Subgroups — C₁₀₉×#3302 #28360556

Order 13952.

386 conjugacy classes of subgroups.

The subgroup lattice has 252 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	#547×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#487×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#485×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#474×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#467×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#462×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#452×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₄×SD₁₆×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	2	yes	—
14	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	#126×C₁₀₉	—	2	2	no	—
19	#124×C₁₀₉	—	2	1	no	—
20	#123×C₁₀₉	—	2	2	no	—
21	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
22	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
23	#121×C₁₀₉	—	2	1	no	—
24	C₄×Q₈×C₁₀₉	—	2	1	no	—
25	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
28	#117×C₁₀₉	C₂²	1	2	yes	—
29	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
30	#107×C₁₀₉	C₂²	1	2	yes	—
31	#106×C₁₀₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
33	#103×C₁₀₉	C₂²	1	2	yes	—
34	#103×C₁₀₉	C₂²	1	2	yes	—
35	#103×C₁₀₉	C₂²	1	1	yes	characteristic
36	#103×C₁₀₉	C₂²	1	2	yes	—
37	#102×C₁₀₉	C₂²	1	1	yes	characteristic
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	2	yes	—
45	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₁₀₉	—	2	2	no	—
47	C₂×D₄×C₁₀₉	D₄	1	1	yes	characteristic
48	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	—	2	1	no	—
50	C₂²×C₄×C₁₀₉	—	2	2	no	—
51	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
52	Q₁₆×C₁₀₉	—	4	2	no	—
53	SD₁₆×C₁₀₉	—	2	2	no	—
54	C₂×C₈×C₁₀₉	—	4	2	no	—
55	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
58	C₄⋊C₄×C₁₀₉	—	2	1	no	—
59	C₄⋊C₄×C₁₀₉	—	2	2	no	—
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	1	yes	characteristic
66	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
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	—	2	1	no	—
72	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
73	C₄²×C₁₀₉	—	2	2	no	—
74	C₄²×C₁₀₉	—	2	1	no	—
75	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₂³×C₁₀₉	—	2	1	no	—
77	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
78	Q₈×C₁₀₉	—	2	1	no	—
79	Q₈×C₁₀₉	—	2	2	no	—
80	Q₈×C₁₀₉	—	2	1	no	—
81	Q₈×C₁₀₉	—	2	2	no	—
82	Q₈×C₁₀₉	—	4	2	no	—
83	D₄×C₁₀₉	—	2	2	no	—
84	D₄×C₁₀₉	—	2	1	no	—
85	D₄×C₁₀₉	—	4	4	no	—
86	D₄×C₁₀₉	#41	1	2	yes	—
87	C₂×C₄×C₁₀₉	—	2	2	no	—
88	C₂×C₄×C₁₀₉	—	2	2	no	—
89	C₂×C₄×C₁₀₉	—	2	1	no	—
90	C₂×C₄×C₁₀₉	—	2	1	no	—
91	C₂×C₄×C₁₀₉	—	2	2	no	—
92	C₂×C₄×C₁₀₉	—	2	1	no	—
93	C₂×C₄×C₁₀₉	—	2	2	no	—
94	C₂×C₄×C₁₀₉	—	4	4	no	—
95	C₂×C₄×C₁₀₉	—	4	2	no	—
96	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
100	C₈×C₁₀₉	—	2	1	no	—
101	C₈×C₁₀₉	—	2	1	no	—
102	C₈×C₁₀₉	—	2	2	no	—
103	C₂²×C₁₀₉	—	2	2	no	—
104	C₂²×C₁₀₉	—	2	2	no	—
105	C₂²×C₁₀₉	—	2	2	no	—
106	C₂²×C₁₀₉	—	2	2	no	—
107	C₂²×C₁₀₉	—	4	1	no	—
108	C₂²×C₁₀₉	—	4	1	no	—
109	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
110	C₄×C₁₀₉	—	2	2	no	—
111	C₄×C₁₀₉	—	2	1	no	—
112	C₄×C₁₀₉	—	4	2	no	—
113	C₄×C₁₀₉	—	4	1	no	—
114	C₄×C₁₀₉	—	4	1	no	—
115	C₄×C₁₀₉	—	4	2	no	—
116	C₄×C₁₀₉	—	4	1	no	—
117	C₄×C₁₀₉	#135	1	2	yes	—
118	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
119	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂×C₁₀₉	—	4	2	no	—
121	C₂×C₁₀₉	—	4	2	no	—
122	C₂×C₁₀₉	#546	1	1	yes	characteristic
123	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
124	C₂×C₁₀₉	#578	1	1	yes	characteristic
125	#3302	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₁₀₉	#3302	1	1	yes	characteristic, direct product factor, Sylow
127	#556	C₂×C₁₀₉	1	1	yes	characteristic, complemented
128	#547	C₂×C₁₀₉	1	1	yes	characteristic
129	#487	C₂×C₁₀₉	1	1	yes	characteristic, complemented
130	#485	C₂×C₁₀₉	1	2	yes	complemented
131	#474	C₂×C₁₀₉	1	1	yes	characteristic, complemented
132	#467	C₂×C₁₀₉	1	2	yes	complemented
133	#462	C₂×C₁₀₉	1	1	yes	characteristic, complemented
134	#452	C₂×C₁₀₉	1	2	yes	complemented
135	C₄×SD₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
136	C₈×D₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
137	#416	C₂×C₁₀₉	1	2	yes	complemented
138	C₂×#41	C₂²×C₁₀₉	1	2	yes	—
139	C₂×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
140	C₂×SD₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
141	C₂²×C₈	C₂²×C₁₀₉	1	2	yes	—
142	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
143	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
144	#126	—	2	2	no	—
145	#124	—	2	1	no	—
146	#123	—	2	2	no	—
147	C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
148	C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
149	#121	—	2	1	no	—
150	C₄×Q₈	—	2	1	no	—
151	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
152	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
153	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
154	#117	C₂²×C₁₀₉	1	2	yes	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	#107	C₂²×C₁₀₉	1	2	yes	—
157	#106	C₂²×C₁₀₉	1	1	yes	characteristic
158	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
159	#103	C₂²×C₁₀₉	1	2	yes	—
160	#103	C₂²×C₁₀₉	1	2	yes	—
161	#103	C₂²×C₁₀₉	1	1	yes	characteristic
162	#103	C₂²×C₁₀₉	1	2	yes	—
163	#102	C₂²×C₁₀₉	1	1	yes	characteristic
164	#102	C₂²×C₁₀₉	1	2	yes	—
165	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
166	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
167	#41	—	2	4	no	—
168	#41	—	4	2	no	—
169	C₂×Q₈	—	2	1	no	—
170	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
171	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	D₄×C₁₀₉	1	1	yes	characteristic
174	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
178	Q₁₆	—	4	2	no	—
179	SD₁₆	—	2	2	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
182	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
183	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
191	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
192	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
193	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	—	2	1	no	—
198	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	1	no	—
201	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
202	C₂³	—	2	1	no	—
203	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
204	Q₈	—	2	1	no	—
205	Q₈	—	2	2	no	—
206	Q₈	—	2	1	no	—
207	Q₈	—	2	2	no	—
208	Q₈	—	4	2	no	—
209	D₄	—	2	2	no	—
210	D₄	—	2	1	no	—
211	D₄	—	4	4	no	—
212	D₄	#41×C₁₀₉	1	2	yes	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
223	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
224	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
225	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
226	C₈	—	2	1	no	—
227	C₈	—	2	1	no	—
228	C₈	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	1	no	—
235	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	1	no	—
240	C₄	—	4	1	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	1	no	—
243	C₄	#135×C₁₀₉	1	2	yes	—
244	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
245	C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
246	C₂	—	4	2	no	—
247	C₂	—	4	2	no	—
248	C₂	#546×C₁₀₉	1	1	yes	characteristic
249	C₂	C₂×#135×C₁₀₉	1	1	yes	characteristic
250	C₂	#578×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	#547×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. #487×C₁₀₉	C₂	semidirect
4. #485×C₁₀₉	C₂	semidirect
5. #474×C₁₀₉	C₂	semidirect
6. #467×C₁₀₉	C₂	semidirect
7. #462×C₁₀₉	C₂	semidirect
8. #452×C₁₀₉	C₂	semidirect
9. C₄×SD₁₆×C₁₀₉	C₂	semidirect
10. C₈×D₄×C₁₀₉	C₂	semidirect
11. #416×C₁₀₉	C₂	semidirect
125. #3302	C₁₀₉	direct (N × H)
126. C₁₀₉	#3302	direct (N × H)
127. #556	C₂×C₁₀₉	semidirect
129. #487	C₂×C₁₀₉	semidirect
130. #485	C₂×C₁₀₉	semidirect
131. #474	C₂×C₁₀₉	semidirect
132. #467	C₂×C₁₀₉	semidirect
133. #462	C₂×C₁₀₉	semidirect
134. #452	C₂×C₁₀₉	semidirect
135. C₄×SD₁₆	C₂×C₁₀₉	semidirect
136. C₈×D₄	C₂×C₁₀₉	semidirect
137. #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
#547×C₁₀₉	2
#487×C₁₀₉	2
#485×C₁₀₉	2
#474×C₁₀₉	2
#467×C₁₀₉	2
#462×C₁₀₉	2
#452×C₁₀₉	2
C₄×SD₁₆×C₁₀₉	2
C₈×D₄×C₁₀₉	2
#416×C₁₀₉	2
#3302	109

Subgroups — C109×#3302 #28360556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3302 #28360556