Subgroups — C₁₀₉×#33420 #42151414

Order 27904.

598 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#131×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2164	C₂	1	4	yes	maximal, complemented
3	C₁₀₉×#2162	C₂	1	4	yes	maximal, complemented
4	C₁₀₉×#2151	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#2113	C₂	1	2	yes	maximal
6	C₄×M₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₁₀₉	C₄	1	1	yes	characteristic
9	C₂×#131×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₄×#41×C₁₀₉	C₄	1	1	yes	characteristic
11	C₂×M₃₂×C₁₀₉	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₁₀₉	C₂²	1	2	yes	—
13	#405×C₁₀₉	C₄	1	2	yes	—
14	#405×C₁₀₉	C₄	1	1	yes	characteristic
15	#405×C₁₀₉	C₂²	1	1	yes	characteristic
16	#405×C₁₀₉	C₂²	1	2	yes	—
17	C₄×M₁₆×C₁₀₉	C₄	1	1	yes	characteristic
18	C₄×M₁₆×C₁₀₉	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
21	C₄⋊C₁₆×C₁₀₉	C₂²	1	8	yes	—
22	C₂²⋊C₁₆×C₁₀₉	C₂²	1	8	yes	—
23	#346×C₁₀₉	C₂²	1	2	yes	—
24	C₄×C₁₆×C₁₀₉	C₂²	1	2	yes	—
25	C₄×C₁₆×C₁₀₉	C₂²	1	2	yes	—
26	C₄×C₁₆×C₁₀₉	C₂²	1	2	yes	—
27	C₂×#41×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	#131×C₁₀₉	—	2	4	no	—
29	#131×C₁₀₉	—	2	4	no	—
30	C₂×M₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₄×Q₈×C₁₀₉	C₈	1	2	yes	—
39	C₄×D₄×C₁₀₉	C₈	1	4	yes	—
40	C₄×D₄×C₁₀₉	C₈	1	2	yes	—
41	#117×C₁₀₉	C₂×C₄	1	2	yes	—
42	#117×C₁₀₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
45	M₃₂×C₁₀₉	—	2	4	no	—
46	C₂×C₁₆×C₁₀₉	—	2	4	no	—
47	C₂×C₁₆×C₁₀₉	C₂³	1	4	yes	—
48	C₂×C₁₆×C₁₀₉	C₂³	1	4	yes	—
49	#97×C₁₀₉	C₂×C₄	1	2	yes	—
50	#97×C₁₀₉	C₂×C₄	1	2	yes	—
51	#97×C₁₀₉	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₁₀₉	C₂³	1	2	yes	—
58	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
59	#41×C₁₀₉	—	2	2	no	—
60	#41×C₁₀₉	—	2	2	no	—
61	C₂×Q₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
62	C₂×D₄×C₁₀₉	C₂×C₈	1	2	yes	—
63	C₂×D₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₉	C₂×C₈	1	4	yes	—
65	C₂²×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	M₁₆×C₁₀₉	—	2	2	no	—
69	M₁₆×C₁₀₉	—	2	2	no	—
70	M₁₆×C₁₀₉	—	2	4	no	—
71	M₁₆×C₁₀₉	—	2	4	no	—
72	C₂×C₈×C₁₀₉	—	2	4	no	—
73	C₂×C₈×C₁₀₉	—	2	4	no	—
74	C₂×C₈×C₁₀₉	M₁₆	1	4	yes	—
75	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₀₉	#41	1	4	yes	—
81	C₂×C₈×C₁₀₉	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₁₀₉	C₂×C₈	1	4	yes	—
83	C₄⋊C₄×C₁₀₉	C₂×C₈	1	2	yes	—
84	C₂²⋊C₄×C₁₀₉	C₂×C₈	1	4	yes	—
85	C₂²⋊C₄×C₁₀₉	C₂×C₈	1	2	yes	—
86	C₄²×C₁₀₉	C₂×C₈	1	4	yes	—
87	C₄²×C₁₀₉	C₂×C₈	1	2	yes	—
88	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
91	C₁₆×C₁₀₉	—	2	4	no	—
92	C₁₆×C₁₀₉	—	2	4	no	—
93	C₂³×C₁₀₉	C₂²×C₈	1	2	yes	—
94	C₂³×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
95	Q₈×C₁₀₉	—	2	2	no	—
96	D₄×C₁₀₉	—	2	4	no	—
97	D₄×C₁₀₉	—	2	2	no	—
98	C₂×C₄×C₁₀₉	—	2	2	no	—
99	C₂×C₄×C₁₀₉	—	2	2	no	—
100	C₂×C₄×C₁₀₉	—	2	4	no	—
101	C₂×C₄×C₁₀₉	—	2	8	no	—
102	C₂×C₄×C₁₀₉	#117	1	2	yes	—
103	C₂×C₄×C₁₀₉	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₁₀₉	C₂²×C₈	1	4	yes	—
105	C₂×C₄×C₁₀₉	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₀₉	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₁₀₉	—	2	2	no	—
111	C₈×C₁₀₉	—	2	2	no	—
112	C₈×C₁₀₉	#117	1	2	yes	—
113	C₈×C₁₀₉	C₂×M₁₆	1	2	yes	—
114	C₈×C₁₀₉	C₂×M₁₆	1	2	yes	—
115	C₈×C₁₀₉	C₂×#41	1	2	yes	—
116	C₂²×C₁₀₉	—	2	2	no	—
117	C₂²×C₁₀₉	—	2	4	no	—
118	C₂²×C₁₀₉	—	2	2	no	—
119	C₂²×C₁₀₉	#431	1	2	yes	—
120	C₂²×C₁₀₉	C₂³×C₈	1	1	yes	characteristic
121	C₄×C₁₀₉	—	2	2	no	—
122	C₄×C₁₀₉	—	2	4	no	—
123	C₄×C₁₀₉	—	2	2	no	—
124	C₄×C₁₀₉	#431	1	2	yes	—
125	C₄×C₁₀₉	#504	1	4	yes	—
126	C₄×C₁₀₉	C₂×#117	1	1	yes	characteristic
127	C₄×C₁₀₉	C₂²×M₁₆	1	1	yes	characteristic
128	#33420	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₁₀₉	—	2	2	no	—
130	C₂×C₁₀₉	—	4	2	no	—
131	C₂×C₁₀₉	C₂×#431	1	1	yes	characteristic
132	C₂×C₁₀₉	C₂×#504	1	2	yes	—
133	C₄×#131	C₂×C₁₀₉	1	1	yes	characteristic
134	#2164	C₂×C₁₀₉	1	4	yes	complemented
135	#2162	C₂×C₁₀₉	1	4	yes	complemented
136	#2151	C₂×C₁₀₉	1	2	yes	complemented
137	#2113	C₂×C₁₀₉	1	2	yes	—
138	C₄×M₃₂	C₂×C₁₀₉	1	1	yes	characteristic, complemented
139	C₂×C₄×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
140	C₁₀₉	#33420	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×#131	C₄×C₁₀₉	1	1	yes	characteristic
142	C₂×#131	C₂²×C₁₀₉	1	1	yes	characteristic
143	C₄×#41	C₄×C₁₀₉	1	1	yes	characteristic
144	C₂×M₃₂	C₂²×C₁₀₉	1	2	yes	—
145	C₂²×C₁₆	C₂²×C₁₀₉	1	2	yes	—
146	#405	C₄×C₁₀₉	1	2	yes	—
147	#405	C₄×C₁₀₉	1	1	yes	characteristic
148	#405	C₂²×C₁₀₉	1	1	yes	characteristic
149	#405	C₂²×C₁₀₉	1	2	yes	—
150	C₄×M₁₆	C₄×C₁₀₉	1	1	yes	characteristic
151	C₄×M₁₆	C₄×C₁₀₉	1	2	yes	—
152	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
154	C₄⋊C₁₆	C₂²×C₁₀₉	1	8	yes	—
155	C₂²⋊C₁₆	C₂²×C₁₀₉	1	8	yes	—
156	#346	C₂²×C₁₀₉	1	2	yes	—
157	C₄×C₁₆	C₂²×C₁₀₉	1	2	yes	—
158	C₄×C₁₆	C₂²×C₁₀₉	1	2	yes	—
159	C₄×C₁₆	C₂²×C₁₀₉	1	2	yes	—
160	C₂×#41	C₂×C₄×C₁₀₉	1	1	yes	characteristic
161	#131	—	2	4	no	—
162	#131	—	2	4	no	—
163	C₂×M₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
164	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
165	C₂×M₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
166	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
167	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
168	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
170	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
171	C₄×Q₈	C₈×C₁₀₉	1	2	yes	—
172	C₄×D₄	C₈×C₁₀₉	1	4	yes	—
173	C₄×D₄	C₈×C₁₀₉	1	2	yes	—
174	#117	C₂×C₄×C₁₀₉	1	2	yes	—
175	#117	C₂×C₄×C₁₀₉	1	1	yes	characteristic
176	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
177	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
178	M₃₂	—	2	4	no	—
179	C₂×C₁₆	—	2	4	no	—
180	C₂×C₁₆	C₂³×C₁₀₉	1	4	yes	—
181	C₂×C₁₆	C₂³×C₁₀₉	1	4	yes	—
182	#97	C₂×C₄×C₁₀₉	1	2	yes	—
183	#97	C₂×C₄×C₁₀₉	1	2	yes	—
184	#97	C₂×C₄×C₁₀₉	1	2	yes	—
185	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
186	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
187	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
188	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
189	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
190	C₄×C₈	C₂³×C₁₀₉	1	2	yes	—
191	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	C₂×Q₈	C₂×C₈×C₁₀₉	1	1	yes	characteristic
195	C₂×D₄	C₂×C₈×C₁₀₉	1	2	yes	—
196	C₂×D₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₈×C₁₀₉	1	4	yes	—
198	C₂²×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
199	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
200	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	M₁₆×C₁₀₉	1	4	yes	—
208	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
209	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
210	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
211	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
212	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
213	C₂×C₈	#41×C₁₀₉	1	4	yes	—
214	C₂×C₈	C₂⁴×C₁₀₉	1	1	yes	characteristic
215	C₄⋊C₄	C₂×C₈×C₁₀₉	1	4	yes	—
216	C₄⋊C₄	C₂×C₈×C₁₀₉	1	2	yes	—
217	C₂²⋊C₄	C₂×C₈×C₁₀₉	1	4	yes	—
218	C₂²⋊C₄	C₂×C₈×C₁₀₉	1	2	yes	—
219	C₄²	C₂×C₈×C₁₀₉	1	4	yes	—
220	C₄²	C₂×C₈×C₁₀₉	1	2	yes	—
221	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
222	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
223	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
224	C₁₆	—	2	4	no	—
225	C₁₆	—	2	4	no	—
226	C₂³	C₂²×C₈×C₁₀₉	1	2	yes	—
227	C₂³	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
228	Q₈	—	2	2	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	#117×C₁₀₉	1	2	yes	—
236	C₂×C₄	C₂²×C₈×C₁₀₉	1	2	yes	—
237	C₂×C₄	C₂²×C₈×C₁₀₉	1	4	yes	—
238	C₂×C₄	C₂²×C₈×C₁₀₉	1	2	yes	—
239	C₂×C₄	C₂²×C₈×C₁₀₉	1	2	yes	—
240	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
241	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
242	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	#117×C₁₀₉	1	2	yes	—
246	C₈	C₂×M₁₆×C₁₀₉	1	2	yes	—
247	C₈	C₂×M₁₆×C₁₀₉	1	2	yes	—
248	C₈	C₂×#41×C₁₀₉	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	#431×C₁₀₉	1	2	yes	—
253	C₂²	C₂³×C₈×C₁₀₉	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	4	no	—
256	C₄	—	2	2	no	—
257	C₄	#431×C₁₀₉	1	2	yes	—
258	C₄	#504×C₁₀₉	1	4	yes	—
259	C₄	C₂×#117×C₁₀₉	1	1	yes	characteristic
260	C₄	C₂²×M₁₆×C₁₀₉	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	4	2	no	—
263	C₂	C₂×#431×C₁₀₉	1	1	yes	characteristic
264	C₂	C₂×#504×C₁₀₉	1	2	yes	—

	subgroup	G/H
center	C₄×C₈×C₁₀₉	C₂³
commutator	C₂²	C₂³×C₈×C₁₀₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₀₉
socle	C₂²×C₁₀₉	C₂³×C₈
prime radical	C₂×#41×C₁₀₉	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₉×#2164	C₂	semidirect
3. C₁₀₉×#2162	C₂	semidirect
4. C₁₀₉×#2151	C₂	semidirect
6. C₄×M₃₂×C₁₀₉	C₂	semidirect
7. C₂×C₄×C₁₆×C₁₀₉	C₂	semidirect
128. #33420	C₁₀₉	direct (N × H)
134. #2164	C₂×C₁₀₉	semidirect
135. #2162	C₂×C₁₀₉	semidirect
136. #2151	C₂×C₁₀₉	semidirect
138. C₄×M₃₂	C₂×C₁₀₉	semidirect
139. C₂×C₄×C₁₆	C₂×C₁₀₉	semidirect
140. C₁₀₉	#33420	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#131×C₁₀₉	2
C₁₀₉×#2164	2
C₁₀₉×#2162	2
C₁₀₉×#2151	2
C₁₀₉×#2113	2
C₄×M₃₂×C₁₀₉	2
C₂×C₄×C₁₆×C₁₀₉	2
#33420	109

Subgroups — C109×#33420 #42151414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#33420 #42151414