Subgroups — C₁₀₉×#19280 #42137274

Order 27904.

766 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₀₉×#3036	C₂	1	2	yes	maximal, complemented
3	C₂×#436×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#1701	C₂	1	4	yes	maximal, complemented
5	C₁₀₉×#1693	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#1687	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#1547	C₂	1	2	yes	maximal, complemented
9	C₂×AGL(1,ℤ/8)×C₁₀₉	—	2	2	no	—
10	C₂²×M₁₆×C₁₀₉	—	2	1	no	—
11	#532×C₁₀₉	C₂²	1	2	yes	—
12	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
13	#500×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
14	#493×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
15	#486×C₁₀₉	C₂²	1	2	yes	complemented
16	#466×C₁₀₉	—	2	1	no	—
17	#466×C₁₀₉	—	2	1	no	—
18	#463×C₁₀₉	—	2	4	no	—
19	#461×C₁₀₉	C₂²	1	4	yes	—
20	#459×C₁₀₉	C₂²	1	4	yes	—
21	C₄×D₈×C₁₀₉	—	2	1	no	—
22	C₄×D₈×C₁₀₉	—	2	1	no	—
23	#436×C₁₀₉	—	2	1	no	—
24	#436×C₁₀₉	—	2	1	no	—
25	#436×C₁₀₉	C₂²	1	2	yes	—
26	#436×C₁₀₉	C₂²	1	2	yes	—
27	#423×C₁₀₉	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
29	#417×C₁₀₉	—	2	2	no	—
30	C₂×C₂²⋊C₈×C₁₀₉	—	2	1	no	—
31	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
32	#334×C₁₀₉	C₂²	1	2	yes	complemented
33	#334×C₁₀₉	C₂²	1	2	yes	complemented
34	#332×C₁₀₉	C₂²	1	4	yes	complemented
35	#331×C₁₀₉	C₂²	1	4	yes	complemented
36	C₂×#41×C₁₀₉	—	2	2	no	—
37	C₂²×D₄×C₁₀₉	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₁₀₉	—	2	1	no	—
39	AGL(1,ℤ/8)×C₁₀₉	—	4	8	no	—
40	C₂×SD₁₆×C₁₀₉	—	2	4	no	—
41	C₂×D₈×C₁₀₉	—	2	4	no	—
42	C₂×D₈×C₁₀₉	—	2	1	no	—
43	C₂×D₈×C₁₀₉	—	2	1	no	—
44	C₂×M₁₆×C₁₀₉	—	2	2	no	—
45	C₂×M₁₆×C₁₀₉	—	2	2	no	—
46	C₂×M₁₆×C₁₀₉	—	2	2	no	—
47	C₂×M₁₆×C₁₀₉	—	4	4	no	—
48	C₂²×C₈×C₁₀₉	—	2	1	no	—
49	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
51	C₄⋊Q₈×C₁₀₉	C₂³	1	2	yes	—
52	#127×C₁₀₉	C₂³	1	2	yes	—
53	#124×C₁₀₉	—	2	2	no	—
54	#121×C₁₀₉	—	2	2	no	—
55	#121×C₁₀₉	—	2	2	no	—
56	C₄×D₄×C₁₀₉	—	2	1	no	—
57	C₄×D₄×C₁₀₉	—	2	2	no	—
58	C₄×D₄×C₁₀₉	—	2	1	no	—
59	C₄×D₄×C₁₀₉	—	2	2	no	—
60	C₄×D₄×C₁₀₉	D₄	1	2	yes	—
61	C₄×D₄×C₁₀₉	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
63	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
64	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
65	#107×C₁₀₉	—	2	1	no	—
66	#107×C₁₀₉	—	2	1	no	—
67	C₄⋊C₈×C₁₀₉	—	2	1	no	—
68	C₄⋊C₈×C₁₀₉	—	2	2	no	—
69	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
70	C₄⋊C₈×C₁₀₉	C₂³	1	4	yes	—
71	#103×C₁₀₉	—	2	4	no	—
72	#102×C₁₀₉	—	2	2	no	—
73	#102×C₁₀₉	—	2	2	no	—
74	#102×C₁₀₉	—	2	4	no	—
75	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
76	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
77	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
78	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
79	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
81	C₂⁴×C₁₀₉	—	2	1	no	—
82	#41×C₁₀₉	—	4	4	no	—
83	#41×C₁₀₉	—	4	4	no	—
84	C₂×Q₈×C₁₀₉	—	2	2	no	—
85	C₂×D₄×C₁₀₉	—	2	1	no	—
86	C₂×D₄×C₁₀₉	—	2	4	no	—
87	C₂×D₄×C₁₀₉	—	2	2	no	—
88	C₂×D₄×C₁₀₉	—	2	2	no	—
89	C₂×D₄×C₁₀₉	—	2	2	no	—
90	C₂×D₄×C₁₀₉	—	4	2	no	—
91	C₂×D₄×C₁₀₉	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₁₀₉	—	2	1	no	—
93	C₂²×C₄×C₁₀₉	—	2	2	no	—
94	C₂²×C₄×C₁₀₉	—	2	2	no	—
95	C₂²×C₄×C₁₀₉	—	4	2	no	—
96	C₂²×C₄×C₁₀₉	—	4	2	no	—
97	C₂²×C₄×C₁₀₉	—	4	2	no	—
98	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
101	SD₁₆×C₁₀₉	—	4	8	no	—
102	D₈×C₁₀₉	—	4	2	no	—
103	D₈×C₁₀₉	—	4	8	no	—
104	D₈×C₁₀₉	—	4	2	no	—
105	M₁₆×C₁₀₉	—	4	4	no	—
106	M₁₆×C₁₀₉	—	4	4	no	—
107	M₁₆×C₁₀₉	—	4	4	no	—
108	C₂×C₈×C₁₀₉	—	2	2	no	—
109	C₂×C₈×C₁₀₉	—	2	2	no	—
110	C₂×C₈×C₁₀₉	—	2	4	no	—
111	C₂×C₈×C₁₀₉	—	4	1	no	—
112	C₂×C₈×C₁₀₉	—	4	2	no	—
113	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
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	C₂×D₄	1	2	yes	—
121	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
122	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
123	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
124	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
125	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₁₀₉	#41	1	1	yes	characteristic
128	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
129	C₂³×C₁₀₉	—	2	2	no	—
130	C₂³×C₁₀₉	—	4	2	no	—
131	C₂³×C₁₀₉	—	4	2	no	—
132	C₂³×C₁₀₉	—	4	2	no	—
133	C₂³×C₁₀₉	—	4	2	no	—
134	C₂³×C₁₀₉	#121	1	1	yes	characteristic
135	Q₈×C₁₀₉	—	4	2	no	—
136	Q₈×C₁₀₉	—	4	2	no	—
137	D₄×C₁₀₉	—	2	4	no	—
138	D₄×C₁₀₉	—	4	2	no	—
139	D₄×C₁₀₉	—	4	2	no	—
140	D₄×C₁₀₉	—	4	2	no	—
141	D₄×C₁₀₉	—	4	2	no	—
142	D₄×C₁₀₉	—	4	2	no	—
143	D₄×C₁₀₉	—	8	2	no	—
144	D₄×C₁₀₉	—	8	4	no	—
145	C₂×C₄×C₁₀₉	—	2	1	no	—
146	C₂×C₄×C₁₀₉	—	2	1	no	—
147	C₂×C₄×C₁₀₉	—	2	2	no	—
148	C₂×C₄×C₁₀₉	—	4	1	no	—
149	C₂×C₄×C₁₀₉	—	4	4	no	—
150	C₂×C₄×C₁₀₉	—	4	2	no	—
151	C₂×C₄×C₁₀₉	—	4	2	no	—
152	C₂×C₄×C₁₀₉	—	4	2	no	—
153	C₂×C₄×C₁₀₉	—	4	2	no	—
154	C₂×C₄×C₁₀₉	—	8	4	no	—
155	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
156	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
157	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
158	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	4	yes	—
159	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
162	C₈×C₁₀₉	—	2	1	no	—
163	C₈×C₁₀₉	—	2	1	no	—
164	C₈×C₁₀₉	—	2	1	no	—
165	C₈×C₁₀₉	—	4	2	no	—
166	C₈×C₁₀₉	—	4	4	no	—
167	C₈×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	complemented
168	C₂²×C₁₀₉	—	2	1	no	—
169	C₂²×C₁₀₉	—	2	1	no	—
170	C₂²×C₁₀₉	—	4	2	no	—
171	C₂²×C₁₀₉	—	4	2	no	—
172	C₂²×C₁₀₉	—	4	2	no	—
173	C₂²×C₁₀₉	—	8	4	no	—
174	C₂²×C₁₀₉	—	8	2	no	—
175	C₂²×C₁₀₉	—	8	2	no	—
176	C₂²×C₁₀₉	—	8	2	no	—
177	C₂²×C₁₀₉	#469	1	2	yes	—
178	C₂²×C₁₀₉	C₂×#121	1	1	yes	characteristic
179	C₄×C₁₀₉	—	2	1	no	—
180	C₄×C₁₀₉	—	2	1	no	—
181	C₄×C₁₀₉	—	2	1	no	—
182	C₄×C₁₀₉	—	4	2	no	—
183	C₄×C₁₀₉	—	8	2	no	—
184	C₄×C₁₀₉	#469	1	2	yes	—
185	C₄×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₄×C₁₀₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
187	C₄×C₁₀₉	#575	1	1	yes	characteristic
188	#19280	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂×C₁₀₉	—	2	2	no	—
190	C₂×C₁₀₉	—	8	2	no	—
191	C₂×C₁₀₉	—	16	2	no	—
192	C₂×C₁₀₉	#3036	1	1	yes	characteristic
193	C₂×C₁₀₉	C₂×#469	1	1	yes	characteristic
194	C₂×C₁₀₉	#3067	1	1	yes	characteristic
195	#3169	C₂×C₁₀₉	1	1	yes	characteristic, complemented
196	#3036	C₂×C₁₀₉	1	2	yes	complemented
197	C₂×#436	C₂×C₁₀₉	1	1	yes	characteristic, complemented
198	#1701	C₂×C₁₀₉	1	4	yes	complemented
199	#1693	C₂×C₁₀₉	1	2	yes	complemented
200	#1687	C₂×C₁₀₉	1	2	yes	complemented
201	#1566	C₂×C₁₀₉	1	1	yes	characteristic, complemented
202	#1547	C₂×C₁₀₉	1	2	yes	complemented
203	C₁₀₉	#19280	1	1	yes	characteristic, direct product factor, Sylow
204	C₂×AGL(1,ℤ/8)	—	2	2	no	—
205	C₂²×M₁₆	—	2	1	no	—
206	#532	C₂²×C₁₀₉	1	2	yes	—
207	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
208	#500	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
209	#493	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
210	#486	C₂²×C₁₀₉	1	2	yes	complemented
211	#466	—	2	1	no	—
212	#466	—	2	1	no	—
213	#463	—	2	4	no	—
214	#461	C₂²×C₁₀₉	1	4	yes	—
215	#459	C₂²×C₁₀₉	1	4	yes	—
216	C₄×D₈	—	2	1	no	—
217	C₄×D₈	—	2	1	no	—
218	#436	—	2	1	no	—
219	#436	—	2	1	no	—
220	#436	C₂²×C₁₀₉	1	2	yes	—
221	#436	C₂²×C₁₀₉	1	2	yes	—
222	#423	C₂²×C₁₀₉	1	2	yes	—
223	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
224	#417	—	2	2	no	—
225	C₂×C₂²⋊C₈	—	2	1	no	—
226	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
227	#334	C₂²×C₁₀₉	1	2	yes	complemented
228	#334	C₂²×C₁₀₉	1	2	yes	complemented
229	#332	C₂²×C₁₀₉	1	4	yes	complemented
230	#331	C₂²×C₁₀₉	1	4	yes	complemented
231	C₂×#41	—	2	2	no	—
232	C₂²×D₄	D₄×C₁₀₉	1	1	yes	characteristic
233	C₂³×C₄	—	2	1	no	—
234	AGL(1,ℤ/8)	—	4	8	no	—
235	C₂×SD₁₆	—	2	4	no	—
236	C₂×D₈	—	2	4	no	—
237	C₂×D₈	—	2	1	no	—
238	C₂×D₈	—	2	1	no	—
239	C₂×M₁₆	—	2	2	no	—
240	C₂×M₁₆	—	2	2	no	—
241	C₂×M₁₆	—	2	2	no	—
242	C₂×M₁₆	—	4	4	no	—
243	C₂²×C₈	—	2	1	no	—
244	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
245	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
246	C₄⋊Q₈	C₂³×C₁₀₉	1	2	yes	—
247	#127	C₂³×C₁₀₉	1	2	yes	—
248	#124	—	2	2	no	—
249	#121	—	2	2	no	—
250	#121	—	2	2	no	—
251	C₄×D₄	—	2	1	no	—
252	C₄×D₄	—	2	2	no	—
253	C₄×D₄	—	2	1	no	—
254	C₄×D₄	—	2	2	no	—
255	C₄×D₄	D₄×C₁₀₉	1	2	yes	—
256	C₄×D₄	C₂³×C₁₀₉	1	2	yes	—
257	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
258	C₂×C₂²⋊C₄	—	2	1	no	—
259	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
260	#107	—	2	1	no	—
261	#107	—	2	1	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	2	2	no	—
264	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
265	C₄⋊C₈	C₂³×C₁₀₉	1	4	yes	—
266	#103	—	2	4	no	—
267	#102	—	2	2	no	—
268	#102	—	2	2	no	—
269	#102	—	2	4	no	—
270	C₂²⋊C₈	—	2	2	no	—
271	C₂²⋊C₈	—	2	2	no	—
272	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
273	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
274	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
275	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
276	C₂⁴	—	2	1	no	—
277	#41	—	4	4	no	—
278	#41	—	4	4	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	2	4	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	4	2	no	—
286	C₂×D₄	C₂×D₄×C₁₀₉	1	2	yes	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	4	2	no	—
291	C₂²×C₄	—	4	2	no	—
292	C₂²×C₄	—	4	2	no	—
293	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
294	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
295	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
296	SD₁₆	—	4	8	no	—
297	D₈	—	4	2	no	—
298	D₈	—	4	8	no	—
299	D₈	—	4	2	no	—
300	M₁₆	—	4	4	no	—
301	M₁₆	—	4	4	no	—
302	M₁₆	—	4	4	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	4	no	—
306	C₂×C₈	—	4	1	no	—
307	C₂×C₈	—	4	2	no	—
308	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
309	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
310	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
311	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	C₂×D₄×C₁₀₉	1	2	yes	—
316	C₂²⋊C₄	—	2	2	no	—
317	C₂²⋊C₄	—	2	2	no	—
318	C₂²⋊C₄	—	4	2	no	—
319	C₂²⋊C₄	—	4	2	no	—
320	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
321	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
322	C₄²	#41×C₁₀₉	1	1	yes	characteristic
323	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
324	C₂³	—	2	2	no	—
325	C₂³	—	4	2	no	—
326	C₂³	—	4	2	no	—
327	C₂³	—	4	2	no	—
328	C₂³	—	4	2	no	—
329	C₂³	#121×C₁₀₉	1	1	yes	characteristic
330	Q₈	—	4	2	no	—
331	Q₈	—	4	2	no	—
332	D₄	—	2	4	no	—
333	D₄	—	4	2	no	—
334	D₄	—	4	2	no	—
335	D₄	—	4	2	no	—
336	D₄	—	4	2	no	—
337	D₄	—	4	2	no	—
338	D₄	—	8	2	no	—
339	D₄	—	8	4	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	4	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	8	4	no	—
350	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
351	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
352	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
353	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	4	yes	—
354	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
356	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	4	2	no	—
361	C₈	—	4	4	no	—
362	C₈	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	complemented
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	4	2	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	8	4	no	—
369	C₂²	—	8	2	no	—
370	C₂²	—	8	2	no	—
371	C₂²	—	8	2	no	—
372	C₂²	#469×C₁₀₉	1	2	yes	—
373	C₂²	C₂×#121×C₁₀₉	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	4	2	no	—
378	C₄	—	8	2	no	—
379	C₄	#469×C₁₀₉	1	2	yes	—
380	C₄	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
381	C₄	C₂×AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
382	C₄	#575×C₁₀₉	1	1	yes	characteristic
383	C₂	—	2	2	no	—
384	C₂	—	8	2	no	—
385	C₂	—	16	2	no	—
386	C₂	C₁₀₉×#3036	1	1	yes	characteristic
387	C₂	C₂×#469×C₁₀₉	1	1	yes	characteristic
388	C₂	C₁₀₉×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₀₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₀₉×#3169	C₂	semidirect
2. C₁₀₉×#3036	C₂	semidirect
3. C₂×#436×C₁₀₉	C₂	semidirect
4. C₁₀₉×#1701	C₂	semidirect
5. C₁₀₉×#1693	C₂	semidirect
6. C₁₀₉×#1687	C₂	semidirect
7. C₁₀₉×#1566	C₂	semidirect
8. C₁₀₉×#1547	C₂	semidirect
13. #500×C₁₀₉	C₂²	semidirect
14. #493×C₁₀₉	C₂²	semidirect
15. #486×C₁₀₉	C₂²	semidirect
32. #334×C₁₀₉	C₂²	semidirect
33. #334×C₁₀₉	C₂²	semidirect
34. #332×C₁₀₉	C₂²	semidirect
35. #331×C₁₀₉	C₂²	semidirect
167. C₈×C₁₀₉	AGL(1,ℤ/8)	semidirect
188. #19280	C₁₀₉	direct (N × H)
195. #3169	C₂×C₁₀₉	semidirect
196. #3036	C₂×C₁₀₉	semidirect
197. C₂×#436	C₂×C₁₀₉	semidirect
198. #1701	C₂×C₁₀₉	semidirect
199. #1693	C₂×C₁₀₉	semidirect
200. #1687	C₂×C₁₀₉	semidirect
201. #1566	C₂×C₁₀₉	semidirect
202. #1547	C₂×C₁₀₉	semidirect
203. C₁₀₉	#19280	direct (N × H)
208. #500	C₂²×C₁₀₉	semidirect
209. #493	C₂²×C₁₀₉	semidirect
210. #486	C₂²×C₁₀₉	semidirect
227. #334	C₂²×C₁₀₉	semidirect
228. #334	C₂²×C₁₀₉	semidirect
229. #332	C₂²×C₁₀₉	semidirect
230. #331	C₂²×C₁₀₉	semidirect
362. C₈	AGL(1,ℤ/8)×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#3169	2
C₁₀₉×#3036	2
C₂×#436×C₁₀₉	2
C₁₀₉×#1701	2
C₁₀₉×#1693	2
C₁₀₉×#1687	2
C₁₀₉×#1566	2
C₁₀₉×#1547	2
#19280	109

Subgroups — C109×#19280 #42137274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#19280 #42137274