Subgroups — C₁₀₉×#3221 #28360475

Order 13952.

388 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#540×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#536×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#496×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#492×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#491×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#488×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#463×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	#461×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	#449×C₁₀₉	C₂	1	2	yes	maximal, complemented
10	#448×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×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
15	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
16	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
17	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
18	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	#127×C₁₀₉	C₂²	1	1	yes	characteristic
21	#125×C₁₀₉	C₂²	1	1	yes	characteristic
22	#124×C₁₀₉	C₂²	1	1	yes	characteristic
23	#124×C₁₀₉	C₂²	1	1	yes	characteristic
24	#123×C₁₀₉	—	2	2	no	—
25	#123×C₁₀₉	—	2	2	no	—
26	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
27	#121×C₁₀₉	—	2	2	no	—
28	#121×C₁₀₉	C₂²	1	2	yes	—
29	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
30	#117×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
32	#103×C₁₀₉	C₂²	1	2	yes	—
33	#103×C₁₀₉	C₂²	1	2	yes	—
34	#103×C₁₀₉	C₂²	1	2	yes	—
35	#102×C₁₀₉	C₂²	1	2	yes	—
36	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
37	#97×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
39	#41×C₁₀₉	—	4	2	no	—
40	C₂×Q₈×C₁₀₉	—	2	1	no	—
41	C₂×Q₈×C₁₀₉	—	4	2	no	—
42	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₀₉	—	2	1	no	—
46	C₂×D₄×C₁₀₉	—	2	2	no	—
47	C₂×D₄×C₁₀₉	—	2	2	no	—
48	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₀₉	—	2	1	no	—
51	C₂²×C₄×C₁₀₉	—	2	1	no	—
52	C₂²×C₄×C₁₀₉	—	2	2	no	—
53	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
54	SD₁₆×C₁₀₉	—	4	2	no	—
55	SD₁₆×C₁₀₉	—	4	2	no	—
56	SD₁₆×C₁₀₉	—	4	2	no	—
57	D₈×C₁₀₉	—	4	2	no	—
58	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
59	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
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	—	2	2	no	—
69	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
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	2	no	—
74	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
75	Q₈×C₁₀₉	—	2	2	no	—
76	Q₈×C₁₀₉	—	2	2	no	—
77	Q₈×C₁₀₉	—	2	2	no	—
78	Q₈×C₁₀₉	—	4	2	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	—	8	2	no	—
84	D₄×C₁₀₉	—	8	2	no	—
85	C₂×C₄×C₁₀₉	—	2	2	no	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	2	1	no	—
88	C₂×C₄×C₁₀₉	—	2	1	no	—
89	C₂×C₄×C₁₀₉	—	2	1	no	—
90	C₂×C₄×C₁₀₉	—	4	2	no	—
91	C₂×C₄×C₁₀₉	—	4	2	no	—
92	C₂×C₄×C₁₀₉	—	8	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	2	no	—
103	C₂²×C₁₀₉	—	4	2	no	—
104	C₂²×C₁₀₉	—	4	2	no	—
105	C₂²×C₁₀₉	—	4	2	no	—
106	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₁₀₉	—	2	1	no	—
108	C₄×C₁₀₉	—	2	2	no	—
109	C₄×C₁₀₉	—	2	2	no	—
110	C₄×C₁₀₉	—	4	1	no	—
111	C₄×C₁₀₉	—	4	1	no	—
112	C₄×C₁₀₉	—	4	1	no	—
113	C₄×C₁₀₉	—	4	2	no	—
114	C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
115	C₂×C₁₀₉	—	4	1	no	—
116	C₂×C₁₀₉	—	8	1	no	—
117	C₂×C₁₀₉	—	8	2	no	—
118	C₂×C₁₀₉	#535	1	1	yes	characteristic
119	C₂×C₁₀₉	#577	1	2	yes	—
120	#3221	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₁₀₉	#3221	1	1	yes	characteristic, direct product factor, Sylow
122	#540	C₂×C₁₀₉	1	1	yes	characteristic, complemented
123	#536	C₂×C₁₀₉	1	1	yes	characteristic, complemented
124	#496	C₂×C₁₀₉	1	1	yes	characteristic, complemented
125	#492	C₂×C₁₀₉	1	1	yes	characteristic, complemented
126	#491	C₂×C₁₀₉	1	1	yes	characteristic, complemented
127	#488	C₂×C₁₀₉	1	1	yes	characteristic, complemented
128	#463	C₂×C₁₀₉	1	2	yes	complemented
129	#461	C₂×C₁₀₉	1	2	yes	complemented
130	#449	C₂×C₁₀₉	1	2	yes	complemented
131	#448	C₂×C₁₀₉	1	2	yes	complemented
132	#433	C₂×C₁₀₉	1	1	yes	characteristic, complemented
133	C₂×#41	C₂²×C₁₀₉	1	1	yes	characteristic
134	C₂²×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
135	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
136	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
137	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
138	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
139	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
140	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
141	#127	C₂²×C₁₀₉	1	1	yes	characteristic
142	#125	C₂²×C₁₀₉	1	1	yes	characteristic
143	#124	C₂²×C₁₀₉	1	1	yes	characteristic
144	#124	C₂²×C₁₀₉	1	1	yes	characteristic
145	#123	—	2	2	no	—
146	#123	—	2	2	no	—
147	C₂²⋊Q₈	—	2	2	no	—
148	#121	—	2	2	no	—
149	#121	C₂²×C₁₀₉	1	2	yes	—
150	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
151	#117	C₂²×C₁₀₉	1	1	yes	characteristic
152	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
153	#103	C₂²×C₁₀₉	1	2	yes	—
154	#103	C₂²×C₁₀₉	1	2	yes	—
155	#103	C₂²×C₁₀₉	1	2	yes	—
156	#102	C₂²×C₁₀₉	1	2	yes	—
157	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
158	#97	C₂²×C₁₀₉	1	1	yes	characteristic
159	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
160	#41	—	4	2	no	—
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
164	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
165	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
170	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
175	SD₁₆	—	4	2	no	—
176	SD₁₆	—	4	2	no	—
177	SD₁₆	—	4	2	no	—
178	D₈	—	4	2	no	—
179	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
180	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
185	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
191	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
192	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
193	C₂³	—	2	1	no	—
194	C₂³	—	2	2	no	—
195	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
196	Q₈	—	2	2	no	—
197	Q₈	—	2	2	no	—
198	Q₈	—	2	2	no	—
199	Q₈	—	4	2	no	—
200	D₄	—	2	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	2	2	no	—
203	D₄	—	4	2	no	—
204	D₄	—	8	2	no	—
205	D₄	—	8	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
215	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
216	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
217	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	2	2	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	2	no	—
227	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	2	no	—
235	C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
236	C₂	—	4	1	no	—
237	C₂	—	8	1	no	—
238	C₂	—	8	2	no	—
239	C₂	#535×C₁₀₉	1	1	yes	characteristic
240	C₂	#577×C₁₀₉	1	2	yes	—

	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 ◁ #117×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #540×C₁₀₉	C₂	semidirect
2. #536×C₁₀₉	C₂	semidirect
3. #496×C₁₀₉	C₂	semidirect
4. #492×C₁₀₉	C₂	semidirect
5. #491×C₁₀₉	C₂	semidirect
6. #488×C₁₀₉	C₂	semidirect
7. #463×C₁₀₉	C₂	semidirect
8. #461×C₁₀₉	C₂	semidirect
9. #449×C₁₀₉	C₂	semidirect
10. #448×C₁₀₉	C₂	semidirect
11. #433×C₁₀₉	C₂	semidirect
120. #3221	C₁₀₉	direct (N × H)
121. C₁₀₉	#3221	direct (N × H)
122. #540	C₂×C₁₀₉	semidirect
123. #536	C₂×C₁₀₉	semidirect
124. #496	C₂×C₁₀₉	semidirect
125. #492	C₂×C₁₀₉	semidirect
126. #491	C₂×C₁₀₉	semidirect
127. #488	C₂×C₁₀₉	semidirect
128. #463	C₂×C₁₀₉	semidirect
129. #461	C₂×C₁₀₉	semidirect
130. #449	C₂×C₁₀₉	semidirect
131. #448	C₂×C₁₀₉	semidirect
132. #433	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#540×C₁₀₉	2
#536×C₁₀₉	2
#496×C₁₀₉	2
#492×C₁₀₉	2
#491×C₁₀₉	2
#488×C₁₀₉	2
#463×C₁₀₉	2
#461×C₁₀₉	2
#449×C₁₀₉	2
#448×C₁₀₉	2
#433×C₁₀₉	2
#3221	109

Subgroups — C109×#3221 #28360475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3221 #28360475