Subgroups — C₁₀₉×#11244 #42129238

Order 27904.

670 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#430×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#423×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#344×C₁₀₉	C₂	1	2	yes	maximal
4	C₂×#339×C₁₀₉	C₂	1	2	yes	maximal
5	C₂×H(3,ℤ/4)×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₀₉	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#117×C₁₀₉	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
11	#430×C₁₀₉	—	2	4	no	—
12	C₂×#108×C₁₀₉	—	2	2	no	—
13	#423×C₁₀₉	—	2	2	no	—
14	#423×C₁₀₉	C₄	1	4	yes	complemented
15	C₂×C₄⋊C₈×C₁₀₉	—	2	2	no	—
16	#344×C₁₀₉	—	2	4	no	—
17	#339×C₁₀₉	—	2	4	no	—
18	H(3,ℤ/4)×C₁₀₉	—	2	2	no	—
19	C₂³×C₄×C₁₀₉	—	2	1	no	—
20	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₁₀₉	—	2	2	no	—
22	C₂×M₁₆×C₁₀₉	—	2	2	no	—
23	C₂×M₁₆×C₁₀₉	—	2	1	no	—
24	C₂×M₁₆×C₁₀₉	—	2	2	no	—
25	C₂×M₁₆×C₁₀₉	—	2	2	no	—
26	C₂×M₁₆×C₁₀₉	—	2	2	no	—
27	C₂×M₁₆×C₁₀₉	—	2	2	no	—
28	C₂×M₁₆×C₁₀₉	—	2	2	no	—
29	C₂×M₁₆×C₁₀₉	—	2	2	no	—
30	C₂×M₁₆×C₁₀₉	—	2	2	no	—
31	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₂×M₁₆×C₁₀₉	D₄	1	4	yes	—
34	C₂×M₁₆×C₁₀₉	D₄	1	2	yes	—
35	C₂²×C₈×C₁₀₉	—	2	1	no	—
36	C₂²×C₈×C₁₀₉	—	2	2	no	—
37	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
38	#117×C₁₀₉	—	2	2	no	—
39	#117×C₁₀₉	—	2	2	no	—
40	#117×C₁₀₉	—	2	2	no	—
41	#117×C₁₀₉	Q₈	1	4	yes	—
42	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
44	C₂×C₄²×C₁₀₉	—	2	1	no	—
45	C₂×C₄²×C₁₀₉	—	2	1	no	—
46	C₂×C₄²×C₁₀₉	—	2	1	no	—
47	C₂×C₄²×C₁₀₉	—	2	2	no	—
48	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
50	C₂×C₄²×C₁₀₉	Q₈	1	1	yes	characteristic
51	C₂×C₄²×C₁₀₉	Q₈	1	1	yes	characteristic
52	#108×C₁₀₉	—	2	8	no	—
53	C₄⋊C₈×C₁₀₉	—	2	8	no	—
54	C₂⁴×C₁₀₉	#41	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₉	—	2	1	no	—
56	C₂²×C₄×C₁₀₉	—	2	2	no	—
57	C₂²×C₄×C₁₀₉	—	2	2	no	—
58	C₂²×C₄×C₁₀₉	—	2	1	no	—
59	C₂²×C₄×C₁₀₉	—	4	4	no	—
60	C₂²×C₄×C₁₀₉	—	4	2	no	—
61	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
69	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
70	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
71	M₁₆×C₁₀₉	—	2	4	no	—
72	M₁₆×C₁₀₉	—	2	4	no	—
73	M₁₆×C₁₀₉	—	2	4	no	—
74	M₁₆×C₁₀₉	—	2	4	no	—
75	M₁₆×C₁₀₉	—	2	4	no	—
76	M₁₆×C₁₀₉	—	4	2	no	—
77	M₁₆×C₁₀₉	—	4	2	no	—
78	M₁₆×C₁₀₉	—	4	2	no	—
79	C₂×C₈×C₁₀₉	—	2	2	no	—
80	C₂×C₈×C₁₀₉	—	2	2	no	—
81	C₂×C₈×C₁₀₉	—	2	2	no	—
82	C₂×C₈×C₁₀₉	—	2	4	no	—
83	C₂×C₈×C₁₀₉	—	4	1	no	—
84	C₂×C₈×C₁₀₉	—	4	2	no	—
85	C₂×C₈×C₁₀₉	—	4	1	no	—
86	C₂×C₈×C₁₀₉	—	4	2	no	—
87	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	4	yes	—
88	C₄⋊C₄×C₁₀₉	—	2	4	no	—
89	C₄⋊C₄×C₁₀₉	—	4	2	no	—
90	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
91	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
92	C₄²×C₁₀₉	—	2	2	no	—
93	C₄²×C₁₀₉	—	2	2	no	—
94	C₄²×C₁₀₉	—	2	1	no	—
95	C₄²×C₁₀₉	—	2	1	no	—
96	C₄²×C₁₀₉	—	2	4	no	—
97	C₄²×C₁₀₉	—	4	2	no	—
98	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
99	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
100	C₂³×C₁₀₉	—	4	2	no	—
101	C₂³×C₁₀₉	#117	1	1	yes	characteristic
102	C₂³×C₁₀₉	#121	1	1	yes	characteristic
103	C₂³×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂³×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
105	C₂³×C₁₀₉	#126	1	2	yes	—
106	C₂×C₄×C₁₀₉	—	2	4	no	—
107	C₂×C₄×C₁₀₉	—	2	2	no	—
108	C₂×C₄×C₁₀₉	—	2	2	no	—
109	C₂×C₄×C₁₀₉	—	4	4	no	—
110	C₂×C₄×C₁₀₉	—	4	4	no	—
111	C₂×C₄×C₁₀₉	—	4	2	no	—
112	C₂×C₄×C₁₀₉	—	4	4	no	—
113	C₂×C₄×C₁₀₉	—	8	4	no	—
114	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
118	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
119	C₂×C₄×C₁₀₉	#121	1	2	yes	—
120	C₂×C₄×C₁₀₉	#121	1	2	yes	—
121	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
122	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	#123	1	2	yes	—
127	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
128	C₂×C₄×C₁₀₉	#125	1	2	yes	—
129	C₂×C₄×C₁₀₉	#125	1	2	yes	—
130	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
131	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
132	C₈×C₁₀₉	—	2	4	no	—
133	C₈×C₁₀₉	—	4	2	no	—
134	C₈×C₁₀₉	—	4	4	no	—
135	C₂²×C₁₀₉	—	2	2	no	—
136	C₂²×C₁₀₉	—	2	4	no	—
137	C₂²×C₁₀₉	—	4	1	no	—
138	C₂²×C₁₀₉	—	4	1	no	—
139	C₂²×C₁₀₉	—	4	2	no	—
140	C₂²×C₁₀₉	#380	1	1	yes	characteristic
141	C₂²×C₁₀₉	#387	1	2	yes	—
142	C₂²×C₁₀₉	#395	1	1	yes	characteristic
143	C₂²×C₁₀₉	#396	1	2	yes	—
144	C₂²×C₁₀₉	#400	1	1	yes	characteristic
145	C₄×C₁₀₉	—	2	4	no	—
146	C₄×C₁₀₉	—	4	2	no	—
147	C₄×C₁₀₉	—	8	4	no	—
148	C₄×C₁₀₉	#386	1	1	yes	characteristic
149	C₄×C₁₀₉	#389	1	2	yes	—
150	C₄×C₁₀₉	#391	1	1	yes	characteristic
151	C₄×C₁₀₉	#394	1	1	yes	characteristic
152	C₄×C₁₀₉	#398	1	2	yes	—
153	C₄×C₁₀₉	#398	1	1	yes	characteristic
154	#11244	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂×C₁₀₉	—	2	1	no	—
156	C₂×C₁₀₉	—	2	2	no	—
157	C₂×C₁₀₉	—	2	2	no	—
158	C₂×C₁₀₉	—	2	1	no	—
159	C₂×C₁₀₉	#1441	1	1	yes	characteristic
160	C₂×C₁₀₉	#2102	1	2	yes	—
161	C₂×#430	C₂×C₁₀₉	1	1	yes	characteristic
162	C₂×#423	C₂×C₁₀₉	1	1	yes	characteristic
163	C₂×#344	C₂×C₁₀₉	1	2	yes	—
164	C₂×#339	C₂×C₁₀₉	1	2	yes	—
165	C₂×H(3,ℤ/4)	C₂×C₁₀₉	1	1	yes	characteristic
166	C₁₀₉	#11244	1	1	yes	characteristic, direct product factor, Sylow
167	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
168	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
169	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
170	C₂×#117	C₂²×C₁₀₉	1	2	yes	—
171	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
172	#430	—	2	4	no	—
173	C₂×#108	—	2	2	no	—
174	#423	—	2	2	no	—
175	#423	C₄×C₁₀₉	1	4	yes	complemented
176	C₂×C₄⋊C₈	—	2	2	no	—
177	#344	—	2	4	no	—
178	#339	—	2	4	no	—
179	H(3,ℤ/4)	—	2	2	no	—
180	C₂³×C₄	—	2	1	no	—
181	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
193	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
194	C₂×M₁₆	D₄×C₁₀₉	1	4	yes	—
195	C₂×M₁₆	D₄×C₁₀₉	1	2	yes	—
196	C₂²×C₈	—	2	1	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	—	2	2	no	—
202	#117	Q₈×C₁₀₉	1	4	yes	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
210	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
211	C₂×C₄²	Q₈×C₁₀₉	1	1	yes	characteristic
212	C₂×C₄²	Q₈×C₁₀₉	1	1	yes	characteristic
213	#108	—	2	8	no	—
214	C₄⋊C₈	—	2	8	no	—
215	C₂⁴	#41×C₁₀₉	1	1	yes	characteristic
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
227	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
229	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
230	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
231	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	4	2	no	—
238	M₁₆	—	4	2	no	—
239	M₁₆	—	4	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	4	yes	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	4	2	no	—
259	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
260	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
261	C₂³	—	4	2	no	—
262	C₂³	#117×C₁₀₉	1	1	yes	characteristic
263	C₂³	#121×C₁₀₉	1	1	yes	characteristic
264	C₂³	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
265	C₂³	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
266	C₂³	#126×C₁₀₉	1	2	yes	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	8	4	no	—
275	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
276	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
277	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
278	C₂×C₄	C₄×Q₈×C₁₀₉	1	2	yes	—
279	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
280	C₂×C₄	#121×C₁₀₉	1	2	yes	—
281	C₂×C₄	#121×C₁₀₉	1	2	yes	—
282	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
283	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
284	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
285	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
286	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
287	C₂×C₄	#123×C₁₀₉	1	2	yes	—
288	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
289	C₂×C₄	#125×C₁₀₉	1	2	yes	—
290	C₂×C₄	#125×C₁₀₉	1	2	yes	—
291	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	2	yes	—
292	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
293	C₈	—	2	4	no	—
294	C₈	—	4	2	no	—
295	C₈	—	4	4	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	2	no	—
301	C₂²	#380×C₁₀₉	1	1	yes	characteristic
302	C₂²	#387×C₁₀₉	1	2	yes	—
303	C₂²	#395×C₁₀₉	1	1	yes	characteristic
304	C₂²	#396×C₁₀₉	1	2	yes	—
305	C₂²	#400×C₁₀₉	1	1	yes	characteristic
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	—	8	4	no	—
309	C₄	#386×C₁₀₉	1	1	yes	characteristic
310	C₄	#389×C₁₀₉	1	2	yes	—
311	C₄	#391×C₁₀₉	1	1	yes	characteristic
312	C₄	#394×C₁₀₉	1	1	yes	characteristic
313	C₄	#398×C₁₀₉	1	2	yes	—
314	C₄	#398×C₁₀₉	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	2	no	—
318	C₂	—	2	1	no	—
319	C₂	C₁₀₉×#1441	1	1	yes	characteristic
320	C₂	C₁₀₉×#2102	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₀₉	#124
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#400
prime radical	C₂⁴×C₁₀₉	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
14. #423×C₁₀₉	C₄	semidirect
154. #11244	C₁₀₉	direct (N × H)
166. C₁₀₉	#11244	direct (N × H)
175. #423	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#430×C₁₀₉	2
C₂×#423×C₁₀₉	2
C₂×#344×C₁₀₉	2
C₂×#339×C₁₀₉	2
C₂×H(3,ℤ/4)×C₁₀₉	2
#11244	109

Subgroups — C109×#11244 #42129238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11244 #42129238