Subgroups — C₁₀₉×#24816 #42142810

Order 27904.

894 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#474×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×SD₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2568	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#2085	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#2068	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#1939	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₀₉×#1877	C₂	1	2	yes	maximal
10	C₁₀₉×#1804	C₂	1	2	yes	maximal
11	C₁₀₉×#1803	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₁₀₉	—	2	1	no	—
18	#474×C₁₀₉	—	2	4	no	—
19	C₄×SD₁₆×C₁₀₉	—	2	4	no	—
20	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
21	C₂×#106×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
24	C₂×#103×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₂×#103×C₁₀₉	C₂²	1	2	yes	—
26	C₂×#102×C₁₀₉	C₂²	1	2	yes	—
27	C₂×#102×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
29	#400×C₁₀₉	—	2	2	no	—
30	#391×C₁₀₉	—	2	1	no	—
31	#391×C₁₀₉	—	2	1	no	—
32	#391×C₁₀₉	C₂²	1	2	yes	—
33	#389×C₁₀₉	C₂²	1	2	yes	—
34	#385×C₁₀₉	—	2	2	no	—
35	#384×C₁₀₉	C₂²	1	1	yes	characteristic
36	#382×C₁₀₉	C₂²	1	2	yes	—
37	#380×C₁₀₉	—	2	1	no	—
38	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
39	#340×C₁₀₉	C₂²	1	2	yes	—
40	#340×C₁₀₉	C₂²	1	2	yes	—
41	#340×C₁₀₉	C₂²	1	2	yes	—
42	#336×C₁₀₉	C₂²	1	2	yes	—
43	C₂²×Q₈×C₁₀₉	—	2	1	no	—
44	C₂²×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₁₀₉	—	2	1	no	—
47	C₂³×C₄×C₁₀₉	—	2	2	no	—
48	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
49	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
50	C₂×SD₁₆×C₁₀₉	Q₈	1	4	yes	complemented
51	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
52	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
54	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
55	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
56	C₄×Q₈×C₁₀₉	—	2	2	no	—
57	C₄×Q₈×C₁₀₉	—	2	2	no	—
58	C₄×D₄×C₁₀₉	—	2	4	no	—
59	C₄×D₄×C₁₀₉	—	2	2	no	—
60	C₄×D₄×C₁₀₉	—	4	2	no	—
61	C₄×D₄×C₁₀₉	D₄	1	4	yes	—
62	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
63	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
64	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
65	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
66	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
67	C₂×C₄⋊C₄×C₁₀₉	—	4	4	no	—
68	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
76	C₂×C₄²×C₁₀₉	—	2	1	no	—
77	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
78	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
79	#107×C₁₀₉	—	2	4	no	—
80	#106×C₁₀₉	—	2	2	no	—
81	C₄⋊C₈×C₁₀₉	—	2	2	no	—
82	C₄⋊C₈×C₁₀₉	—	4	2	no	—
83	#103×C₁₀₉	—	2	2	no	—
84	#103×C₁₀₉	—	4	2	no	—
85	#102×C₁₀₉	—	2	2	no	—
86	#102×C₁₀₉	—	2	4	no	—
87	C₄×C₈×C₁₀₉	—	2	2	no	—
88	#95×C₁₀₉	—	2	2	no	—
89	#95×C₁₀₉	—	2	2	no	—
90	#95×C₁₀₉	—	2	2	no	—
91	#95×C₁₀₉	—	2	2	no	—
92	#95×C₁₀₉	—	2	2	no	—
93	#95×C₁₀₉	D₄	1	2	yes	—
94	C₂⁴×C₁₀₉	—	2	1	no	—
95	C₂×Q₈×C₁₀₉	—	2	1	no	—
96	C₂×Q₈×C₁₀₉	—	2	1	no	—
97	C₂×Q₈×C₁₀₉	—	2	1	no	—
98	C₂×Q₈×C₁₀₉	—	2	1	no	—
99	C₂×Q₈×C₁₀₉	—	2	1	no	—
100	C₂×Q₈×C₁₀₉	—	4	2	no	—
101	C₂×Q₈×C₁₀₉	—	4	2	no	—
102	C₂×Q₈×C₁₀₉	—	4	2	no	—
103	C₂×Q₈×C₁₀₉	C₂×Q₈	1	2	yes	—
104	C₂×D₄×C₁₀₉	—	2	1	no	—
105	C₂×D₄×C₁₀₉	—	2	1	no	—
106	C₂×D₄×C₁₀₉	—	2	1	no	—
107	C₂×D₄×C₁₀₉	C₂×D₄	1	2	yes	—
108	C₂×D₄×C₁₀₉	C₂×Q₈	1	2	yes	—
109	C₂×D₄×C₁₀₉	#41	1	2	yes	—
110	C₂²×C₄×C₁₀₉	—	2	1	no	—
111	C₂²×C₄×C₁₀₉	—	2	1	no	—
112	C₂²×C₄×C₁₀₉	—	2	4	no	—
113	C₂²×C₄×C₁₀₉	—	2	1	no	—
114	C₂²×C₄×C₁₀₉	—	2	2	no	—
115	C₂²×C₄×C₁₀₉	—	2	2	no	—
116	C₂²×C₄×C₁₀₉	—	4	4	no	—
117	C₂²×C₄×C₁₀₉	—	4	4	no	—
118	C₂²×C₄×C₁₀₉	—	4	2	no	—
119	C₂²×C₄×C₁₀₉	—	4	2	no	—
120	C₂²×C₄×C₁₀₉	—	8	2	no	—
121	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
126	SD₁₆×C₁₀₉	—	2	8	no	—
127	C₂×C₈×C₁₀₉	—	2	1	no	—
128	C₂×C₈×C₁₀₉	—	2	1	no	—
129	C₂×C₈×C₁₀₉	—	2	1	no	—
130	C₂×C₈×C₁₀₉	—	2	2	no	—
131	C₂×C₈×C₁₀₉	—	2	2	no	—
132	C₂×C₈×C₁₀₉	—	2	1	no	—
133	C₂×C₈×C₁₀₉	—	2	2	no	—
134	C₂×C₈×C₁₀₉	—	2	1	no	—
135	C₂×C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
136	C₄⋊C₄×C₁₀₉	—	2	2	no	—
137	C₄⋊C₄×C₁₀₉	—	2	2	no	—
138	C₄⋊C₄×C₁₀₉	—	2	1	no	—
139	C₄⋊C₄×C₁₀₉	—	2	1	no	—
140	C₄⋊C₄×C₁₀₉	—	2	2	no	—
141	C₄⋊C₄×C₁₀₉	—	2	2	no	—
142	C₄⋊C₄×C₁₀₉	—	2	1	no	—
143	C₄⋊C₄×C₁₀₉	—	2	2	no	—
144	C₄⋊C₄×C₁₀₉	—	2	2	no	—
145	C₄⋊C₄×C₁₀₉	—	4	4	no	—
146	C₄⋊C₄×C₁₀₉	—	4	2	no	—
147	C₄⋊C₄×C₁₀₉	—	4	2	no	—
148	C₄⋊C₄×C₁₀₉	—	4	2	no	—
149	C₄⋊C₄×C₁₀₉	—	8	1	no	—
150	C₄⋊C₄×C₁₀₉	C₂×D₄	1	2	yes	—
151	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
152	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
153	C₄²×C₁₀₉	—	2	2	no	—
154	C₄²×C₁₀₉	—	2	2	no	—
155	C₄²×C₁₀₉	—	4	2	no	—
156	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
157	C₄²×C₁₀₉	#41	1	2	yes	—
158	C₂³×C₁₀₉	—	2	2	no	—
159	C₂³×C₁₀₉	—	2	2	no	—
160	C₂³×C₁₀₉	—	2	2	no	—
161	C₂³×C₁₀₉	—	4	1	no	—
162	C₂³×C₁₀₉	—	4	1	no	—
163	C₂³×C₁₀₉	—	4	1	no	—
164	C₂³×C₁₀₉	—	4	1	no	—
165	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
166	Q₈×C₁₀₉	—	2	2	no	—
167	Q₈×C₁₀₉	—	2	2	no	—
168	Q₈×C₁₀₉	—	4	1	no	—
169	Q₈×C₁₀₉	—	4	1	no	—
170	Q₈×C₁₀₉	—	8	4	no	—
171	D₄×C₁₀₉	—	2	2	no	—
172	D₄×C₁₀₉	—	2	2	no	—
173	D₄×C₁₀₉	—	2	2	no	—
174	D₄×C₁₀₉	C₂²⋊Q₈	1	4	yes	—
175	C₂×C₄×C₁₀₉	—	2	1	no	—
176	C₂×C₄×C₁₀₉	—	2	1	no	—
177	C₂×C₄×C₁₀₉	—	2	2	no	—
178	C₂×C₄×C₁₀₉	—	2	2	no	—
179	C₂×C₄×C₁₀₉	—	2	2	no	—
180	C₂×C₄×C₁₀₉	—	2	2	no	—
181	C₂×C₄×C₁₀₉	—	2	2	no	—
182	C₂×C₄×C₁₀₉	—	2	2	no	—
183	C₂×C₄×C₁₀₉	—	2	2	no	—
184	C₂×C₄×C₁₀₉	—	2	2	no	—
185	C₂×C₄×C₁₀₉	—	4	1	no	—
186	C₂×C₄×C₁₀₉	—	4	1	no	—
187	C₂×C₄×C₁₀₉	—	4	1	no	—
188	C₂×C₄×C₁₀₉	—	4	1	no	—
189	C₂×C₄×C₁₀₉	—	4	8	no	—
190	C₂×C₄×C₁₀₉	—	4	1	no	—
191	C₂×C₄×C₁₀₉	—	4	2	no	—
192	C₂×C₄×C₁₀₉	—	4	2	no	—
193	C₂×C₄×C₁₀₉	—	4	2	no	—
194	C₂×C₄×C₁₀₉	—	4	2	no	—
195	C₂×C₄×C₁₀₉	—	4	2	no	—
196	C₂×C₄×C₁₀₉	—	4	2	no	—
197	C₂×C₄×C₁₀₉	—	4	1	no	—
198	C₂×C₄×C₁₀₉	—	4	1	no	—
199	C₂×C₄×C₁₀₉	—	8	8	no	—
200	C₂×C₄×C₁₀₉	#121	1	2	yes	—
201	C₂×C₄×C₁₀₉	#121	1	2	yes	—
202	C₂×C₄×C₁₀₉	#135	1	2	yes	—
203	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
204	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
206	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
207	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
208	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
209	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
210	C₈×C₁₀₉	—	2	2	no	—
211	C₈×C₁₀₉	—	4	1	no	—
212	C₈×C₁₀₉	—	4	2	no	—
213	C₂²×C₁₀₉	—	2	4	no	—
214	C₂²×C₁₀₉	—	4	2	no	—
215	C₂²×C₁₀₉	—	4	2	no	—
216	C₂²×C₁₀₉	—	4	2	no	—
217	C₂²×C₁₀₉	—	4	2	no	—
218	C₂²×C₁₀₉	—	4	2	no	—
219	C₂²×C₁₀₉	—	4	2	no	—
220	C₂²×C₁₀₉	C₂×#121	1	1	yes	characteristic
221	C₂²×C₁₀₉	#546	1	1	yes	characteristic
222	C₂²×C₁₀₉	D₄×Q₈	1	1	yes	characteristic
223	C₂²×C₁₀₉	C₂×#135	1	1	yes	characteristic
224	C₂²×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
225	C₂²×C₁₀₉	#577	1	1	yes	characteristic
226	C₂²×C₁₀₉	#578	1	1	yes	characteristic
227	C₄×C₁₀₉	—	2	2	no	—
228	C₄×C₁₀₉	—	4	2	no	—
229	C₄×C₁₀₉	—	4	2	no	—
230	C₄×C₁₀₉	—	4	2	no	—
231	C₄×C₁₀₉	—	4	2	no	—
232	C₄×C₁₀₉	—	8	2	no	—
233	C₄×C₁₀₉	—	8	1	no	—
234	C₄×C₁₀₉	—	8	2	no	—
235	C₄×C₁₀₉	#463	1	4	yes	—
236	C₄×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
237	C₄×C₁₀₉	#537	1	1	yes	characteristic
238	C₄×C₁₀₉	#552	1	1	yes	characteristic
239	C₄×C₁₀₉	#554	1	1	yes	characteristic
240	#24816	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
241	C₂×C₁₀₉	—	4	4	no	—
242	C₂×C₁₀₉	#2417	1	1	yes	characteristic
243	C₂×C₁₀₉	C₂×#463	1	1	yes	characteristic
244	C₂×C₁₀₉	#3046	1	1	yes	characteristic
245	C₂×C₁₀₉	#3302	1	1	yes	characteristic
246	C₂×C₁₀₉	#3324	1	1	yes	characteristic
247	C₂×C₁₀₉	#3381	1	1	yes	characteristic
248	C₂×C₁₀₉	#3386	1	1	yes	characteristic
249	C₂×#474	C₂×C₁₀₉	1	1	yes	characteristic
250	C₂×C₄×SD₁₆	C₂×C₁₀₉	1	1	yes	characteristic
251	#2568	C₂×C₁₀₉	1	1	yes	characteristic, complemented
252	D₄×C₄⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic
253	#2085	C₂×C₁₀₉	1	2	yes	complemented
254	#2068	C₂×C₁₀₉	1	2	yes	complemented
255	#1990	C₂×C₁₀₉	1	1	yes	characteristic, complemented
256	#1939	C₂×C₁₀₉	1	1	yes	characteristic, complemented
257	#1877	C₂×C₁₀₉	1	2	yes	—
258	#1804	C₂×C₁₀₉	1	2	yes	—
259	#1803	C₂×C₁₀₉	1	1	yes	characteristic, complemented
260	C₁₀₉	#24816	1	1	yes	characteristic, direct product factor, Sylow
261	C₂²×SD₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
262	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
263	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
264	C₂×C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
265	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
266	C₂²×C₄⋊C₄	—	2	1	no	—
267	#474	—	2	4	no	—
268	C₄×SD₁₆	—	2	4	no	—
269	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
270	C₂×#106	C₂²×C₁₀₉	1	1	yes	characteristic
271	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
272	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
273	C₂×#103	C₂²×C₁₀₉	1	1	yes	characteristic
274	C₂×#103	C₂²×C₁₀₉	1	2	yes	—
275	C₂×#102	C₂²×C₁₀₉	1	1	yes	characteristic
276	C₂×#102	C₂²×C₁₀₉	1	2	yes	—
277	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
278	#400	—	2	2	no	—
279	#391	—	2	1	no	—
280	#391	—	2	1	no	—
281	#391	C₂²×C₁₀₉	1	2	yes	—
282	#389	C₂²×C₁₀₉	1	2	yes	—
283	#385	—	2	2	no	—
284	#384	C₂²×C₁₀₉	1	1	yes	characteristic
285	#382	C₂²×C₁₀₉	1	2	yes	—
286	#380	—	2	1	no	—
287	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
288	#340	C₂²×C₁₀₉	1	2	yes	—
289	#340	C₂²×C₁₀₉	1	2	yes	—
290	#340	C₂²×C₁₀₉	1	2	yes	—
291	#336	C₂²×C₁₀₉	1	2	yes	—
292	C₂²×Q₈	—	2	1	no	—
293	C₂²×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
294	C₂²×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
295	C₂³×C₄	—	2	1	no	—
296	C₂³×C₄	—	2	2	no	—
297	C₂×SD₁₆	—	2	2	no	—
298	C₂×SD₁₆	—	2	2	no	—
299	C₂×SD₁₆	Q₈×C₁₀₉	1	4	yes	complemented
300	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
301	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
302	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
303	C₄⋊Q₈	—	2	2	no	—
304	C₄⋊Q₈	—	2	2	no	—
305	C₄×Q₈	—	2	2	no	—
306	C₄×Q₈	—	2	2	no	—
307	C₄×D₄	—	2	4	no	—
308	C₄×D₄	—	2	2	no	—
309	C₄×D₄	—	4	2	no	—
310	C₄×D₄	D₄×C₁₀₉	1	4	yes	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	—	2	2	no	—
313	C₂×C₄⋊C₄	—	2	1	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	—	2	1	no	—
316	C₂×C₄⋊C₄	—	4	4	no	—
317	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
318	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
319	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
320	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
321	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
322	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
323	C₂×C₂²⋊C₄	—	2	1	no	—
324	C₂×C₂²⋊C₄	—	2	2	no	—
325	C₂×C₄²	—	2	1	no	—
326	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
327	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
328	#107	—	2	4	no	—
329	#106	—	2	2	no	—
330	C₄⋊C₈	—	2	2	no	—
331	C₄⋊C₈	—	4	2	no	—
332	#103	—	2	2	no	—
333	#103	—	4	2	no	—
334	#102	—	2	2	no	—
335	#102	—	2	4	no	—
336	C₄×C₈	—	2	2	no	—
337	#95	—	2	2	no	—
338	#95	—	2	2	no	—
339	#95	—	2	2	no	—
340	#95	—	2	2	no	—
341	#95	—	2	2	no	—
342	#95	D₄×C₁₀₉	1	2	yes	—
343	C₂⁴	—	2	1	no	—
344	C₂×Q₈	—	2	1	no	—
345	C₂×Q₈	—	2	1	no	—
346	C₂×Q₈	—	2	1	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	4	2	no	—
350	C₂×Q₈	—	4	2	no	—
351	C₂×Q₈	—	4	2	no	—
352	C₂×Q₈	C₂×Q₈×C₁₀₉	1	2	yes	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	C₂×D₄×C₁₀₉	1	2	yes	—
357	C₂×D₄	C₂×Q₈×C₁₀₉	1	2	yes	—
358	C₂×D₄	#41×C₁₀₉	1	2	yes	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	4	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	4	4	no	—
366	C₂²×C₄	—	4	4	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	—	4	2	no	—
369	C₂²×C₄	—	8	2	no	—
370	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
371	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
372	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
373	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
374	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
375	SD₁₆	—	2	8	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	C₂×Q₈×C₁₀₉	1	2	yes	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	4	4	no	—
395	C₄⋊C₄	—	4	2	no	—
396	C₄⋊C₄	—	4	2	no	—
397	C₄⋊C₄	—	4	2	no	—
398	C₄⋊C₄	—	8	1	no	—
399	C₄⋊C₄	C₂×D₄×C₁₀₉	1	2	yes	—
400	C₂²⋊C₄	—	2	4	no	—
401	C₂²⋊C₄	—	4	4	no	—
402	C₄²	—	2	2	no	—
403	C₄²	—	2	2	no	—
404	C₄²	—	4	2	no	—
405	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
406	C₄²	#41×C₁₀₉	1	2	yes	—
407	C₂³	—	2	2	no	—
408	C₂³	—	2	2	no	—
409	C₂³	—	2	2	no	—
410	C₂³	—	4	1	no	—
411	C₂³	—	4	1	no	—
412	C₂³	—	4	1	no	—
413	C₂³	—	4	1	no	—
414	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
415	Q₈	—	2	2	no	—
416	Q₈	—	2	2	no	—
417	Q₈	—	4	1	no	—
418	Q₈	—	4	1	no	—
419	Q₈	—	8	4	no	—
420	D₄	—	2	2	no	—
421	D₄	—	2	2	no	—
422	D₄	—	2	2	no	—
423	D₄	C₂²⋊Q₈×C₁₀₉	1	4	yes	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	8	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	8	8	no	—
449	C₂×C₄	#121×C₁₀₉	1	2	yes	—
450	C₂×C₄	#121×C₁₀₉	1	2	yes	—
451	C₂×C₄	#135×C₁₀₉	1	2	yes	—
452	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
453	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
454	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
455	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
456	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
457	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
458	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
459	C₈	—	2	2	no	—
460	C₈	—	4	1	no	—
461	C₈	—	4	2	no	—
462	C₂²	—	2	4	no	—
463	C₂²	—	4	2	no	—
464	C₂²	—	4	2	no	—
465	C₂²	—	4	2	no	—
466	C₂²	—	4	2	no	—
467	C₂²	—	4	2	no	—
468	C₂²	—	4	2	no	—
469	C₂²	C₂×#121×C₁₀₉	1	1	yes	characteristic
470	C₂²	#546×C₁₀₉	1	1	yes	characteristic
471	C₂²	D₄×Q₈×C₁₀₉	1	1	yes	characteristic
472	C₂²	C₂×#135×C₁₀₉	1	1	yes	characteristic
473	C₂²	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
474	C₂²	#577×C₁₀₉	1	1	yes	characteristic
475	C₂²	#578×C₁₀₉	1	1	yes	characteristic
476	C₄	—	2	2	no	—
477	C₄	—	4	2	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	2	no	—
480	C₄	—	4	2	no	—
481	C₄	—	8	2	no	—
482	C₄	—	8	1	no	—
483	C₄	—	8	2	no	—
484	C₄	#463×C₁₀₉	1	4	yes	—
485	C₄	C₂×C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
486	C₄	#537×C₁₀₉	1	1	yes	characteristic
487	C₄	#552×C₁₀₉	1	1	yes	characteristic
488	C₄	#554×C₁₀₉	1	1	yes	characteristic
489	C₂	—	4	4	no	—
490	C₂	C₁₀₉×#2417	1	1	yes	characteristic
491	C₂	C₂×#463×C₁₀₉	1	1	yes	characteristic
492	C₂	C₁₀₉×#3046	1	1	yes	characteristic
493	C₂	C₁₀₉×#3302	1	1	yes	characteristic
494	C₂	C₁₀₉×#3324	1	1	yes	characteristic
495	C₂	C₁₀₉×#3381	1	1	yes	characteristic
496	C₂	C₁₀₉×#3386	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#2568	C₂	semidirect
5. C₁₀₉×#2085	C₂	semidirect
6. C₁₀₉×#2068	C₂	semidirect
7. C₁₀₉×#1990	C₂	semidirect
8. C₁₀₉×#1939	C₂	semidirect
11. C₁₀₉×#1803	C₂	semidirect
50. C₂×SD₁₆×C₁₀₉	Q₈	semidirect
240. #24816	C₁₀₉	direct (N × H)
251. #2568	C₂×C₁₀₉	semidirect
253. #2085	C₂×C₁₀₉	semidirect
254. #2068	C₂×C₁₀₉	semidirect
255. #1990	C₂×C₁₀₉	semidirect
256. #1939	C₂×C₁₀₉	semidirect
259. #1803	C₂×C₁₀₉	semidirect
260. C₁₀₉	#24816	direct (N × H)
299. C₂×SD₁₆	Q₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#474×C₁₀₉	2
C₂×C₄×SD₁₆×C₁₀₉	2
C₁₀₉×#2568	2
D₄×C₄⋊C₄×C₁₀₉	2
C₁₀₉×#2085	2
C₁₀₉×#2068	2
C₁₀₉×#1990	2
C₁₀₉×#1939	2
C₁₀₉×#1877	2
C₁₀₉×#1804	2
C₁₀₉×#1803	2
#24816	109

Subgroups — C109×#24816 #42142810

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#24816 #42142810