Subgroups — C₁₀₉×#3042 #28360296

Order 13952.

494 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×AGL(1,ℤ/8)×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×SD₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#536×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#518×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#463×C₁₀₉	C₂	1	4	yes	maximal, complemented
6	#461×C₁₀₉	C₂	1	4	yes	maximal, complemented
7	#424×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#419×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#416×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
13	AGL(1,ℤ/8)×C₁₀₉	—	2	4	no	—
14	C₂×SD₁₆×C₁₀₉	—	2	4	no	—
15	C₂×SD₁₆×C₁₀₉	C₂²	1	4	yes	—
16	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
17	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
18	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	complemented
21	#124×C₁₀₉	C₂²	1	2	yes	complemented
22	#123×C₁₀₉	—	2	2	no	—
23	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
24	C₄×D₄×C₁₀₉	—	2	2	no	—
25	C₄×D₄×C₁₀₉	C₂²	1	4	yes	—
26	#117×C₁₀₉	C₂²	1	1	yes	characteristic
27	#117×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
29	C₄⋊C₈×C₁₀₉	C₂²	1	4	yes	complemented
30	#103×C₁₀₉	C₂²	1	2	yes	complemented
31	#103×C₁₀₉	C₂²	1	4	yes	complemented
32	#102×C₁₀₉	C₂²	1	2	yes	complemented
33	C₂⁴×C₁₀₉	—	2	1	no	—
34	#41×C₁₀₉	—	2	2	no	—
35	#41×C₁₀₉	—	2	2	no	—
36	C₂×Q₈×C₁₀₉	—	2	1	no	—
37	C₂×Q₈×C₁₀₉	—	2	2	no	—
38	C₂×Q₈×C₁₀₉	—	2	2	no	—
39	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
40	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₁₀₉	—	2	1	no	—
42	C₂×D₄×C₁₀₉	—	2	1	no	—
43	C₂×D₄×C₁₀₉	—	2	2	no	—
44	C₂×D₄×C₁₀₉	D₄	1	4	yes	—
45	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₀₉	—	2	1	no	—
48	C₂²×C₄×C₁₀₉	—	2	1	no	—
49	C₂²×C₄×C₁₀₉	—	2	4	no	—
50	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
51	SD₁₆×C₁₀₉	—	2	8	no	—
52	SD₁₆×C₁₀₉	—	2	4	no	—
53	D₈×C₁₀₉	—	2	4	no	—
54	M₁₆×C₁₀₉	—	2	2	no	—
55	C₂×C₈×C₁₀₉	—	2	2	no	—
56	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
57	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
58	C₄⋊C₄×C₁₀₉	—	2	4	no	—
59	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
60	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
61	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
62	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
63	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
64	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
65	C₄²×C₁₀₉	—	2	1	no	—
66	C₄²×C₁₀₉	C₂³	1	2	yes	—
67	C₂³×C₁₀₉	—	2	2	no	—
68	C₂³×C₁₀₉	—	2	4	no	—
69	C₂³×C₁₀₉	—	2	1	no	—
70	C₂³×C₁₀₉	—	8	2	no	—
71	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
72	Q₈×C₁₀₉	—	2	2	no	—
73	Q₈×C₁₀₉	—	2	1	no	—
74	Q₈×C₁₀₉	—	2	2	no	—
75	Q₈×C₁₀₉	—	2	1	no	—
76	Q₈×C₁₀₉	—	4	2	no	—
77	D₄×C₁₀₉	—	2	4	no	—
78	D₄×C₁₀₉	—	2	1	no	—
79	D₄×C₁₀₉	—	2	2	no	—
80	D₄×C₁₀₉	—	2	1	no	—
81	D₄×C₁₀₉	—	4	2	no	—
82	D₄×C₁₀₉	C₂×D₄	1	4	yes	—
83	C₂×C₄×C₁₀₉	—	2	2	no	—
84	C₂×C₄×C₁₀₉	—	2	2	no	—
85	C₂×C₄×C₁₀₉	—	2	1	no	—
86	C₂×C₄×C₁₀₉	—	4	2	no	—
87	C₂×C₄×C₁₀₉	—	4	2	no	—
88	C₂×C₄×C₁₀₉	—	8	4	no	—
89	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
90	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₁₀₉	#41	1	2	yes	—
93	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
94	C₈×C₁₀₉	—	2	2	no	—
95	C₈×C₁₀₉	—	2	2	no	—
96	C₂²×C₁₀₉	—	2	2	no	—
97	C₂²×C₁₀₉	—	2	4	no	—
98	C₂²×C₁₀₉	—	4	4	no	—
99	C₂²×C₁₀₉	—	4	1	no	—
100	C₂²×C₁₀₉	—	4	2	no	—
101	C₂²×C₁₀₉	—	8	4	no	—
102	C₂²×C₁₀₉	#121	1	2	yes	—
103	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
104	C₄×C₁₀₉	—	2	4	no	—
105	C₄×C₁₀₉	—	4	1	no	—
106	C₄×C₁₀₉	—	4	2	no	—
107	C₄×C₁₀₉	—	4	2	no	—
108	C₄×C₁₀₉	#121	1	2	yes	—
109	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
110	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
111	C₂×C₁₀₉	—	2	2	no	—
112	C₂×C₁₀₉	—	4	4	no	—
113	C₂×C₁₀₉	—	8	1	no	—
114	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
115	C₂×C₁₀₉	#577	1	2	yes	—
116	#3042	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₁₀₉	#3042	1	1	yes	characteristic, direct product factor, Sylow
118	C₂×AGL(1,ℤ/8)	C₂×C₁₀₉	1	1	yes	characteristic
119	C₂²×SD₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
120	#536	C₂×C₁₀₉	1	1	yes	characteristic, complemented
121	#518	C₂×C₁₀₉	1	1	yes	characteristic, complemented
122	#463	C₂×C₁₀₉	1	4	yes	complemented
123	#461	C₂×C₁₀₉	1	4	yes	complemented
124	#424	C₂×C₁₀₉	1	1	yes	characteristic, complemented
125	#419	C₂×C₁₀₉	1	1	yes	characteristic, complemented
126	#416	C₂×C₁₀₉	1	1	yes	characteristic, complemented
127	C₂×#41	C₂²×C₁₀₉	1	1	yes	characteristic
128	C₂²×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
129	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
130	AGL(1,ℤ/8)	—	2	4	no	—
131	C₂×SD₁₆	—	2	4	no	—
132	C₂×SD₁₆	C₂²×C₁₀₉	1	4	yes	—
133	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
134	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
135	C₂×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
136	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
137	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	complemented
138	#124	C₂²×C₁₀₉	1	2	yes	complemented
139	#123	—	2	2	no	—
140	C₂²⋊Q₈	—	2	2	no	—
141	C₄×D₄	—	2	2	no	—
142	C₄×D₄	C₂²×C₁₀₉	1	4	yes	—
143	#117	C₂²×C₁₀₉	1	1	yes	characteristic
144	#117	C₂²×C₁₀₉	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	—	2	2	no	—
146	C₄⋊C₈	C₂²×C₁₀₉	1	4	yes	complemented
147	#103	C₂²×C₁₀₉	1	2	yes	complemented
148	#103	C₂²×C₁₀₉	1	4	yes	complemented
149	#102	C₂²×C₁₀₉	1	2	yes	complemented
150	C₂⁴	—	2	1	no	—
151	#41	—	2	2	no	—
152	#41	—	2	2	no	—
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	2	2	no	—
156	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
157	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
158	C₂×D₄	—	2	1	no	—
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	D₄×C₁₀₉	1	4	yes	—
162	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
163	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
168	SD₁₆	—	2	8	no	—
169	SD₁₆	—	2	4	no	—
170	D₈	—	2	4	no	—
171	M₁₆	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
174	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
177	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
178	C₂²⋊C₄	—	2	1	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₂²⋊C₄	—	2	4	no	—
181	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
182	C₄²	—	2	1	no	—
183	C₄²	C₂³×C₁₀₉	1	2	yes	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	4	no	—
186	C₂³	—	2	1	no	—
187	C₂³	—	8	2	no	—
188	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
189	Q₈	—	2	2	no	—
190	Q₈	—	2	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	2	1	no	—
193	Q₈	—	4	2	no	—
194	D₄	—	2	4	no	—
195	D₄	—	2	1	no	—
196	D₄	—	2	2	no	—
197	D₄	—	2	1	no	—
198	D₄	—	4	2	no	—
199	D₄	C₂×D₄×C₁₀₉	1	4	yes	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	8	4	no	—
206	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
207	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
208	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
209	C₂×C₄	#41×C₁₀₉	1	2	yes	—
210	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	4	4	no	—
216	C₂²	—	4	1	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	8	4	no	—
219	C₂²	#121×C₁₀₉	1	2	yes	—
220	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	4	1	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	2	no	—
225	C₄	#121×C₁₀₉	1	2	yes	—
226	C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
227	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
228	C₂	—	2	2	no	—
229	C₂	—	4	4	no	—
230	C₂	—	8	1	no	—
231	C₂	C₂×#121×C₁₀₉	1	1	yes	characteristic
232	C₂	#577×C₁₀₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	C₂×AGL(1,ℤ/8)×C₁₀₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ #117×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×SD₁₆×C₁₀₉	C₂	semidirect
3. #536×C₁₀₉	C₂	semidirect
4. #518×C₁₀₉	C₂	semidirect
5. #463×C₁₀₉	C₂	semidirect
6. #461×C₁₀₉	C₂	semidirect
7. #424×C₁₀₉	C₂	semidirect
8. #419×C₁₀₉	C₂	semidirect
9. #416×C₁₀₉	C₂	semidirect
20. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
21. #124×C₁₀₉	C₂²	semidirect
29. C₄⋊C₈×C₁₀₉	C₂²	semidirect
30. #103×C₁₀₉	C₂²	semidirect
31. #103×C₁₀₉	C₂²	semidirect
32. #102×C₁₀₉	C₂²	semidirect
116. #3042	C₁₀₉	direct (N × H)
117. C₁₀₉	#3042	direct (N × H)
119. C₂²×SD₁₆	C₂×C₁₀₉	semidirect
120. #536	C₂×C₁₀₉	semidirect
121. #518	C₂×C₁₀₉	semidirect
122. #463	C₂×C₁₀₉	semidirect
123. #461	C₂×C₁₀₉	semidirect
124. #424	C₂×C₁₀₉	semidirect
125. #419	C₂×C₁₀₉	semidirect
126. #416	C₂×C₁₀₉	semidirect
137. C₄⋊Q₈	C₂²×C₁₀₉	semidirect
138. #124	C₂²×C₁₀₉	semidirect
146. C₄⋊C₈	C₂²×C₁₀₉	semidirect
147. #103	C₂²×C₁₀₉	semidirect
148. #103	C₂²×C₁₀₉	semidirect
149. #102	C₂²×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×AGL(1,ℤ/8)×C₁₀₉	2
C₂²×SD₁₆×C₁₀₉	2
#536×C₁₀₉	2
#518×C₁₀₉	2
#463×C₁₀₉	2
#461×C₁₀₉	2
#424×C₁₀₉	2
#419×C₁₀₉	2
#416×C₁₀₉	2
#3042	109

Subgroups — C109×#3042 #28360296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3042 #28360296