Subgroups — C₁₀₉×#21644 #42139638

Order 27904.

890 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#486×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2331	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#2328	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1993	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1961	C₂	1	1	yes	maximal, characteristic
6	C₁₀₉×#1936	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1934	C₂	1	2	yes	maximal, complemented
8	C₁₀₉×#1835	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₀₉×#1762	C₂	1	2	yes	maximal, complemented
10	C₄×#102×C₁₀₉	C₂	1	2	yes	maximal, complemented
11	C₁₀₉×#1726	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×#127×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
15	#486×C₁₀₉	C₄	1	8	yes	complemented
16	C₂×#102×C₁₀₉	C₂²	1	2	yes	—
17	C₂×#102×C₁₀₉	C₂²	1	2	yes	—
18	C₂×#102×C₁₀₉	C₂²	1	2	yes	—
19	C₂×#102×C₁₀₉	C₂²	1	2	yes	—
20	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
26	#391×C₁₀₉	—	2	2	no	—
27	#390×C₁₀₉	C₂²	1	2	yes	—
28	#389×C₁₀₉	C₂²	1	2	yes	—
29	#387×C₁₀₉	—	2	2	no	—
30	#386×C₁₀₉	—	2	2	no	—
31	#386×C₁₀₉	—	2	2	no	—
32	#382×C₁₀₉	C₂²	1	1	yes	characteristic
33	#382×C₁₀₉	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
35	#376×C₁₀₉	C₂²	1	1	yes	characteristic
36	#340×C₁₀₉	C₂²	1	2	yes	—
37	#340×C₁₀₉	C₂²	1	2	yes	—
38	#340×C₁₀₉	C₂²	1	2	yes	—
39	#340×C₁₀₉	C₂²	1	2	yes	—
40	C₂²×Q₈×C₁₀₉	—	2	1	no	—
41	C₂²×D₄×C₁₀₉	—	2	1	no	—
42	C₂²×D₄×C₁₀₉	—	2	1	no	—
43	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
44	C₂³×C₄×C₁₀₉	—	2	2	no	—
45	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
50	C₄⋊Q₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
51	#127×C₁₀₉	—	2	2	no	—
52	#127×C₁₀₉	C₂×C₄	1	4	yes	—
53	C₄×D₄×C₁₀₉	—	2	4	no	—
54	C₄×D₄×C₁₀₉	—	2	4	no	—
55	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
57	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
60	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
63	C₂×C₄²×C₁₀₉	—	2	2	no	—
64	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
67	#102×C₁₀₉	—	2	4	no	—
68	#102×C₁₀₉	—	2	4	no	—
69	#102×C₁₀₉	C₂×C₄	1	8	yes	complemented
70	#102×C₁₀₉	C₂×C₄	1	8	yes	complemented
71	#97×C₁₀₉	—	2	2	no	—
72	#97×C₁₀₉	—	2	2	no	—
73	#97×C₁₀₉	—	2	2	no	—
74	C₄×C₈×C₁₀₉	—	2	2	no	—
75	C₄×C₈×C₁₀₉	—	2	2	no	—
76	C₄×C₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
77	#95×C₁₀₉	—	2	2	no	—
78	#95×C₁₀₉	—	2	1	no	—
79	#95×C₁₀₉	—	2	1	no	—
80	#95×C₁₀₉	—	2	1	no	—
81	#95×C₁₀₉	—	2	1	no	—
82	#95×C₁₀₉	—	2	2	no	—
83	#95×C₁₀₉	D₄	1	2	yes	—
84	#95×C₁₀₉	D₄	1	2	yes	—
85	C₂⁴×C₁₀₉	—	2	2	no	—
86	C₂×Q₈×C₁₀₉	—	2	4	no	—
87	C₂×Q₈×C₁₀₉	—	8	2	no	—
88	C₂×D₄×C₁₀₉	—	2	4	no	—
89	C₂×D₄×C₁₀₉	—	2	4	no	—
90	C₂×D₄×C₁₀₉	—	2	2	no	—
91	C₂×D₄×C₁₀₉	—	2	2	no	—
92	C₂×D₄×C₁₀₉	—	2	2	no	—
93	C₂×D₄×C₁₀₉	—	2	2	no	—
94	C₂×D₄×C₁₀₉	—	2	2	no	—
95	C₂×D₄×C₁₀₉	—	8	2	no	—
96	C₂×D₄×C₁₀₉	—	8	2	no	—
97	C₂×D₄×C₁₀₉	C₂²×C₄	1	4	yes	—
98	C₂²×C₄×C₁₀₉	—	2	2	no	—
99	C₂²×C₄×C₁₀₉	—	2	2	no	—
100	C₂²×C₄×C₁₀₉	—	2	2	no	—
101	C₂²×C₄×C₁₀₉	—	4	4	no	—
102	C₂²×C₄×C₁₀₉	—	8	2	no	—
103	C₂²×C₄×C₁₀₉	—	8	2	no	—
104	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
109	C₂×C₈×C₁₀₉	—	2	1	no	—
110	C₂×C₈×C₁₀₉	—	2	1	no	—
111	C₂×C₈×C₁₀₉	—	2	1	no	—
112	C₂×C₈×C₁₀₉	—	2	1	no	—
113	C₂×C₈×C₁₀₉	—	2	1	no	—
114	C₂×C₈×C₁₀₉	—	2	1	no	—
115	C₂×C₈×C₁₀₉	—	2	1	no	—
116	C₂×C₈×C₁₀₉	—	2	1	no	—
117	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₁₀₉	#41	1	2	yes	—
120	C₂×C₈×C₁₀₉	#41	1	2	yes	—
121	C₄⋊C₄×C₁₀₉	—	2	4	no	—
122	C₄⋊C₄×C₁₀₉	—	2	2	no	—
123	C₄⋊C₄×C₁₀₉	—	2	2	no	—
124	C₄⋊C₄×C₁₀₉	—	2	2	no	—
125	C₄⋊C₄×C₁₀₉	—	2	2	no	—
126	C₄⋊C₄×C₁₀₉	—	2	2	no	—
127	C₄⋊C₄×C₁₀₉	—	8	2	no	—
128	C₄⋊C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
129	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
130	C₂²⋊C₄×C₁₀₉	—	8	2	no	—
131	C₂²⋊C₄×C₁₀₉	—	8	2	no	—
132	C₄²×C₁₀₉	—	2	1	no	—
133	C₄²×C₁₀₉	—	2	1	no	—
134	C₄²×C₁₀₉	—	8	2	no	—
135	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
136	C₄²×C₁₀₉	#41	1	2	yes	—
137	C₄²×C₁₀₉	#41	1	2	yes	—
138	C₄²×C₁₀₉	#41	1	2	yes	—
139	C₂³×C₁₀₉	—	2	4	no	—
140	C₂³×C₁₀₉	—	4	2	no	—
141	C₂³×C₁₀₉	—	4	2	no	—
142	C₂³×C₁₀₉	—	4	2	no	—
143	C₂³×C₁₀₉	—	4	2	no	—
144	C₂³×C₁₀₉	—	4	2	no	—
145	C₂³×C₁₀₉	—	4	2	no	—
146	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
147	Q₈×C₁₀₉	—	8	4	no	—
148	D₄×C₁₀₉	—	2	4	no	—
149	D₄×C₁₀₉	—	2	4	no	—
150	D₄×C₁₀₉	—	4	2	no	—
151	D₄×C₁₀₉	—	4	2	no	—
152	D₄×C₁₀₉	—	8	4	no	—
153	D₄×C₁₀₉	—	8	4	no	—
154	C₂×C₄×C₁₀₉	—	2	2	no	—
155	C₂×C₄×C₁₀₉	—	2	2	no	—
156	C₂×C₄×C₁₀₉	—	2	2	no	—
157	C₂×C₄×C₁₀₉	—	2	2	no	—
158	C₂×C₄×C₁₀₉	—	2	4	no	—
159	C₂×C₄×C₁₀₉	—	2	2	no	—
160	C₂×C₄×C₁₀₉	—	2	2	no	—
161	C₂×C₄×C₁₀₉	—	2	2	no	—
162	C₂×C₄×C₁₀₉	—	4	2	no	—
163	C₂×C₄×C₁₀₉	—	4	2	no	—
164	C₂×C₄×C₁₀₉	—	4	2	no	—
165	C₂×C₄×C₁₀₉	—	4	2	no	—
166	C₂×C₄×C₁₀₉	—	4	2	no	—
167	C₂×C₄×C₁₀₉	—	4	2	no	—
168	C₂×C₄×C₁₀₉	—	4	2	no	—
169	C₂×C₄×C₁₀₉	—	4	2	no	—
170	C₂×C₄×C₁₀₉	—	8	8	no	—
171	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
172	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
173	C₂×C₄×C₁₀₉	#124	1	4	yes	—
174	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
175	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
176	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
177	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
178	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
179	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
180	C₈×C₁₀₉	—	2	2	no	—
181	C₈×C₁₀₉	—	2	2	no	—
182	C₈×C₁₀₉	—	2	2	no	—
183	C₈×C₁₀₉	—	2	2	no	—
184	C₂²×C₁₀₉	—	4	4	no	—
185	C₂²×C₁₀₉	—	4	4	no	—
186	C₂²×C₁₀₉	—	4	4	no	—
187	C₂²×C₁₀₉	—	8	2	no	—
188	C₂²×C₁₀₉	—	8	2	no	—
189	C₂²×C₁₀₉	—	8	2	no	—
190	C₂²×C₁₀₉	—	8	2	no	—
191	C₂²×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂²×C₁₀₉	C₂×#124	1	1	yes	characteristic
193	C₂²×C₁₀₉	#532	1	1	yes	characteristic
194	C₂²×C₁₀₉	#576	1	2	yes	—
195	C₂²×C₁₀₉	#577	1	2	yes	—
196	C₄×C₁₀₉	—	2	4	no	—
197	C₄×C₁₀₉	—	2	2	no	—
198	C₄×C₁₀₉	—	2	2	no	—
199	C₄×C₁₀₉	—	4	2	no	—
200	C₄×C₁₀₉	—	4	4	no	—
201	C₄×C₁₀₉	—	8	2	no	—
202	C₄×C₁₀₉	—	8	2	no	—
203	C₄×C₁₀₉	C₄×#41	1	1	yes	characteristic
204	C₄×C₁₀₉	C₄×#41	1	1	yes	characteristic
205	C₄×C₁₀₉	#529	1	1	yes	characteristic
206	C₄×C₁₀₉	#529	1	1	yes	characteristic
207	#21644	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂×C₁₀₉	—	8	4	no	—
209	C₂×C₁₀₉	C₄×#124	1	1	yes	characteristic
210	C₂×C₁₀₉	#2948	1	1	yes	characteristic
211	C₂×C₁₀₉	#2949	1	1	yes	characteristic
212	C₂×C₁₀₉	#3136	1	2	yes	—
213	C₂×C₁₀₉	#3179	1	2	yes	—
214	C₂×#486	C₂×C₁₀₉	1	1	yes	characteristic
215	#2331	C₂×C₁₀₉	1	1	yes	characteristic, complemented
216	#2328	C₂×C₁₀₉	1	1	yes	characteristic
217	#1993	C₂×C₁₀₉	1	1	yes	characteristic, complemented
218	#1961	C₂×C₁₀₉	1	1	yes	characteristic
219	#1936	C₂×C₁₀₉	1	2	yes	complemented
220	#1934	C₂×C₁₀₉	1	2	yes	complemented
221	#1835	C₂×C₁₀₉	1	1	yes	characteristic, complemented
222	#1762	C₂×C₁₀₉	1	2	yes	complemented
223	C₄×#102	C₂×C₁₀₉	1	2	yes	complemented
224	#1726	C₂×C₁₀₉	1	1	yes	characteristic, complemented
225	C₁₀₉	#21644	1	1	yes	characteristic, direct product factor, Sylow
226	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
227	C₂×#127	C₂²×C₁₀₉	1	1	yes	characteristic
228	C₂×C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
229	#486	C₄×C₁₀₉	1	8	yes	complemented
230	C₂×#102	C₂²×C₁₀₉	1	2	yes	—
231	C₂×#102	C₂²×C₁₀₉	1	2	yes	—
232	C₂×#102	C₂²×C₁₀₉	1	2	yes	—
233	C₂×#102	C₂²×C₁₀₉	1	2	yes	—
234	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
235	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
236	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
237	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
238	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
239	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
240	#391	—	2	2	no	—
241	#390	C₂²×C₁₀₉	1	2	yes	—
242	#389	C₂²×C₁₀₉	1	2	yes	—
243	#387	—	2	2	no	—
244	#386	—	2	2	no	—
245	#386	—	2	2	no	—
246	#382	C₂²×C₁₀₉	1	1	yes	characteristic
247	#382	C₂²×C₁₀₉	1	1	yes	characteristic
248	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
249	#376	C₂²×C₁₀₉	1	1	yes	characteristic
250	#340	C₂²×C₁₀₉	1	2	yes	—
251	#340	C₂²×C₁₀₉	1	2	yes	—
252	#340	C₂²×C₁₀₉	1	2	yes	—
253	#340	C₂²×C₁₀₉	1	2	yes	—
254	C₂²×Q₈	—	2	1	no	—
255	C₂²×D₄	—	2	1	no	—
256	C₂²×D₄	—	2	1	no	—
257	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
258	C₂³×C₄	—	2	2	no	—
259	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
260	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
261	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
263	C₄⋊Q₈	—	2	2	no	—
264	C₄⋊Q₈	C₂×C₄×C₁₀₉	1	4	yes	complemented
265	#127	—	2	2	no	—
266	#127	C₂×C₄×C₁₀₉	1	4	yes	—
267	C₄×D₄	—	2	4	no	—
268	C₄×D₄	—	2	4	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	2	2	no	—
271	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
272	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
273	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
274	C₂×C₂²⋊C₄	—	2	2	no	—
275	C₂×C₂²⋊C₄	—	2	2	no	—
276	C₂×C₂²⋊C₄	—	2	2	no	—
277	C₂×C₄²	—	2	2	no	—
278	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
279	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
280	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
281	#102	—	2	4	no	—
282	#102	—	2	4	no	—
283	#102	C₂×C₄×C₁₀₉	1	8	yes	complemented
284	#102	C₂×C₄×C₁₀₉	1	8	yes	complemented
285	#97	—	2	2	no	—
286	#97	—	2	2	no	—
287	#97	—	2	2	no	—
288	C₄×C₈	—	2	2	no	—
289	C₄×C₈	—	2	2	no	—
290	C₄×C₈	C₂×C₄×C₁₀₉	1	4	yes	complemented
291	#95	—	2	2	no	—
292	#95	—	2	1	no	—
293	#95	—	2	1	no	—
294	#95	—	2	1	no	—
295	#95	—	2	1	no	—
296	#95	—	2	2	no	—
297	#95	D₄×C₁₀₉	1	2	yes	—
298	#95	D₄×C₁₀₉	1	2	yes	—
299	C₂⁴	—	2	2	no	—
300	C₂×Q₈	—	2	4	no	—
301	C₂×Q₈	—	8	2	no	—
302	C₂×D₄	—	2	4	no	—
303	C₂×D₄	—	2	4	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	8	2	no	—
310	C₂×D₄	—	8	2	no	—
311	C₂×D₄	C₂²×C₄×C₁₀₉	1	4	yes	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	4	4	no	—
316	C₂²×C₄	—	8	2	no	—
317	C₂²×C₄	—	8	2	no	—
318	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
319	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
321	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
322	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
332	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
333	C₂×C₈	#41×C₁₀₉	1	2	yes	—
334	C₂×C₈	#41×C₁₀₉	1	2	yes	—
335	C₄⋊C₄	—	2	4	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	8	2	no	—
342	C₄⋊C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
343	C₂²⋊C₄	—	4	4	no	—
344	C₂²⋊C₄	—	8	2	no	—
345	C₂²⋊C₄	—	8	2	no	—
346	C₄²	—	2	1	no	—
347	C₄²	—	2	1	no	—
348	C₄²	—	8	2	no	—
349	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
350	C₄²	#41×C₁₀₉	1	2	yes	—
351	C₄²	#41×C₁₀₉	1	2	yes	—
352	C₄²	#41×C₁₀₉	1	2	yes	—
353	C₂³	—	2	4	no	—
354	C₂³	—	4	2	no	—
355	C₂³	—	4	2	no	—
356	C₂³	—	4	2	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	4	2	no	—
359	C₂³	—	4	2	no	—
360	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
361	Q₈	—	8	4	no	—
362	D₄	—	2	4	no	—
363	D₄	—	2	4	no	—
364	D₄	—	4	2	no	—
365	D₄	—	4	2	no	—
366	D₄	—	8	4	no	—
367	D₄	—	8	4	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	8	8	no	—
385	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
386	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
387	C₂×C₄	#124×C₁₀₉	1	4	yes	—
388	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
389	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
390	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
391	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
392	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
393	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
394	C₈	—	2	2	no	—
395	C₈	—	2	2	no	—
396	C₈	—	2	2	no	—
397	C₈	—	2	2	no	—
398	C₂²	—	4	4	no	—
399	C₂²	—	4	4	no	—
400	C₂²	—	4	4	no	—
401	C₂²	—	8	2	no	—
402	C₂²	—	8	2	no	—
403	C₂²	—	8	2	no	—
404	C₂²	—	8	2	no	—
405	C₂²	C₂×C₄×D₄×C₁₀₉	1	1	yes	characteristic
406	C₂²	C₂×#124×C₁₀₉	1	1	yes	characteristic
407	C₂²	#532×C₁₀₉	1	1	yes	characteristic
408	C₂²	#576×C₁₀₉	1	2	yes	—
409	C₂²	#577×C₁₀₉	1	2	yes	—
410	C₄	—	2	2	no	—
411	C₄	—	2	2	no	—
412	C₄	—	2	4	no	—
413	C₄	—	4	4	no	—
414	C₄	—	4	2	no	—
415	C₄	—	8	2	no	—
416	C₄	—	8	2	no	—
417	C₄	C₄×#41×C₁₀₉	1	1	yes	characteristic
418	C₄	C₄×#41×C₁₀₉	1	1	yes	characteristic
419	C₄	#529×C₁₀₉	1	1	yes	characteristic
420	C₄	#529×C₁₀₉	1	1	yes	characteristic
421	C₂	—	8	4	no	—
422	C₂	C₄×#124×C₁₀₉	1	1	yes	characteristic
423	C₂	C₁₀₉×#2948	1	1	yes	characteristic
424	C₂	C₁₀₉×#2949	1	1	yes	characteristic
425	C₂	C₁₀₉×#3136	1	2	yes	—
426	C₂	C₁₀₉×#3179	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
prime radical	C₂×#127×C₁₀₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ #376×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₉×#2331	C₂	semidirect
4. C₁₀₉×#1993	C₂	semidirect
6. C₁₀₉×#1936	C₂	semidirect
7. C₁₀₉×#1934	C₂	semidirect
8. C₁₀₉×#1835	C₂	semidirect
9. C₁₀₉×#1762	C₂	semidirect
10. C₄×#102×C₁₀₉	C₂	semidirect
11. C₁₀₉×#1726	C₂	semidirect
15. #486×C₁₀₉	C₄	semidirect
50. C₄⋊Q₈×C₁₀₉	C₂×C₄	semidirect
69. #102×C₁₀₉	C₂×C₄	semidirect
70. #102×C₁₀₉	C₂×C₄	semidirect
76. C₄×C₈×C₁₀₉	C₂×C₄	semidirect
207. #21644	C₁₀₉	direct (N × H)
215. #2331	C₂×C₁₀₉	semidirect
217. #1993	C₂×C₁₀₉	semidirect
219. #1936	C₂×C₁₀₉	semidirect
220. #1934	C₂×C₁₀₉	semidirect
221. #1835	C₂×C₁₀₉	semidirect
222. #1762	C₂×C₁₀₉	semidirect
223. C₄×#102	C₂×C₁₀₉	semidirect
224. #1726	C₂×C₁₀₉	semidirect
225. C₁₀₉	#21644	direct (N × H)
229. #486	C₄×C₁₀₉	semidirect
264. C₄⋊Q₈	C₂×C₄×C₁₀₉	semidirect
283. #102	C₂×C₄×C₁₀₉	semidirect
284. #102	C₂×C₄×C₁₀₉	semidirect
290. C₄×C₈	C₂×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×#486×C₁₀₉	2
C₁₀₉×#2331	2
C₁₀₉×#2328	2
C₁₀₉×#1993	2
C₁₀₉×#1961	2
C₁₀₉×#1936	2
C₁₀₉×#1934	2
C₁₀₉×#1835	2
C₁₀₉×#1762	2
C₄×#102×C₁₀₉	2
C₁₀₉×#1726	2
#21644	109

Subgroups — C109×#21644 #42139638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#21644 #42139638