Subgroups — C₁₀₉×#3108 #28360362

Order 13952.

390 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#529×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#483×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#480×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#479×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#464×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#451×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#448×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₄×Q₁₆×C₁₀₉	C₂	1	2	yes	maximal, complemented
10	C₄×SD₁₆×C₁₀₉	C₂	1	2	yes	maximal, complemented
11	#431×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
15	#126×C₁₀₉	—	2	1	no	—
16	#125×C₁₀₉	C₂²	1	1	yes	characteristic
17	#124×C₁₀₉	C₂²	1	1	yes	characteristic
18	#123×C₁₀₉	—	2	1	no	—
19	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	#121×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₁₀₉	—	2	1	no	—
22	C₄×Q₈×C₁₀₉	—	2	1	no	—
23	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
24	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
25	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₀₉	—	2	1	no	—
27	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
28	#117×C₁₀₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₁₀₉	—	2	1	no	—
32	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
33	#107×C₁₀₉	C₂²	1	2	yes	—
34	#106×C₁₀₉	C₂²	1	2	yes	—
35	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
36	#103×C₁₀₉	C₂²	1	2	yes	—
37	#103×C₁₀₉	C₂²	1	2	yes	—
38	#103×C₁₀₉	C₂²	1	2	yes	—
39	#102×C₁₀₉	C₂²	1	2	yes	—
40	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
43	C₂×Q₈×C₁₀₉	—	2	4	no	—
44	C₂×Q₈×C₁₀₉	—	2	1	no	—
45	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
47	C₂×D₄×C₁₀₉	—	2	1	no	—
48	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	—	2	1	no	—
50	C₂²×C₄×C₁₀₉	—	2	1	no	—
51	C₂²×C₄×C₁₀₉	—	2	1	no	—
52	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
55	Q₁₆×C₁₀₉	—	2	4	no	—
56	SD₁₆×C₁₀₉	—	2	4	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	1	no	—
62	C₄⋊C₄×C₁₀₉	—	2	1	no	—
63	C₄⋊C₄×C₁₀₉	—	2	1	no	—
64	C₄⋊C₄×C₁₀₉	—	2	1	no	—
65	C₄⋊C₄×C₁₀₉	—	2	1	no	—
66	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
67	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
70	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
71	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
72	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
73	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
74	C₄²×C₁₀₉	—	2	2	no	—
75	C₄²×C₁₀₉	—	2	2	no	—
76	C₄²×C₁₀₉	—	2	1	no	—
77	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
78	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
79	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
80	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
81	C₂³×C₁₀₉	—	2	1	no	—
82	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₁₀₉	—	2	4	no	—
84	Q₈×C₁₀₉	—	2	2	no	—
85	Q₈×C₁₀₉	—	2	2	no	—
86	Q₈×C₁₀₉	#41	1	4	yes	—
87	D₄×C₁₀₉	—	2	2	no	—
88	D₄×C₁₀₉	—	4	2	no	—
89	C₂×C₄×C₁₀₉	—	2	4	no	—
90	C₂×C₄×C₁₀₉	—	2	2	no	—
91	C₂×C₄×C₁₀₉	—	2	2	no	—
92	C₂×C₄×C₁₀₉	—	2	2	no	—
93	C₂×C₄×C₁₀₉	—	2	2	no	—
94	C₂×C₄×C₁₀₉	—	2	2	no	—
95	C₂×C₄×C₁₀₉	—	2	1	no	—
96	C₂×C₄×C₁₀₉	—	2	1	no	—
97	C₂×C₄×C₁₀₉	—	2	1	no	—
98	C₂×C₄×C₁₀₉	—	4	2	no	—
99	C₂×C₄×C₁₀₉	—	4	2	no	—
100	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
106	C₈×C₁₀₉	—	2	2	no	—
107	C₈×C₁₀₉	—	2	2	no	—
108	C₂²×C₁₀₉	—	2	1	no	—
109	C₂²×C₁₀₉	—	2	1	no	—
110	C₂²×C₁₀₉	—	4	1	no	—
111	C₂²×C₁₀₉	—	4	1	no	—
112	C₂²×C₁₀₉	—	4	1	no	—
113	C₂²×C₁₀₉	#135	1	2	yes	—
114	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₁₀₉	—	2	4	no	—
116	C₄×C₁₀₉	—	2	1	no	—
117	C₄×C₁₀₉	—	2	1	no	—
118	C₄×C₁₀₉	—	2	1	no	—
119	C₄×C₁₀₉	—	2	1	no	—
120	C₄×C₁₀₉	—	4	1	no	—
121	C₄×C₁₀₉	—	4	2	no	—
122	C₄×C₁₀₉	—	4	1	no	—
123	C₄×C₁₀₉	—	4	1	no	—
124	C₄×C₁₀₉	#137	1	2	yes	—
125	C₄×C₁₀₉	C₂×#41	1	2	yes	—
126	C₂×C₁₀₉	—	2	2	no	—
127	C₂×C₁₀₉	—	8	1	no	—
128	C₂×C₁₀₉	#525	1	1	yes	characteristic
129	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
130	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
131	#3108	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₁₀₉	#3108	1	1	yes	characteristic, direct product factor, Sylow
133	#529	C₂×C₁₀₉	1	1	yes	characteristic
134	C₂×C₄×Q₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
135	#483	C₂×C₁₀₉	1	1	yes	characteristic, complemented
136	#480	C₂×C₁₀₉	1	1	yes	characteristic
137	#479	C₂×C₁₀₉	1	2	yes	complemented
138	#464	C₂×C₁₀₉	1	2	yes	complemented
139	#451	C₂×C₁₀₉	1	1	yes	characteristic, complemented
140	#448	C₂×C₁₀₉	1	1	yes	characteristic
141	C₄×Q₁₆	C₂×C₁₀₉	1	2	yes	complemented
142	C₄×SD₁₆	C₂×C₁₀₉	1	2	yes	complemented
143	#431	C₂×C₁₀₉	1	1	yes	characteristic, complemented
144	C₂²×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
145	C₂×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
146	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
147	#126	—	2	1	no	—
148	#125	C₂²×C₁₀₉	1	1	yes	characteristic
149	#124	C₂²×C₁₀₉	1	1	yes	characteristic
150	#123	—	2	1	no	—
151	C₂²⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
152	#121	C₂²×C₁₀₉	1	1	yes	characteristic
153	C₄×Q₈	—	2	1	no	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
156	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
157	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
160	#117	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
165	#107	C₂²×C₁₀₉	1	2	yes	—
166	#106	C₂²×C₁₀₉	1	2	yes	—
167	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
168	#103	C₂²×C₁₀₉	1	2	yes	—
169	#103	C₂²×C₁₀₉	1	2	yes	—
170	#103	C₂²×C₁₀₉	1	2	yes	—
171	#102	C₂²×C₁₀₉	1	2	yes	—
172	C₂²⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
173	C₂²⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
174	C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	2	1	no	—
177	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
178	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
185	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
186	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
187	Q₁₆	—	2	4	no	—
188	SD₁₆	—	2	4	no	—
189	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
190	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
199	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
200	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
201	C₂²⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
210	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
211	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
212	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
213	C₂³	—	2	1	no	—
214	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	Q₈	—	2	2	no	—
217	Q₈	—	2	2	no	—
218	Q₈	#41×C₁₀₉	1	4	yes	—
219	D₄	—	2	2	no	—
220	D₄	—	4	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
233	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
234	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
235	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
237	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	#135×C₁₀₉	1	2	yes	—
246	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
247	C₄	—	2	4	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	1	no	—
256	C₄	#137×C₁₀₉	1	2	yes	—
257	C₄	C₂×#41×C₁₀₉	1	2	yes	—
258	C₂	—	2	2	no	—
259	C₂	—	8	1	no	—
260	C₂	#525×C₁₀₉	1	1	yes	characteristic
261	C₂	C₂×#135×C₁₀₉	1	1	yes	characteristic
262	C₂	C₂×#137×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#121×C₁₀₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₈×C₁₀₉	C₂	semidirect
3. #483×C₁₀₉	C₂	semidirect
5. #479×C₁₀₉	C₂	semidirect
6. #464×C₁₀₉	C₂	semidirect
7. #451×C₁₀₉	C₂	semidirect
9. C₄×Q₁₆×C₁₀₉	C₂	semidirect
10. C₄×SD₁₆×C₁₀₉	C₂	semidirect
11. #431×C₁₀₉	C₂	semidirect
131. #3108	C₁₀₉	direct (N × H)
132. C₁₀₉	#3108	direct (N × H)
134. C₂×C₄×Q₈	C₂×C₁₀₉	semidirect
135. #483	C₂×C₁₀₉	semidirect
137. #479	C₂×C₁₀₉	semidirect
138. #464	C₂×C₁₀₉	semidirect
139. #451	C₂×C₁₀₉	semidirect
141. C₄×Q₁₆	C₂×C₁₀₉	semidirect
142. C₄×SD₁₆	C₂×C₁₀₉	semidirect
143. #431	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#529×C₁₀₉	2
C₂×C₄×Q₈×C₁₀₉	2
#483×C₁₀₉	2
#480×C₁₀₉	2
#479×C₁₀₉	2
#464×C₁₀₉	2
#451×C₁₀₉	2
#448×C₁₀₉	2
C₄×Q₁₆×C₁₀₉	2
C₄×SD₁₆×C₁₀₉	2
#431×C₁₀₉	2
#3108	109

Subgroups — C109×#3108 #28360362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3108 #28360362