Subgroups — C₁₀₉×#3036 #28360290

Order 13952.

548 conjugacy classes of subgroups.

The subgroup lattice has 260 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	2	yes	maximal, complemented
2	#532×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#463×C₁₀₉	C₂	1	4	yes	maximal, complemented
5	#461×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#459×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#423×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#417×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₂×#41×C₁₀₉	C₂²	1	2	yes	—
10	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₀₉	—	2	1	no	—
12	AGL(1,ℤ/8)×C₁₀₉	—	2	8	no	—
13	C₂×SD₁₆×C₁₀₉	C₂²	1	4	yes	—
14	C₂×D₈×C₁₀₉	C₂²	1	4	yes	—
15	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
17	#127×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
18	#124×C₁₀₉	C₂²	1	2	yes	complemented
19	#121×C₁₀₉	—	2	1	no	—
20	#121×C₁₀₉	—	2	1	no	—
21	C₄×D₄×C₁₀₉	—	2	1	no	—
22	C₄×D₄×C₁₀₉	—	2	1	no	—
23	C₄×D₄×C₁₀₉	—	2	1	no	—
24	C₄×D₄×C₁₀₉	—	2	1	no	—
25	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
26	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
29	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
30	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	complemented
31	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	complemented
32	#103×C₁₀₉	C₂²	1	4	yes	complemented
33	#102×C₁₀₉	C₂²	1	4	yes	complemented
34	C₂⁴×C₁₀₉	—	2	1	no	—
35	#41×C₁₀₉	—	2	4	no	—
36	#41×C₁₀₉	—	2	4	no	—
37	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
38	C₂×D₄×C₁₀₉	—	2	2	no	—
39	C₂×D₄×C₁₀₉	—	2	1	no	—
40	C₂×D₄×C₁₀₉	—	2	1	no	—
41	C₂×D₄×C₁₀₉	—	2	1	no	—
42	C₂×D₄×C₁₀₉	—	2	2	no	—
43	C₂×D₄×C₁₀₉	D₄	1	4	yes	complemented
44	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
45	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₀₉	—	2	1	no	—
47	C₂²×C₄×C₁₀₉	—	2	2	no	—
48	C₂²×C₄×C₁₀₉	—	2	2	no	—
49	C₂²×C₄×C₁₀₉	—	2	2	no	—
50	C₂²×C₄×C₁₀₉	—	4	4	no	—
51	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
54	SD₁₆×C₁₀₉	—	2	8	no	—
55	D₈×C₁₀₉	—	2	8	no	—
56	M₁₆×C₁₀₉	—	2	4	no	—
57	C₂×C₈×C₁₀₉	C₂³	1	4	yes	—
58	C₄⋊C₄×C₁₀₉	—	2	1	no	—
59	C₄⋊C₄×C₁₀₉	—	2	1	no	—
60	C₄⋊C₄×C₁₀₉	—	2	1	no	—
61	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
62	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
63	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
64	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
65	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
66	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
67	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
68	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
70	C₂³×C₁₀₉	—	2	4	no	—
71	C₂³×C₁₀₉	—	2	2	no	—
72	C₂³×C₁₀₉	—	2	2	no	—
73	C₂³×C₁₀₉	—	4	2	no	—
74	C₂³×C₁₀₉	—	4	2	no	—
75	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
76	Q₈×C₁₀₉	—	2	2	no	—
77	Q₈×C₁₀₉	—	2	2	no	—
78	D₄×C₁₀₉	—	2	4	no	—
79	D₄×C₁₀₉	—	2	2	no	—
80	D₄×C₁₀₉	—	2	2	no	—
81	D₄×C₁₀₉	—	2	2	no	—
82	D₄×C₁₀₉	—	4	2	no	—
83	D₄×C₁₀₉	—	4	4	no	—
84	D₄×C₁₀₉	—	4	2	no	—
85	D₄×C₁₀₉	C₂×D₄	1	4	yes	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	2	2	no	—
88	C₂×C₄×C₁₀₉	—	2	2	no	—
89	C₂×C₄×C₁₀₉	—	4	1	no	—
90	C₂×C₄×C₁₀₉	—	4	4	no	—
91	C₂×C₄×C₁₀₉	—	4	2	no	—
92	C₂×C₄×C₁₀₉	—	4	4	no	—
93	C₂×C₄×C₁₀₉	—	4	4	no	—
94	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	#41	1	2	yes	—
101	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
102	C₈×C₁₀₉	—	2	4	no	—
103	C₂²×C₁₀₉	—	2	1	no	—
104	C₂²×C₁₀₉	—	2	1	no	—
105	C₂²×C₁₀₉	—	2	4	no	—
106	C₂²×C₁₀₉	—	4	8	no	—
107	C₂²×C₁₀₉	—	4	2	no	—
108	C₂²×C₁₀₉	—	4	2	no	—
109	C₂²×C₁₀₉	—	4	2	no	—
110	C₂²×C₁₀₉	—	4	2	no	—
111	C₂²×C₁₀₉	—	4	2	no	—
112	C₂²×C₁₀₉	#121	1	2	yes	—
113	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
114	C₄×C₁₀₉	—	2	1	no	—
115	C₄×C₁₀₉	—	2	1	no	—
116	C₄×C₁₀₉	—	2	1	no	—
117	C₄×C₁₀₉	—	4	2	no	—
118	C₄×C₁₀₉	—	4	2	no	—
119	C₄×C₁₀₉	#121	1	2	yes	—
120	C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	complemented
121	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
122	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
123	C₂×C₁₀₉	—	2	2	no	—
124	C₂×C₁₀₉	—	4	4	no	—
125	C₂×C₁₀₉	—	8	2	no	—
126	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
127	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₁₀₉	#575	1	1	yes	characteristic
129	#3036	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₁₀₉	#3036	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×AGL(1,ℤ/8)	C₂×C₁₀₉	1	2	yes	complemented
132	#532	C₂×C₁₀₉	1	1	yes	characteristic, complemented
133	C₂×C₄×D₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
134	#463	C₂×C₁₀₉	1	4	yes	complemented
135	#461	C₂×C₁₀₉	1	2	yes	complemented
136	#459	C₂×C₁₀₉	1	2	yes	complemented
137	#423	C₂×C₁₀₉	1	1	yes	characteristic, complemented
138	#417	C₂×C₁₀₉	1	2	yes	complemented
139	C₂×#41	C₂²×C₁₀₉	1	2	yes	—
140	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
141	C₂³×C₄	—	2	1	no	—
142	AGL(1,ℤ/8)	—	2	8	no	—
143	C₂×SD₁₆	C₂²×C₁₀₉	1	4	yes	—
144	C₂×D₈	C₂²×C₁₀₉	1	4	yes	—
145	C₂×M₁₆	C₂²×C₁₀₉	1	2	yes	—
146	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
147	#127	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
148	#124	C₂²×C₁₀₉	1	2	yes	complemented
149	#121	—	2	1	no	—
150	#121	—	2	1	no	—
151	C₄×D₄	—	2	1	no	—
152	C₄×D₄	—	2	1	no	—
153	C₄×D₄	—	2	1	no	—
154	C₄×D₄	—	2	1	no	—
155	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
156	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
157	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
158	C₂×C₂²⋊C₄	—	2	1	no	—
159	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
160	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	complemented
161	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	complemented
162	#103	C₂²×C₁₀₉	1	4	yes	complemented
163	#102	C₂²×C₁₀₉	1	4	yes	complemented
164	C₂⁴	—	2	1	no	—
165	#41	—	2	4	no	—
166	#41	—	2	4	no	—
167	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	D₄×C₁₀₉	1	4	yes	complemented
174	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
175	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	4	4	no	—
181	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
182	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
183	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
184	SD₁₆	—	2	8	no	—
185	D₈	—	2	8	no	—
186	M₁₆	—	2	4	no	—
187	C₂×C₈	C₂³×C₁₀₉	1	4	yes	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
197	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
198	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
199	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
200	C₂³	—	2	4	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	4	2	no	—
204	C₂³	—	4	2	no	—
205	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
206	Q₈	—	2	2	no	—
207	Q₈	—	2	2	no	—
208	D₄	—	2	4	no	—
209	D₄	—	2	2	no	—
210	D₄	—	2	2	no	—
211	D₄	—	2	2	no	—
212	D₄	—	4	2	no	—
213	D₄	—	4	4	no	—
214	D₄	—	4	2	no	—
215	D₄	C₂×D₄×C₁₀₉	1	4	yes	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
225	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
226	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
227	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
228	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
229	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
230	C₂×C₄	#41×C₁₀₉	1	2	yes	—
231	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
232	C₈	—	2	4	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	4	8	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	#121×C₁₀₉	1	2	yes	—
243	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	#121×C₁₀₉	1	2	yes	—
250	C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	complemented
251	C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
252	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
253	C₂	—	2	2	no	—
254	C₂	—	4	4	no	—
255	C₂	—	8	2	no	—
256	C₂	C₂×#121×C₁₀₉	1	1	yes	characteristic
257	C₂	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
258	C₂	#575×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. C₂×AGL(1,ℤ/8)×C₁₀₉	C₂	semidirect
2. #532×C₁₀₉	C₂	semidirect
3. C₂×C₄×D₄×C₁₀₉	C₂	semidirect
4. #463×C₁₀₉	C₂	semidirect
5. #461×C₁₀₉	C₂	semidirect
6. #459×C₁₀₉	C₂	semidirect
7. #423×C₁₀₉	C₂	semidirect
8. #417×C₁₀₉	C₂	semidirect
16. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
17. #127×C₁₀₉	C₂²	semidirect
18. #124×C₁₀₉	C₂²	semidirect
30. C₄⋊C₈×C₁₀₉	C₂²	semidirect
31. C₄⋊C₈×C₁₀₉	C₂²	semidirect
32. #103×C₁₀₉	C₂²	semidirect
33. #102×C₁₀₉	C₂²	semidirect
43. C₂×D₄×C₁₀₉	D₄	semidirect
120. C₄×C₁₀₉	AGL(1,ℤ/8)	semidirect
129. #3036	C₁₀₉	direct (N × H)
130. C₁₀₉	#3036	direct (N × H)
131. C₂×AGL(1,ℤ/8)	C₂×C₁₀₉	semidirect
132. #532	C₂×C₁₀₉	semidirect
133. C₂×C₄×D₄	C₂×C₁₀₉	semidirect
134. #463	C₂×C₁₀₉	semidirect
135. #461	C₂×C₁₀₉	semidirect
136. #459	C₂×C₁₀₉	semidirect
137. #423	C₂×C₁₀₉	semidirect
138. #417	C₂×C₁₀₉	semidirect
146. C₄⋊Q₈	C₂²×C₁₀₉	semidirect
147. #127	C₂²×C₁₀₉	semidirect
148. #124	C₂²×C₁₀₉	semidirect
160. C₄⋊C₈	C₂²×C₁₀₉	semidirect
161. C₄⋊C₈	C₂²×C₁₀₉	semidirect
162. #103	C₂²×C₁₀₉	semidirect
163. #102	C₂²×C₁₀₉	semidirect
173. C₂×D₄	D₄×C₁₀₉	semidirect
250. C₄	AGL(1,ℤ/8)×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
#532×C₁₀₉	2
C₂×C₄×D₄×C₁₀₉	2
#463×C₁₀₉	2
#461×C₁₀₉	2
#459×C₁₀₉	2
#423×C₁₀₉	2
#417×C₁₀₉	2
#3036	109

Subgroups — C109×#3036 #28360290

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3036 #28360290