Subgroups — C₁₀₉×#3360 #28360614

Order 13952.

350 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	#556×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#554×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#488×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#487×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#479×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#476×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#463×C₁₀₉	C₂	1	2	yes	maximal
8	#461×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×SD₁₆×C₁₀₉	C₂	1	2	yes	maximal, complemented
11	C₄×D₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
12	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
13	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	#126×C₁₀₉	—	2	2	no	—
17	#125×C₁₀₉	C₂²	1	2	yes	—
18	#124×C₁₀₉	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
20	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
21	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
22	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
23	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
26	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
27	#117×C₁₀₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
29	#107×C₁₀₉	C₂²	1	1	yes	characteristic
30	#106×C₁₀₉	C₂²	1	2	yes	—
31	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
33	#103×C₁₀₉	C₂²	1	2	yes	—
34	#103×C₁₀₉	C₂²	1	2	yes	—
35	#103×C₁₀₉	C₂²	1	2	yes	—
36	#102×C₁₀₉	C₂²	1	1	yes	characteristic
37	#102×C₁₀₉	C₂²	1	1	yes	characteristic
38	#102×C₁₀₉	C₂²	1	2	yes	—
39	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
41	C₂×Q₈×C₁₀₉	—	2	1	no	—
42	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
44	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₉	—	2	1	no	—
47	C₂²×C₄×C₁₀₉	—	2	1	no	—
48	C₂²×C₄×C₁₀₉	—	2	2	no	—
49	SD₁₆×C₁₀₉	—	2	4	no	—
50	D₈×C₁₀₉	—	2	2	no	—
51	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
53	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₁₀₉	—	2	2	no	—
55	C₄⋊C₄×C₁₀₉	—	2	1	no	—
56	C₄⋊C₄×C₁₀₉	—	2	1	no	—
57	C₄⋊C₄×C₁₀₉	—	2	2	no	—
58	C₄⋊C₄×C₁₀₉	—	2	2	no	—
59	C₄⋊C₄×C₁₀₉	—	2	1	no	—
60	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
62	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
63	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
66	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
67	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
68	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
69	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
70	C₄²×C₁₀₉	—	2	2	no	—
71	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₄²×C₁₀₉	C₂³	1	2	yes	—
73	C₂³×C₁₀₉	—	2	1	no	—
74	C₂³×C₁₀₉	—	2	1	no	—
75	Q₈×C₁₀₉	—	2	2	no	—
76	Q₈×C₁₀₉	—	2	2	no	—
77	Q₈×C₁₀₉	—	2	2	no	—
78	Q₈×C₁₀₉	—	8	1	no	—
79	D₄×C₁₀₉	—	2	1	no	—
80	D₄×C₁₀₉	—	2	1	no	—
81	D₄×C₁₀₉	—	2	1	no	—
82	D₄×C₁₀₉	#41	1	2	yes	—
83	C₂×C₄×C₁₀₉	—	2	1	no	—
84	C₂×C₄×C₁₀₉	—	2	2	no	—
85	C₂×C₄×C₁₀₉	—	2	1	no	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	4	2	no	—
88	C₂×C₄×C₁₀₉	—	4	2	no	—
89	C₂×C₄×C₁₀₉	—	8	2	no	—
90	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
95	C₈×C₁₀₉	—	2	1	no	—
96	C₈×C₁₀₉	—	2	2	no	—
97	C₈×C₁₀₉	—	2	1	no	—
98	C₂²×C₁₀₉	—	2	2	no	—
99	C₂²×C₁₀₉	—	4	1	no	—
100	C₂²×C₁₀₉	—	4	1	no	—
101	C₂²×C₁₀₉	—	4	1	no	—
102	C₂²×C₁₀₉	—	4	1	no	—
103	C₂²×C₁₀₉	—	4	1	no	—
104	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₁₀₉	—	2	2	no	—
106	C₄×C₁₀₉	—	2	2	no	—
107	C₄×C₁₀₉	—	2	1	no	—
108	C₄×C₁₀₉	—	4	2	no	—
109	C₄×C₁₀₉	—	4	1	no	—
110	C₄×C₁₀₉	—	4	1	no	—
111	C₄×C₁₀₉	—	4	2	no	—
112	C₄×C₁₀₉	#135	1	2	yes	—
113	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
114	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
115	C₂×C₁₀₉	—	4	2	no	—
116	C₂×C₁₀₉	—	8	1	no	—
117	C₂×C₁₀₉	#548	1	1	yes	characteristic
118	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
119	C₂×C₁₀₉	#577	1	1	yes	characteristic
120	#3360	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₁₀₉	#3360	1	1	yes	characteristic, direct product factor, Sylow
122	#556	C₂×C₁₀₉	1	1	yes	characteristic, complemented
123	#554	C₂×C₁₀₉	1	1	yes	characteristic, complemented
124	#488	C₂×C₁₀₉	1	2	yes	complemented
125	#487	C₂×C₁₀₉	1	1	yes	characteristic, complemented
126	#479	C₂×C₁₀₉	1	2	yes	complemented
127	#476	C₂×C₁₀₉	1	1	yes	characteristic, complemented
128	#463	C₂×C₁₀₉	1	2	yes	—
129	#461	C₂×C₁₀₉	1	1	yes	characteristic, complemented
130	C₈×Q₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
131	C₄×SD₁₆	C₂×C₁₀₉	1	2	yes	complemented
132	C₄×D₈	C₂×C₁₀₉	1	1	yes	characteristic
133	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
134	C₂×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
135	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
136	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
137	#126	—	2	2	no	—
138	#125	C₂²×C₁₀₉	1	2	yes	—
139	#124	C₂²×C₁₀₉	1	2	yes	—
140	C₂²⋊Q₈	—	2	1	no	—
141	C₂²⋊Q₈	—	2	1	no	—
142	C₂²⋊Q₈	—	2	2	no	—
143	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
144	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
145	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
146	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
147	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
148	#117	—	2	1	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	#107	C₂²×C₁₀₉	1	1	yes	characteristic
151	#106	C₂²×C₁₀₉	1	2	yes	—
152	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
153	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
154	#103	C₂²×C₁₀₉	1	2	yes	—
155	#103	C₂²×C₁₀₉	1	2	yes	—
156	#103	C₂²×C₁₀₉	1	2	yes	—
157	#102	C₂²×C₁₀₉	1	1	yes	characteristic
158	#102	C₂²×C₁₀₉	1	1	yes	characteristic
159	#102	C₂²×C₁₀₉	1	2	yes	—
160	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
161	C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
162	C₂×Q₈	—	2	1	no	—
163	C₂×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
164	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
165	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
166	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	2	2	no	—
170	SD₁₆	—	2	4	no	—
171	D₈	—	2	2	no	—
172	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
174	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
182	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
183	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
184	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
185	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	2	1	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
193	C₄²	C₂³×C₁₀₉	1	2	yes	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	1	no	—
196	Q₈	—	2	2	no	—
197	Q₈	—	2	2	no	—
198	Q₈	—	2	2	no	—
199	Q₈	—	8	1	no	—
200	D₄	—	2	1	no	—
201	D₄	—	2	1	no	—
202	D₄	—	2	1	no	—
203	D₄	#41×C₁₀₉	1	2	yes	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
212	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
213	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
214	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
215	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
216	C₈	—	2	1	no	—
217	C₈	—	2	2	no	—
218	C₈	—	2	1	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	1	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	2	no	—
233	C₄	#135×C₁₀₉	1	2	yes	—
234	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
235	C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
236	C₂	—	4	2	no	—
237	C₂	—	8	1	no	—
238	C₂	#548×C₁₀₉	1	1	yes	characteristic
239	C₂	C₂×#135×C₁₀₉	1	1	yes	characteristic
240	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	C₂×D₈×C₁₀₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄×Q₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #556×C₁₀₉	C₂	semidirect
2. #554×C₁₀₉	C₂	semidirect
3. #488×C₁₀₉	C₂	semidirect
4. #487×C₁₀₉	C₂	semidirect
5. #479×C₁₀₉	C₂	semidirect
6. #476×C₁₀₉	C₂	semidirect
8. #461×C₁₀₉	C₂	semidirect
9. C₈×Q₈×C₁₀₉	C₂	semidirect
10. C₄×SD₁₆×C₁₀₉	C₂	semidirect
120. #3360	C₁₀₉	direct (N × H)
121. C₁₀₉	#3360	direct (N × H)
122. #556	C₂×C₁₀₉	semidirect
123. #554	C₂×C₁₀₉	semidirect
124. #488	C₂×C₁₀₉	semidirect
125. #487	C₂×C₁₀₉	semidirect
126. #479	C₂×C₁₀₉	semidirect
127. #476	C₂×C₁₀₉	semidirect
129. #461	C₂×C₁₀₉	semidirect
130. C₈×Q₈	C₂×C₁₀₉	semidirect
131. C₄×SD₁₆	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
#554×C₁₀₉	2
#488×C₁₀₉	2
#487×C₁₀₉	2
#479×C₁₀₉	2
#476×C₁₀₉	2
#463×C₁₀₉	2
#461×C₁₀₉	2
C₈×Q₈×C₁₀₉	2
C₄×SD₁₆×C₁₀₉	2
C₄×D₈×C₁₀₉	2
#3360	109

Subgroups — C109×#3360 #28360614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3360 #28360614