Subgroups — C₁₀₉×#18615 #42136609

Order 27904.

930 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#3506	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#3237	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#3182	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#3101	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1650	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#1642	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1638	C₂	1	2	yes	maximal, complemented
8	C₁₀₉×#1631	C₂	1	2	yes	maximal, complemented
9	C₁₀₉×#1553	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₀₉×#1490	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₀₉×#1485	C₂	1	1	yes	maximal, characteristic, complemented
12	#559×C₁₀₉	—	2	2	no	—
13	#556×C₁₀₉	C₂²	1	2	yes	—
14	#555×C₁₀₉	—	2	1	no	—
15	#553×C₁₀₉	—	2	2	no	—
16	#550×C₁₀₉	C₂²	1	2	yes	—
17	#547×C₁₀₉	—	2	1	no	—
18	#538×C₁₀₉	—	2	2	no	—
19	#535×C₁₀₉	—	2	1	no	—
20	#533×C₁₀₉	C₂²	1	1	yes	characteristic
21	#533×C₁₀₉	C₂²	1	1	yes	characteristic
22	#532×C₁₀₉	C₂²	1	1	yes	characteristic
23	#525×C₁₀₉	—	2	2	no	—
24	#520×C₁₀₉	—	2	1	no	—
25	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
26	#501×C₁₀₉	—	2	1	no	—
27	#500×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
28	#492×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
29	#491×C₁₀₉	—	2	1	no	—
30	#487×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
31	#486×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
32	#485×C₁₀₉	—	2	2	no	—
33	#483×C₁₀₉	—	2	1	no	—
34	#477×C₁₀₉	C₂²	1	2	yes	complemented
35	#476×C₁₀₉	C₂²	1	2	yes	—
36	#475×C₁₀₉	C₂²	1	2	yes	complemented
37	#468×C₁₀₉	—	2	2	no	—
38	#461×C₁₀₉	C₂²	1	2	yes	—
39	#450×C₁₀₉	—	2	1	no	—
40	#440×C₁₀₉	—	2	2	no	—
41	C₄×SD₁₆×C₁₀₉	—	2	2	no	—
42	#432×C₁₀₉	C₂²	1	1	yes	characteristic
43	#431×C₁₀₉	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
45	#332×C₁₀₉	C₂²	1	2	yes	complemented
46	#331×C₁₀₉	C₂²	1	2	yes	complemented
47	Q₈⋊C₈×C₁₀₉	C₂²	1	2	yes	complemented
48	Q₈⋊C₈×C₁₀₉	C₂²	1	2	yes	complemented
49	D₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
50	D₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
51	C₂×#41×C₁₀₉	—	2	2	no	—
52	C₂×#41×C₁₀₉	—	2	1	no	—
53	C₂²×D₄×C₁₀₉	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₁₀₉	—	2	1	no	—
55	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
56	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
57	C₂×M₁₆×C₁₀₉	—	2	1	no	—
58	C₄⋊Q₈×C₁₀₉	D₄	1	1	yes	characteristic
59	C₄⋊Q₈×C₁₀₉	D₄	1	1	yes	characteristic
60	C₄⋊Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
62	C₄⋊Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
63	#127×C₁₀₉	—	2	2	no	—
64	#127×C₁₀₉	C₂³	1	1	yes	characteristic
65	#126×C₁₀₉	—	2	4	no	—
66	#125×C₁₀₉	—	2	1	no	—
67	#125×C₁₀₉	—	2	1	no	—
68	#124×C₁₀₉	—	2	4	no	—
69	#124×C₁₀₉	—	2	2	no	—
70	#124×C₁₀₉	—	2	1	no	—
71	#123×C₁₀₉	—	2	4	no	—
72	#123×C₁₀₉	—	2	4	no	—
73	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
74	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
75	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
76	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
77	C₂²⋊Q₈×C₁₀₉	D₄	1	2	yes	—
78	#121×C₁₀₉	—	2	2	no	—
79	#121×C₁₀₉	—	2	2	no	—
80	#121×C₁₀₉	—	2	4	no	—
81	#121×C₁₀₉	—	2	1	no	—
82	#121×C₁₀₉	—	2	2	no	—
83	#121×C₁₀₉	D₄	1	2	yes	—
84	#120×C₁₀₉	—	2	4	no	—
85	C₄×Q₈×C₁₀₉	—	2	2	no	—
86	C₄×Q₈×C₁₀₉	—	2	1	no	—
87	C₄×Q₈×C₁₀₉	—	2	1	no	—
88	C₄×Q₈×C₁₀₉	—	2	1	no	—
89	C₄×Q₈×C₁₀₉	C₂³	1	2	yes	—
90	C₄×D₄×C₁₀₉	—	2	4	no	—
91	C₄×D₄×C₁₀₉	—	2	4	no	—
92	C₄×D₄×C₁₀₉	—	2	2	no	—
93	C₄×D₄×C₁₀₉	—	2	1	no	—
94	C₄×D₄×C₁₀₉	—	2	1	no	—
95	C₄×D₄×C₁₀₉	—	2	1	no	—
96	C₄×D₄×C₁₀₉	D₄	1	2	yes	—
97	C₄×D₄×C₁₀₉	D₄	1	2	yes	—
98	C₄×D₄×C₁₀₉	C₂³	1	2	yes	—
99	#117×C₁₀₉	—	2	2	no	—
100	#117×C₁₀₉	—	2	1	no	—
101	#117×C₁₀₉	—	2	2	no	—
102	#117×C₁₀₉	—	2	1	no	—
103	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
104	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
105	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
106	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
107	#107×C₁₀₉	—	2	2	no	—
108	#107×C₁₀₉	—	2	2	no	—
109	#106×C₁₀₉	—	2	2	no	—
110	#106×C₁₀₉	—	2	2	no	—
111	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
112	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
113	#103×C₁₀₉	—	2	2	no	—
114	#103×C₁₀₉	—	2	2	no	—
115	#103×C₁₀₉	—	2	2	no	—
116	#103×C₁₀₉	—	2	2	no	—
117	#102×C₁₀₉	—	2	2	no	—
118	#102×C₁₀₉	—	2	2	no	—
119	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
120	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
121	#97×C₁₀₉	—	2	1	no	—
122	#97×C₁₀₉	—	2	1	no	—
123	C₄×C₈×C₁₀₉	—	2	1	no	—
124	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
126	C₂⁴×C₁₀₉	—	2	1	no	—
127	#41×C₁₀₉	—	4	4	no	—
128	#41×C₁₀₉	—	8	4	no	—
129	C₂×Q₈×C₁₀₉	—	2	1	no	—
130	C₂×Q₈×C₁₀₉	—	2	1	no	—
131	C₂×Q₈×C₁₀₉	—	2	1	no	—
132	C₂×Q₈×C₁₀₉	—	2	1	no	—
133	C₂×Q₈×C₁₀₉	C₂×D₄	1	2	yes	—
134	C₂×D₄×C₁₀₉	—	2	2	no	—
135	C₂×D₄×C₁₀₉	—	2	4	no	—
136	C₂×D₄×C₁₀₉	—	2	4	no	—
137	C₂×D₄×C₁₀₉	—	2	1	no	—
138	C₂×D₄×C₁₀₉	—	2	1	no	—
139	C₂×D₄×C₁₀₉	—	2	1	no	—
140	C₂×D₄×C₁₀₉	—	4	2	no	—
141	C₂×D₄×C₁₀₉	C₂×D₄	1	2	yes	—
142	C₂×D₄×C₁₀₉	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₁₀₉	—	2	4	no	—
144	C₂²×C₄×C₁₀₉	—	2	1	no	—
145	C₂²×C₄×C₁₀₉	—	2	2	no	—
146	C₂²×C₄×C₁₀₉	—	2	2	no	—
147	C₂²×C₄×C₁₀₉	—	2	2	no	—
148	C₂²×C₄×C₁₀₉	—	8	2	no	—
149	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
152	SD₁₆×C₁₀₉	—	4	4	no	—
153	SD₁₆×C₁₀₉	—	4	4	no	—
154	M₁₆×C₁₀₉	—	4	2	no	—
155	C₂×C₈×C₁₀₉	—	2	1	no	—
156	C₂×C₈×C₁₀₉	—	2	2	no	—
157	C₂×C₈×C₁₀₉	—	2	1	no	—
158	C₂×C₈×C₁₀₉	—	2	2	no	—
159	C₄⋊C₄×C₁₀₉	—	2	4	no	—
160	C₄⋊C₄×C₁₀₉	—	2	1	no	—
161	C₄⋊C₄×C₁₀₉	—	2	1	no	—
162	C₄⋊C₄×C₁₀₉	—	2	1	no	—
163	C₄⋊C₄×C₁₀₉	—	2	2	no	—
164	C₄⋊C₄×C₁₀₉	—	2	1	no	—
165	C₄⋊C₄×C₁₀₉	—	2	1	no	—
166	C₄⋊C₄×C₁₀₉	—	2	2	no	—
167	C₄⋊C₄×C₁₀₉	—	2	1	no	—
168	C₄⋊C₄×C₁₀₉	—	2	1	no	—
169	C₄⋊C₄×C₁₀₉	—	2	1	no	—
170	C₄⋊C₄×C₁₀₉	—	2	1	no	—
171	C₄⋊C₄×C₁₀₉	—	4	1	no	—
172	C₄⋊C₄×C₁₀₉	—	4	1	no	—
173	C₄⋊C₄×C₁₀₉	C₂×D₄	1	2	yes	—
174	C₄⋊C₄×C₁₀₉	C₂×D₄	1	2	yes	—
175	C₄⋊C₄×C₁₀₉	C₂×D₄	1	2	yes	—
176	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
177	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
178	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
179	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
180	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
181	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
182	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
183	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
184	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
185	C₄²×C₁₀₉	—	2	4	no	—
186	C₄²×C₁₀₉	—	2	2	no	—
187	C₄²×C₁₀₉	—	4	1	no	—
188	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
189	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
191	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
192	C₂³×C₁₀₉	—	2	2	no	—
193	C₂³×C₁₀₉	—	2	2	no	—
194	C₂³×C₁₀₉	—	4	2	no	—
195	C₂³×C₁₀₉	—	8	1	no	—
196	C₂³×C₁₀₉	—	8	1	no	—
197	C₂³×C₁₀₉	#120	1	1	yes	characteristic
198	Q₈×C₁₀₉	—	2	4	no	—
199	Q₈×C₁₀₉	—	4	2	no	—
200	Q₈×C₁₀₉	—	4	2	no	—
201	Q₈×C₁₀₉	—	4	2	no	—
202	Q₈×C₁₀₉	—	8	1	no	—
203	D₄×C₁₀₉	—	2	2	no	—
204	D₄×C₁₀₉	—	2	2	no	—
205	D₄×C₁₀₉	—	4	2	no	—
206	D₄×C₁₀₉	—	4	2	no	—
207	D₄×C₁₀₉	—	4	4	no	—
208	D₄×C₁₀₉	—	4	2	no	—
209	D₄×C₁₀₉	—	4	2	no	—
210	D₄×C₁₀₉	—	8	4	no	—
211	D₄×C₁₀₉	—	8	4	no	—
212	C₂×C₄×C₁₀₉	—	2	2	no	—
213	C₂×C₄×C₁₀₉	—	2	2	no	—
214	C₂×C₄×C₁₀₉	—	2	2	no	—
215	C₂×C₄×C₁₀₉	—	2	2	no	—
216	C₂×C₄×C₁₀₉	—	2	2	no	—
217	C₂×C₄×C₁₀₉	—	2	2	no	—
218	C₂×C₄×C₁₀₉	—	4	1	no	—
219	C₂×C₄×C₁₀₉	—	4	4	no	—
220	C₂×C₄×C₁₀₉	—	4	4	no	—
221	C₂×C₄×C₁₀₉	—	4	1	no	—
222	C₂×C₄×C₁₀₉	—	4	1	no	—
223	C₂×C₄×C₁₀₉	—	8	2	no	—
224	C₂×C₄×C₁₀₉	—	8	1	no	—
225	C₂×C₄×C₁₀₉	—	8	4	no	—
226	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
227	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
228	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
229	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
232	C₈×C₁₀₉	—	2	2	no	—
233	C₈×C₁₀₉	—	4	1	no	—
234	C₈×C₁₀₉	—	4	2	no	—
235	C₈×C₁₀₉	—	4	2	no	—
236	C₂²×C₁₀₉	—	2	1	no	—
237	C₂²×C₁₀₉	—	2	1	no	—
238	C₂²×C₁₀₉	—	2	1	no	—
239	C₂²×C₁₀₉	—	4	2	no	—
240	C₂²×C₁₀₉	—	4	2	no	—
241	C₂²×C₁₀₉	—	4	2	no	—
242	C₂²×C₁₀₉	—	4	2	no	—
243	C₂²×C₁₀₉	—	4	2	no	—
244	C₂²×C₁₀₉	—	4	2	no	—
245	C₂²×C₁₀₉	—	8	4	no	—
246	C₂²×C₁₀₉	C₂×#120	1	1	yes	characteristic
247	C₄×C₁₀₉	—	2	1	no	—
248	C₄×C₁₀₉	—	2	1	no	—
249	C₄×C₁₀₉	—	2	1	no	—
250	C₄×C₁₀₉	—	2	1	no	—
251	C₄×C₁₀₉	—	4	2	no	—
252	C₄×C₁₀₉	—	4	2	no	—
253	C₄×C₁₀₉	—	4	2	no	—
254	C₄×C₁₀₉	—	4	2	no	—
255	C₄×C₁₀₉	—	8	1	no	—
256	C₄×C₁₀₉	—	8	1	no	—
257	C₄×C₁₀₉	#575	1	2	yes	—
258	C₄×C₁₀₉	#577	1	2	yes	—
259	#18615	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
260	C₂×C₁₀₉	—	4	1	no	—
261	C₂×C₁₀₉	—	8	2	no	—
262	C₂×C₁₀₉	—	8	2	no	—
263	C₂×C₁₀₉	#3005	1	1	yes	characteristic
264	C₂×C₁₀₉	#3009	1	1	yes	characteristic
265	C₂×C₁₀₉	#3018	1	1	yes	characteristic
266	#3506	C₂×C₁₀₉	1	1	yes	characteristic
267	#3237	C₂×C₁₀₉	1	1	yes	characteristic, complemented
268	#3182	C₂×C₁₀₉	1	1	yes	characteristic, complemented
269	#3101	C₂×C₁₀₉	1	1	yes	characteristic, complemented
270	#1650	C₂×C₁₀₉	1	2	yes	complemented
271	#1642	C₂×C₁₀₉	1	2	yes	complemented
272	#1638	C₂×C₁₀₉	1	2	yes	complemented
273	#1631	C₂×C₁₀₉	1	2	yes	complemented
274	#1553	C₂×C₁₀₉	1	1	yes	characteristic, complemented
275	#1490	C₂×C₁₀₉	1	1	yes	characteristic, complemented
276	#1485	C₂×C₁₀₉	1	1	yes	characteristic, complemented
277	C₁₀₉	#18615	1	1	yes	characteristic, direct product factor, Sylow
278	#559	—	2	2	no	—
279	#556	C₂²×C₁₀₉	1	2	yes	—
280	#555	—	2	1	no	—
281	#553	—	2	2	no	—
282	#550	C₂²×C₁₀₉	1	2	yes	—
283	#547	—	2	1	no	—
284	#538	—	2	2	no	—
285	#535	—	2	1	no	—
286	#533	C₂²×C₁₀₉	1	1	yes	characteristic
287	#533	C₂²×C₁₀₉	1	1	yes	characteristic
288	#532	C₂²×C₁₀₉	1	1	yes	characteristic
289	#525	—	2	2	no	—
290	#520	—	2	1	no	—
291	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
292	#501	—	2	1	no	—
293	#500	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
294	#492	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
295	#491	—	2	1	no	—
296	#487	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
297	#486	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
298	#485	—	2	2	no	—
299	#483	—	2	1	no	—
300	#477	C₂²×C₁₀₉	1	2	yes	complemented
301	#476	C₂²×C₁₀₉	1	2	yes	—
302	#475	C₂²×C₁₀₉	1	2	yes	complemented
303	#468	—	2	2	no	—
304	#461	C₂²×C₁₀₉	1	2	yes	—
305	#450	—	2	1	no	—
306	#440	—	2	2	no	—
307	C₄×SD₁₆	—	2	2	no	—
308	#432	C₂²×C₁₀₉	1	1	yes	characteristic
309	#431	C₂²×C₁₀₉	1	1	yes	characteristic
310	C₄×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
311	#332	C₂²×C₁₀₉	1	2	yes	complemented
312	#331	C₂²×C₁₀₉	1	2	yes	complemented
313	Q₈⋊C₈	C₂²×C₁₀₉	1	2	yes	complemented
314	Q₈⋊C₈	C₂²×C₁₀₉	1	2	yes	complemented
315	D₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
316	D₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
317	C₂×#41	—	2	2	no	—
318	C₂×#41	—	2	1	no	—
319	C₂²×D₄	D₄×C₁₀₉	1	1	yes	characteristic
320	C₂³×C₄	—	2	1	no	—
321	C₂×SD₁₆	—	2	2	no	—
322	C₂×SD₁₆	—	2	2	no	—
323	C₂×M₁₆	—	2	1	no	—
324	C₄⋊Q₈	D₄×C₁₀₉	1	1	yes	characteristic
325	C₄⋊Q₈	D₄×C₁₀₉	1	1	yes	characteristic
326	C₄⋊Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
327	C₄⋊Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
328	C₄⋊Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
329	#127	—	2	2	no	—
330	#127	C₂³×C₁₀₉	1	1	yes	characteristic
331	#126	—	2	4	no	—
332	#125	—	2	1	no	—
333	#125	—	2	1	no	—
334	#124	—	2	4	no	—
335	#124	—	2	2	no	—
336	#124	—	2	1	no	—
337	#123	—	2	4	no	—
338	#123	—	2	4	no	—
339	C₂²⋊Q₈	—	2	2	no	—
340	C₂²⋊Q₈	—	2	1	no	—
341	C₂²⋊Q₈	—	2	1	no	—
342	C₂²⋊Q₈	—	2	1	no	—
343	C₂²⋊Q₈	D₄×C₁₀₉	1	2	yes	—
344	#121	—	2	2	no	—
345	#121	—	2	2	no	—
346	#121	—	2	4	no	—
347	#121	—	2	1	no	—
348	#121	—	2	2	no	—
349	#121	D₄×C₁₀₉	1	2	yes	—
350	#120	—	2	4	no	—
351	C₄×Q₈	—	2	2	no	—
352	C₄×Q₈	—	2	1	no	—
353	C₄×Q₈	—	2	1	no	—
354	C₄×Q₈	—	2	1	no	—
355	C₄×Q₈	C₂³×C₁₀₉	1	2	yes	—
356	C₄×D₄	—	2	4	no	—
357	C₄×D₄	—	2	4	no	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	—	2	1	no	—
360	C₄×D₄	—	2	1	no	—
361	C₄×D₄	—	2	1	no	—
362	C₄×D₄	D₄×C₁₀₉	1	2	yes	—
363	C₄×D₄	D₄×C₁₀₉	1	2	yes	—
364	C₄×D₄	C₂³×C₁₀₉	1	2	yes	—
365	#117	—	2	2	no	—
366	#117	—	2	1	no	—
367	#117	—	2	2	no	—
368	#117	—	2	1	no	—
369	C₂×C₄⋊C₄	—	2	2	no	—
370	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
371	C₂×C₂²⋊C₄	—	2	1	no	—
372	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
373	#107	—	2	2	no	—
374	#107	—	2	2	no	—
375	#106	—	2	2	no	—
376	#106	—	2	2	no	—
377	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
378	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
379	#103	—	2	2	no	—
380	#103	—	2	2	no	—
381	#103	—	2	2	no	—
382	#103	—	2	2	no	—
383	#102	—	2	2	no	—
384	#102	—	2	2	no	—
385	C₂²⋊C₈	—	2	1	no	—
386	C₂²⋊C₈	—	2	1	no	—
387	#97	—	2	1	no	—
388	#97	—	2	1	no	—
389	C₄×C₈	—	2	1	no	—
390	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
391	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
392	C₂⁴	—	2	1	no	—
393	#41	—	4	4	no	—
394	#41	—	8	4	no	—
395	C₂×Q₈	—	2	1	no	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	—	2	1	no	—
398	C₂×Q₈	—	2	1	no	—
399	C₂×Q₈	C₂×D₄×C₁₀₉	1	2	yes	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	2	1	no	—
404	C₂×D₄	—	2	1	no	—
405	C₂×D₄	—	2	1	no	—
406	C₂×D₄	—	4	2	no	—
407	C₂×D₄	C₂×D₄×C₁₀₉	1	2	yes	—
408	C₂×D₄	C₂×D₄×C₁₀₉	1	2	yes	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	8	2	no	—
415	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
416	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
417	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
418	SD₁₆	—	4	4	no	—
419	SD₁₆	—	4	4	no	—
420	M₁₆	—	4	2	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	1	no	—
424	C₂×C₈	—	2	2	no	—
425	C₄⋊C₄	—	2	4	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	2	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	—	4	1	no	—
438	C₄⋊C₄	—	4	1	no	—
439	C₄⋊C₄	C₂×D₄×C₁₀₉	1	2	yes	—
440	C₄⋊C₄	C₂×D₄×C₁₀₉	1	2	yes	—
441	C₄⋊C₄	C₂×D₄×C₁₀₉	1	2	yes	—
442	C₂²⋊C₄	—	2	4	no	—
443	C₂²⋊C₄	—	2	4	no	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	2	2	no	—
446	C₂²⋊C₄	—	2	2	no	—
447	C₂²⋊C₄	—	2	2	no	—
448	C₂²⋊C₄	—	2	2	no	—
449	C₂²⋊C₄	—	2	2	no	—
450	C₂²⋊C₄	—	4	1	no	—
451	C₄²	—	2	4	no	—
452	C₄²	—	2	2	no	—
453	C₄²	—	4	1	no	—
454	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
455	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
456	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
457	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
458	C₂³	—	2	2	no	—
459	C₂³	—	2	2	no	—
460	C₂³	—	4	2	no	—
461	C₂³	—	8	1	no	—
462	C₂³	—	8	1	no	—
463	C₂³	#120×C₁₀₉	1	1	yes	characteristic
464	Q₈	—	2	4	no	—
465	Q₈	—	4	2	no	—
466	Q₈	—	4	2	no	—
467	Q₈	—	4	2	no	—
468	Q₈	—	8	1	no	—
469	D₄	—	2	2	no	—
470	D₄	—	2	2	no	—
471	D₄	—	4	2	no	—
472	D₄	—	4	2	no	—
473	D₄	—	4	4	no	—
474	D₄	—	4	2	no	—
475	D₄	—	4	2	no	—
476	D₄	—	8	4	no	—
477	D₄	—	8	4	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	4	no	—
486	C₂×C₄	—	4	4	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	8	2	no	—
490	C₂×C₄	—	8	1	no	—
491	C₂×C₄	—	8	4	no	—
492	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
493	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
494	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
495	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
496	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
497	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
498	C₈	—	2	2	no	—
499	C₈	—	4	1	no	—
500	C₈	—	4	2	no	—
501	C₈	—	4	2	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	2	1	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	4	2	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	—	4	2	no	—
511	C₂²	—	8	4	no	—
512	C₂²	C₂×#120×C₁₀₉	1	1	yes	characteristic
513	C₄	—	2	1	no	—
514	C₄	—	2	1	no	—
515	C₄	—	2	1	no	—
516	C₄	—	2	1	no	—
517	C₄	—	4	2	no	—
518	C₄	—	4	2	no	—
519	C₄	—	4	2	no	—
520	C₄	—	4	2	no	—
521	C₄	—	8	1	no	—
522	C₄	—	8	1	no	—
523	C₄	#575×C₁₀₉	1	2	yes	—
524	C₄	#577×C₁₀₉	1	2	yes	—
525	C₂	—	4	1	no	—
526	C₂	—	8	2	no	—
527	C₂	—	8	2	no	—
528	C₂	C₁₀₉×#3005	1	1	yes	characteristic
529	C₂	C₁₀₉×#3009	1	1	yes	characteristic
530	C₂	C₁₀₉×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₀₉
prime radical	C₁₀₉×#3506	C₂
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
2. C₁₀₉×#3237	C₂	semidirect
3. C₁₀₉×#3182	C₂	semidirect
4. C₁₀₉×#3101	C₂	semidirect
5. C₁₀₉×#1650	C₂	semidirect
6. C₁₀₉×#1642	C₂	semidirect
7. C₁₀₉×#1638	C₂	semidirect
8. C₁₀₉×#1631	C₂	semidirect
9. C₁₀₉×#1553	C₂	semidirect
10. C₁₀₉×#1490	C₂	semidirect
11. C₁₀₉×#1485	C₂	semidirect
27. #500×C₁₀₉	C₂²	semidirect
28. #492×C₁₀₉	C₂²	semidirect
30. #487×C₁₀₉	C₂²	semidirect
31. #486×C₁₀₉	C₂²	semidirect
34. #477×C₁₀₉	C₂²	semidirect
36. #475×C₁₀₉	C₂²	semidirect
45. #332×C₁₀₉	C₂²	semidirect
46. #331×C₁₀₉	C₂²	semidirect
47. Q₈⋊C₈×C₁₀₉	C₂²	semidirect
48. Q₈⋊C₈×C₁₀₉	C₂²	semidirect
259. #18615	C₁₀₉	direct (N × H)
267. #3237	C₂×C₁₀₉	semidirect
268. #3182	C₂×C₁₀₉	semidirect
269. #3101	C₂×C₁₀₉	semidirect
270. #1650	C₂×C₁₀₉	semidirect
271. #1642	C₂×C₁₀₉	semidirect
272. #1638	C₂×C₁₀₉	semidirect
273. #1631	C₂×C₁₀₉	semidirect
274. #1553	C₂×C₁₀₉	semidirect
275. #1490	C₂×C₁₀₉	semidirect
276. #1485	C₂×C₁₀₉	semidirect
277. C₁₀₉	#18615	direct (N × H)
293. #500	C₂²×C₁₀₉	semidirect
294. #492	C₂²×C₁₀₉	semidirect
296. #487	C₂²×C₁₀₉	semidirect
297. #486	C₂²×C₁₀₉	semidirect
300. #477	C₂²×C₁₀₉	semidirect
302. #475	C₂²×C₁₀₉	semidirect
311. #332	C₂²×C₁₀₉	semidirect
312. #331	C₂²×C₁₀₉	semidirect
313. Q₈⋊C₈	C₂²×C₁₀₉	semidirect
314. Q₈⋊C₈	C₂²×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#3506	2
C₁₀₉×#3237	2
C₁₀₉×#3182	2
C₁₀₉×#3101	2
C₁₀₉×#1650	2
C₁₀₉×#1642	2
C₁₀₉×#1638	2
C₁₀₉×#1631	2
C₁₀₉×#1553	2
C₁₀₉×#1490	2
C₁₀₉×#1485	2
#18615	109

Subgroups — C109×#18615 #42136609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#18615 #42136609