Subgroups — C₁₀₉×#18611 #42136605

Order 27904.

1106 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#3497	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#3235	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#3182	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#3104	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1649	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#1641	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1637	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×#1491	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₀₉×#1484	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₁₀₉	—	2	1	no	—
13	#550×C₁₀₉	—	2	1	no	—
14	#550×C₁₀₉	C₂²	1	2	yes	—
15	D₄×Q₈×C₁₀₉	—	2	1	no	—
16	D₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
17	#548×C₁₀₉	—	2	2	no	—
18	#548×C₁₀₉	—	2	2	no	—
19	#537×C₁₀₉	—	2	2	no	—
20	#537×C₁₀₉	—	2	1	no	—
21	#533×C₁₀₉	C₂²	1	1	yes	characteristic
22	#532×C₁₀₉	—	2	2	no	—
23	#532×C₁₀₉	C₂²	1	1	yes	characteristic
24	#532×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
26	#501×C₁₀₉	—	2	1	no	—
27	#500×C₁₀₉	C₂²	1	1	yes	characteristic
28	#492×C₁₀₉	C₂²	1	1	yes	characteristic
29	#490×C₁₀₉	—	2	1	no	—
30	#487×C₁₀₉	C₂²	1	1	yes	characteristic
31	#486×C₁₀₉	C₂²	1	1	yes	characteristic
32	#484×C₁₀₉	—	2	1	no	—
33	#482×C₁₀₉	—	2	2	no	—
34	#477×C₁₀₉	C₂²	1	2	yes	—
35	#476×C₁₀₉	C₂²	1	2	yes	—
36	#474×C₁₀₉	C₂²	1	2	yes	—
37	#468×C₁₀₉	—	2	2	no	—
38	#462×C₁₀₉	C₂²	1	2	yes	—
39	#451×C₁₀₉	—	2	1	no	—
40	#440×C₁₀₉	—	2	2	no	—
41	C₄×Q₁₆×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	—
46	#331×C₁₀₉	C₂²	1	2	yes	—
47	Q₈⋊C₈×C₁₀₉	C₂²	1	2	yes	—
48	Q₈⋊C₈×C₁₀₉	C₂²	1	2	yes	—
49	D₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
50	D₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
51	2₋¹⁺⁴×C₁₀₉	—	4	8	no	—
52	C₂×#41×C₁₀₉	—	2	4	no	—
53	C₂×#41×C₁₀₉	—	2	2	no	—
54	C₂×#41×C₁₀₉	—	2	2	no	—
55	C₂×#41×C₁₀₉	—	2	1	no	—
56	C₂×#41×C₁₀₉	—	2	1	no	—
57	C₂²×Q₈×C₁₀₉	—	2	2	no	—
58	C₂²×Q₈×C₁₀₉	—	2	2	no	—
59	C₂²×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
60	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
61	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
62	C₂×M₁₆×C₁₀₉	—	2	1	no	—
63	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
64	C₄⋊Q₈×C₁₀₉	D₄	1	1	yes	characteristic
65	C₄⋊Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
66	C₄⋊Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
68	#127×C₁₀₉	—	2	2	no	—
69	#127×C₁₀₉	D₄	1	1	yes	characteristic
70	#127×C₁₀₉	C₂³	1	1	yes	characteristic
71	#125×C₁₀₉	—	2	1	no	—
72	#124×C₁₀₉	—	2	4	no	—
73	#124×C₁₀₉	—	2	1	no	—
74	#124×C₁₀₉	—	2	1	no	—
75	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
76	C₂²⋊Q₈×C₁₀₉	—	2	4	no	—
77	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
78	C₂²⋊Q₈×C₁₀₉	—	2	4	no	—
79	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
80	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
81	#121×C₁₀₉	—	2	4	no	—
82	#121×C₁₀₉	—	2	2	no	—
83	#121×C₁₀₉	—	2	4	no	—
84	#121×C₁₀₉	—	2	4	no	—
85	#121×C₁₀₉	—	2	1	no	—
86	#121×C₁₀₉	—	2	1	no	—
87	#121×C₁₀₉	—	2	2	no	—
88	#121×C₁₀₉	D₄	1	2	yes	—
89	#121×C₁₀₉	D₄	1	2	yes	—
90	C₄×Q₈×C₁₀₉	—	2	1	no	—
91	C₄×Q₈×C₁₀₉	—	2	1	no	—
92	C₄×Q₈×C₁₀₉	—	2	1	no	—
93	C₄×Q₈×C₁₀₉	—	2	1	no	—
94	C₄×Q₈×C₁₀₉	D₄	1	2	yes	—
95	C₄×Q₈×C₁₀₉	C₂³	1	2	yes	—
96	C₄×D₄×C₁₀₉	—	2	4	no	—
97	C₄×D₄×C₁₀₉	—	2	2	no	—
98	C₄×D₄×C₁₀₉	—	2	2	no	—
99	C₄×D₄×C₁₀₉	—	2	1	no	—
100	C₄×D₄×C₁₀₉	—	2	1	no	—
101	C₄×D₄×C₁₀₉	D₄	1	2	yes	—
102	C₄×D₄×C₁₀₉	C₂³	1	2	yes	—
103	#117×C₁₀₉	—	2	1	no	—
104	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
105	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
106	C₂×C₄²×C₁₀₉	—	2	1	no	—
107	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
108	#107×C₁₀₉	—	2	2	no	—
109	#107×C₁₀₉	—	2	2	no	—
110	#107×C₁₀₉	—	2	2	no	—
111	#106×C₁₀₉	—	2	2	no	—
112	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
113	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
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	#102×C₁₀₉	—	2	2	no	—
120	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
121	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
122	#97×C₁₀₉	—	2	1	no	—
123	#97×C₁₀₉	—	2	1	no	—
124	C₄×C₈×C₁₀₉	—	2	1	no	—
125	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
126	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
127	#41×C₁₀₉	—	4	16	no	—
128	#41×C₁₀₉	—	4	4	no	—
129	#41×C₁₀₉	—	4	4	no	—
130	#41×C₁₀₉	—	4	8	no	—
131	#41×C₁₀₉	—	4	4	no	—
132	#41×C₁₀₉	—	4	4	no	—
133	C₂×Q₈×C₁₀₉	—	2	4	no	—
134	C₂×Q₈×C₁₀₉	—	2	4	no	—
135	C₂×Q₈×C₁₀₉	—	2	1	no	—
136	C₂×Q₈×C₁₀₉	—	2	1	no	—
137	C₂×Q₈×C₁₀₉	—	2	1	no	—
138	C₂×Q₈×C₁₀₉	—	2	1	no	—
139	C₂×Q₈×C₁₀₉	—	2	1	no	—
140	C₂×Q₈×C₁₀₉	—	4	8	no	—
141	C₂×Q₈×C₁₀₉	—	4	4	no	—
142	C₂×Q₈×C₁₀₉	—	4	4	no	—
143	C₂×Q₈×C₁₀₉	C₂×D₄	1	2	yes	—
144	C₂×D₄×C₁₀₉	—	2	4	no	—
145	C₂×D₄×C₁₀₉	—	2	2	no	—
146	C₂×D₄×C₁₀₉	—	2	2	no	—
147	C₂×D₄×C₁₀₉	—	2	4	no	—
148	C₂×D₄×C₁₀₉	—	2	1	no	—
149	C₂×D₄×C₁₀₉	—	2	1	no	—
150	C₂×D₄×C₁₀₉	—	2	1	no	—
151	C₂×D₄×C₁₀₉	—	2	1	no	—
152	C₂×D₄×C₁₀₉	C₂×D₄	1	2	yes	—
153	C₂×D₄×C₁₀₉	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₁₀₉	—	2	1	no	—
155	C₂²×C₄×C₁₀₉	—	2	4	no	—
156	C₂²×C₄×C₁₀₉	—	2	4	no	—
157	C₂²×C₄×C₁₀₉	—	2	1	no	—
158	C₂²×C₄×C₁₀₉	—	2	2	no	—
159	C₂²×C₄×C₁₀₉	—	2	2	no	—
160	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
163	Q₁₆×C₁₀₉	—	4	4	no	—
164	SD₁₆×C₁₀₉	—	4	4	no	—
165	M₁₆×C₁₀₉	—	4	2	no	—
166	C₂×C₈×C₁₀₉	—	2	1	no	—
167	C₂×C₈×C₁₀₉	—	2	2	no	—
168	C₂×C₈×C₁₀₉	—	2	1	no	—
169	C₂×C₈×C₁₀₉	—	2	2	no	—
170	C₄⋊C₄×C₁₀₉	—	2	4	no	—
171	C₄⋊C₄×C₁₀₉	—	2	2	no	—
172	C₄⋊C₄×C₁₀₉	—	2	1	no	—
173	C₄⋊C₄×C₁₀₉	—	2	1	no	—
174	C₄⋊C₄×C₁₀₉	—	2	2	no	—
175	C₄⋊C₄×C₁₀₉	—	2	1	no	—
176	C₄⋊C₄×C₁₀₉	—	2	1	no	—
177	C₄⋊C₄×C₁₀₉	—	2	1	no	—
178	C₄⋊C₄×C₁₀₉	—	2	1	no	—
179	C₄⋊C₄×C₁₀₉	—	2	1	no	—
180	C₄⋊C₄×C₁₀₉	—	4	1	no	—
181	C₄⋊C₄×C₁₀₉	—	4	1	no	—
182	C₄⋊C₄×C₁₀₉	C₂×D₄	1	2	yes	—
183	C₄⋊C₄×C₁₀₉	C₂×D₄	1	2	yes	—
184	C₄⋊C₄×C₁₀₉	C₂×D₄	1	2	yes	—
185	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
186	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
187	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
188	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
189	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
190	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
191	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
192	C₄²×C₁₀₉	—	2	2	no	—
193	C₄²×C₁₀₉	—	2	2	no	—
194	C₄²×C₁₀₉	—	4	1	no	—
195	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
196	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
197	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
199	C₂³×C₁₀₉	—	2	2	no	—
200	C₂³×C₁₀₉	—	2	2	no	—
201	C₂³×C₁₀₉	#120	1	1	yes	characteristic
202	Q₈×C₁₀₉	—	2	4	no	—
203	Q₈×C₁₀₉	—	2	2	no	—
204	Q₈×C₁₀₉	—	4	8	no	—
205	Q₈×C₁₀₉	—	4	2	no	—
206	Q₈×C₁₀₉	—	4	2	no	—
207	Q₈×C₁₀₉	—	4	2	no	—
208	Q₈×C₁₀₉	—	4	2	no	—
209	Q₈×C₁₀₉	AGL(1,ℤ/8)	1	4	yes	—
210	D₄×C₁₀₉	—	2	4	no	—
211	D₄×C₁₀₉	—	4	2	no	—
212	D₄×C₁₀₉	—	4	8	no	—
213	D₄×C₁₀₉	—	4	2	no	—
214	D₄×C₁₀₉	—	4	4	no	—
215	D₄×C₁₀₉	—	4	4	no	—
216	D₄×C₁₀₉	—	4	2	no	—
217	D₄×C₁₀₉	—	4	2	no	—
218	D₄×C₁₀₉	—	8	1	no	—
219	D₄×C₁₀₉	—	8	4	no	—
220	C₂×C₄×C₁₀₉	—	2	4	no	—
221	C₂×C₄×C₁₀₉	—	2	2	no	—
222	C₂×C₄×C₁₀₉	—	2	2	no	—
223	C₂×C₄×C₁₀₉	—	2	2	no	—
224	C₂×C₄×C₁₀₉	—	2	2	no	—
225	C₂×C₄×C₁₀₉	—	2	2	no	—
226	C₂×C₄×C₁₀₉	—	2	2	no	—
227	C₂×C₄×C₁₀₉	—	4	1	no	—
228	C₂×C₄×C₁₀₉	—	4	8	no	—
229	C₂×C₄×C₁₀₉	—	4	8	no	—
230	C₂×C₄×C₁₀₉	—	4	4	no	—
231	C₂×C₄×C₁₀₉	—	4	2	no	—
232	C₂×C₄×C₁₀₉	—	4	2	no	—
233	C₂×C₄×C₁₀₉	—	4	1	no	—
234	C₂×C₄×C₁₀₉	—	4	1	no	—
235	C₂×C₄×C₁₀₉	—	8	1	no	—
236	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
237	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
238	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
239	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
242	C₈×C₁₀₉	—	2	2	no	—
243	C₈×C₁₀₉	—	4	1	no	—
244	C₈×C₁₀₉	—	4	2	no	—
245	C₈×C₁₀₉	—	4	2	no	—
246	C₂²×C₁₀₉	—	2	1	no	—
247	C₂²×C₁₀₉	—	2	1	no	—
248	C₂²×C₁₀₉	—	2	1	no	—
249	C₂²×C₁₀₉	—	4	2	no	—
250	C₂²×C₁₀₉	—	4	2	no	—
251	C₂²×C₁₀₉	—	4	2	no	—
252	C₂²×C₁₀₉	—	4	2	no	—
253	C₂²×C₁₀₉	—	4	2	no	—
254	C₂²×C₁₀₉	—	4	2	no	—
255	C₂²×C₁₀₉	C₂×#120	1	1	yes	characteristic
256	C₄×C₁₀₉	—	2	4	no	—
257	C₄×C₁₀₉	—	2	1	no	—
258	C₄×C₁₀₉	—	2	1	no	—
259	C₄×C₁₀₉	—	2	1	no	—
260	C₄×C₁₀₉	—	2	1	no	—
261	C₄×C₁₀₉	—	4	2	no	—
262	C₄×C₁₀₉	—	4	2	no	—
263	C₄×C₁₀₉	—	4	2	no	—
264	C₄×C₁₀₉	—	8	1	no	—
265	C₄×C₁₀₉	—	8	1	no	—
266	C₄×C₁₀₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
267	C₄×C₁₀₉	#578	1	2	yes	—
268	#18611	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
269	C₂×C₁₀₉	—	4	1	no	—
270	C₂×C₁₀₉	—	8	2	no	—
271	C₂×C₁₀₉	—	8	2	no	—
272	C₂×C₁₀₉	#3003	1	1	yes	characteristic
273	C₂×C₁₀₉	#3011	1	1	yes	characteristic
274	C₂×C₁₀₉	#3018	1	1	yes	characteristic
275	#3497	C₂×C₁₀₉	1	1	yes	characteristic
276	#3235	C₂×C₁₀₉	1	1	yes	characteristic, complemented
277	#3182	C₂×C₁₀₉	1	1	yes	characteristic, complemented
278	#3104	C₂×C₁₀₉	1	1	yes	characteristic, complemented
279	#1649	C₂×C₁₀₉	1	2	yes	complemented
280	#1641	C₂×C₁₀₉	1	2	yes	complemented
281	#1637	C₂×C₁₀₉	1	2	yes	complemented
282	#1631	C₂×C₁₀₉	1	2	yes	complemented
283	#1553	C₂×C₁₀₉	1	1	yes	characteristic, complemented
284	#1491	C₂×C₁₀₉	1	1	yes	characteristic, complemented
285	#1484	C₂×C₁₀₉	1	1	yes	characteristic, complemented
286	C₁₀₉	#18611	1	1	yes	characteristic, direct product factor, Sylow
287	C₂×2₋¹⁺⁴	—	2	1	no	—
288	#550	—	2	1	no	—
289	#550	C₂²×C₁₀₉	1	2	yes	—
290	D₄×Q₈	—	2	1	no	—
291	D₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
292	#548	—	2	2	no	—
293	#548	—	2	2	no	—
294	#537	—	2	2	no	—
295	#537	—	2	1	no	—
296	#533	C₂²×C₁₀₉	1	1	yes	characteristic
297	#532	—	2	2	no	—
298	#532	C₂²×C₁₀₉	1	1	yes	characteristic
299	#532	C₂²×C₁₀₉	1	1	yes	characteristic
300	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
301	#501	—	2	1	no	—
302	#500	C₂²×C₁₀₉	1	1	yes	characteristic
303	#492	C₂²×C₁₀₉	1	1	yes	characteristic
304	#490	—	2	1	no	—
305	#487	C₂²×C₁₀₉	1	1	yes	characteristic
306	#486	C₂²×C₁₀₉	1	1	yes	characteristic
307	#484	—	2	1	no	—
308	#482	—	2	2	no	—
309	#477	C₂²×C₁₀₉	1	2	yes	—
310	#476	C₂²×C₁₀₉	1	2	yes	—
311	#474	C₂²×C₁₀₉	1	2	yes	—
312	#468	—	2	2	no	—
313	#462	C₂²×C₁₀₉	1	2	yes	—
314	#451	—	2	1	no	—
315	#440	—	2	2	no	—
316	C₄×Q₁₆	—	2	2	no	—
317	#432	C₂²×C₁₀₉	1	1	yes	characteristic
318	#431	C₂²×C₁₀₉	1	1	yes	characteristic
319	C₄×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
320	#332	C₂²×C₁₀₉	1	2	yes	—
321	#331	C₂²×C₁₀₉	1	2	yes	—
322	Q₈⋊C₈	C₂²×C₁₀₉	1	2	yes	—
323	Q₈⋊C₈	C₂²×C₁₀₉	1	2	yes	—
324	D₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
325	D₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
326	2₋¹⁺⁴	—	4	8	no	—
327	C₂×#41	—	2	4	no	—
328	C₂×#41	—	2	2	no	—
329	C₂×#41	—	2	2	no	—
330	C₂×#41	—	2	1	no	—
331	C₂×#41	—	2	1	no	—
332	C₂²×Q₈	—	2	2	no	—
333	C₂²×Q₈	—	2	2	no	—
334	C₂²×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
335	C₂×Q₁₆	—	2	2	no	—
336	C₂×SD₁₆	—	2	2	no	—
337	C₂×M₁₆	—	2	1	no	—
338	C₄⋊Q₈	—	2	2	no	—
339	C₄⋊Q₈	D₄×C₁₀₉	1	1	yes	characteristic
340	C₄⋊Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
341	C₄⋊Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
342	C₄⋊Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
343	#127	—	2	2	no	—
344	#127	D₄×C₁₀₉	1	1	yes	characteristic
345	#127	C₂³×C₁₀₉	1	1	yes	characteristic
346	#125	—	2	1	no	—
347	#124	—	2	4	no	—
348	#124	—	2	1	no	—
349	#124	—	2	1	no	—
350	C₂²⋊Q₈	—	2	2	no	—
351	C₂²⋊Q₈	—	2	4	no	—
352	C₂²⋊Q₈	—	2	2	no	—
353	C₂²⋊Q₈	—	2	4	no	—
354	C₂²⋊Q₈	—	2	1	no	—
355	C₂²⋊Q₈	—	2	1	no	—
356	#121	—	2	4	no	—
357	#121	—	2	2	no	—
358	#121	—	2	4	no	—
359	#121	—	2	4	no	—
360	#121	—	2	1	no	—
361	#121	—	2	1	no	—
362	#121	—	2	2	no	—
363	#121	D₄×C₁₀₉	1	2	yes	—
364	#121	D₄×C₁₀₉	1	2	yes	—
365	C₄×Q₈	—	2	1	no	—
366	C₄×Q₈	—	2	1	no	—
367	C₄×Q₈	—	2	1	no	—
368	C₄×Q₈	—	2	1	no	—
369	C₄×Q₈	D₄×C₁₀₉	1	2	yes	—
370	C₄×Q₈	C₂³×C₁₀₉	1	2	yes	—
371	C₄×D₄	—	2	4	no	—
372	C₄×D₄	—	2	2	no	—
373	C₄×D₄	—	2	2	no	—
374	C₄×D₄	—	2	1	no	—
375	C₄×D₄	—	2	1	no	—
376	C₄×D₄	D₄×C₁₀₉	1	2	yes	—
377	C₄×D₄	C₂³×C₁₀₉	1	2	yes	—
378	#117	—	2	1	no	—
379	C₂×C₄⋊C₄	—	2	1	no	—
380	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
381	C₂×C₄²	—	2	1	no	—
382	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
383	#107	—	2	2	no	—
384	#107	—	2	2	no	—
385	#107	—	2	2	no	—
386	#106	—	2	2	no	—
387	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
388	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
389	#103	—	2	2	no	—
390	#103	—	2	2	no	—
391	#103	—	2	2	no	—
392	#102	—	2	2	no	—
393	#102	—	2	2	no	—
394	#102	—	2	2	no	—
395	C₂²⋊C₈	—	2	1	no	—
396	C₂²⋊C₈	—	2	1	no	—
397	#97	—	2	1	no	—
398	#97	—	2	1	no	—
399	C₄×C₈	—	2	1	no	—
400	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
401	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
402	#41	—	4	16	no	—
403	#41	—	4	4	no	—
404	#41	—	4	4	no	—
405	#41	—	4	8	no	—
406	#41	—	4	4	no	—
407	#41	—	4	4	no	—
408	C₂×Q₈	—	2	4	no	—
409	C₂×Q₈	—	2	4	no	—
410	C₂×Q₈	—	2	1	no	—
411	C₂×Q₈	—	2	1	no	—
412	C₂×Q₈	—	2	1	no	—
413	C₂×Q₈	—	2	1	no	—
414	C₂×Q₈	—	2	1	no	—
415	C₂×Q₈	—	4	8	no	—
416	C₂×Q₈	—	4	4	no	—
417	C₂×Q₈	—	4	4	no	—
418	C₂×Q₈	C₂×D₄×C₁₀₉	1	2	yes	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	4	no	—
423	C₂×D₄	—	2	1	no	—
424	C₂×D₄	—	2	1	no	—
425	C₂×D₄	—	2	1	no	—
426	C₂×D₄	—	2	1	no	—
427	C₂×D₄	C₂×D₄×C₁₀₉	1	2	yes	—
428	C₂×D₄	C₂×D₄×C₁₀₉	1	2	yes	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	2	no	—
435	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
436	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
437	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
438	Q₁₆	—	4	4	no	—
439	SD₁₆	—	4	4	no	—
440	M₁₆	—	4	2	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	—	2	1	no	—
444	C₂×C₈	—	2	2	no	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	4	1	no	—
456	C₄⋊C₄	—	4	1	no	—
457	C₄⋊C₄	C₂×D₄×C₁₀₉	1	2	yes	—
458	C₄⋊C₄	C₂×D₄×C₁₀₉	1	2	yes	—
459	C₄⋊C₄	C₂×D₄×C₁₀₉	1	2	yes	—
460	C₂²⋊C₄	—	2	4	no	—
461	C₂²⋊C₄	—	2	4	no	—
462	C₂²⋊C₄	—	2	2	no	—
463	C₂²⋊C₄	—	2	2	no	—
464	C₂²⋊C₄	—	2	2	no	—
465	C₂²⋊C₄	—	2	2	no	—
466	C₂²⋊C₄	—	4	1	no	—
467	C₄²	—	2	2	no	—
468	C₄²	—	2	2	no	—
469	C₄²	—	4	1	no	—
470	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
471	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
472	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
473	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
474	C₂³	—	2	2	no	—
475	C₂³	—	2	2	no	—
476	C₂³	#120×C₁₀₉	1	1	yes	characteristic
477	Q₈	—	2	4	no	—
478	Q₈	—	2	2	no	—
479	Q₈	—	4	8	no	—
480	Q₈	—	4	2	no	—
481	Q₈	—	4	2	no	—
482	Q₈	—	4	2	no	—
483	Q₈	—	4	2	no	—
484	Q₈	AGL(1,ℤ/8)×C₁₀₉	1	4	yes	—
485	D₄	—	2	4	no	—
486	D₄	—	4	2	no	—
487	D₄	—	4	8	no	—
488	D₄	—	4	2	no	—
489	D₄	—	4	4	no	—
490	D₄	—	4	4	no	—
491	D₄	—	4	2	no	—
492	D₄	—	4	2	no	—
493	D₄	—	8	1	no	—
494	D₄	—	8	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	8	no	—
504	C₂×C₄	—	4	8	no	—
505	C₂×C₄	—	4	4	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	4	1	no	—
509	C₂×C₄	—	4	1	no	—
510	C₂×C₄	—	8	1	no	—
511	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
512	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
513	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
514	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
515	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
516	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
517	C₈	—	2	2	no	—
518	C₈	—	4	1	no	—
519	C₈	—	4	2	no	—
520	C₈	—	4	2	no	—
521	C₂²	—	2	1	no	—
522	C₂²	—	2	1	no	—
523	C₂²	—	2	1	no	—
524	C₂²	—	4	2	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	4	2	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	2	no	—
530	C₂²	C₂×#120×C₁₀₉	1	1	yes	characteristic
531	C₄	—	2	4	no	—
532	C₄	—	2	1	no	—
533	C₄	—	2	1	no	—
534	C₄	—	2	1	no	—
535	C₄	—	2	1	no	—
536	C₄	—	4	2	no	—
537	C₄	—	4	2	no	—
538	C₄	—	4	2	no	—
539	C₄	—	8	1	no	—
540	C₄	—	8	1	no	—
541	C₄	C₂×AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
542	C₄	#578×C₁₀₉	1	2	yes	—
543	C₂	—	4	1	no	—
544	C₂	—	8	2	no	—
545	C₂	—	8	2	no	—
546	C₂	C₁₀₉×#3003	1	1	yes	characteristic
547	C₂	C₁₀₉×#3011	1	1	yes	characteristic
548	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×#3497	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×#3235	C₂	semidirect
3. C₁₀₉×#3182	C₂	semidirect
4. C₁₀₉×#3104	C₂	semidirect
5. C₁₀₉×#1649	C₂	semidirect
6. C₁₀₉×#1641	C₂	semidirect
7. C₁₀₉×#1637	C₂	semidirect
8. C₁₀₉×#1631	C₂	semidirect
9. C₁₀₉×#1553	C₂	semidirect
10. C₁₀₉×#1491	C₂	semidirect
11. C₁₀₉×#1484	C₂	semidirect
268. #18611	C₁₀₉	direct (N × H)
276. #3235	C₂×C₁₀₉	semidirect
277. #3182	C₂×C₁₀₉	semidirect
278. #3104	C₂×C₁₀₉	semidirect
279. #1649	C₂×C₁₀₉	semidirect
280. #1641	C₂×C₁₀₉	semidirect
281. #1637	C₂×C₁₀₉	semidirect
282. #1631	C₂×C₁₀₉	semidirect
283. #1553	C₂×C₁₀₉	semidirect
284. #1491	C₂×C₁₀₉	semidirect
285. #1484	C₂×C₁₀₉	semidirect
286. C₁₀₉	#18611	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×#3497	2
C₁₀₉×#3235	2
C₁₀₉×#3182	2
C₁₀₉×#3104	2
C₁₀₉×#1649	2
C₁₀₉×#1641	2
C₁₀₉×#1637	2
C₁₀₉×#1631	2
C₁₀₉×#1553	2
C₁₀₉×#1491	2
C₁₀₉×#1484	2
#18611	109

Subgroups — C109×#18611 #42136605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#18611 #42136605