Subgroups — C₁₀₉×#32519 #42150513

Order 27904.

558 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2509	C₂	1	2	yes	maximal
2	C₁₀₉×#2088	C₂	1	2	yes	maximal
3	C₁₀₉×#2088	C₂	1	2	yes	maximal
4	C₁₀₉×#2085	C₂	1	2	yes	maximal
5	C₁₀₉×#1949	C₂	1	2	yes	maximal
6	C₁₀₉×#1944	C₂	1	1	yes	maximal, characteristic
7	C₁₀₉×#1944	C₂	1	1	yes	maximal, characteristic
8	C₁₀₉×#1940	C₂	1	1	yes	maximal, characteristic
9	C₁₀₉×#1920	C₂	1	2	yes	maximal
10	C₂×#107×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#107×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×#106×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×#106×C₁₀₉	C₂²	1	2	yes	—
14	C₂×#106×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
18	#400×C₁₀₉	—	2	2	no	—
19	#400×C₁₀₉	—	2	2	no	—
20	#389×C₁₀₉	—	2	2	no	—
21	#389×C₁₀₉	C₂²	1	2	yes	—
22	#389×C₁₀₉	C₂²	1	2	yes	—
23	#389×C₁₀₉	C₂²	1	2	yes	—
24	#389×C₁₀₉	C₂²	1	2	yes	—
25	#387×C₁₀₉	—	2	2	no	—
26	#382×C₁₀₉	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
29	#340×C₁₀₉	C₂²	1	2	yes	—
30	#340×C₁₀₉	C₂²	1	2	yes	—
31	#340×C₁₀₉	C₂²	1	2	yes	—
32	#340×C₁₀₉	C₂²	1	2	yes	—
33	#340×C₁₀₉	C₂²	1	2	yes	—
34	#336×C₁₀₉	C₂²	1	2	yes	—
35	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
47	C₂×C₄²×C₁₀₉	—	2	2	no	—
48	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
50	#107×C₁₀₉	—	4	1	no	—
51	#107×C₁₀₉	—	4	1	no	—
52	#106×C₁₀₉	—	2	2	no	—
53	#106×C₁₀₉	—	4	2	no	—
54	#106×C₁₀₉	Q₈	1	4	yes	—
55	C₄⋊C₈×C₁₀₉	—	2	4	no	—
56	C₄⋊C₈×C₁₀₉	—	4	1	no	—
57	#97×C₁₀₉	—	4	1	no	—
58	#95×C₁₀₉	—	2	2	no	—
59	#95×C₁₀₉	—	2	2	no	—
60	#95×C₁₀₉	—	2	2	no	—
61	#95×C₁₀₉	—	2	2	no	—
62	#95×C₁₀₉	—	2	2	no	—
63	#95×C₁₀₉	—	2	2	no	—
64	#95×C₁₀₉	—	2	2	no	—
65	#95×C₁₀₉	D₄	1	1	yes	characteristic
66	#95×C₁₀₉	D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₉	—	2	2	no	—
68	C₂²×C₄×C₁₀₉	—	2	2	no	—
69	C₂²×C₄×C₁₀₉	—	2	2	no	—
70	C₂²×C₄×C₁₀₉	—	2	2	no	—
71	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₁₀₉	—	2	1	no	—
77	C₂×C₈×C₁₀₉	—	2	1	no	—
78	C₂×C₈×C₁₀₉	—	2	1	no	—
79	C₂×C₈×C₁₀₉	—	2	1	no	—
80	C₂×C₈×C₁₀₉	—	2	2	no	—
81	C₂×C₈×C₁₀₉	—	2	2	no	—
82	C₂×C₈×C₁₀₉	—	2	2	no	—
83	C₂×C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₁₀₉	#41	1	2	yes	—
85	C₄⋊C₄×C₁₀₉	—	2	2	no	—
86	C₄⋊C₄×C₁₀₉	—	2	2	no	—
87	C₄⋊C₄×C₁₀₉	—	2	2	no	—
88	C₄⋊C₄×C₁₀₉	—	2	2	no	—
89	C₄⋊C₄×C₁₀₉	—	2	2	no	—
90	C₄⋊C₄×C₁₀₉	—	2	2	no	—
91	C₄⋊C₄×C₁₀₉	—	2	2	no	—
92	C₄⋊C₄×C₁₀₉	—	4	2	no	—
93	C₄⋊C₄×C₁₀₉	—	4	4	no	—
94	C₄⋊C₄×C₁₀₉	—	4	4	no	—
95	C₄⋊C₄×C₁₀₉	—	8	2	no	—
96	C₄⋊C₄×C₁₀₉	—	8	2	no	—
97	C₄⋊C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
98	C₄²×C₁₀₉	—	2	2	no	—
99	C₄²×C₁₀₉	—	2	1	no	—
100	C₄²×C₁₀₉	—	2	1	no	—
101	C₄²×C₁₀₉	—	2	2	no	—
102	C₄²×C₁₀₉	—	8	2	no	—
103	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₀₉	—	2	1	no	—
105	C₂×C₄×C₁₀₉	—	2	1	no	—
106	C₂×C₄×C₁₀₉	—	2	2	no	—
107	C₂×C₄×C₁₀₉	—	2	2	no	—
108	C₂×C₄×C₁₀₉	—	2	4	no	—
109	C₂×C₄×C₁₀₉	—	2	2	no	—
110	C₂×C₄×C₁₀₉	—	2	2	no	—
111	C₂×C₄×C₁₀₉	—	2	2	no	—
112	C₂×C₄×C₁₀₉	—	2	2	no	—
113	C₂×C₄×C₁₀₉	—	2	2	no	—
114	C₂×C₄×C₁₀₉	—	2	2	no	—
115	C₂×C₄×C₁₀₉	—	4	2	no	—
116	C₂×C₄×C₁₀₉	—	4	2	no	—
117	C₂×C₄×C₁₀₉	—	4	2	no	—
118	C₂×C₄×C₁₀₉	—	4	2	no	—
119	C₂×C₄×C₁₀₉	—	4	2	no	—
120	C₂×C₄×C₁₀₉	—	4	2	no	—
121	C₂×C₄×C₁₀₉	—	4	2	no	—
122	C₂×C₄×C₁₀₉	—	4	2	no	—
123	C₂×C₄×C₁₀₉	—	4	2	no	—
124	C₂×C₄×C₁₀₉	—	4	2	no	—
125	C₂×C₄×C₁₀₉	—	4	2	no	—
126	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
130	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
131	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
132	C₈×C₁₀₉	—	2	2	no	—
133	C₈×C₁₀₉	—	2	2	no	—
134	C₈×C₁₀₉	—	4	2	no	—
135	C₂²×C₁₀₉	#537	1	1	yes	characteristic
136	C₂²×C₁₀₉	#547	1	1	yes	characteristic
137	C₂²×C₁₀₉	D₄×Q₈	1	1	yes	characteristic
138	C₂²×C₁₀₉	#575	1	2	yes	—
139	C₂²×C₁₀₉	#577	1	2	yes	—
140	C₄×C₁₀₉	—	4	2	no	—
141	C₄×C₁₀₉	—	4	2	no	—
142	C₄×C₁₀₉	—	4	4	no	—
143	C₄×C₁₀₉	—	4	1	no	—
144	C₄×C₁₀₉	—	4	1	no	—
145	C₄×C₁₀₉	—	8	2	no	—
146	C₄×C₁₀₉	—	8	2	no	—
147	C₄×C₁₀₉	—	8	2	no	—
148	C₄×C₁₀₉	#539	1	1	yes	characteristic
149	C₄×C₁₀₉	#544	1	1	yes	characteristic
150	C₄×C₁₀₉	#557	1	1	yes	characteristic
151	C₄×C₁₀₉	#564	1	1	yes	characteristic
152	#32519	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂×C₁₀₉	#2726	1	1	yes	characteristic
154	C₂×C₁₀₉	#3256	1	1	yes	characteristic
155	C₂×C₁₀₉	#3263	1	1	yes	characteristic
156	C₂×C₁₀₉	#3350	1	2	yes	—
157	C₂×C₁₀₉	#3388	1	2	yes	—
158	#2509	C₂×C₁₀₉	1	2	yes	—
159	#2088	C₂×C₁₀₉	1	2	yes	—
160	#2088	C₂×C₁₀₉	1	2	yes	—
161	#2085	C₂×C₁₀₉	1	2	yes	—
162	#1949	C₂×C₁₀₉	1	2	yes	—
163	#1944	C₂×C₁₀₉	1	1	yes	characteristic
164	#1944	C₂×C₁₀₉	1	1	yes	characteristic
165	#1940	C₂×C₁₀₉	1	1	yes	characteristic
166	#1920	C₂×C₁₀₉	1	2	yes	—
167	C₁₀₉	#32519	1	1	yes	characteristic, direct product factor, Sylow
168	C₂×#107	C₂²×C₁₀₉	1	1	yes	characteristic
169	C₂×#107	C₂²×C₁₀₉	1	1	yes	characteristic
170	C₂×#106	C₂²×C₁₀₉	1	1	yes	characteristic
171	C₂×#106	C₂²×C₁₀₉	1	2	yes	—
172	C₂×#106	C₂²×C₁₀₉	1	1	yes	characteristic
173	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
174	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
175	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
176	#400	—	2	2	no	—
177	#400	—	2	2	no	—
178	#389	—	2	2	no	—
179	#389	C₂²×C₁₀₉	1	2	yes	—
180	#389	C₂²×C₁₀₉	1	2	yes	—
181	#389	C₂²×C₁₀₉	1	2	yes	—
182	#389	C₂²×C₁₀₉	1	2	yes	—
183	#387	—	2	2	no	—
184	#382	C₂²×C₁₀₉	1	2	yes	—
185	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
186	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
187	#340	C₂²×C₁₀₉	1	2	yes	—
188	#340	C₂²×C₁₀₉	1	2	yes	—
189	#340	C₂²×C₁₀₉	1	2	yes	—
190	#340	C₂²×C₁₀₉	1	2	yes	—
191	#340	C₂²×C₁₀₉	1	2	yes	—
192	#336	C₂²×C₁₀₉	1	2	yes	—
193	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
194	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
207	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
208	#107	—	4	1	no	—
209	#107	—	4	1	no	—
210	#106	—	2	2	no	—
211	#106	—	4	2	no	—
212	#106	Q₈×C₁₀₉	1	4	yes	—
213	C₄⋊C₈	—	2	4	no	—
214	C₄⋊C₈	—	4	1	no	—
215	#97	—	4	1	no	—
216	#95	—	2	2	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	D₄×C₁₀₉	1	1	yes	characteristic
224	#95	D₄×C₁₀₉	1	1	yes	characteristic
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
232	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
233	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	C₂×Q₈×C₁₀₉	1	2	yes	—
242	C₂×C₈	#41×C₁₀₉	1	2	yes	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	8	2	no	—
254	C₄⋊C₄	—	8	2	no	—
255	C₄⋊C₄	C₂×Q₈×C₁₀₉	1	4	yes	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	8	2	no	—
261	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
285	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
286	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
287	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
288	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
289	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	2	2	no	—
292	C₈	—	4	2	no	—
293	C₂²	#537×C₁₀₉	1	1	yes	characteristic
294	C₂²	#547×C₁₀₉	1	1	yes	characteristic
295	C₂²	D₄×Q₈×C₁₀₉	1	1	yes	characteristic
296	C₂²	#575×C₁₀₉	1	2	yes	—
297	C₂²	#577×C₁₀₉	1	2	yes	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	4	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	#539×C₁₀₉	1	1	yes	characteristic
307	C₄	#544×C₁₀₉	1	1	yes	characteristic
308	C₄	#557×C₁₀₉	1	1	yes	characteristic
309	C₄	#564×C₁₀₉	1	1	yes	characteristic
310	C₂	C₁₀₉×#2726	1	1	yes	characteristic
311	C₂	C₁₀₉×#3256	1	1	yes	characteristic
312	C₂	C₁₀₉×#3263	1	1	yes	characteristic
313	C₂	C₁₀₉×#3350	1	2	yes	—
314	C₂	C₁₀₉×#3388	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₄×C₄⋊C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
152. #32519	C₁₀₉	direct (N × H)
167. C₁₀₉	#32519	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×#2509	2
C₁₀₉×#2088	2
C₁₀₉×#2085	2
C₁₀₉×#1949	2
C₁₀₉×#1944	2
C₁₀₉×#1940	2
C₁₀₉×#1920	2
#32519	109

Subgroups — C109×#32519 #42150513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#32519 #42150513