Subgroups — C₁₀₉×#12784 #42130778

Order 27904.

478 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊C₁₆×C₁₀₉	C₂	1	2	yes	maximal
4	C₁₀₉×#1374	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆×C₁₀₉	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×#117×C₁₀₉	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₁₀₉	C₂²	1	4	yes	—
9	#405×C₁₀₉	—	2	4	no	—
10	C₂×#97×C₁₀₉	C₄	1	1	yes	characteristic
11	C₂×#97×C₁₀₉	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₁₀₉	—	2	4	no	—
15	C₂²⋊C₁₆×C₁₀₉	—	2	4	no	—
16	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂×M₁₆×C₁₀₉	—	2	2	no	—
18	C₂×M₁₆×C₁₀₉	—	2	2	no	—
19	C₂×M₁₆×C₁₀₉	—	2	2	no	—
20	C₂²×C₈×C₁₀₉	—	2	2	no	—
21	C₂²×C₈×C₁₀₉	—	2	2	no	—
22	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
26	C₂²×C₈×C₁₀₉	Q₈	1	2	yes	—
27	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
28	#117×C₁₀₉	—	2	2	no	—
29	#117×C₁₀₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₀₉	C₈	1	2	yes	—
31	C₂×C₂²⋊C₄×C₁₀₉	C₈	1	2	yes	—
32	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₁₀₉	—	2	4	no	—
35	C₂×C₁₆×C₁₀₉	—	2	4	no	—
36	C₂×C₁₆×C₁₀₉	—	2	4	no	—
37	#97×C₁₀₉	—	2	2	no	—
38	#97×C₁₀₉	—	2	2	no	—
39	C₄×C₈×C₁₀₉	—	2	2	no	—
40	C₄×C₈×C₁₀₉	D₄	1	4	yes	—
41	C₂⁴×C₁₀₉	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₉	—	2	1	no	—
43	C₂²×C₄×C₁₀₉	—	2	1	no	—
44	C₂²×C₄×C₁₀₉	—	2	1	no	—
45	C₂²×C₄×C₁₀₉	—	2	1	no	—
46	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₀₉	M₁₆	1	2	yes	—
51	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
52	M₁₆×C₁₀₉	—	2	8	no	—
53	C₂×C₈×C₁₀₉	—	2	1	no	—
54	C₂×C₈×C₁₀₉	—	2	1	no	—
55	C₂×C₈×C₁₀₉	—	2	1	no	—
56	C₂×C₈×C₁₀₉	—	2	1	no	—
57	C₂×C₈×C₁₀₉	—	2	1	no	—
58	C₂×C₈×C₁₀₉	—	2	1	no	—
59	C₂×C₈×C₁₀₉	—	2	4	no	—
60	C₂×C₈×C₁₀₉	—	2	4	no	—
61	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₀₉	M₁₆	1	2	yes	—
64	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₁₀₉	#41	1	2	yes	—
68	C₄⋊C₄×C₁₀₉	—	2	4	no	—
69	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
70	C₄²×C₁₀₉	—	2	1	no	—
71	C₄²×C₁₀₉	—	2	1	no	—
72	C₄²×C₁₀₉	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₁₀₉	M₁₆	1	2	yes	—
74	C₁₆×C₁₀₉	—	4	4	no	—
75	C₂³×C₁₀₉	—	2	1	no	—
76	C₂³×C₁₀₉	—	2	1	no	—
77	C₂³×C₁₀₉	—	2	4	no	—
78	C₂³×C₁₀₉	C₄⋊C₈	1	2	yes	—
79	C₂³×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₀₉	—	2	1	no	—
81	C₂×C₄×C₁₀₉	—	2	1	no	—
82	C₂×C₄×C₁₀₉	—	2	4	no	—
83	C₂×C₄×C₁₀₉	—	2	2	no	—
84	C₂×C₄×C₁₀₉	—	2	2	no	—
85	C₂×C₄×C₁₀₉	—	2	2	no	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	2	2	no	—
88	C₂×C₄×C₁₀₉	—	2	4	no	—
89	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	4	yes	—
90	C₂×C₄×C₁₀₉	C₄⋊C₈	1	2	yes	—
91	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
94	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
97	C₈×C₁₀₉	—	2	2	no	—
98	C₈×C₁₀₉	—	2	2	no	—
99	C₈×C₁₀₉	—	2	2	no	—
100	C₈×C₁₀₉	#121	1	2	yes	—
101	C₈×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
102	C₂²×C₁₀₉	—	2	2	no	—
103	C₂²×C₁₀₉	—	2	4	no	—
104	C₂²×C₁₀₉	—	4	2	no	—
105	C₂²×C₁₀₉	—	4	2	no	—
106	C₂²×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
107	C₂²×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
108	C₂²×C₁₀₉	#431	1	1	yes	characteristic
109	C₂²×C₁₀₉	#504	1	4	yes	—
110	C₄×C₁₀₉	—	2	2	no	—
111	C₄×C₁₀₉	—	2	4	no	—
112	C₄×C₁₀₉	—	4	2	no	—
113	C₄×C₁₀₉	#380	1	1	yes	characteristic
114	C₄×C₁₀₉	#407	1	1	yes	characteristic
115	C₄×C₁₀₉	#423	1	1	yes	characteristic
116	C₄×C₁₀₉	#432	1	1	yes	characteristic
117	#12784	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂×C₁₀₉	—	4	2	no	—
119	C₂×C₁₀₉	#1797	1	1	yes	characteristic
120	C₂×C₁₀₉	#2113	1	2	yes	—
121	C₂×C₁₀₉	#2151	1	2	yes	—
122	C₂×C₁₀₉	#2164	1	2	yes	—
123	C₂×#405	C₂×C₁₀₉	1	1	yes	characteristic
124	C₂×C₄⋊C₁₆	C₂×C₁₀₉	1	2	yes	complemented
125	C₂×C₂²⋊C₁₆	C₂×C₁₀₉	1	2	yes	—
126	#1374	C₂×C₁₀₉	1	2	yes	complemented
127	C₁₀₉	#12784	1	1	yes	characteristic, direct product factor, Sylow
128	C₂²×M₁₆	C₄×C₁₀₉	1	1	yes	characteristic
129	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
130	C₂×#117	C₄×C₁₀₉	1	1	yes	characteristic
131	C₂²×C₁₆	C₂²×C₁₀₉	1	4	yes	—
132	#405	—	2	4	no	—
133	C₂×#97	C₄×C₁₀₉	1	1	yes	characteristic
134	C₂×#97	C₄×C₁₀₉	1	1	yes	characteristic
135	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
136	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
137	C₄⋊C₁₆	—	2	4	no	—
138	C₂²⋊C₁₆	—	2	4	no	—
139	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	2	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
146	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
147	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
148	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
149	C₂²×C₈	Q₈×C₁₀₉	1	2	yes	—
150	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
151	#117	—	2	2	no	—
152	#117	—	2	2	no	—
153	C₂×C₄⋊C₄	C₈×C₁₀₉	1	2	yes	—
154	C₂×C₂²⋊C₄	C₈×C₁₀₉	1	2	yes	—
155	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
156	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
157	C₂×C₁₆	—	2	4	no	—
158	C₂×C₁₆	—	2	4	no	—
159	C₂×C₁₆	—	2	4	no	—
160	#97	—	2	2	no	—
161	#97	—	2	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	D₄×C₁₀₉	1	4	yes	—
164	C₂⁴	C₂×C₈×C₁₀₉	1	1	yes	characteristic
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
170	C₂²×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
171	C₂²×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
172	C₂²×C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
173	C₂²×C₄	M₁₆×C₁₀₉	1	2	yes	—
174	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
175	M₁₆	—	2	8	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
185	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
186	C₂×C₈	M₁₆×C₁₀₉	1	2	yes	—
187	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
188	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
189	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
190	C₂×C₈	#41×C₁₀₉	1	2	yes	—
191	C₄⋊C₄	—	2	4	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	C₂²⋊C₄×C₁₀₉	1	2	yes	—
196	C₄²	M₁₆×C₁₀₉	1	2	yes	—
197	C₁₆	—	4	4	no	—
198	C₂³	—	2	4	no	—
199	C₂³	—	2	1	no	—
200	C₂³	—	2	1	no	—
201	C₂³	C₄⋊C₈×C₁₀₉	1	2	yes	—
202	C₂³	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	4	yes	—
213	C₂×C₄	C₄⋊C₈×C₁₀₉	1	2	yes	—
214	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
216	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
217	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
218	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
219	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	#121×C₁₀₉	1	2	yes	—
224	C₈	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
230	C₂²	C₂×C₄⋊C₈×C₁₀₉	1	1	yes	characteristic
231	C₂²	#431×C₁₀₉	1	1	yes	characteristic
232	C₂²	#504×C₁₀₉	1	4	yes	—
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	2	no	—
236	C₄	#380×C₁₀₉	1	1	yes	characteristic
237	C₄	#407×C₁₀₉	1	1	yes	characteristic
238	C₄	#423×C₁₀₉	1	1	yes	characteristic
239	C₄	#432×C₁₀₉	1	1	yes	characteristic
240	C₂	—	4	2	no	—
241	C₂	C₁₀₉×#1797	1	1	yes	characteristic
242	C₂	C₁₀₉×#2113	1	2	yes	—
243	C₂	C₁₀₉×#2151	1	2	yes	—
244	C₂	C₁₀₉×#2164	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₁₀₉	C₂³
commutator	C₂³	C₂²×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂³×C₁₀₉	C₂²×C₈
prime radical	C₂⁴×C₁₀₉	C₂×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊C₁₆×C₁₀₉	C₂	semidirect
4. C₁₀₉×#1374	C₂	semidirect
117. #12784	C₁₀₉	direct (N × H)
124. C₂×C₄⋊C₁₆	C₂×C₁₀₉	semidirect
126. #1374	C₂×C₁₀₉	semidirect
127. C₁₀₉	#12784	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#405×C₁₀₉	2
C₂×C₄⋊C₁₆×C₁₀₉	2
C₂×C₂²⋊C₁₆×C₁₀₉	2
C₁₀₉×#1374	2
#12784	109

Subgroups — C109×#12784 #42130778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12784 #42130778