Subgroups — C₁₀₉×#12985 #42130979

Order 27904.

510 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#435×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#349×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#2112	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1377	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₁₀₉	C₄	1	1	yes	characteristic
9	C₂×M₃₂×C₁₀₉	—	2	1	no	—
10	C₂×M₃₂×C₁₀₉	—	2	4	no	—
11	C₂×M₃₂×C₁₀₉	C₂²	1	2	yes	—
12	C₂×M₃₂×C₁₀₉	C₂²	1	2	yes	—
13	C₂²×C₁₆×C₁₀₉	—	2	1	no	—
14	#435×C₁₀₉	—	2	1	no	—
15	#435×C₁₀₉	—	2	1	no	—
16	#435×C₁₀₉	—	2	1	no	—
17	#435×C₁₀₉	—	2	1	no	—
18	C₂×C₄⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
19	C₂×C₂²⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
21	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
22	#349×C₁₀₉	—	2	4	no	—
23	C₂²⋊C₁₆×C₁₀₉	—	2	1	no	—
24	C₂²⋊C₁₆×C₁₀₉	—	2	1	no	—
25	C₂²×D₄×C₁₀₉	C₈	1	1	yes	characteristic
26	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₁₀₉	—	2	2	no	—
29	C₂×M₁₆×C₁₀₉	—	2	1	no	—
30	C₂×M₁₆×C₁₀₉	—	2	1	no	—
31	C₂×M₁₆×C₁₀₉	—	2	2	no	—
32	C₂²×C₈×C₁₀₉	—	2	2	no	—
33	C₂²×C₈×C₁₀₉	—	2	2	no	—
34	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₄×D₄×C₁₀₉	—	2	1	no	—
43	C₄×D₄×C₁₀₉	—	2	1	no	—
44	C₄×D₄×C₁₀₉	—	2	1	no	—
45	C₄×D₄×C₁₀₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₁₀₉	C₈	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₁₀₉	C₈	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₁₀₉	C₈	1	1	yes	characteristic
49	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
50	M₃₂×C₁₀₉	—	2	4	no	—
51	M₃₂×C₁₀₉	—	2	2	no	—
52	M₃₂×C₁₀₉	—	2	2	no	—
53	M₃₂×C₁₀₉	—	2	2	no	—
54	M₃₂×C₁₀₉	D₄	1	4	yes	complemented
55	C₂×C₁₆×C₁₀₉	—	2	4	no	—
56	C₂×C₁₆×C₁₀₉	—	2	2	no	—
57	C₂×C₁₆×C₁₀₉	—	2	2	no	—
58	C₄⋊C₈×C₁₀₉	—	2	1	no	—
59	C₄⋊C₈×C₁₀₉	—	2	1	no	—
60	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
61	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
62	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
63	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
64	#97×C₁₀₉	—	2	1	no	—
65	#97×C₁₀₉	—	2	1	no	—
66	C₂⁴×C₁₀₉	C₂×C₈	1	1	yes	characteristic
67	C₂⁴×C₁₀₉	C₂×C₈	1	1	yes	characteristic
68	C₂×D₄×C₁₀₉	—	2	1	no	—
69	C₂×D₄×C₁₀₉	—	2	1	no	—
70	C₂×D₄×C₁₀₉	—	4	2	no	—
71	C₂²×C₄×C₁₀₉	—	2	2	no	—
72	C₂²×C₄×C₁₀₉	—	2	2	no	—
73	C₂²×C₄×C₁₀₉	—	2	1	no	—
74	C₂²×C₄×C₁₀₉	—	2	2	no	—
75	C₂²×C₄×C₁₀₉	—	2	1	no	—
76	C₂²×C₄×C₁₀₉	—	2	2	no	—
77	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
84	C₂²×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
85	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
86	M₁₆×C₁₀₉	—	4	1	no	—
87	M₁₆×C₁₀₉	—	4	2	no	—
88	M₁₆×C₁₀₉	—	4	1	no	—
89	C₂×C₈×C₁₀₉	—	2	2	no	—
90	C₂×C₈×C₁₀₉	—	2	1	no	—
91	C₂×C₈×C₁₀₉	—	2	1	no	—
92	C₂×C₈×C₁₀₉	—	2	2	no	—
93	C₂×C₈×C₁₀₉	—	2	2	no	—
94	C₂×C₈×C₁₀₉	—	2	1	no	—
95	C₂×C₈×C₁₀₉	—	2	1	no	—
96	C₂×C₈×C₁₀₉	—	2	1	no	—
97	C₂×C₈×C₁₀₉	—	2	1	no	—
98	C₂×C₈×C₁₀₉	—	2	2	no	—
99	C₂×C₈×C₁₀₉	—	2	2	no	—
100	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₀₉	#41	1	2	yes	—
106	C₄⋊C₄×C₁₀₉	—	2	1	no	—
107	C₄⋊C₄×C₁₀₉	—	2	1	no	—
108	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
109	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
110	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
111	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
112	C₄²×C₁₀₉	—	2	1	no	—
113	C₄²×C₁₀₉	—	2	1	no	—
114	C₁₆×C₁₀₉	—	2	4	no	—
115	C₁₆×C₁₀₉	—	4	2	no	—
116	C₂³×C₁₀₉	—	2	2	no	—
117	C₂³×C₁₀₉	—	2	2	no	—
118	C₂³×C₁₀₉	—	2	1	no	—
119	C₂³×C₁₀₉	—	2	2	no	—
120	C₂³×C₁₀₉	—	4	1	no	—
121	C₂³×C₁₀₉	—	4	1	no	—
122	C₂³×C₁₀₉	—	4	1	no	—
123	C₂³×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
124	C₂³×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
125	C₂³×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
126	D₄×C₁₀₉	—	4	2	no	—
127	D₄×C₁₀₉	—	4	2	no	—
128	C₂×C₄×C₁₀₉	—	2	2	no	—
129	C₂×C₄×C₁₀₉	—	2	1	no	—
130	C₂×C₄×C₁₀₉	—	2	1	no	—
131	C₂×C₄×C₁₀₉	—	2	1	no	—
132	C₂×C₄×C₁₀₉	—	2	1	no	—
133	C₂×C₄×C₁₀₉	—	2	2	no	—
134	C₂×C₄×C₁₀₉	—	2	2	no	—
135	C₂×C₄×C₁₀₉	—	2	1	no	—
136	C₂×C₄×C₁₀₉	—	2	1	no	—
137	C₂×C₄×C₁₀₉	—	2	1	no	—
138	C₂×C₄×C₁₀₉	—	2	2	no	—
139	C₂×C₄×C₁₀₉	—	4	1	no	—
140	C₂×C₄×C₁₀₉	—	4	1	no	—
141	C₂×C₄×C₁₀₉	—	4	2	no	—
142	C₂×C₄×C₁₀₉	—	4	1	no	—
143	C₂×C₄×C₁₀₉	—	4	1	no	—
144	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
145	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
146	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
148	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
149	C₂×C₄×C₁₀₉	#131	1	2	yes	—
150	C₈×C₁₀₉	—	2	1	no	—
151	C₈×C₁₀₉	—	2	1	no	—
152	C₈×C₁₀₉	—	4	1	no	—
153	C₈×C₁₀₉	—	4	1	no	—
154	C₈×C₁₀₉	#120	1	1	yes	characteristic
155	C₈×C₁₀₉	#121	1	2	yes	—
156	C₈×C₁₀₉	#123	1	1	yes	characteristic
157	C₂²×C₁₀₉	—	2	1	no	—
158	C₂²×C₁₀₉	—	2	1	no	—
159	C₂²×C₁₀₉	—	2	1	no	—
160	C₂²×C₁₀₉	—	2	1	no	—
161	C₂²×C₁₀₉	—	2	1	no	—
162	C₂²×C₁₀₉	—	2	1	no	—
163	C₂²×C₁₀₉	—	2	1	no	—
164	C₂²×C₁₀₉	—	4	4	no	—
165	C₂²×C₁₀₉	—	4	1	no	—
166	C₂²×C₁₀₉	—	4	1	no	—
167	C₂²×C₁₀₉	—	4	1	no	—
168	C₂²×C₁₀₉	—	8	2	no	—
169	C₂²×C₁₀₉	#349	1	2	yes	complemented
170	C₂²×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
171	C₂²×C₁₀₉	C₈×D₄	1	2	yes	—
172	C₄×C₁₀₉	—	2	1	no	—
173	C₄×C₁₀₉	—	2	1	no	—
174	C₄×C₁₀₉	—	4	1	no	—
175	C₄×C₁₀₉	—	4	1	no	—
176	C₄×C₁₀₉	—	4	1	no	—
177	C₄×C₁₀₉	#386	1	1	yes	characteristic
178	C₄×C₁₀₉	#408	1	1	yes	characteristic
179	C₄×C₁₀₉	#435	1	2	yes	—
180	#12985	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂×C₁₀₉	—	2	2	no	—
182	C₂×C₁₀₉	—	2	2	no	—
183	C₂×C₁₀₉	—	4	1	no	—
184	C₂×C₁₀₉	—	8	1	no	—
185	C₂×C₁₀₉	#1869	1	1	yes	characteristic
186	C₂×C₁₀₉	C₂×#349	1	1	yes	characteristic
187	C₂×C₁₀₉	#2115	1	1	yes	characteristic
188	C₂²×M₃₂	C₂×C₁₀₉	1	1	yes	characteristic, complemented
189	C₂×#435	C₂×C₁₀₉	1	1	yes	characteristic
190	C₂×#349	C₂×C₁₀₉	1	2	yes	complemented
191	#2112	C₂×C₁₀₉	1	1	yes	characteristic, complemented
192	#1377	C₂×C₁₀₉	1	2	yes	complemented
193	C₁₀₉	#12985	1	1	yes	characteristic, direct product factor, Sylow
194	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
195	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
196	C₂×C₄×D₄	C₄×C₁₀₉	1	1	yes	characteristic
197	C₂×M₃₂	—	2	1	no	—
198	C₂×M₃₂	—	2	4	no	—
199	C₂×M₃₂	C₂²×C₁₀₉	1	2	yes	—
200	C₂×M₃₂	C₂²×C₁₀₉	1	2	yes	—
201	C₂²×C₁₆	—	2	1	no	—
202	#435	—	2	1	no	—
203	#435	—	2	1	no	—
204	#435	—	2	1	no	—
205	#435	—	2	1	no	—
206	C₂×C₄⋊C₈	C₄×C₁₀₉	1	1	yes	characteristic
207	C₂×C₂²⋊C₈	C₄×C₁₀₉	1	1	yes	characteristic
208	C₂×C₂²⋊C₈	C₄×C₁₀₉	1	1	yes	characteristic
209	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
210	#349	—	2	4	no	—
211	C₂²⋊C₁₆	—	2	1	no	—
212	C₂²⋊C₁₆	—	2	1	no	—
213	C₂²×D₄	C₈×C₁₀₉	1	1	yes	characteristic
214	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
215	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
216	C₂×M₁₆	—	2	2	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	—	2	2	no	—
220	C₂²×C₈	—	2	2	no	—
221	C₂²×C₈	—	2	2	no	—
222	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
223	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
224	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
225	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
226	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
227	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
228	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
229	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
230	C₄×D₄	—	2	1	no	—
231	C₄×D₄	—	2	1	no	—
232	C₄×D₄	—	2	1	no	—
233	C₄×D₄	—	2	1	no	—
234	C₂×C₄⋊C₄	C₈×C₁₀₉	1	1	yes	characteristic
235	C₂×C₂²⋊C₄	C₈×C₁₀₉	1	1	yes	characteristic
236	C₂×C₂²⋊C₄	C₈×C₁₀₉	1	1	yes	characteristic
237	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
238	M₃₂	—	2	4	no	—
239	M₃₂	—	2	2	no	—
240	M₃₂	—	2	2	no	—
241	M₃₂	—	2	2	no	—
242	M₃₂	D₄×C₁₀₉	1	4	yes	complemented
243	C₂×C₁₆	—	2	4	no	—
244	C₂×C₁₆	—	2	2	no	—
245	C₂×C₁₆	—	2	2	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₂²⋊C₈	—	2	1	no	—
249	C₂²⋊C₈	—	2	1	no	—
250	C₂²⋊C₈	—	2	1	no	—
251	C₂²⋊C₈	—	2	1	no	—
252	#97	—	2	1	no	—
253	#97	—	2	1	no	—
254	C₂⁴	C₂×C₈×C₁₀₉	1	1	yes	characteristic
255	C₂⁴	C₂×C₈×C₁₀₉	1	1	yes	characteristic
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	1	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
266	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
271	C₂²×C₄	M₁₆×C₁₀₉	1	1	yes	characteristic
272	C₂²×C₄	M₁₆×C₁₀₉	1	1	yes	characteristic
273	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
274	M₁₆	—	4	1	no	—
275	M₁₆	—	4	2	no	—
276	M₁₆	—	4	1	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
289	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
290	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
291	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
292	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
293	C₂×C₈	#41×C₁₀₉	1	2	yes	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₂²⋊C₄	—	2	1	no	—
297	C₂²⋊C₄	—	2	1	no	—
298	C₂²⋊C₄	—	2	1	no	—
299	C₂²⋊C₄	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	2	1	no	—
302	C₁₆	—	2	4	no	—
303	C₁₆	—	4	2	no	—
304	C₂³	—	2	2	no	—
305	C₂³	—	2	2	no	—
306	C₂³	—	2	1	no	—
307	C₂³	—	2	2	no	—
308	C₂³	—	4	1	no	—
309	C₂³	—	4	1	no	—
310	C₂³	—	4	1	no	—
311	C₂³	C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
312	C₂³	C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
313	C₂³	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
314	D₄	—	4	2	no	—
315	D₄	—	4	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
333	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
334	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
335	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
336	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
337	C₂×C₄	#131×C₁₀₉	1	2	yes	—
338	C₈	—	2	1	no	—
339	C₈	—	2	1	no	—
340	C₈	—	4	1	no	—
341	C₈	—	4	1	no	—
342	C₈	#120×C₁₀₉	1	1	yes	characteristic
343	C₈	#121×C₁₀₉	1	2	yes	—
344	C₈	#123×C₁₀₉	1	1	yes	characteristic
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	4	4	no	—
353	C₂²	—	4	1	no	—
354	C₂²	—	4	1	no	—
355	C₂²	—	4	1	no	—
356	C₂²	—	8	2	no	—
357	C₂²	#349×C₁₀₉	1	2	yes	complemented
358	C₂²	C₂×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
359	C₂²	C₈×D₄×C₁₀₉	1	2	yes	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	4	1	no	—
363	C₄	—	4	1	no	—
364	C₄	—	4	1	no	—
365	C₄	#386×C₁₀₉	1	1	yes	characteristic
366	C₄	#408×C₁₀₉	1	1	yes	characteristic
367	C₄	#435×C₁₀₉	1	2	yes	—
368	C₂	—	2	2	no	—
369	C₂	—	2	2	no	—
370	C₂	—	4	1	no	—
371	C₂	—	8	1	no	—
372	C₂	C₁₀₉×#1869	1	1	yes	characteristic
373	C₂	C₂×#349×C₁₀₉	1	1	yes	characteristic
374	C₂	C₁₀₉×#2115	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₁₀₉	C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂³×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×M₃₂×C₁₀₉	C₂	semidirect
3. C₂×#349×C₁₀₉	C₂	semidirect
4. C₁₀₉×#2112	C₂	semidirect
5. C₁₀₉×#1377	C₂	semidirect
54. M₃₂×C₁₀₉	D₄	semidirect
169. C₂²×C₁₀₉	#349	semidirect
180. #12985	C₁₀₉	direct (N × H)
188. C₂²×M₃₂	C₂×C₁₀₉	semidirect
190. C₂×#349	C₂×C₁₀₉	semidirect
191. #2112	C₂×C₁₀₉	semidirect
192. #1377	C₂×C₁₀₉	semidirect
193. C₁₀₉	#12985	direct (N × H)
242. M₃₂	D₄×C₁₀₉	semidirect
357. C₂²	#349×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₁₀₉	2
C₂×#435×C₁₀₉	2
C₂×#349×C₁₀₉	2
C₁₀₉×#2112	2
C₁₀₉×#1377	2
#12985	109

Subgroups — C109×#12985 #42130979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12985 #42130979