no	—
162	C₄²	D₄×C₁₀₉	1	4	yes	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	2	no	—
169	C₂³	#41×C₁₀₉	1	4	yes	—
170	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
171	Q₈	—	2	8	no	—
172	D₄	—	2	8	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	4	8	no	—
186	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
187	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
188	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
189	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
190	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
191	C₂×C₄	#41×C₁₀₉	1	2	yes	—
192	C₂²	—	2	4	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	4	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	2	no	—
199	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
200	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
201	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
202	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
203	C₂²	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	4	no	—
210	C₄	#121×C₁₀₉	1	4	yes	—
211	C₂	—	4	4	no	—
212	C₂	C₂×#121×C₁₀₉	1	1	yes	characteristic
213	C₂	#525×C₁₀₉	1	1	yes	characteristic
214	C₂	#536×C₁₀₉	1	1	yes	characteristic
215	C₂	#548×C₁₀₉	1	2	yes	—
216	C₂	#552×C₁₀₉	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊Q₈×C₁₀₉	C₂	semidirect
2. C₂×C₄×Q₈×C₁₀₉	C₂	semidirect
4. #398×C₁₀₉	C₂	semidirect
6. #388×C₁₀₉	C₂	semidirect
7. #382×C₁₀₉	C₂	semidirect
8. #380×C₁₀₉	C₂	semidirect
31. C₂×Q₈×C₁₀₉	D₄	semidirect
108. #2536	C₁₀₉	direct (N × H)
109. C₁₀₉	#2536	direct (N × H)
110. C₂×C₂²⋊Q₈	C₂×C₁₀₉	semidirect
111. C₂×C₄×Q₈	C₂×C₁₀₉	semidirect
113. #398	C₂×C₁₀₉	semidirect
115. #388	C₂×C₁₀₉	semidirect
116. #382	C₂×C₁₀₉	semidirect
117. #380	C₂×C₁₀₉	semidirect
140. C₂×Q₈	D₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Q₈×C₁₀₉	2
C₂×C₄×Q₈×C₁₀₉	2
C₂×C₄×D₄×C₁₀₉	2
#398×C₁₀₉	2
#394×C₁₀₉	2
#388×C₁₀₉	2
#382×C₁₀₉	2
#380×C₁₀₉	2
#2536	109

Subgroups — C109×#2536 #28359790

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2536 #28359790