Subgroups — C₁₀₉×#3222 #28360476

Order 13952.

396 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#541×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#535×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#497×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#492×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#490×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#488×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#464×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	#460×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	#452×C₁₀₉	C₂	1	2	yes	maximal, complemented
10	#450×C₁₀₉	C₂	1	2	yes	maximal, complemented
11	#433×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	complemented
15	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
16	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
17	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	complemented
18	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
19	#127×C₁₀₉	C₂²	1	1	yes	characteristic
20	#127×C₁₀₉	C₂²	1	1	yes	characteristic
21	#126×C₁₀₉	—	2	2	no	—
22	#125×C₁₀₉	—	2	2	no	—
23	#125×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
24	#124×C₁₀₉	C₂²	1	1	yes	characteristic
25	#124×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
27	#121×C₁₀₉	—	2	2	no	—
28	#120×C₁₀₉	—	2	2	no	—
29	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	complemented
30	#117×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	complemented
32	#103×C₁₀₉	C₂²	1	2	yes	complemented
33	#102×C₁₀₉	C₂²	1	2	yes	—
34	#102×C₁₀₉	C₂²	1	2	yes	—
35	#102×C₁₀₉	C₂²	1	2	yes	complemented
36	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
37	#97×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
39	C₂⁴×C₁₀₉	—	2	1	no	—
40	#41×C₁₀₉	—	4	2	no	—
41	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
43	C₂×D₄×C₁₀₉	—	2	1	no	—
44	C₂×D₄×C₁₀₉	—	2	1	no	—
45	C₂×D₄×C₁₀₉	—	2	2	no	—
46	C₂×D₄×C₁₀₉	—	2	2	no	—
47	C₂×D₄×C₁₀₉	—	4	2	no	—
48	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₀₉	—	2	1	no	—
52	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
53	Q₁₆×C₁₀₉	—	4	2	no	—
54	SD₁₆×C₁₀₉	—	4	2	no	—
55	SD₁₆×C₁₀₉	—	4	2	no	—
56	SD₁₆×C₁₀₉	—	4	2	no	—
57	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
58	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
59	C₄⋊C₄×C₁₀₉	—	2	2	no	—
60	C₄⋊C₄×C₁₀₉	—	2	2	no	—
61	C₄⋊C₄×C₁₀₉	—	2	2	no	—
62	C₄⋊C₄×C₁₀₉	—	2	2	no	—
63	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
64	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
65	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
66	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
67	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
68	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
69	C₄²×C₁₀₉	—	2	2	no	—
70	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
71	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₂³×C₁₀₉	—	2	1	no	—
73	C₂³×C₁₀₉	—	2	1	no	—
74	C₂³×C₁₀₉	—	2	1	no	—
75	C₂³×C₁₀₉	—	4	2	no	—
76	C₂³×C₁₀₉	—	8	2	no	—
77	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
78	Q₈×C₁₀₉	—	2	2	no	—
79	Q₈×C₁₀₉	—	2	2	no	—
80	Q₈×C₁₀₉	—	2	2	no	—
81	D₄×C₁₀₉	—	2	2	no	—
82	D₄×C₁₀₉	—	2	2	no	—
83	D₄×C₁₀₉	—	2	2	no	—
84	D₄×C₁₀₉	—	4	2	no	—
85	D₄×C₁₀₉	—	4	2	no	—
86	D₄×C₁₀₉	—	8	2	no	—
87	D₄×C₁₀₉	—	8	2	no	—
88	C₂×C₄×C₁₀₉	—	2	2	no	—
89	C₂×C₄×C₁₀₉	—	2	2	no	—
90	C₂×C₄×C₁₀₉	—	2	2	no	—
91	C₂×C₄×C₁₀₉	—	2	1	no	—
92	C₂×C₄×C₁₀₉	—	4	2	no	—
93	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
97	C₈×C₁₀₉	—	2	2	no	—
98	C₈×C₁₀₉	—	2	2	no	—
99	C₂²×C₁₀₉	—	2	1	no	—
100	C₂²×C₁₀₉	—	2	2	no	—
101	C₂²×C₁₀₉	—	4	1	no	—
102	C₂²×C₁₀₉	—	4	1	no	—
103	C₂²×C₁₀₉	—	4	1	no	—
104	C₂²×C₁₀₉	—	4	2	no	—
105	C₂²×C₁₀₉	—	4	2	no	—
106	C₂²×C₁₀₉	—	4	2	no	—
107	C₂²×C₁₀₉	—	8	4	no	—
108	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
109	C₄×C₁₀₉	—	2	1	no	—
110	C₄×C₁₀₉	—	2	2	no	—
111	C₄×C₁₀₉	—	2	2	no	—
112	C₄×C₁₀₉	—	4	1	no	—
113	C₄×C₁₀₉	—	4	2	no	—
114	C₄×C₁₀₉	—	4	2	no	—
115	C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
116	C₂×C₁₀₉	—	4	1	no	—
117	C₂×C₁₀₉	—	8	1	no	—
118	C₂×C₁₀₉	—	8	1	no	—
119	C₂×C₁₀₉	—	8	1	no	—
120	C₂×C₁₀₉	#535	1	1	yes	characteristic
121	C₂×C₁₀₉	#577	1	2	yes	—
122	#3222	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₁₀₉	#3222	1	1	yes	characteristic, direct product factor, Sylow
124	#541	C₂×C₁₀₉	1	1	yes	characteristic, complemented
125	#535	C₂×C₁₀₉	1	1	yes	characteristic
126	#497	C₂×C₁₀₉	1	1	yes	characteristic, complemented
127	#492	C₂×C₁₀₉	1	1	yes	characteristic, complemented
128	#490	C₂×C₁₀₉	1	1	yes	characteristic, complemented
129	#488	C₂×C₁₀₉	1	1	yes	characteristic, complemented
130	#464	C₂×C₁₀₉	1	2	yes	complemented
131	#460	C₂×C₁₀₉	1	2	yes	complemented
132	#452	C₂×C₁₀₉	1	2	yes	complemented
133	#450	C₂×C₁₀₉	1	2	yes	complemented
134	#433	C₂×C₁₀₉	1	1	yes	characteristic, complemented
135	C₂×#41	C₂²×C₁₀₉	1	1	yes	characteristic
136	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
137	C₂×Q₁₆	C₂²×C₁₀₉	1	2	yes	complemented
138	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
139	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
140	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	complemented
141	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
142	#127	C₂²×C₁₀₉	1	1	yes	characteristic
143	#127	C₂²×C₁₀₉	1	1	yes	characteristic
144	#126	—	2	2	no	—
145	#125	—	2	2	no	—
146	#125	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
147	#124	C₂²×C₁₀₉	1	1	yes	characteristic
148	#124	C₂²×C₁₀₉	1	1	yes	characteristic
149	C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
150	#121	—	2	2	no	—
151	#120	—	2	2	no	—
152	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	complemented
153	#117	C₂²×C₁₀₉	1	1	yes	characteristic
154	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	complemented
155	#103	C₂²×C₁₀₉	1	2	yes	complemented
156	#102	C₂²×C₁₀₉	1	2	yes	—
157	#102	C₂²×C₁₀₉	1	2	yes	—
158	#102	C₂²×C₁₀₉	1	2	yes	complemented
159	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
160	#97	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
161	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
162	C₂⁴	—	2	1	no	—
163	#41	—	4	2	no	—
164	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
165	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
172	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
173	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
176	Q₁₆	—	4	2	no	—
177	SD₁₆	—	4	2	no	—
178	SD₁₆	—	4	2	no	—
179	SD₁₆	—	4	2	no	—
180	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
181	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
187	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
192	C₄²	—	2	2	no	—
193	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
194	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	1	no	—
198	C₂³	—	4	2	no	—
199	C₂³	—	8	2	no	—
200	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
201	Q₈	—	2	2	no	—
202	Q₈	—	2	2	no	—
203	Q₈	—	2	2	no	—
204	D₄	—	2	2	no	—
205	D₄	—	2	2	no	—
206	D₄	—	2	2	no	—
207	D₄	—	4	2	no	—
208	D₄	—	4	2	no	—
209	D₄	—	8	2	no	—
210	D₄	—	8	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
217	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
218	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
219	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	8	4	no	—
231	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
239	C₂	—	4	1	no	—
240	C₂	—	8	1	no	—
241	C₂	—	8	1	no	—
242	C₂	—	8	1	no	—
243	C₂	#535×C₁₀₉	1	1	yes	characteristic
244	C₂	#577×C₁₀₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	#535×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
1. #541×C₁₀₉	C₂	semidirect
3. #497×C₁₀₉	C₂	semidirect
4. #492×C₁₀₉	C₂	semidirect
5. #490×C₁₀₉	C₂	semidirect
6. #488×C₁₀₉	C₂	semidirect
7. #464×C₁₀₉	C₂	semidirect
8. #460×C₁₀₉	C₂	semidirect
9. #452×C₁₀₉	C₂	semidirect
10. #450×C₁₀₉	C₂	semidirect
11. #433×C₁₀₉	C₂	semidirect
14. C₂×Q₁₆×C₁₀₉	C₂²	semidirect
17. C₂×SD₁₆×C₁₀₉	C₂²	semidirect
18. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
23. #125×C₁₀₉	C₂²	semidirect
29. C₄×Q₈×C₁₀₉	C₂²	semidirect
31. C₄⋊C₈×C₁₀₉	C₂²	semidirect
32. #103×C₁₀₉	C₂²	semidirect
35. #102×C₁₀₉	C₂²	semidirect
37. #97×C₁₀₉	C₂²	semidirect
38. C₄×C₈×C₁₀₉	C₂²	semidirect
122. #3222	C₁₀₉	direct (N × H)
123. C₁₀₉	#3222	direct (N × H)
124. #541	C₂×C₁₀₉	semidirect
126. #497	C₂×C₁₀₉	semidirect
127. #492	C₂×C₁₀₉	semidirect
128. #490	C₂×C₁₀₉	semidirect
129. #488	C₂×C₁₀₉	semidirect
130. #464	C₂×C₁₀₉	semidirect
131. #460	C₂×C₁₀₉	semidirect
132. #452	C₂×C₁₀₉	semidirect
133. #450	C₂×C₁₀₉	semidirect
134. #433	C₂×C₁₀₉	semidirect
137. C₂×Q₁₆	C₂²×C₁₀₉	semidirect
140. C₂×SD₁₆	C₂²×C₁₀₉	semidirect
141. C₄⋊Q₈	C₂²×C₁₀₉	semidirect
146. #125	C₂²×C₁₀₉	semidirect
152. C₄×Q₈	C₂²×C₁₀₉	semidirect
154. C₄⋊C₈	C₂²×C₁₀₉	semidirect
155. #103	C₂²×C₁₀₉	semidirect
158. #102	C₂²×C₁₀₉	semidirect
160. #97	C₂²×C₁₀₉	semidirect
161. C₄×C₈	C₂²×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#541×C₁₀₉	2
#535×C₁₀₉	2
#497×C₁₀₉	2
#492×C₁₀₉	2
#490×C₁₀₉	2
#488×C₁₀₉	2
#464×C₁₀₉	2
#460×C₁₀₉	2
#452×C₁₀₉	2
#450×C₁₀₉	2
#433×C₁₀₉	2
#3222	109

Subgroups — C109×#3222 #28360476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3222 #28360476