Subgroups — C₂×C₁₀₉×#1350 #42129702

Order 27904.

446 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#467×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#341×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1350	C₂	1	4	yes	maximal, direct product factor
5	C₂²×Q₁₆×C₁₀₉	C₄	1	1	yes	characteristic
6	C₂²×M₁₆×C₁₀₉	—	2	1	no	—
7	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
9	C₂²×C₁₆×C₁₀₉	—	2	1	no	—
10	#467×C₁₀₉	—	2	1	no	—
11	#467×C₁₀₉	—	2	1	no	—
12	#467×C₁₀₉	C₄	1	2	yes	—
13	#467×C₁₀₉	C₂²	1	2	yes	—
14	C₂×#107×C₁₀₉	C₄	1	1	yes	characteristic
15	C₂×#103×C₁₀₉	—	2	1	no	—
16	C₂²⋊C₁₆×C₁₀₉	—	2	1	no	—
17	C₂²⋊C₁₆×C₁₀₉	C₂²	1	2	yes	—
18	#341×C₁₀₉	—	2	1	no	—
19	#341×C₁₀₉	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₀₉	—	2	1	no	—
21	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
22	C₂×Q₁₆×C₁₀₉	—	2	1	no	—
23	C₂×Q₁₆×C₁₀₉	—	4	2	no	—
24	C₂×Q₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₁₀₉	—	2	4	no	—
26	C₂×M₁₆×C₁₀₉	—	2	2	no	—
27	C₂×M₁₆×C₁₀₉	—	4	2	no	—
28	C₂×M₁₆×C₁₀₉	—	4	1	no	—
29	C₂²×C₈×C₁₀₉	—	4	2	no	—
30	C₂²×C₈×C₁₀₉	—	4	1	no	—
31	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
38	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
39	C₂²⋊Q₈×C₁₀₉	—	4	1	no	—
40	C₂²⋊Q₈×C₁₀₉	—	4	1	no	—
41	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
42	C₂×C₄⋊C₄×C₁₀₉	—	4	1	no	—
43	C₂×C₂²⋊C₄×C₁₀₉	—	4	1	no	—
44	C₂×C₁₆×C₁₀₉	—	2	2	no	—
45	C₂×C₁₆×C₁₀₉	—	4	2	no	—
46	#107×C₁₀₉	—	2	1	no	—
47	#107×C₁₀₉	C₂×C₄	1	2	yes	—
48	#103×C₁₀₉	—	2	2	no	—
49	#103×C₁₀₉	—	2	2	no	—
50	C₂⁴×C₁₀₉	SD₁₆	1	1	yes	characteristic
51	C₂×Q₈×C₁₀₉	—	2	2	no	—
52	C₂×Q₈×C₁₀₉	—	4	1	no	—
53	C₂×Q₈×C₁₀₉	—	4	2	no	—
54	C₂×Q₈×C₁₀₉	—	4	2	no	—
55	C₂²×C₄×C₁₀₉	—	2	2	no	—
56	C₂²×C₄×C₁₀₉	—	2	2	no	—
57	C₂²×C₄×C₁₀₉	—	4	1	no	—
58	C₂²×C₄×C₁₀₉	—	4	1	no	—
59	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₉	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₉	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
65	Q₁₆×C₁₀₉	—	4	2	no	—
66	Q₁₆×C₁₀₉	—	4	2	no	—
67	M₁₆×C₁₀₉	—	2	4	no	—
68	M₁₆×C₁₀₉	—	4	4	no	—
69	M₁₆×C₁₀₉	—	4	2	no	—
70	C₂×C₈×C₁₀₉	—	2	2	no	—
71	C₂×C₈×C₁₀₉	—	2	2	no	—
72	C₂×C₈×C₁₀₉	—	4	1	no	—
73	C₂×C₈×C₁₀₉	—	4	1	no	—
74	C₂×C₈×C₁₀₉	—	4	1	no	—
75	C₂×C₈×C₁₀₉	—	4	1	no	—
76	C₂×C₈×C₁₀₉	—	4	4	no	—
77	C₂×C₈×C₁₀₉	—	4	2	no	—
78	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
82	C₄⋊C₄×C₁₀₉	—	2	2	no	—
83	C₄⋊C₄×C₁₀₉	—	4	1	no	—
84	C₄⋊C₄×C₁₀₉	—	4	2	no	—
85	C₄⋊C₄×C₁₀₉	—	4	2	no	—
86	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
87	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
88	C₁₆×C₁₀₉	—	4	2	no	—
89	C₂³×C₁₀₉	—	2	2	no	—
90	C₂³×C₁₀₉	—	2	2	no	—
91	C₂³×C₁₀₉	—	2	2	no	—
92	C₂³×C₁₀₉	#102	1	1	yes	characteristic
93	C₂³×C₁₀₉	#102	1	1	yes	characteristic
94	C₂³×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
95	Q₈×C₁₀₉	—	4	2	no	—
96	Q₈×C₁₀₉	—	4	2	no	—
97	Q₈×C₁₀₉	—	8	2	no	—
98	C₂×C₄×C₁₀₉	—	2	2	no	—
99	C₂×C₄×C₁₀₉	—	2	2	no	—
100	C₂×C₄×C₁₀₉	—	2	2	no	—
101	C₂×C₄×C₁₀₉	—	2	1	no	—
102	C₂×C₄×C₁₀₉	—	2	1	no	—
103	C₂×C₄×C₁₀₉	—	2	1	no	—
104	C₂×C₄×C₁₀₉	—	2	1	no	—
105	C₂×C₄×C₁₀₉	—	4	2	no	—
106	C₂×C₄×C₁₀₉	—	4	2	no	—
107	C₂×C₄×C₁₀₉	—	8	2	no	—
108	C₂×C₄×C₁₀₉	—	8	2	no	—
109	C₂×C₄×C₁₀₉	#99	1	2	yes	—
110	C₂×C₄×C₁₀₉	#102	1	1	yes	characteristic
111	C₂×C₄×C₁₀₉	#102	1	1	yes	characteristic
112	C₂×C₄×C₁₀₉	#104	1	2	yes	—
113	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
115	C₈×C₁₀₉	—	2	1	no	—
116	C₈×C₁₀₉	—	2	1	no	—
117	C₈×C₁₀₉	—	2	1	no	—
118	C₈×C₁₀₉	—	2	1	no	—
119	C₈×C₁₀₉	—	4	4	no	—
120	C₂²×C₁₀₉	—	2	1	no	—
121	C₂²×C₁₀₉	—	2	1	no	—
122	C₂²×C₁₀₉	—	2	1	no	—
123	C₂²×C₁₀₉	—	2	1	no	—
124	C₂²×C₁₀₉	—	2	1	no	—
125	C₂²×C₁₀₉	—	2	1	no	—
126	C₂²×C₁₀₉	—	4	4	no	—
127	C₂²×C₁₀₉	#327	1	2	yes	—
128	C₂²×C₁₀₉	#359	1	2	yes	—
129	C₂²×C₁₀₉	#362	1	2	yes	—
130	C₂²×C₁₀₉	C₂×#102	1	1	yes	characteristic
131	C₄×C₁₀₉	—	2	1	no	—
132	C₄×C₁₀₉	—	2	1	no	—
133	C₄×C₁₀₉	—	8	2	no	—
134	C₄×C₁₀₉	—	8	2	no	—
135	C₄×C₁₀₉	#327	1	2	yes	—
136	C₄×C₁₀₉	C₂×#99	1	1	yes	characteristic
137	C₄×C₁₀₉	C₂×#104	1	1	yes	characteristic
138	C₂×#1350	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂×C₁₀₉	—	4	1	no	—
140	C₂×C₁₀₉	—	4	1	no	—
141	C₂×C₁₀₉	#1350	1	4	yes	direct product factor
142	C₂×C₁₀₉	C₂×#327	1	1	yes	characteristic
143	C₂×C₁₀₉	C₂×#359	1	1	yes	characteristic
144	C₂×C₁₀₉	C₂×#362	1	1	yes	characteristic
145	C₂×#467	C₂×C₁₀₉	1	1	yes	characteristic
146	C₂×C₂²⋊C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
147	C₂×#341	C₂×C₁₀₉	1	1	yes	characteristic
148	#1350	C₂×C₁₀₉	1	4	yes	direct product factor
149	C₁₀₉	C₂×#1350	1	1	yes	characteristic, direct product factor, Sylow
150	C₂²×Q₁₆	C₄×C₁₀₉	1	1	yes	characteristic
151	C₂²×M₁₆	—	2	1	no	—
152	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
153	C₂×C₂²⋊Q₈	—	2	1	no	—
154	C₂²×C₁₆	—	2	1	no	—
155	#467	—	2	1	no	—
156	#467	—	2	1	no	—
157	#467	C₄×C₁₀₉	1	2	yes	—
158	#467	C₂²×C₁₀₉	1	2	yes	—
159	C₂×#107	C₄×C₁₀₉	1	1	yes	characteristic
160	C₂×#103	—	2	1	no	—
161	C₂²⋊C₁₆	—	2	1	no	—
162	C₂²⋊C₁₆	C₂²×C₁₀₉	1	2	yes	—
163	#341	—	2	1	no	—
164	#341	C₂²×C₁₀₉	1	2	yes	—
165	C₂²×Q₈	—	2	1	no	—
166	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
167	C₂×Q₁₆	—	2	1	no	—
168	C₂×Q₁₆	—	4	2	no	—
169	C₂×Q₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	4	2	no	—
173	C₂×M₁₆	—	4	1	no	—
174	C₂²×C₈	—	4	2	no	—
175	C₂²×C₈	—	4	1	no	—
176	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
177	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
178	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
179	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
180	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
182	C₂²⋊Q₈	—	2	2	no	—
183	C₂²⋊Q₈	—	2	2	no	—
184	C₂²⋊Q₈	—	4	1	no	—
185	C₂²⋊Q₈	—	4	1	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	4	1	no	—
188	C₂×C₂²⋊C₄	—	4	1	no	—
189	C₂×C₁₆	—	2	2	no	—
190	C₂×C₁₆	—	4	2	no	—
191	#107	—	2	1	no	—
192	#107	C₂×C₄×C₁₀₉	1	2	yes	—
193	#103	—	2	2	no	—
194	#103	—	2	2	no	—
195	C₂⁴	SD₁₆×C₁₀₉	1	1	yes	characteristic
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂×Q₈	—	4	2	no	—
199	C₂×Q₈	—	4	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	—	4	1	no	—
204	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
205	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
206	C₂²×C₄	D₈×C₁₀₉	1	1	yes	characteristic
207	C₂²×C₄	D₈×C₁₀₉	1	1	yes	characteristic
208	C₂²×C₄	SD₁₆×C₁₀₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
210	Q₁₆	—	4	2	no	—
211	Q₁₆	—	4	2	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	4	4	no	—
214	M₁₆	—	4	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
226	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₂²⋊C₄	—	4	2	no	—
233	C₁₆	—	4	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	#102×C₁₀₉	1	1	yes	characteristic
238	C₂³	#102×C₁₀₉	1	1	yes	characteristic
239	C₂³	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
240	Q₈	—	4	2	no	—
241	Q₈	—	4	2	no	—
242	Q₈	—	8	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	#99×C₁₀₉	1	2	yes	—
255	C₂×C₄	#102×C₁₀₉	1	1	yes	characteristic
256	C₂×C₄	#102×C₁₀₉	1	1	yes	characteristic
257	C₂×C₄	#104×C₁₀₉	1	2	yes	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
259	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	4	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	4	4	no	—
272	C₂²	#327×C₁₀₉	1	2	yes	—
273	C₂²	#359×C₁₀₉	1	2	yes	—
274	C₂²	#362×C₁₀₉	1	2	yes	—
275	C₂²	C₂×#102×C₁₀₉	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	#327×C₁₀₉	1	2	yes	—
281	C₄	C₂×#99×C₁₀₉	1	1	yes	characteristic
282	C₄	C₂×#104×C₁₀₉	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	4	1	no	—
285	C₂	C₁₀₉×#1350	1	4	yes	direct product factor
286	C₂	C₂×#327×C₁₀₉	1	1	yes	characteristic
287	C₂	C₂×#359×C₁₀₉	1	1	yes	characteristic
288	C₂	C₂×#362×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₉	#102
commutator	C₂×C₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
prime radical	C₂⁴×C₁₀₉	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ C₂³×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₀₉×#1350	C₂	direct (N × H)
138. C₂×#1350	C₁₀₉	direct (N × H)
141. C₂×C₁₀₉	#1350	direct (N × H)
148. #1350	C₂×C₁₀₉	direct (N × H)
149. C₁₀₉	C₂×#1350	direct (N × H)
285. C₂	C₁₀₉×#1350	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×#467×C₁₀₉	2
C₂×C₂²⋊C₁₆×C₁₀₉	2
C₂×#341×C₁₀₉	2
C₁₀₉×#1350	2
C₂×#1350	109

Subgroups — C2×C109×#1350 #42129702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₉×#1350 #42129702