Subgroups — C₁₀₉×#18650 #42136644

Order 27904.

1146 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×#3499	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#3226	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#500×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#3039	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1649	C₂	1	4	yes	maximal, complemented
6	C₁₀₉×#1647	C₂	1	4	yes	maximal, complemented
7	C₁₀₉×#1546	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#1492	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₀₉×#1476	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×2₋¹⁺⁴×C₁₀₉	—	2	1	no	—
11	C₂×#137×C₁₀₉	—	2	1	no	—
12	D₄×Q₈×C₁₀₉	—	2	1	no	—
13	D₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
14	#547×C₁₀₉	—	2	2	no	—
15	#547×C₁₀₉	—	2	1	no	—
16	#547×C₁₀₉	—	2	2	no	—
17	#547×C₁₀₉	C₂²	1	2	yes	—
18	#536×C₁₀₉	—	2	2	no	—
19	#536×C₁₀₉	—	2	1	no	—
20	#532×C₁₀₉	—	2	1	no	—
21	#532×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Q₈×C₁₀₉	—	2	1	no	—
23	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₄×#41×C₁₀₉	C₂²	1	1	yes	characteristic
26	#500×C₁₀₉	—	2	1	no	—
27	#500×C₁₀₉	—	2	2	no	—
28	#500×C₁₀₉	C₂²	1	4	yes	—
29	#486×C₁₀₉	—	2	1	no	—
30	#480×C₁₀₉	—	2	2	no	—
31	#474×C₁₀₉	C₂²	1	4	yes	—
32	#462×C₁₀₉	—	2	1	no	—
33	#462×C₁₀₉	C₂²	1	2	yes	—
34	#461×C₁₀₉	—	2	1	no	—
35	#461×C₁₀₉	C₂²	1	2	yes	—
36	#431×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₂×#107×C₁₀₉	—	2	2	no	—
38	#423×C₁₀₉	C₂²	1	1	yes	characteristic
39	#419×C₁₀₉	—	2	1	no	—
40	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	#332×C₁₀₉	C₂²	1	4	yes	—
42	Q₈⋊C₈×C₁₀₉	C₂²	1	2	yes	—
43	D₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
44	D₄⋊C₈×C₁₀₉	C₂²	1	4	yes	—
45	2₋¹⁺⁴×C₁₀₉	—	4	4	no	—
46	2₋¹⁺⁴×C₁₀₉	—	4	4	no	—
47	C₂×#41×C₁₀₉	—	2	2	no	—
48	C₂×#41×C₁₀₉	—	2	4	no	—
49	C₂×#41×C₁₀₉	—	2	2	no	—
50	C₂×#41×C₁₀₉	—	2	1	no	—
51	C₂×#41×C₁₀₉	D₄	1	1	yes	characteristic
52	C₂²×Q₈×C₁₀₉	—	2	2	no	—
53	C₂²×Q₈×C₁₀₉	—	2	2	no	—
54	C₂²×Q₈×C₁₀₉	—	2	1	no	—
55	C₂²×Q₈×C₁₀₉	—	2	1	no	—
56	#137×C₁₀₉	—	4	4	no	—
57	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
58	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
59	C₂×M₁₆×C₁₀₉	—	2	1	no	—
60	C₂²×C₈×C₁₀₉	—	2	1	no	—
61	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
62	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
63	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
64	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
65	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
66	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
67	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
68	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
69	C₄⋊Q₈×C₁₀₉	D₄	1	1	yes	characteristic
70	C₄⋊Q₈×C₁₀₉	D₄	1	2	yes	—
71	C₄⋊Q₈×C₁₀₉	C₂³	1	2	yes	—
72	C₄⋊Q₈×C₁₀₉	C₂³	1	2	yes	—
73	#127×C₁₀₉	D₄	1	1	yes	characteristic
74	#124×C₁₀₉	—	2	2	no	—
75	#124×C₁₀₉	—	2	1	no	—
76	#123×C₁₀₉	—	2	4	no	—
77	#123×C₁₀₉	—	2	4	no	—
78	#123×C₁₀₉	—	2	4	no	—
79	C₂²⋊Q₈×C₁₀₉	—	2	4	no	—
80	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
81	C₂²⋊Q₈×C₁₀₉	—	2	4	no	—
82	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
83	#121×C₁₀₉	—	2	2	no	—
84	#121×C₁₀₉	—	2	2	no	—
85	#121×C₁₀₉	—	2	1	no	—
86	#121×C₁₀₉	D₄	1	2	yes	—
87	C₄×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
88	C₄×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
89	C₄×D₄×C₁₀₉	—	2	2	no	—
90	C₄×D₄×C₁₀₉	—	2	2	no	—
91	C₄×D₄×C₁₀₉	—	2	1	no	—
92	C₄×D₄×C₁₀₉	—	2	1	no	—
93	C₄×D₄×C₁₀₉	—	2	1	no	—
94	C₄×D₄×C₁₀₉	D₄	1	1	yes	characteristic
95	C₄×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
96	C₄×D₄×C₁₀₉	C₂³	1	2	yes	—
97	#117×C₁₀₉	—	2	1	no	—
98	#117×C₁₀₉	D₄	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
100	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
101	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
102	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
103	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
104	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
105	C₂×C₄²×C₁₀₉	—	2	1	no	—
106	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
107	#107×C₁₀₉	—	2	8	no	—
108	#107×C₁₀₉	—	2	4	no	—
109	C₄⋊C₈×C₁₀₉	—	2	1	no	—
110	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
111	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
112	#103×C₁₀₉	—	2	2	no	—
113	#103×C₁₀₉	—	2	2	no	—
114	#102×C₁₀₉	—	2	4	no	—
115	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
116	C₄×C₈×C₁₀₉	—	2	1	no	—
117	C₄×C₈×C₁₀₉	—	2	1	no	—
118	C₄×C₈×C₁₀₉	C₂³	1	2	yes	—
119	#41×C₁₀₉	—	2	2	no	—
120	#41×C₁₀₉	—	4	8	no	—
121	#41×C₁₀₉	—	4	16	no	—
122	#41×C₁₀₉	—	4	4	no	—
123	#41×C₁₀₉	—	4	4	no	—
124	#41×C₁₀₉	—	4	2	no	—
125	#41×C₁₀₉	—	4	2	no	—
126	#41×C₁₀₉	D₈	1	4	yes	—
127	C₂×Q₈×C₁₀₉	—	2	4	no	—
128	C₂×Q₈×C₁₀₉	—	2	2	no	—
129	C₂×Q₈×C₁₀₉	—	2	1	no	—
130	C₂×Q₈×C₁₀₉	—	2	2	no	—
131	C₂×Q₈×C₁₀₉	—	2	2	no	—
132	C₂×Q₈×C₁₀₉	—	4	8	no	—
133	C₂×Q₈×C₁₀₉	—	4	4	no	—
134	C₂×Q₈×C₁₀₉	—	4	4	no	—
135	C₂×Q₈×C₁₀₉	—	4	1	no	—
136	C₂×Q₈×C₁₀₉	—	4	2	no	—
137	C₂×Q₈×C₁₀₉	—	4	2	no	—
138	C₂×Q₈×C₁₀₉	—	4	2	no	—
139	C₂×Q₈×C₁₀₉	—	4	2	no	—
140	C₂×Q₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
141	C₂×D₄×C₁₀₉	—	2	1	no	—
142	C₂×D₄×C₁₀₉	—	2	4	no	—
143	C₂×D₄×C₁₀₉	—	2	2	no	—
144	C₂×D₄×C₁₀₉	—	2	1	no	—
145	C₂×D₄×C₁₀₉	—	2	1	no	—
146	C₂×D₄×C₁₀₉	C₂×D₄	1	2	yes	—
147	C₂×D₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₁₀₉	—	2	1	no	—
149	C₂²×C₄×C₁₀₉	—	2	4	no	—
150	C₂²×C₄×C₁₀₉	—	2	4	no	—
151	C₂²×C₄×C₁₀₉	—	2	2	no	—
152	C₂²×C₄×C₁₀₉	—	2	1	no	—
153	C₂²×C₄×C₁₀₉	—	2	1	no	—
154	C₂²×C₄×C₁₀₉	—	2	2	no	—
155	C₂²×C₄×C₁₀₉	—	4	1	no	—
156	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
159	Q₁₆×C₁₀₉	—	4	4	no	—
160	SD₁₆×C₁₀₉	—	4	4	no	—
161	M₁₆×C₁₀₉	—	4	2	no	—
162	C₂×C₈×C₁₀₉	—	2	4	no	—
163	C₂×C₈×C₁₀₉	—	2	2	no	—
164	C₂×C₈×C₁₀₉	—	2	2	no	—
165	C₂×C₈×C₁₀₉	—	2	2	no	—
166	C₄⋊C₄×C₁₀₉	—	2	4	no	—
167	C₄⋊C₄×C₁₀₉	—	2	4	no	—
168	C₄⋊C₄×C₁₀₉	—	2	1	no	—
169	C₄⋊C₄×C₁₀₉	—	2	2	no	—
170	C₄⋊C₄×C₁₀₉	—	2	2	no	—
171	C₄⋊C₄×C₁₀₉	—	2	1	no	—
172	C₄⋊C₄×C₁₀₉	—	2	2	no	—
173	C₄⋊C₄×C₁₀₉	—	2	1	no	—
174	C₄⋊C₄×C₁₀₉	—	2	2	no	—
175	C₄⋊C₄×C₁₀₉	—	2	2	no	—
176	C₄⋊C₄×C₁₀₉	—	4	1	no	—
177	C₄⋊C₄×C₁₀₉	—	4	1	no	—
178	C₄⋊C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
179	C₄⋊C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
180	C₄⋊C₄×C₁₀₉	C₂×D₄	1	2	yes	—
181	C₄⋊C₄×C₁₀₉	C₂×D₄	1	4	yes	—
182	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
183	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
184	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
185	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
186	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
187	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
188	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
189	C₄²×C₁₀₉	—	2	1	no	—
190	C₄²×C₁₀₉	—	2	1	no	—
191	C₄²×C₁₀₉	—	2	1	no	—
192	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
196	C₂³×C₁₀₉	—	2	1	no	—
197	C₂³×C₁₀₉	—	2	2	no	—
198	C₂³×C₁₀₉	#120	1	1	yes	characteristic
199	Q₈×C₁₀₉	—	2	1	no	—
200	Q₈×C₁₀₉	—	4	8	no	—
201	Q₈×C₁₀₉	—	4	2	no	—
202	Q₈×C₁₀₉	—	4	2	no	—
203	Q₈×C₁₀₉	—	4	2	no	—
204	Q₈×C₁₀₉	—	4	4	no	—
205	Q₈×C₁₀₉	—	4	2	no	—
206	Q₈×C₁₀₉	—	4	2	no	—
207	Q₈×C₁₀₉	—	8	2	no	—
208	Q₈×C₁₀₉	C₂×D₈	1	2	yes	—
209	D₄×C₁₀₉	—	2	4	no	—
210	D₄×C₁₀₉	—	2	1	no	—
211	D₄×C₁₀₉	—	2	4	no	—
212	D₄×C₁₀₉	—	4	8	no	—
213	D₄×C₁₀₉	—	4	4	no	—
214	D₄×C₁₀₉	—	4	2	no	—
215	D₄×C₁₀₉	—	4	2	no	—
216	D₄×C₁₀₉	C₂×D₈	1	2	yes	—
217	C₂×C₄×C₁₀₉	—	2	1	no	—
218	C₂×C₄×C₁₀₉	—	2	2	no	—
219	C₂×C₄×C₁₀₉	—	2	4	no	—
220	C₂×C₄×C₁₀₉	—	2	2	no	—
221	C₂×C₄×C₁₀₉	—	2	1	no	—
222	C₂×C₄×C₁₀₉	—	2	1	no	—
223	C₂×C₄×C₁₀₉	—	2	4	no	—
224	C₂×C₄×C₁₀₉	—	2	2	no	—
225	C₂×C₄×C₁₀₉	—	4	8	no	—
226	C₂×C₄×C₁₀₉	—	4	8	no	—
227	C₂×C₄×C₁₀₉	—	4	4	no	—
228	C₂×C₄×C₁₀₉	—	4	2	no	—
229	C₂×C₄×C₁₀₉	—	4	2	no	—
230	C₂×C₄×C₁₀₉	—	4	2	no	—
231	C₂×C₄×C₁₀₉	—	4	2	no	—
232	C₂×C₄×C₁₀₉	—	4	2	no	—
233	C₂×C₄×C₁₀₉	—	8	2	no	—
234	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
235	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
236	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
237	C₂×C₄×C₁₀₉	C₂×D₈	1	2	yes	—
238	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
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	4	no	—
243	C₈×C₁₀₉	—	4	2	no	—
244	C₈×C₁₀₉	—	4	2	no	—
245	C₂²×C₁₀₉	—	2	1	no	—
246	C₂²×C₁₀₉	—	2	1	no	—
247	C₂²×C₁₀₉	—	2	2	no	—
248	C₂²×C₁₀₉	—	4	1	no	—
249	C₂²×C₁₀₉	—	4	1	no	—
250	C₂²×C₁₀₉	—	4	2	no	—
251	C₂²×C₁₀₉	—	4	2	no	—
252	C₂²×C₁₀₉	—	4	2	no	—
253	C₂²×C₁₀₉	#447	1	2	yes	—
254	C₂²×C₁₀₉	C₂×#120	1	1	yes	characteristic
255	C₄×C₁₀₉	—	2	2	no	—
256	C₄×C₁₀₉	—	2	1	no	—
257	C₄×C₁₀₉	—	2	1	no	—
258	C₄×C₁₀₉	—	4	4	no	—
259	C₄×C₁₀₉	—	4	2	no	—
260	C₄×C₁₀₉	—	4	1	no	—
261	C₄×C₁₀₉	—	4	1	no	—
262	C₄×C₁₀₉	—	8	2	no	—
263	C₄×C₁₀₉	#447	1	2	yes	—
264	C₄×C₁₀₉	#456	1	2	yes	—
265	C₄×C₁₀₉	C₂²×D₈	1	1	yes	characteristic
266	C₄×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
267	C₄×C₁₀₉	#578	1	2	yes	—
268	#18650	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
269	C₂×C₁₀₉	—	2	2	no	—
270	C₂×C₁₀₉	—	4	2	no	—
271	C₂×C₁₀₉	—	8	2	no	—
272	C₂×C₁₀₉	C₂×#447	1	1	yes	characteristic
273	C₂×C₁₀₉	#3011	1	1	yes	characteristic
274	C₂×C₁₀₉	C₂×#456	1	1	yes	characteristic
275	#3499	C₂×C₁₀₉	1	1	yes	characteristic
276	#3226	C₂×C₁₀₉	1	1	yes	characteristic, complemented
277	C₂×#500	C₂×C₁₀₉	1	1	yes	characteristic, complemented
278	#3039	C₂×C₁₀₉	1	1	yes	characteristic, complemented
279	#1649	C₂×C₁₀₉	1	4	yes	complemented
280	#1647	C₂×C₁₀₉	1	4	yes	complemented
281	#1546	C₂×C₁₀₉	1	1	yes	characteristic, complemented
282	#1492	C₂×C₁₀₉	1	1	yes	characteristic, complemented
283	#1476	C₂×C₁₀₉	1	1	yes	characteristic, complemented
284	C₁₀₉	#18650	1	1	yes	characteristic, direct product factor, Sylow
285	C₂×2₋¹⁺⁴	—	2	1	no	—
286	C₂×#137	—	2	1	no	—
287	D₄×Q₈	—	2	1	no	—
288	D₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
289	#547	—	2	2	no	—
290	#547	—	2	1	no	—
291	#547	—	2	2	no	—
292	#547	C₂²×C₁₀₉	1	2	yes	—
293	#536	—	2	2	no	—
294	#536	—	2	1	no	—
295	#532	—	2	1	no	—
296	#532	C₂²×C₁₀₉	1	1	yes	characteristic
297	C₂×C₄⋊Q₈	—	2	1	no	—
298	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
299	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
300	C₄×#41	C₂²×C₁₀₉	1	1	yes	characteristic
301	#500	—	2	2	no	—
302	#500	—	2	1	no	—
303	#500	C₂²×C₁₀₉	1	4	yes	—
304	#486	—	2	1	no	—
305	#480	—	2	2	no	—
306	#474	C₂²×C₁₀₉	1	4	yes	—
307	#462	—	2	1	no	—
308	#462	C₂²×C₁₀₉	1	2	yes	—
309	#461	—	2	1	no	—
310	#461	C₂²×C₁₀₉	1	2	yes	—
311	#431	C₂²×C₁₀₉	1	1	yes	characteristic
312	C₂×#107	—	2	2	no	—
313	#423	C₂²×C₁₀₉	1	1	yes	characteristic
314	#419	—	2	1	no	—
315	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
316	#332	C₂²×C₁₀₉	1	4	yes	—
317	Q₈⋊C₈	C₂²×C₁₀₉	1	2	yes	—
318	D₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
319	D₄⋊C₈	C₂²×C₁₀₉	1	4	yes	—
320	2₋¹⁺⁴	—	4	4	no	—
321	2₋¹⁺⁴	—	4	4	no	—
322	C₂×#41	—	2	2	no	—
323	C₂×#41	—	2	4	no	—
324	C₂×#41	—	2	2	no	—
325	C₂×#41	—	2	1	no	—
326	C₂×#41	D₄×C₁₀₉	1	1	yes	characteristic
327	C₂²×Q₈	—	2	2	no	—
328	C₂²×Q₈	—	2	2	no	—
329	C₂²×Q₈	—	2	1	no	—
330	C₂²×Q₈	—	2	1	no	—
331	#137	—	4	4	no	—
332	C₂×Q₁₆	—	2	2	no	—
333	C₂×SD₁₆	—	2	2	no	—
334	C₂×M₁₆	—	2	1	no	—
335	C₂²×C₈	—	2	1	no	—
336	C₄⋊Q₈	—	2	2	no	—
337	C₄⋊Q₈	—	2	2	no	—
338	C₄⋊Q₈	—	2	2	no	—
339	C₄⋊Q₈	—	2	1	no	—
340	C₄⋊Q₈	—	2	1	no	—
341	C₄⋊Q₈	—	2	1	no	—
342	C₄⋊Q₈	—	2	1	no	—
343	C₄⋊Q₈	—	2	1	no	—
344	C₄⋊Q₈	D₄×C₁₀₉	1	1	yes	characteristic
345	C₄⋊Q₈	D₄×C₁₀₉	1	2	yes	—
346	C₄⋊Q₈	C₂³×C₁₀₉	1	2	yes	—
347	C₄⋊Q₈	C₂³×C₁₀₉	1	2	yes	—
348	#127	D₄×C₁₀₉	1	1	yes	characteristic
349	#124	—	2	2	no	—
350	#124	—	2	1	no	—
351	#123	—	2	4	no	—
352	#123	—	2	4	no	—
353	#123	—	2	4	no	—
354	C₂²⋊Q₈	—	2	4	no	—
355	C₂²⋊Q₈	—	2	2	no	—
356	C₂²⋊Q₈	—	2	4	no	—
357	C₂²⋊Q₈	—	2	2	no	—
358	#121	—	2	2	no	—
359	#121	—	2	2	no	—
360	#121	—	2	1	no	—
361	#121	D₄×C₁₀₉	1	2	yes	—
362	C₄×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
363	C₄×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
364	C₄×D₄	—	2	2	no	—
365	C₄×D₄	—	2	2	no	—
366	C₄×D₄	—	2	1	no	—
367	C₄×D₄	—	2	1	no	—
368	C₄×D₄	—	2	1	no	—
369	C₄×D₄	D₄×C₁₀₉	1	1	yes	characteristic
370	C₄×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
371	C₄×D₄	C₂³×C₁₀₉	1	2	yes	—
372	#117	—	2	1	no	—
373	#117	D₄×C₁₀₉	1	1	yes	characteristic
374	C₂×C₄⋊C₄	—	2	2	no	—
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	—	2	1	no	—
378	C₂×C₄⋊C₄	—	2	1	no	—
379	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
380	C₂×C₄²	—	2	1	no	—
381	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
382	#107	—	2	8	no	—
383	#107	—	2	4	no	—
384	C₄⋊C₈	—	2	1	no	—
385	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
386	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
387	#103	—	2	2	no	—
388	#103	—	2	2	no	—
389	#102	—	2	4	no	—
390	C₂²⋊C₈	—	2	1	no	—
391	C₄×C₈	—	2	1	no	—
392	C₄×C₈	—	2	1	no	—
393	C₄×C₈	C₂³×C₁₀₉	1	2	yes	—
394	#41	—	2	2	no	—
395	#41	—	4	8	no	—
396	#41	—	4	16	no	—
397	#41	—	4	4	no	—
398	#41	—	4	4	no	—
399	#41	—	4	2	no	—
400	#41	—	4	2	no	—
401	#41	D₈×C₁₀₉	1	4	yes	—
402	C₂×Q₈	—	2	4	no	—
403	C₂×Q₈	—	2	2	no	—
404	C₂×Q₈	—	2	1	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	—	4	8	no	—
408	C₂×Q₈	—	4	4	no	—
409	C₂×Q₈	—	4	4	no	—
410	C₂×Q₈	—	4	1	no	—
411	C₂×Q₈	—	4	2	no	—
412	C₂×Q₈	—	4	2	no	—
413	C₂×Q₈	—	4	2	no	—
414	C₂×Q₈	—	4	2	no	—
415	C₂×Q₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	—	2	4	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	C₂×D₄×C₁₀₉	1	2	yes	—
422	C₂×D₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	4	1	no	—
431	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
433	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
434	Q₁₆	—	4	4	no	—
435	SD₁₆	—	4	4	no	—
436	M₁₆	—	4	2	no	—
437	C₂×C₈	—	2	4	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	—	2	2	no	—
441	C₄⋊C₄	—	2	4	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	4	1	no	—
452	C₄⋊C₄	—	4	1	no	—
453	C₄⋊C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
454	C₄⋊C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
455	C₄⋊C₄	C₂×D₄×C₁₀₉	1	2	yes	—
456	C₄⋊C₄	C₂×D₄×C₁₀₉	1	4	yes	—
457	C₂²⋊C₄	—	2	4	no	—
458	C₂²⋊C₄	—	2	4	no	—
459	C₂²⋊C₄	—	2	1	no	—
460	C₂²⋊C₄	—	2	1	no	—
461	C₂²⋊C₄	—	2	2	no	—
462	C₂²⋊C₄	—	2	1	no	—
463	C₂²⋊C₄	—	2	2	no	—
464	C₄²	—	2	1	no	—
465	C₄²	—	2	1	no	—
466	C₄²	—	2	1	no	—
467	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
468	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
469	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
470	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
471	C₂³	—	2	1	no	—
472	C₂³	—	2	2	no	—
473	C₂³	#120×C₁₀₉	1	1	yes	characteristic
474	Q₈	—	2	1	no	—
475	Q₈	—	4	8	no	—
476	Q₈	—	4	2	no	—
477	Q₈	—	4	2	no	—
478	Q₈	—	4	2	no	—
479	Q₈	—	4	4	no	—
480	Q₈	—	4	2	no	—
481	Q₈	—	4	2	no	—
482	Q₈	—	8	2	no	—
483	Q₈	C₂×D₈×C₁₀₉	1	2	yes	—
484	D₄	—	2	4	no	—
485	D₄	—	2	1	no	—
486	D₄	—	2	4	no	—
487	D₄	—	4	8	no	—
488	D₄	—	4	4	no	—
489	D₄	—	4	2	no	—
490	D₄	—	4	2	no	—
491	D₄	C₂×D₈×C₁₀₉	1	2	yes	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	4	8	no	—
501	C₂×C₄	—	4	8	no	—
502	C₂×C₄	—	4	4	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	8	2	no	—
509	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
510	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
511	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
512	C₂×C₄	C₂×D₈×C₁₀₉	1	2	yes	—
513	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
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	4	no	—
518	C₈	—	4	2	no	—
519	C₈	—	4	2	no	—
520	C₂²	—	2	1	no	—
521	C₂²	—	2	1	no	—
522	C₂²	—	2	2	no	—
523	C₂²	—	4	1	no	—
524	C₂²	—	4	1	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	4	2	no	—
528	C₂²	#447×C₁₀₉	1	2	yes	—
529	C₂²	C₂×#120×C₁₀₉	1	1	yes	characteristic
530	C₄	—	2	2	no	—
531	C₄	—	2	1	no	—
532	C₄	—	2	1	no	—
533	C₄	—	4	4	no	—
534	C₄	—	4	2	no	—
535	C₄	—	4	1	no	—
536	C₄	—	4	1	no	—
537	C₄	—	8	2	no	—
538	C₄	#447×C₁₀₉	1	2	yes	—
539	C₄	#456×C₁₀₉	1	2	yes	—
540	C₄	C₂²×D₈×C₁₀₉	1	1	yes	characteristic
541	C₄	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
542	C₄	#578×C₁₀₉	1	2	yes	—
543	C₂	—	2	2	no	—
544	C₂	—	4	2	no	—
545	C₂	—	8	2	no	—
546	C₂	C₂×#447×C₁₀₉	1	1	yes	characteristic
547	C₂	C₁₀₉×#3011	1	1	yes	characteristic
548	C₂	C₂×#456×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₀₉
prime radical	C₁₀₉×#3499	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×#3226	C₂	semidirect
3. C₂×#500×C₁₀₉	C₂	semidirect
4. C₁₀₉×#3039	C₂	semidirect
5. C₁₀₉×#1649	C₂	semidirect
6. C₁₀₉×#1647	C₂	semidirect
7. C₁₀₉×#1546	C₂	semidirect
8. C₁₀₉×#1492	C₂	semidirect
9. C₁₀₉×#1476	C₂	semidirect
268. #18650	C₁₀₉	direct (N × H)
276. #3226	C₂×C₁₀₉	semidirect
277. C₂×#500	C₂×C₁₀₉	semidirect
278. #3039	C₂×C₁₀₉	semidirect
279. #1649	C₂×C₁₀₉	semidirect
280. #1647	C₂×C₁₀₉	semidirect
281. #1546	C₂×C₁₀₉	semidirect
282. #1492	C₂×C₁₀₉	semidirect
283. #1476	C₂×C₁₀₉	semidirect
284. C₁₀₉	#18650	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×#3499	2
C₁₀₉×#3226	2
C₂×#500×C₁₀₉	2
C₁₀₉×#3039	2
C₁₀₉×#1649	2
C₁₀₉×#1647	2
C₁₀₉×#1546	2
C₁₀₉×#1492	2
C₁₀₉×#1476	2
#18650	109

Subgroups — C109×#18650 #42136644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#18650 #42136644