Subgroups — C₁₀₉×#10048 #42128042

Order 27904.

746 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#492×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#459×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2563	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#1982	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#333×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₀₉×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₁₀₉	—	2	2	no	—
9	C₂²×D₈×C₁₀₉	—	2	1	no	—
10	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#127×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
13	#492×C₁₀₉	—	2	2	no	—
14	#492×C₁₀₉	—	2	2	no	—
15	#459×C₁₀₉	—	2	2	no	—
16	#459×C₁₀₉	—	2	2	no	—
17	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×#102×C₁₀₉	—	2	1	no	—
20	C₂×#102×C₁₀₉	—	2	2	no	—
21	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
22	#393×C₁₀₉	—	2	2	no	—
23	#390×C₁₀₉	—	2	1	no	—
24	#382×C₁₀₉	—	2	1	no	—
25	#380×C₁₀₉	—	2	2	no	—
26	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
27	#333×C₁₀₉	—	2	2	no	—
28	D₄⋊C₈×C₁₀₉	—	2	2	no	—
29	C₂²×Q₈×C₁₀₉	D₄	1	2	yes	—
30	C₂²×D₄×C₁₀₉	—	2	1	no	—
31	C₂²×D₄×C₁₀₉	—	2	1	no	—
32	C₂²×D₄×C₁₀₉	—	2	1	no	—
33	C₂²×D₄×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₁₀₉	—	2	1	no	—
35	C₂×SD₁₆×C₁₀₉	—	2	8	no	—
36	C₂×SD₁₆×C₁₀₉	—	4	8	no	—
37	C₂×D₈×C₁₀₉	—	2	4	no	—
38	C₂×D₈×C₁₀₉	—	4	2	no	—
39	C₂×D₈×C₁₀₉	—	4	2	no	—
40	C₂²×C₈×C₁₀₉	—	2	1	no	—
41	C₂²×C₈×C₁₀₉	—	2	1	no	—
42	C₂²×C₈×C₁₀₉	—	2	1	no	—
43	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
44	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
45	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
46	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
47	#127×C₁₀₉	—	2	1	no	—
48	#127×C₁₀₉	—	2	1	no	—
49	#127×C₁₀₉	—	2	1	no	—
50	#127×C₁₀₉	D₄	1	2	yes	—
51	C₄×D₄×C₁₀₉	—	2	1	no	—
52	C₄×D₄×C₁₀₉	—	2	1	no	—
53	C₄×D₄×C₁₀₉	—	2	1	no	—
54	C₄×D₄×C₁₀₉	—	2	1	no	—
55	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
57	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
60	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
61	C₂×C₄²×C₁₀₉	—	2	1	no	—
62	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₁₀₉	—	2	1	no	—
64	C₄⋊C₈×C₁₀₉	—	2	1	no	—
65	C₄⋊C₈×C₁₀₉	—	2	1	no	—
66	C₄⋊C₈×C₁₀₉	D₄	1	2	yes	—
67	#102×C₁₀₉	—	2	4	no	—
68	#102×C₁₀₉	—	4	4	no	—
69	C₄×C₈×C₁₀₉	—	2	1	no	—
70	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
71	#95×C₁₀₉	—	2	4	no	—
72	#95×C₁₀₉	—	2	1	no	—
73	C₂⁴×C₁₀₉	—	2	1	no	—
74	C₂⁴×C₁₀₉	—	4	1	no	—
75	C₂×Q₈×C₁₀₉	—	2	2	no	—
76	C₂×Q₈×C₁₀₉	—	2	2	no	—
77	C₂×Q₈×C₁₀₉	—	4	4	no	—
78	C₂×D₄×C₁₀₉	—	2	1	no	—
79	C₂×D₄×C₁₀₉	—	2	2	no	—
80	C₂×D₄×C₁₀₉	—	2	2	no	—
81	C₂×D₄×C₁₀₉	—	2	1	no	—
82	C₂×D₄×C₁₀₉	—	2	2	no	—
83	C₂×D₄×C₁₀₉	—	2	1	no	—
84	C₂×D₄×C₁₀₉	—	2	1	no	—
85	C₂×D₄×C₁₀₉	—	2	1	no	—
86	C₂×D₄×C₁₀₉	—	4	4	no	—
87	C₂×D₄×C₁₀₉	—	4	1	no	—
88	C₂×D₄×C₁₀₉	—	4	1	no	—
89	C₂×D₄×C₁₀₉	—	4	1	no	—
90	C₂×D₄×C₁₀₉	—	4	1	no	—
91	C₂×D₄×C₁₀₉	—	4	1	no	—
92	C₂×D₄×C₁₀₉	—	4	2	no	—
93	C₂×D₄×C₁₀₉	—	4	2	no	—
94	C₂×D₄×C₁₀₉	—	4	1	no	—
95	C₂×D₄×C₁₀₉	—	4	1	no	—
96	C₂×D₄×C₁₀₉	SD₁₆	1	2	yes	complemented
97	C₂²×C₄×C₁₀₉	—	2	1	no	—
98	C₂²×C₄×C₁₀₉	—	2	2	no	—
99	C₂²×C₄×C₁₀₉	—	2	2	no	—
100	C₂²×C₄×C₁₀₉	—	4	1	no	—
101	C₂²×C₄×C₁₀₉	—	4	1	no	—
102	C₂²×C₄×C₁₀₉	—	4	2	no	—
103	C₂²×C₄×C₁₀₉	—	4	2	no	—
104	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
107	SD₁₆×C₁₀₉	—	4	16	no	—
108	D₈×C₁₀₉	—	4	8	no	—
109	C₂×C₈×C₁₀₉	—	2	4	no	—
110	C₂×C₈×C₁₀₉	—	2	2	no	—
111	C₂×C₈×C₁₀₉	—	2	2	no	—
112	C₂×C₈×C₁₀₉	—	4	1	no	—
113	C₂×C₈×C₁₀₉	—	4	1	no	—
114	C₂×C₈×C₁₀₉	—	4	1	no	—
115	C₂×C₈×C₁₀₉	—	4	2	no	—
116	C₄⋊C₄×C₁₀₉	—	2	1	no	—
117	C₄⋊C₄×C₁₀₉	—	2	1	no	—
118	C₄⋊C₄×C₁₀₉	—	2	2	no	—
119	C₄⋊C₄×C₁₀₉	—	2	2	no	—
120	C₄⋊C₄×C₁₀₉	—	4	1	no	—
121	C₄⋊C₄×C₁₀₉	—	4	1	no	—
122	C₄⋊C₄×C₁₀₉	—	4	1	no	—
123	C₄⋊C₄×C₁₀₉	—	4	1	no	—
124	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
125	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
126	C₂²⋊C₄×C₁₀₉	—	8	4	no	—
127	C₄²×C₁₀₉	—	4	1	no	—
128	C₄²×C₁₀₉	—	4	1	no	—
129	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₁₀₉	#41	1	1	yes	characteristic
133	C₂³×C₁₀₉	—	2	2	no	—
134	C₂³×C₁₀₉	—	4	1	no	—
135	C₂³×C₁₀₉	—	4	2	no	—
136	C₂³×C₁₀₉	—	4	1	no	—
137	C₂³×C₁₀₉	—	4	2	no	—
138	C₂³×C₁₀₉	—	4	2	no	—
139	C₂³×C₁₀₉	—	8	1	no	—
140	C₂³×C₁₀₉	—	8	1	no	—
141	C₂³×C₁₀₉	—	8	2	no	—
142	C₂³×C₁₀₉	—	8	2	no	—
143	C₂³×C₁₀₉	#120	1	1	yes	characteristic
144	Q₈×C₁₀₉	—	4	4	no	—
145	Q₈×C₁₀₉	—	8	2	no	—
146	D₄×C₁₀₉	—	2	2	no	—
147	D₄×C₁₀₉	—	2	2	no	—
148	D₄×C₁₀₉	—	4	4	no	—
149	D₄×C₁₀₉	—	4	2	no	—
150	D₄×C₁₀₉	—	4	1	no	—
151	D₄×C₁₀₉	—	4	2	no	—
152	D₄×C₁₀₉	—	4	4	no	—
153	D₄×C₁₀₉	—	4	1	no	—
154	D₄×C₁₀₉	—	8	2	no	—
155	D₄×C₁₀₉	—	8	1	no	—
156	D₄×C₁₀₉	—	8	1	no	—
157	D₄×C₁₀₉	—	8	1	no	—
158	D₄×C₁₀₉	—	8	1	no	—
159	C₂×C₄×C₁₀₉	—	2	2	no	—
160	C₂×C₄×C₁₀₉	—	4	1	no	—
161	C₂×C₄×C₁₀₉	—	4	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	4	no	—
167	C₂×C₄×C₁₀₉	—	8	2	no	—
168	C₂×C₄×C₁₀₉	—	8	1	no	—
169	C₂×C₄×C₁₀₉	—	8	1	no	—
170	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
171	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
172	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
173	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
174	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
175	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
176	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
177	C₂×C₄×C₁₀₉	#135	1	1	yes	characteristic
178	C₂×C₄×C₁₀₉	#135	1	2	yes	—
179	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
181	C₂×C₄×C₁₀₉	#137	1	1	yes	characteristic
182	C₈×C₁₀₉	—	2	4	no	—
183	C₈×C₁₀₉	—	4	2	no	—
184	C₈×C₁₀₉	—	8	1	no	—
185	C₂²×C₁₀₉	—	4	4	no	—
186	C₂²×C₁₀₉	—	4	2	no	—
187	C₂²×C₁₀₉	—	8	2	no	—
188	C₂²×C₁₀₉	—	8	1	no	—
189	C₂²×C₁₀₉	—	8	1	no	—
190	C₂²×C₁₀₉	—	8	2	no	—
191	C₂²×C₁₀₉	—	8	2	no	—
192	C₂²×C₁₀₉	—	8	2	no	—
193	C₂²×C₁₀₉	—	16	2	no	—
194	C₂²×C₁₀₉	—	16	2	no	—
195	C₂²×C₁₀₉	#392	1	1	yes	characteristic
196	C₂²×C₁₀₉	#448	1	1	yes	characteristic
197	C₂²×C₁₀₉	#449	1	2	yes	—
198	C₂²×C₁₀₉	#449	1	1	yes	characteristic
199	C₂²×C₁₀₉	#453	1	2	yes	—
200	C₄×C₁₀₉	—	2	1	no	—
201	C₄×C₁₀₉	—	2	1	no	—
202	C₄×C₁₀₉	—	8	2	no	—
203	C₄×C₁₀₉	—	8	2	no	—
204	C₄×C₁₀₉	—	8	1	no	—
205	C₄×C₁₀₉	#450	1	1	yes	characteristic
206	C₄×C₁₀₉	#452	1	1	yes	characteristic
207	C₄×C₁₀₉	#463	1	2	yes	—
208	C₄×C₁₀₉	#465	1	1	yes	characteristic
209	C₄×C₁₀₉	#468	1	1	yes	characteristic
210	C₄×C₁₀₉	#495	1	1	yes	characteristic
211	C₄×C₁₀₉	#496	1	1	yes	characteristic
212	#10048	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂×C₁₀₉	—	8	2	no	—
214	C₂×C₁₀₉	—	16	2	no	—
215	C₂×C₁₀₉	#1622	1	2	yes	—
216	C₂×C₁₀₉	#1624	1	2	yes	—
217	C₂×C₁₀₉	#2000	1	1	yes	characteristic
218	C₂×C₁₀₉	#2002	1	1	yes	characteristic
219	C₂×C₁₀₉	#2012	1	1	yes	characteristic
220	C₂×#492	C₂×C₁₀₉	1	1	yes	characteristic, complemented
221	C₂×#459	C₂×C₁₀₉	1	1	yes	characteristic
222	#2563	C₂×C₁₀₉	1	1	yes	characteristic, complemented
223	#1982	C₂×C₁₀₉	1	1	yes	characteristic, complemented
224	C₂×#333	C₂×C₁₀₉	1	1	yes	characteristic, complemented
225	C₂×D₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
226	#1279	C₂×C₁₀₉	1	1	yes	characteristic, complemented
227	C₁₀₉	#10048	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²×SD₁₆	—	2	2	no	—
229	C₂²×D₈	—	2	1	no	—
230	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
231	C₂×#127	C₂²×C₁₀₉	1	1	yes	characteristic
232	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
233	#492	—	2	2	no	—
234	#492	—	2	2	no	—
235	#459	—	2	2	no	—
236	#459	—	2	2	no	—
237	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
238	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
239	C₂×#102	—	2	1	no	—
240	C₂×#102	—	2	2	no	—
241	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
242	#393	—	2	2	no	—
243	#390	—	2	1	no	—
244	#382	—	2	1	no	—
245	#380	—	2	2	no	—
246	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
247	#333	—	2	2	no	—
248	D₄⋊C₈	—	2	2	no	—
249	C₂²×Q₈	D₄×C₁₀₉	1	2	yes	—
250	C₂²×D₄	—	2	1	no	—
251	C₂²×D₄	—	2	1	no	—
252	C₂²×D₄	—	2	1	no	—
253	C₂²×D₄	D₄×C₁₀₉	1	1	yes	characteristic
254	C₂³×C₄	—	2	1	no	—
255	C₂×SD₁₆	—	2	8	no	—
256	C₂×SD₁₆	—	4	8	no	—
257	C₂×D₈	—	2	4	no	—
258	C₂×D₈	—	4	2	no	—
259	C₂×D₈	—	4	2	no	—
260	C₂²×C₈	—	2	1	no	—
261	C₂²×C₈	—	2	1	no	—
262	C₂²×C₈	—	2	1	no	—
263	C₄⋊Q₈	—	2	1	no	—
264	C₄⋊Q₈	—	2	1	no	—
265	C₄⋊Q₈	—	2	1	no	—
266	C₄⋊Q₈	—	2	1	no	—
267	#127	—	2	1	no	—
268	#127	—	2	1	no	—
269	#127	—	2	1	no	—
270	#127	D₄×C₁₀₉	1	2	yes	—
271	C₄×D₄	—	2	1	no	—
272	C₄×D₄	—	2	1	no	—
273	C₄×D₄	—	2	1	no	—
274	C₄×D₄	—	2	1	no	—
275	C₂×C₄⋊C₄	—	2	1	no	—
276	C₂×C₄⋊C₄	—	2	1	no	—
277	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
278	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
279	C₂×C₂²⋊C₄	—	2	1	no	—
280	C₂×C₂²⋊C₄	—	2	4	no	—
281	C₂×C₄²	—	2	1	no	—
282	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	—	2	1	no	—
285	C₄⋊C₈	—	2	1	no	—
286	C₄⋊C₈	D₄×C₁₀₉	1	2	yes	—
287	#102	—	2	4	no	—
288	#102	—	4	4	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
291	#95	—	2	4	no	—
292	#95	—	2	1	no	—
293	C₂⁴	—	2	1	no	—
294	C₂⁴	—	4	1	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	4	4	no	—
298	C₂×D₄	—	2	1	no	—
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	2	2	no	—
301	C₂×D₄	—	2	1	no	—
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	2	1	no	—
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	—	2	1	no	—
306	C₂×D₄	—	4	4	no	—
307	C₂×D₄	—	4	1	no	—
308	C₂×D₄	—	4	1	no	—
309	C₂×D₄	—	4	1	no	—
310	C₂×D₄	—	4	1	no	—
311	C₂×D₄	—	4	1	no	—
312	C₂×D₄	—	4	2	no	—
313	C₂×D₄	—	4	2	no	—
314	C₂×D₄	—	4	1	no	—
315	C₂×D₄	—	4	1	no	—
316	C₂×D₄	SD₁₆×C₁₀₉	1	2	yes	complemented
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	4	1	no	—
321	C₂²×C₄	—	4	1	no	—
322	C₂²×C₄	—	4	2	no	—
323	C₂²×C₄	—	4	2	no	—
324	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
325	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
326	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
327	SD₁₆	—	4	16	no	—
328	D₈	—	4	8	no	—
329	C₂×C₈	—	2	4	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	4	1	no	—
333	C₂×C₈	—	4	1	no	—
334	C₂×C₈	—	4	1	no	—
335	C₂×C₈	—	4	2	no	—
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	4	1	no	—
341	C₄⋊C₄	—	4	1	no	—
342	C₄⋊C₄	—	4	1	no	—
343	C₄⋊C₄	—	4	1	no	—
344	C₂²⋊C₄	—	4	1	no	—
345	C₂²⋊C₄	—	4	1	no	—
346	C₂²⋊C₄	—	8	4	no	—
347	C₄²	—	4	1	no	—
348	C₄²	—	4	1	no	—
349	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
350	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
351	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
352	C₄²	#41×C₁₀₉	1	1	yes	characteristic
353	C₂³	—	2	2	no	—
354	C₂³	—	4	1	no	—
355	C₂³	—	4	2	no	—
356	C₂³	—	4	1	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	4	2	no	—
359	C₂³	—	8	1	no	—
360	C₂³	—	8	1	no	—
361	C₂³	—	8	2	no	—
362	C₂³	—	8	2	no	—
363	C₂³	#120×C₁₀₉	1	1	yes	characteristic
364	Q₈	—	4	4	no	—
365	Q₈	—	8	2	no	—
366	D₄	—	2	2	no	—
367	D₄	—	2	2	no	—
368	D₄	—	4	4	no	—
369	D₄	—	4	2	no	—
370	D₄	—	4	1	no	—
371	D₄	—	4	2	no	—
372	D₄	—	4	4	no	—
373	D₄	—	4	1	no	—
374	D₄	—	8	2	no	—
375	D₄	—	8	1	no	—
376	D₄	—	8	1	no	—
377	D₄	—	8	1	no	—
378	D₄	—	8	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	4	no	—
387	C₂×C₄	—	8	2	no	—
388	C₂×C₄	—	8	1	no	—
389	C₂×C₄	—	8	1	no	—
390	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
391	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
392	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
393	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
394	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
395	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
396	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
397	C₂×C₄	#135×C₁₀₉	1	1	yes	characteristic
398	C₂×C₄	#135×C₁₀₉	1	2	yes	—
399	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
400	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
401	C₂×C₄	#137×C₁₀₉	1	1	yes	characteristic
402	C₈	—	2	4	no	—
403	C₈	—	4	2	no	—
404	C₈	—	8	1	no	—
405	C₂²	—	4	2	no	—
406	C₂²	—	4	4	no	—
407	C₂²	—	8	1	no	—
408	C₂²	—	8	1	no	—
409	C₂²	—	8	2	no	—
410	C₂²	—	8	2	no	—
411	C₂²	—	8	2	no	—
412	C₂²	—	8	2	no	—
413	C₂²	—	16	2	no	—
414	C₂²	—	16	2	no	—
415	C₂²	#392×C₁₀₉	1	1	yes	characteristic
416	C₂²	#448×C₁₀₉	1	1	yes	characteristic
417	C₂²	#449×C₁₀₉	1	1	yes	characteristic
418	C₂²	#449×C₁₀₉	1	2	yes	—
419	C₂²	#453×C₁₀₉	1	2	yes	—
420	C₄	—	2	1	no	—
421	C₄	—	2	1	no	—
422	C₄	—	8	2	no	—
423	C₄	—	8	1	no	—
424	C₄	—	8	2	no	—
425	C₄	#450×C₁₀₉	1	1	yes	characteristic
426	C₄	#452×C₁₀₉	1	1	yes	characteristic
427	C₄	#463×C₁₀₉	1	2	yes	—
428	C₄	#465×C₁₀₉	1	1	yes	characteristic
429	C₄	#468×C₁₀₉	1	1	yes	characteristic
430	C₄	#495×C₁₀₉	1	1	yes	characteristic
431	C₄	#496×C₁₀₉	1	1	yes	characteristic
432	C₂	—	8	2	no	—
433	C₂	—	16	2	no	—
434	C₂	C₁₀₉×#1622	1	2	yes	—
435	C₂	C₁₀₉×#1624	1	2	yes	—
436	C₂	C₁₀₉×#2000	1	1	yes	characteristic
437	C₂	C₁₀₉×#2002	1	1	yes	characteristic
438	C₂	C₁₀₉×#2012	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₉	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	C₂×#459×C₁₀₉	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#492×C₁₀₉	C₂	semidirect
3. C₁₀₉×#2563	C₂	semidirect
4. C₁₀₉×#1982	C₂	semidirect
5. C₂×#333×C₁₀₉	C₂	semidirect
6. C₂×D₄⋊C₈×C₁₀₉	C₂	semidirect
7. C₁₀₉×#1279	C₂	semidirect
96. C₂×D₄×C₁₀₉	SD₁₆	semidirect
212. #10048	C₁₀₉	direct (N × H)
220. C₂×#492	C₂×C₁₀₉	semidirect
222. #2563	C₂×C₁₀₉	semidirect
223. #1982	C₂×C₁₀₉	semidirect
224. C₂×#333	C₂×C₁₀₉	semidirect
225. C₂×D₄⋊C₈	C₂×C₁₀₉	semidirect
226. #1279	C₂×C₁₀₉	semidirect
227. C₁₀₉	#10048	direct (N × H)
316. C₂×D₄	SD₁₆×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#492×C₁₀₉	2
C₂×#459×C₁₀₉	2
C₁₀₉×#2563	2
C₁₀₉×#1982	2
C₂×#333×C₁₀₉	2
C₂×D₄⋊C₈×C₁₀₉	2
C₁₀₉×#1279	2
#10048	109

Subgroups — C109×#10048 #42128042

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10048 #42128042