Subgroups — C₁₀₉×#3304 #28360558

Order 13952.

396 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#556×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#550×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#486×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#482×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#475×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#465×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#460×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#452×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₄×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#416×C₁₀₉	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₁₀₉	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
15	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	#127×C₁₀₉	—	2	1	no	—
18	#127×C₁₀₉	C₂²	1	1	yes	characteristic
19	#126×C₁₀₉	—	2	2	no	—
20	#124×C₁₀₉	—	2	1	no	—
21	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
22	#121×C₁₀₉	—	2	2	no	—
23	#121×C₁₀₉	C₂²	1	2	yes	—
24	C₄×Q₈×C₁₀₉	—	2	1	no	—
25	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₀₉	—	2	1	no	—
28	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
29	#117×C₁₀₉	C₂²	1	2	yes	—
30	#107×C₁₀₉	C₂²	1	1	yes	characteristic
31	#106×C₁₀₉	C₂²	1	2	yes	—
32	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
33	#103×C₁₀₉	C₂²	1	1	yes	characteristic
34	#103×C₁₀₉	C₂²	1	2	yes	—
35	#103×C₁₀₉	C₂²	1	1	yes	characteristic
36	#102×C₁₀₉	C₂²	1	2	yes	—
37	#102×C₁₀₉	C₂²	1	2	yes	—
38	#102×C₁₀₉	C₂²	1	2	yes	—
39	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
40	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	#41×C₁₀₉	—	2	4	no	—
42	#41×C₁₀₉	—	4	2	no	—
43	C₂×Q₈×C₁₀₉	—	2	1	no	—
44	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₁₀₉	—	2	2	no	—
47	C₂×D₄×C₁₀₉	—	2	2	no	—
48	C₂×D₄×C₁₀₉	—	2	1	no	—
49	C₂×D₄×C₁₀₉	—	2	1	no	—
50	C₂×D₄×C₁₀₉	D₄	1	1	yes	characteristic
51	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₀₉	—	2	2	no	—
53	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
54	Q₁₆×C₁₀₉	—	2	2	no	—
55	SD₁₆×C₁₀₉	—	4	2	no	—
56	C₂×C₈×C₁₀₉	—	4	2	no	—
57	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
60	C₄⋊C₄×C₁₀₉	—	2	2	no	—
61	C₄⋊C₄×C₁₀₉	—	2	1	no	—
62	C₄⋊C₄×C₁₀₉	—	2	1	no	—
63	C₄⋊C₄×C₁₀₉	—	2	1	no	—
64	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
66	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
69	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
70	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
71	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
72	C₄²×C₁₀₉	—	2	2	no	—
73	C₄²×C₁₀₉	—	2	1	no	—
74	C₄²×C₁₀₉	—	2	1	no	—
75	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₂³×C₁₀₉	—	2	2	no	—
77	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
78	Q₈×C₁₀₉	—	2	1	no	—
79	Q₈×C₁₀₉	—	2	1	no	—
80	Q₈×C₁₀₉	—	2	1	no	—
81	Q₈×C₁₀₉	—	4	2	no	—
82	Q₈×C₁₀₉	#41	1	2	yes	—
83	D₄×C₁₀₉	—	2	2	no	—
84	D₄×C₁₀₉	—	2	2	no	—
85	D₄×C₁₀₉	—	2	2	no	—
86	D₄×C₁₀₉	—	4	4	no	—
87	D₄×C₁₀₉	—	4	2	no	—
88	D₄×C₁₀₉	—	8	1	no	—
89	D₄×C₁₀₉	—	8	2	no	—
90	C₂×C₄×C₁₀₉	—	2	1	no	—
91	C₂×C₄×C₁₀₉	—	2	2	no	—
92	C₂×C₄×C₁₀₉	—	2	2	no	—
93	C₂×C₄×C₁₀₉	—	2	1	no	—
94	C₂×C₄×C₁₀₉	—	2	1	no	—
95	C₂×C₄×C₁₀₉	—	2	2	no	—
96	C₂×C₄×C₁₀₉	—	2	1	no	—
97	C₂×C₄×C₁₀₉	—	4	4	no	—
98	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
102	C₈×C₁₀₉	—	2	1	no	—
103	C₈×C₁₀₉	—	2	1	no	—
104	C₈×C₁₀₉	—	2	2	no	—
105	C₂²×C₁₀₉	—	2	2	no	—
106	C₂²×C₁₀₉	—	2	2	no	—
107	C₂²×C₁₀₉	—	2	2	no	—
108	C₂²×C₁₀₉	—	4	2	no	—
109	C₂²×C₁₀₉	—	4	2	no	—
110	C₂²×C₁₀₉	—	4	2	no	—
111	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
112	C₄×C₁₀₉	—	2	2	no	—
113	C₄×C₁₀₉	—	2	2	no	—
114	C₄×C₁₀₉	—	2	1	no	—
115	C₄×C₁₀₉	—	4	1	no	—
116	C₄×C₁₀₉	—	4	2	no	—
117	C₄×C₁₀₉	—	4	1	no	—
118	C₄×C₁₀₉	—	4	1	no	—
119	C₄×C₁₀₉	#135	1	2	yes	—
120	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
121	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂×C₁₀₉	—	4	2	no	—
123	C₂×C₁₀₉	—	8	2	no	—
124	C₂×C₁₀₉	#546	1	1	yes	characteristic
125	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
126	C₂×C₁₀₉	#577	1	1	yes	characteristic
127	#3304	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₁₀₉	#3304	1	1	yes	characteristic, direct product factor, Sylow
129	#556	C₂×C₁₀₉	1	1	yes	characteristic, complemented
130	#550	C₂×C₁₀₉	1	1	yes	characteristic
131	#486	C₂×C₁₀₉	1	1	yes	characteristic, complemented
132	#482	C₂×C₁₀₉	1	2	yes	complemented
133	#475	C₂×C₁₀₉	1	1	yes	characteristic, complemented
134	#465	C₂×C₁₀₉	1	2	yes	complemented
135	#460	C₂×C₁₀₉	1	1	yes	characteristic, complemented
136	#452	C₂×C₁₀₉	1	2	yes	complemented
137	C₄×Q₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
138	C₈×D₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
139	#416	C₂×C₁₀₉	1	2	yes	complemented
140	C₂×#41	C₂²×C₁₀₉	1	2	yes	—
141	C₂×Q₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
142	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
143	C₂²×C₈	C₂²×C₁₀₉	1	2	yes	—
144	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
145	#127	—	2	1	no	—
146	#127	C₂²×C₁₀₉	1	1	yes	characteristic
147	#126	—	2	2	no	—
148	#124	—	2	1	no	—
149	C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
150	#121	—	2	2	no	—
151	#121	C₂²×C₁₀₉	1	2	yes	—
152	C₄×Q₈	—	2	1	no	—
153	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
154	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
155	C₄×D₄	—	2	1	no	—
156	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
157	#117	C₂²×C₁₀₉	1	2	yes	—
158	#107	C₂²×C₁₀₉	1	1	yes	characteristic
159	#106	C₂²×C₁₀₉	1	2	yes	—
160	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
161	#103	C₂²×C₁₀₉	1	1	yes	characteristic
162	#103	C₂²×C₁₀₉	1	2	yes	—
163	#103	C₂²×C₁₀₉	1	1	yes	characteristic
164	#102	C₂²×C₁₀₉	1	2	yes	—
165	#102	C₂²×C₁₀₉	1	2	yes	—
166	#102	C₂²×C₁₀₉	1	2	yes	—
167	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
168	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
169	#41	—	2	4	no	—
170	#41	—	4	2	no	—
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
173	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	D₄×C₁₀₉	1	1	yes	characteristic
179	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
182	Q₁₆	—	2	2	no	—
183	SD₁₆	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
186	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
187	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
193	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
194	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
195	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
196	C₂²⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	—	2	2	no	—
198	C₂²⋊C₄	—	2	2	no	—
199	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
204	C₂³	—	2	2	no	—
205	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
206	Q₈	—	2	1	no	—
207	Q₈	—	2	1	no	—
208	Q₈	—	2	1	no	—
209	Q₈	—	4	2	no	—
210	Q₈	#41×C₁₀₉	1	2	yes	—
211	D₄	—	2	2	no	—
212	D₄	—	2	2	no	—
213	D₄	—	2	2	no	—
214	D₄	—	4	4	no	—
215	D₄	—	4	2	no	—
216	D₄	—	8	1	no	—
217	D₄	—	8	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
227	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
228	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
229	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	1	no	—
246	C₄	—	4	1	no	—
247	C₄	#135×C₁₀₉	1	2	yes	—
248	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
249	C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
250	C₂	—	4	2	no	—
251	C₂	—	8	2	no	—
252	C₂	#546×C₁₀₉	1	1	yes	characteristic
253	C₂	C₂×#135×C₁₀₉	1	1	yes	characteristic
254	C₂	#577×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	#550×C₁₀₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄×D₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #556×C₁₀₉	C₂	semidirect
3. #486×C₁₀₉	C₂	semidirect
4. #482×C₁₀₉	C₂	semidirect
5. #475×C₁₀₉	C₂	semidirect
6. #465×C₁₀₉	C₂	semidirect
7. #460×C₁₀₉	C₂	semidirect
8. #452×C₁₀₉	C₂	semidirect
9. C₄×Q₁₆×C₁₀₉	C₂	semidirect
10. C₈×D₄×C₁₀₉	C₂	semidirect
11. #416×C₁₀₉	C₂	semidirect
127. #3304	C₁₀₉	direct (N × H)
128. C₁₀₉	#3304	direct (N × H)
129. #556	C₂×C₁₀₉	semidirect
131. #486	C₂×C₁₀₉	semidirect
132. #482	C₂×C₁₀₉	semidirect
133. #475	C₂×C₁₀₉	semidirect
134. #465	C₂×C₁₀₉	semidirect
135. #460	C₂×C₁₀₉	semidirect
136. #452	C₂×C₁₀₉	semidirect
137. C₄×Q₁₆	C₂×C₁₀₉	semidirect
138. C₈×D₄	C₂×C₁₀₉	semidirect
139. #416	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#556×C₁₀₉	2
#550×C₁₀₉	2
#486×C₁₀₉	2
#482×C₁₀₉	2
#475×C₁₀₉	2
#465×C₁₀₉	2
#460×C₁₀₉	2
#452×C₁₀₉	2
C₄×Q₁₆×C₁₀₉	2
C₈×D₄×C₁₀₉	2
#416×C₁₀₉	2
#3304	109

Subgroups — C109×#3304 #28360558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3304 #28360558