Subgroups — C₁₀₉×#11228 #42129222

Order 27904.

898 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#389×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#340×C₁₀₉	C₂	1	2	yes	maximal
4	C₂×#340×C₁₀₉	C₂	1	2	yes	maximal
5	C₁₀₉×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₀₉	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₁₀₉	—	2	1	no	—
8	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₁₀₉	—	2	4	no	—
13	C₂×C₄⋊C₈×C₁₀₉	—	2	2	no	—
14	#389×C₁₀₉	—	2	2	no	—
15	#389×C₁₀₉	C₄	1	4	yes	complemented
16	C₂×#95×C₁₀₉	—	2	1	no	—
17	C₂×#95×C₁₀₉	—	2	1	no	—
18	C₂×#95×C₁₀₉	—	2	1	no	—
19	#340×C₁₀₉	—	2	4	no	—
20	#340×C₁₀₉	—	2	4	no	—
21	C₂³×C₄×C₁₀₉	—	2	1	no	—
22	C₂³×C₄×C₁₀₉	—	2	1	no	—
23	C₂³×C₄×C₁₀₉	—	2	2	no	—
24	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₁₀₉	—	2	2	no	—
28	C₂²×C₈×C₁₀₉	—	2	2	no	—
29	C₂²×C₈×C₁₀₉	—	2	2	no	—
30	C₂²×C₈×C₁₀₉	—	2	2	no	—
31	C₂²×C₈×C₁₀₉	—	2	2	no	—
32	C₂²×C₈×C₁₀₉	—	2	2	no	—
33	C₂²×C₈×C₁₀₉	D₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
44	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₀₉	—	4	4	no	—
47	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₁₀₉	Q₈	1	4	yes	—
51	C₂×C₄⋊C₄×C₁₀₉	Q₈	1	2	yes	—
52	C₂×C₄²×C₁₀₉	—	2	2	no	—
53	C₂×C₄²×C₁₀₉	—	2	2	no	—
54	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
56	C₄⋊C₈×C₁₀₉	—	2	8	no	—
57	#95×C₁₀₉	—	2	4	no	—
58	#95×C₁₀₉	—	4	2	no	—
59	#95×C₁₀₉	—	4	2	no	—
60	C₂⁴×C₁₀₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₉	—	2	2	no	—
62	C₂²×C₄×C₁₀₉	—	2	2	no	—
63	C₂²×C₄×C₁₀₉	—	2	2	no	—
64	C₂²×C₄×C₁₀₉	—	2	2	no	—
65	C₂²×C₄×C₁₀₉	—	2	2	no	—
66	C₂²×C₄×C₁₀₉	—	2	2	no	—
67	C₂²×C₄×C₁₀₉	—	2	1	no	—
68	C₂²×C₄×C₁₀₉	—	2	2	no	—
69	C₂²×C₄×C₁₀₉	—	2	1	no	—
70	C₂²×C₄×C₁₀₉	—	2	2	no	—
71	C₂²×C₄×C₁₀₉	—	2	4	no	—
72	C₂²×C₄×C₁₀₉	—	4	2	no	—
73	C₂²×C₄×C₁₀₉	—	4	2	no	—
74	C₂²×C₄×C₁₀₉	—	4	2	no	—
75	C₂²×C₄×C₁₀₉	—	4	2	no	—
76	C₂²×C₄×C₁₀₉	—	4	2	no	—
77	C₂²×C₄×C₁₀₉	—	4	2	no	—
78	C₂²×C₄×C₁₀₉	—	4	2	no	—
79	C₂²×C₄×C₁₀₉	—	4	2	no	—
80	C₂²×C₄×C₁₀₉	—	4	2	no	—
81	C₂²×C₄×C₁₀₉	—	4	1	no	—
82	C₂²×C₄×C₁₀₉	—	4	2	no	—
83	C₂²×C₄×C₁₀₉	—	4	1	no	—
84	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
85	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
86	C₂²×C₄×C₁₀₉	SD₁₆	1	2	yes	—
87	C₂²×C₄×C₁₀₉	Q₁₆	1	2	yes	—
88	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
94	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
95	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
96	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
97	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
98	C₂×C₈×C₁₀₉	—	2	4	no	—
99	C₂×C₈×C₁₀₉	—	2	4	no	—
100	C₂×C₈×C₁₀₉	—	2	4	no	—
101	C₂×C₈×C₁₀₉	—	4	2	no	—
102	C₂×C₈×C₁₀₉	—	4	2	no	—
103	C₂×C₈×C₁₀₉	—	4	2	no	—
104	C₂×C₈×C₁₀₉	—	4	2	no	—
105	C₄⋊C₄×C₁₀₉	—	2	2	no	—
106	C₄⋊C₄×C₁₀₉	—	2	2	no	—
107	C₄⋊C₄×C₁₀₉	—	2	2	no	—
108	C₄⋊C₄×C₁₀₉	—	2	4	no	—
109	C₄⋊C₄×C₁₀₉	—	2	4	no	—
110	C₄⋊C₄×C₁₀₉	—	2	4	no	—
111	C₄⋊C₄×C₁₀₉	—	4	8	no	—
112	C₄⋊C₄×C₁₀₉	—	4	1	no	—
113	C₄⋊C₄×C₁₀₉	—	4	2	no	—
114	C₄⋊C₄×C₁₀₉	—	4	1	no	—
115	C₄⋊C₄×C₁₀₉	—	4	2	no	—
116	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
117	C₄²×C₁₀₉	—	2	4	no	—
118	C₄²×C₁₀₉	—	2	2	no	—
119	C₄²×C₁₀₉	—	2	2	no	—
120	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂³×C₁₀₉	#121	1	1	yes	characteristic
122	C₂³×C₁₀₉	#121	1	2	yes	—
123	C₂³×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂³×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
125	C₂³×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
126	C₂³×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
127	C₂³×C₁₀₉	#135	1	2	yes	—
128	C₂³×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂³×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
130	C₂³×C₁₀₉	#137	1	1	yes	characteristic
131	C₂×C₄×C₁₀₉	—	2	2	no	—
132	C₂×C₄×C₁₀₉	—	2	2	no	—
133	C₂×C₄×C₁₀₉	—	2	4	no	—
134	C₂×C₄×C₁₀₉	—	2	4	no	—
135	C₂×C₄×C₁₀₉	—	2	4	no	—
136	C₂×C₄×C₁₀₉	—	2	2	no	—
137	C₂×C₄×C₁₀₉	—	2	2	no	—
138	C₂×C₄×C₁₀₉	—	2	2	no	—
139	C₂×C₄×C₁₀₉	—	4	2	no	—
140	C₂×C₄×C₁₀₉	—	4	2	no	—
141	C₂×C₄×C₁₀₉	—	4	4	no	—
142	C₂×C₄×C₁₀₉	—	4	2	no	—
143	C₂×C₄×C₁₀₉	—	4	2	no	—
144	C₂×C₄×C₁₀₉	—	4	4	no	—
145	C₂×C₄×C₁₀₉	—	4	2	no	—
146	C₂×C₄×C₁₀₉	—	4	4	no	—
147	C₂×C₄×C₁₀₉	—	4	2	no	—
148	C₂×C₄×C₁₀₉	—	4	2	no	—
149	C₂×C₄×C₁₀₉	—	4	4	no	—
150	C₂×C₄×C₁₀₉	—	4	4	no	—
151	C₂×C₄×C₁₀₉	—	8	2	no	—
152	C₂×C₄×C₁₀₉	—	8	2	no	—
153	C₂×C₄×C₁₀₉	—	8	2	no	—
154	C₂×C₄×C₁₀₉	—	8	2	no	—
155	C₂×C₄×C₁₀₉	—	8	2	no	—
156	C₂×C₄×C₁₀₉	—	8	2	no	—
157	C₂×C₄×C₁₀₉	#103	1	4	yes	—
158	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
160	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
161	C₂×C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
162	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
163	C₂×C₄×C₁₀₉	#121	1	2	yes	—
164	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
165	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
166	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
167	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
168	C₂×C₄×C₁₀₉	#123	1	2	yes	—
169	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
170	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
171	C₂×C₄×C₁₀₉	#125	1	2	yes	—
172	C₂×C₄×C₁₀₉	#125	1	2	yes	—
173	C₂×C₄×C₁₀₉	#126	1	2	yes	—
174	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
175	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
176	C₈×C₁₀₉	—	4	4	no	—
177	C₂²×C₁₀₉	#380	1	1	yes	characteristic
178	C₂²×C₁₀₉	#386	1	1	yes	characteristic
179	C₂²×C₁₀₉	#391	1	1	yes	characteristic
180	C₂²×C₁₀₉	#396	1	2	yes	—
181	C₂²×C₁₀₉	#398	1	2	yes	—
182	C₂²×C₁₀₉	C₂×#103	1	1	yes	characteristic
183	C₂²×C₁₀₉	#416	1	1	yes	characteristic
184	C₂²×C₁₀₉	#417	1	1	yes	characteristic
185	C₂²×C₁₀₉	#418	1	1	yes	characteristic
186	C₂²×C₁₀₉	#460	1	1	yes	characteristic
187	C₂²×C₁₀₉	#462	1	1	yes	characteristic
188	C₂²×C₁₀₉	#463	1	2	yes	—
189	C₂²×C₁₀₉	#465	1	2	yes	—
190	C₂²×C₁₀₉	#467	1	2	yes	—
191	C₂²×C₁₀₉	#468	1	2	yes	—
192	C₂²×C₁₀₉	#469	1	2	yes	—
193	C₂²×C₁₀₉	#475	1	1	yes	characteristic
194	C₂²×C₁₀₉	#476	1	2	yes	—
195	C₂²×C₁₀₉	#477	1	1	yes	characteristic
196	C₂²×C₁₀₉	#477	1	2	yes	—
197	C₂²×C₁₀₉	#478	1	2	yes	—
198	C₂²×C₁₀₉	#478	1	2	yes	—
199	C₂²×C₁₀₉	#479	1	2	yes	—
200	C₄×C₁₀₉	—	2	4	no	—
201	C₄×C₁₀₉	—	4	4	no	—
202	C₄×C₁₀₉	—	4	2	no	—
203	C₄×C₁₀₉	—	8	4	no	—
204	C₄×C₁₀₉	—	8	2	no	—
205	C₄×C₁₀₉	#387	1	2	yes	—
206	C₄×C₁₀₉	#389	1	2	yes	—
207	C₄×C₁₀₉	#394	1	1	yes	characteristic
208	C₄×C₁₀₉	#395	1	1	yes	characteristic
209	C₄×C₁₀₉	#398	1	1	yes	characteristic
210	C₄×C₁₀₉	#400	1	1	yes	characteristic
211	#11228	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂×C₁₀₉	#1441	1	1	yes	characteristic
213	C₂×C₁₀₉	#1803	1	1	yes	characteristic
214	C₂×C₁₀₉	#1804	1	1	yes	characteristic
215	C₂×C₁₀₉	#1892	1	1	yes	characteristic
216	C₂×C₁₀₉	#1895	1	1	yes	characteristic
217	C₂×C₁₀₉	#1987	1	1	yes	characteristic
218	C₂×C₁₀₉	#1989	1	1	yes	characteristic
219	C₂×C₁₀₉	#2065	1	2	yes	—
220	C₂×C₁₀₉	#2068	1	2	yes	—
221	C₂×C₁₀₉	#2087	1	2	yes	—
222	C₂×C₁₀₉	#2088	1	2	yes	—
223	C₂²×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic
224	C₂×#389	C₂×C₁₀₉	1	1	yes	characteristic
225	C₂×#340	C₂×C₁₀₉	1	2	yes	—
226	C₂×#340	C₂×C₁₀₉	1	2	yes	—
227	C₄⋊#95	C₂×C₁₀₉	1	1	yes	characteristic
228	C₁₀₉	#11228	1	1	yes	characteristic, direct product factor, Sylow
229	C₂³×C₈	C₂²×C₁₀₉	1	2	yes	—
230	C₂²×C₄⋊C₄	—	2	1	no	—
231	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
232	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
233	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
234	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
235	C₂×C₄⋊C₈	—	2	4	no	—
236	C₂×C₄⋊C₈	—	2	2	no	—
237	#389	—	2	2	no	—
238	#389	C₄×C₁₀₉	1	4	yes	complemented
239	C₂×#95	—	2	1	no	—
240	C₂×#95	—	2	1	no	—
241	C₂×#95	—	2	1	no	—
242	#340	—	2	4	no	—
243	#340	—	2	4	no	—
244	C₂³×C₄	—	2	1	no	—
245	C₂³×C₄	—	2	1	no	—
246	C₂³×C₄	—	2	2	no	—
247	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
248	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
249	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
250	C₂²×C₈	—	2	2	no	—
251	C₂²×C₈	—	2	2	no	—
252	C₂²×C₈	—	2	2	no	—
253	C₂²×C₈	—	2	2	no	—
254	C₂²×C₈	—	2	2	no	—
255	C₂²×C₈	—	2	2	no	—
256	C₂²×C₈	D₄×C₁₀₉	1	4	yes	—
257	C₂×C₄⋊C₄	—	2	1	no	—
258	C₂×C₄⋊C₄	—	2	1	no	—
259	C₂×C₄⋊C₄	—	2	1	no	—
260	C₂×C₄⋊C₄	—	2	2	no	—
261	C₂×C₄⋊C₄	—	2	2	no	—
262	C₂×C₄⋊C₄	—	2	2	no	—
263	C₂×C₄⋊C₄	—	2	2	no	—
264	C₂×C₄⋊C₄	—	2	2	no	—
265	C₂×C₄⋊C₄	—	2	2	no	—
266	C₂×C₄⋊C₄	—	2	4	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	—	4	4	no	—
270	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
271	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
272	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
273	C₂×C₄⋊C₄	Q₈×C₁₀₉	1	4	yes	—
274	C₂×C₄⋊C₄	Q₈×C₁₀₉	1	2	yes	—
275	C₂×C₄²	—	2	2	no	—
276	C₂×C₄²	—	2	2	no	—
277	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
278	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
279	C₄⋊C₈	—	2	8	no	—
280	#95	—	2	4	no	—
281	#95	—	4	2	no	—
282	#95	—	4	2	no	—
283	C₂⁴	C₂×D₄×C₁₀₉	1	1	yes	characteristic
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	2	1	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	1	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	4	2	no	—
296	C₂²×C₄	—	4	2	no	—
297	C₂²×C₄	—	4	2	no	—
298	C₂²×C₄	—	4	2	no	—
299	C₂²×C₄	—	4	2	no	—
300	C₂²×C₄	—	4	2	no	—
301	C₂²×C₄	—	4	2	no	—
302	C₂²×C₄	—	4	2	no	—
303	C₂²×C₄	—	4	2	no	—
304	C₂²×C₄	—	4	1	no	—
305	C₂²×C₄	—	4	2	no	—
306	C₂²×C₄	—	4	1	no	—
307	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
308	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
309	C₂²×C₄	SD₁₆×C₁₀₉	1	2	yes	—
310	C₂²×C₄	Q₁₆×C₁₀₉	1	2	yes	—
311	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
312	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
314	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
315	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
316	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
317	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
318	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
319	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
320	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
321	C₂×C₈	—	2	4	no	—
322	C₂×C₈	—	2	4	no	—
323	C₂×C₈	—	2	4	no	—
324	C₂×C₈	—	4	2	no	—
325	C₂×C₈	—	4	2	no	—
326	C₂×C₈	—	4	2	no	—
327	C₂×C₈	—	4	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	4	no	—
332	C₄⋊C₄	—	2	4	no	—
333	C₄⋊C₄	—	2	4	no	—
334	C₄⋊C₄	—	4	8	no	—
335	C₄⋊C₄	—	4	1	no	—
336	C₄⋊C₄	—	4	2	no	—
337	C₄⋊C₄	—	4	1	no	—
338	C₄⋊C₄	—	4	2	no	—
339	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
340	C₄²	—	2	4	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	2	no	—
343	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
344	C₂³	#121×C₁₀₉	1	1	yes	characteristic
345	C₂³	#121×C₁₀₉	1	2	yes	—
346	C₂³	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
347	C₂³	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
348	C₂³	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
349	C₂³	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
350	C₂³	#135×C₁₀₉	1	2	yes	—
351	C₂³	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
352	C₂³	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
353	C₂³	#137×C₁₀₉	1	1	yes	characteristic
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	2	4	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	4	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	4	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	4	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	4	no	—
373	C₂×C₄	—	4	4	no	—
374	C₂×C₄	—	8	2	no	—
375	C₂×C₄	—	8	2	no	—
376	C₂×C₄	—	8	2	no	—
377	C₂×C₄	—	8	2	no	—
378	C₂×C₄	—	8	2	no	—
379	C₂×C₄	—	8	2	no	—
380	C₂×C₄	#103×C₁₀₉	1	4	yes	—
381	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
382	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
383	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
384	C₂×C₄	C₄×Q₈×C₁₀₉	1	2	yes	—
385	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
386	C₂×C₄	#121×C₁₀₉	1	2	yes	—
387	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
388	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
389	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
390	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
391	C₂×C₄	#123×C₁₀₉	1	2	yes	—
392	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
393	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
394	C₂×C₄	#125×C₁₀₉	1	2	yes	—
395	C₂×C₄	#125×C₁₀₉	1	2	yes	—
396	C₂×C₄	#126×C₁₀₉	1	2	yes	—
397	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	2	yes	—
398	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
399	C₈	—	4	4	no	—
400	C₂²	#380×C₁₀₉	1	1	yes	characteristic
401	C₂²	#386×C₁₀₉	1	1	yes	characteristic
402	C₂²	#391×C₁₀₉	1	1	yes	characteristic
403	C₂²	#396×C₁₀₉	1	2	yes	—
404	C₂²	#398×C₁₀₉	1	2	yes	—
405	C₂²	C₂×#103×C₁₀₉	1	1	yes	characteristic
406	C₂²	#416×C₁₀₉	1	1	yes	characteristic
407	C₂²	#417×C₁₀₉	1	1	yes	characteristic
408	C₂²	#418×C₁₀₉	1	1	yes	characteristic
409	C₂²	#460×C₁₀₉	1	1	yes	characteristic
410	C₂²	#462×C₁₀₉	1	1	yes	characteristic
411	C₂²	#463×C₁₀₉	1	2	yes	—
412	C₂²	#465×C₁₀₉	1	2	yes	—
413	C₂²	#467×C₁₀₉	1	2	yes	—
414	C₂²	#468×C₁₀₉	1	2	yes	—
415	C₂²	#469×C₁₀₉	1	2	yes	—
416	C₂²	#475×C₁₀₉	1	1	yes	characteristic
417	C₂²	#476×C₁₀₉	1	2	yes	—
418	C₂²	#477×C₁₀₉	1	2	yes	—
419	C₂²	#477×C₁₀₉	1	1	yes	characteristic
420	C₂²	#478×C₁₀₉	1	2	yes	—
421	C₂²	#478×C₁₀₉	1	2	yes	—
422	C₂²	#479×C₁₀₉	1	2	yes	—
423	C₄	—	2	4	no	—
424	C₄	—	4	4	no	—
425	C₄	—	4	2	no	—
426	C₄	—	8	4	no	—
427	C₄	—	8	2	no	—
428	C₄	#387×C₁₀₉	1	2	yes	—
429	C₄	#389×C₁₀₉	1	2	yes	—
430	C₄	#394×C₁₀₉	1	1	yes	characteristic
431	C₄	#395×C₁₀₉	1	1	yes	characteristic
432	C₄	#398×C₁₀₉	1	1	yes	characteristic
433	C₄	#400×C₁₀₉	1	1	yes	characteristic
434	C₂	C₁₀₉×#1441	1	1	yes	characteristic
435	C₂	C₁₀₉×#1803	1	1	yes	characteristic
436	C₂	C₁₀₉×#1804	1	1	yes	characteristic
437	C₂	C₁₀₉×#1892	1	1	yes	characteristic
438	C₂	C₁₀₉×#1895	1	1	yes	characteristic
439	C₂	C₁₀₉×#1987	1	1	yes	characteristic
440	C₂	C₁₀₉×#1989	1	1	yes	characteristic
441	C₂	C₁₀₉×#2065	1	2	yes	—
442	C₂	C₁₀₉×#2068	1	2	yes	—
443	C₂	C₁₀₉×#2087	1	2	yes	—
444	C₂	C₁₀₉×#2088	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₀₉	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₀₉ ◁ C₂²×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
15. #389×C₁₀₉	C₄	semidirect
211. #11228	C₁₀₉	direct (N × H)
228. C₁₀₉	#11228	direct (N × H)
238. #389	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₈×C₁₀₉	2
C₂×#389×C₁₀₉	2
C₂×#340×C₁₀₉	2
C₁₀₉×C₄⋊#95	2
#11228	109

Subgroups — C109×#11228 #42129222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11228 #42129222