Subgroups — C₁₀₉×#24813 #42142807

Order 27904.

894 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#475×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×SD₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2542	C₂	1	1	yes	maximal, characteristic
4	Q₈×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#2084	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#2065	C₂	1	2	yes	maximal
7	C₁₀₉×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#1939	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₀₉×#1880	C₂	1	2	yes	maximal
10	C₁₀₉×#1803	C₂	1	2	yes	maximal, complemented
11	C₁₀₉×#1802	C₂	1	1	yes	maximal, characteristic
12	C₂²×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
15	C₂×C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
16	C₂×C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
17	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	#475×C₁₀₉	—	2	4	no	—
20	C₄×SD₁₆×C₁₀₉	—	2	4	no	—
21	C₂×#106×C₁₀₉	C₂²	1	2	yes	—
22	C₂×#106×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
25	C₂×#103×C₁₀₉	C₂²	1	2	yes	—
26	C₂×#103×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂×#102×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×#102×C₁₀₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
30	#399×C₁₀₉	—	2	2	no	—
31	#390×C₁₀₉	C₂²	1	2	yes	—
32	#389×C₁₀₉	—	2	2	no	—
33	#389×C₁₀₉	C₂²	1	2	yes	—
34	#384×C₁₀₉	—	2	1	no	—
35	#384×C₁₀₉	C₂²	1	1	yes	characteristic
36	#384×C₁₀₉	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×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	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
48	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
49	C₂×SD₁₆×C₁₀₉	Q₈	1	4	yes	complemented
50	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
51	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
54	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
55	C₂²⋊Q₈×C₁₀₉	—	4	4	no	—
56	C₂²⋊Q₈×C₁₀₉	—	4	4	no	—
57	C₄×Q₈×C₁₀₉	—	2	2	no	—
58	C₄×Q₈×C₁₀₉	—	2	4	no	—
59	C₄×Q₈×C₁₀₉	—	4	2	no	—
60	C₄×Q₈×C₁₀₉	D₄	1	4	yes	—
61	C₄×D₄×C₁₀₉	—	2	2	no	—
62	C₄×D₄×C₁₀₉	—	2	2	no	—
63	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
64	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
65	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
66	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
67	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
68	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
69	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
70	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
71	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
77	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
78	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
80	C₂×C₄²×C₁₀₉	—	2	1	no	—
81	C₂×C₄²×C₁₀₉	—	2	2	no	—
82	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
83	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
84	#106×C₁₀₉	—	2	2	no	—
85	#106×C₁₀₉	—	2	4	no	—
86	C₄⋊C₈×C₁₀₉	—	2	2	no	—
87	C₄⋊C₈×C₁₀₉	—	4	2	no	—
88	#103×C₁₀₉	—	2	2	no	—
89	#103×C₁₀₉	—	2	4	no	—
90	#102×C₁₀₉	—	2	2	no	—
91	#102×C₁₀₉	—	4	2	no	—
92	C₄×C₈×C₁₀₉	—	2	2	no	—
93	#95×C₁₀₉	—	2	2	no	—
94	#95×C₁₀₉	D₄	1	2	yes	—
95	C₂⁴×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	—	4	2	no	—
100	C₂×Q₈×C₁₀₉	—	4	2	no	—
101	C₂×Q₈×C₁₀₉	—	4	2	no	—
102	C₂×Q₈×C₁₀₉	C₂×D₄	1	2	yes	—
103	C₂×Q₈×C₁₀₉	C₂×Q₈	1	2	yes	—
104	C₂×Q₈×C₁₀₉	#41	1	2	yes	—
105	C₂×D₄×C₁₀₉	—	2	1	no	—
106	C₂×D₄×C₁₀₉	—	2	1	no	—
107	C₂×D₄×C₁₀₉	—	2	1	no	—
108	C₂×D₄×C₁₀₉	—	2	1	no	—
109	C₂×D₄×C₁₀₉	—	2	1	no	—
110	C₂×D₄×C₁₀₉	C₂×Q₈	1	2	yes	—
111	C₂²×C₄×C₁₀₉	—	2	1	no	—
112	C₂²×C₄×C₁₀₉	—	2	1	no	—
113	C₂²×C₄×C₁₀₉	—	2	2	no	—
114	C₂²×C₄×C₁₀₉	—	2	2	no	—
115	C₂²×C₄×C₁₀₉	—	2	1	no	—
116	C₂²×C₄×C₁₀₉	—	4	2	no	—
117	C₂²×C₄×C₁₀₉	—	4	2	no	—
118	C₂²×C₄×C₁₀₉	—	4	2	no	—
119	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
124	SD₁₆×C₁₀₉	—	2	8	no	—
125	C₂×C₈×C₁₀₉	—	2	1	no	—
126	C₂×C₈×C₁₀₉	—	2	1	no	—
127	C₂×C₈×C₁₀₉	—	2	1	no	—
128	C₂×C₈×C₁₀₉	—	2	2	no	—
129	C₂×C₈×C₁₀₉	—	2	2	no	—
130	C₂×C₈×C₁₀₉	—	2	1	no	—
131	C₂×C₈×C₁₀₉	—	2	2	no	—
132	C₂×C₈×C₁₀₉	—	2	1	no	—
133	C₂×C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
134	C₄⋊C₄×C₁₀₉	—	2	4	no	—
135	C₄⋊C₄×C₁₀₉	—	2	2	no	—
136	C₄⋊C₄×C₁₀₉	—	2	1	no	—
137	C₄⋊C₄×C₁₀₉	—	2	1	no	—
138	C₄⋊C₄×C₁₀₉	—	2	2	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	2	no	—
143	C₄⋊C₄×C₁₀₉	—	4	4	no	—
144	C₄⋊C₄×C₁₀₉	—	4	4	no	—
145	C₄⋊C₄×C₁₀₉	—	4	2	no	—
146	C₄⋊C₄×C₁₀₉	—	4	4	no	—
147	C₄⋊C₄×C₁₀₉	—	4	2	no	—
148	C₄⋊C₄×C₁₀₉	—	8	1	no	—
149	C₄⋊C₄×C₁₀₉	—	8	2	no	—
150	C₄⋊C₄×C₁₀₉	SD₁₆	1	4	yes	complemented
151	C₄⋊C₄×C₁₀₉	C₂×D₄	1	2	yes	—
152	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
153	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
154	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
155	C₄²×C₁₀₉	—	2	4	no	—
156	C₄²×C₁₀₉	—	2	2	no	—
157	C₄²×C₁₀₉	—	2	2	no	—
158	C₄²×C₁₀₉	—	4	4	no	—
159	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
160	C₄²×C₁₀₉	#41	1	2	yes	—
161	C₂³×C₁₀₉	—	2	2	no	—
162	C₂³×C₁₀₉	—	4	1	no	—
163	C₂³×C₁₀₉	—	4	1	no	—
164	C₂³×C₁₀₉	—	4	1	no	—
165	C₂³×C₁₀₉	—	4	1	no	—
166	C₂³×C₁₀₉	—	4	1	no	—
167	C₂³×C₁₀₉	—	4	1	no	—
168	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
169	Q₈×C₁₀₉	—	2	2	no	—
170	Q₈×C₁₀₉	—	2	2	no	—
171	Q₈×C₁₀₉	—	2	2	no	—
172	Q₈×C₁₀₉	—	8	4	no	—
173	Q₈×C₁₀₉	C₂²⋊Q₈	1	4	yes	complemented
174	D₄×C₁₀₉	—	2	2	no	—
175	D₄×C₁₀₉	—	2	2	no	—
176	D₄×C₁₀₉	—	4	1	no	—
177	D₄×C₁₀₉	—	4	1	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	1	no	—
183	C₂×C₄×C₁₀₉	—	2	2	no	—
184	C₂×C₄×C₁₀₉	—	2	1	no	—
185	C₂×C₄×C₁₀₉	—	2	2	no	—
186	C₂×C₄×C₁₀₉	—	2	2	no	—
187	C₂×C₄×C₁₀₉	—	2	2	no	—
188	C₂×C₄×C₁₀₉	—	2	2	no	—
189	C₂×C₄×C₁₀₉	—	4	1	no	—
190	C₂×C₄×C₁₀₉	—	4	1	no	—
191	C₂×C₄×C₁₀₉	—	4	1	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	1	no	—
196	C₂×C₄×C₁₀₉	—	4	2	no	—
197	C₂×C₄×C₁₀₉	—	4	1	no	—
198	C₂×C₄×C₁₀₉	—	4	2	no	—
199	C₂×C₄×C₁₀₉	—	4	2	no	—
200	C₂×C₄×C₁₀₉	—	8	4	no	—
201	C₂×C₄×C₁₀₉	#121	1	2	yes	—
202	C₂×C₄×C₁₀₉	#121	1	2	yes	—
203	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	4	yes	—
204	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
205	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
206	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
208	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
209	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
211	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
212	C₈×C₁₀₉	—	2	2	no	—
213	C₈×C₁₀₉	—	4	1	no	—
214	C₈×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	—	8	1	no	—
219	C₂²×C₁₀₉	—	8	1	no	—
220	C₂²×C₁₀₉	—	8	1	no	—
221	C₂²×C₁₀₉	—	8	1	no	—
222	C₂²×C₁₀₉	C₂×#121	1	1	yes	characteristic
223	C₂²×C₁₀₉	#546	1	1	yes	characteristic
224	C₂²×C₁₀₉	D₄×Q₈	1	1	yes	characteristic
225	C₂²×C₁₀₉	C₂²×SD₁₆	1	1	yes	characteristic
226	C₂²×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
227	C₂²×C₁₀₉	#577	1	1	yes	characteristic
228	C₂²×C₁₀₉	#578	1	1	yes	characteristic
229	C₄×C₁₀₉	—	2	4	no	—
230	C₄×C₁₀₉	—	2	4	no	—
231	C₄×C₁₀₉	—	2	2	no	—
232	C₄×C₁₀₉	—	4	2	no	—
233	C₄×C₁₀₉	—	4	2	no	—
234	C₄×C₁₀₉	—	8	2	no	—
235	C₄×C₁₀₉	—	8	1	no	—
236	C₄×C₁₀₉	—	8	2	no	—
237	C₄×C₁₀₉	#460	1	4	yes	—
238	C₄×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
239	C₄×C₁₀₉	#537	1	1	yes	characteristic
240	C₄×C₁₀₉	#552	1	1	yes	characteristic
241	C₄×C₁₀₉	#554	1	1	yes	characteristic
242	#24813	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
243	C₂×C₁₀₉	—	8	2	no	—
244	C₂×C₁₀₉	#2417	1	1	yes	characteristic
245	C₂×C₁₀₉	C₂×#460	1	1	yes	characteristic
246	C₂×C₁₀₉	#3046	1	1	yes	characteristic
247	C₂×C₁₀₉	#3298	1	1	yes	characteristic
248	C₂×C₁₀₉	#3332	1	1	yes	characteristic
249	C₂×C₁₀₉	Q₈×SD₁₆	1	1	yes	characteristic
250	C₂×C₁₀₉	#3386	1	1	yes	characteristic
251	C₂×#475	C₂×C₁₀₉	1	1	yes	characteristic, complemented
252	C₂×C₄×SD₁₆	C₂×C₁₀₉	1	1	yes	characteristic
253	#2542	C₂×C₁₀₉	1	1	yes	characteristic
254	Q₈×C₄⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
255	#2084	C₂×C₁₀₉	1	2	yes	complemented
256	#2065	C₂×C₁₀₉	1	2	yes	—
257	#1990	C₂×C₁₀₉	1	1	yes	characteristic, complemented
258	#1939	C₂×C₁₀₉	1	1	yes	characteristic, complemented
259	#1880	C₂×C₁₀₉	1	2	yes	—
260	#1803	C₂×C₁₀₉	1	2	yes	complemented
261	#1802	C₂×C₁₀₉	1	1	yes	characteristic
262	C₁₀₉	#24813	1	1	yes	characteristic, direct product factor, Sylow
263	C₂²×SD₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
264	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
265	C₂×C₂²⋊Q₈	—	2	1	no	—
266	C₂×C₂²⋊Q₈	—	2	1	no	—
267	C₂×C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
268	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
269	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
270	#475	—	2	4	no	—
271	C₄×SD₁₆	—	2	4	no	—
272	C₂×#106	C₂²×C₁₀₉	1	2	yes	—
273	C₂×#106	C₂²×C₁₀₉	1	1	yes	characteristic
274	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
275	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
276	C₂×#103	C₂²×C₁₀₉	1	2	yes	—
277	C₂×#103	C₂²×C₁₀₉	1	1	yes	characteristic
278	C₂×#102	C₂²×C₁₀₉	1	1	yes	characteristic
279	C₂×#102	C₂²×C₁₀₉	1	2	yes	—
280	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
281	#399	—	2	2	no	—
282	#390	C₂²×C₁₀₉	1	2	yes	—
283	#389	—	2	2	no	—
284	#389	C₂²×C₁₀₉	1	2	yes	—
285	#384	—	2	1	no	—
286	#384	C₂²×C₁₀₉	1	1	yes	characteristic
287	#384	C₂²×C₁₀₉	1	2	yes	—
288	C₄×C₄⋊C₄	—	2	1	no	—
289	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
290	#340	C₂²×C₁₀₉	1	2	yes	—
291	#340	C₂²×C₁₀₉	1	2	yes	—
292	#340	C₂²×C₁₀₉	1	2	yes	—
293	#336	C₂²×C₁₀₉	1	2	yes	—
294	C₂²×Q₈	—	2	1	no	—
295	C₂²×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
296	C₂²×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
297	C₂³×C₄	—	2	1	no	—
298	C₂×SD₁₆	—	2	2	no	—
299	C₂×SD₁₆	—	2	2	no	—
300	C₂×SD₁₆	Q₈×C₁₀₉	1	4	yes	complemented
301	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
302	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
303	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
304	C₄⋊Q₈	—	2	2	no	—
305	C₄⋊Q₈	—	2	2	no	—
306	C₂²⋊Q₈	—	4	4	no	—
307	C₂²⋊Q₈	—	4	4	no	—
308	C₄×Q₈	—	2	2	no	—
309	C₄×Q₈	—	2	4	no	—
310	C₄×Q₈	—	4	2	no	—
311	C₄×Q₈	D₄×C₁₀₉	1	4	yes	—
312	C₄×D₄	—	2	2	no	—
313	C₄×D₄	—	2	2	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	—	2	2	no	—
316	C₂×C₄⋊C₄	—	2	2	no	—
317	C₂×C₄⋊C₄	—	2	2	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	2	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
323	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
325	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
326	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
327	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
328	C₂×C₂²⋊C₄	—	2	1	no	—
329	C₂×C₂²⋊C₄	—	2	2	no	—
330	C₂×C₂²⋊C₄	—	2	2	no	—
331	C₂×C₄²	—	2	1	no	—
332	C₂×C₄²	—	2	2	no	—
333	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
334	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
335	#106	—	2	2	no	—
336	#106	—	2	4	no	—
337	C₄⋊C₈	—	2	2	no	—
338	C₄⋊C₈	—	4	2	no	—
339	#103	—	2	2	no	—
340	#103	—	2	4	no	—
341	#102	—	2	2	no	—
342	#102	—	4	2	no	—
343	C₄×C₈	—	2	2	no	—
344	#95	—	2	2	no	—
345	#95	D₄×C₁₀₉	1	2	yes	—
346	C₂⁴	—	2	1	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	1	no	—
350	C₂×Q₈	—	4	2	no	—
351	C₂×Q₈	—	4	2	no	—
352	C₂×Q₈	—	4	2	no	—
353	C₂×Q₈	C₂×D₄×C₁₀₉	1	2	yes	—
354	C₂×Q₈	C₂×Q₈×C₁₀₉	1	2	yes	—
355	C₂×Q₈	#41×C₁₀₉	1	2	yes	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	2	1	no	—
358	C₂×D₄	—	2	1	no	—
359	C₂×D₄	—	2	1	no	—
360	C₂×D₄	—	2	1	no	—
361	C₂×D₄	C₂×Q₈×C₁₀₉	1	2	yes	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	—	4	2	no	—
369	C₂²×C₄	—	4	2	no	—
370	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
371	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
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	4	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	1	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	4	4	no	—
395	C₄⋊C₄	—	4	4	no	—
396	C₄⋊C₄	—	4	2	no	—
397	C₄⋊C₄	—	4	4	no	—
398	C₄⋊C₄	—	4	2	no	—
399	C₄⋊C₄	—	8	1	no	—
400	C₄⋊C₄	—	8	2	no	—
401	C₄⋊C₄	SD₁₆×C₁₀₉	1	4	yes	complemented
402	C₄⋊C₄	C₂×D₄×C₁₀₉	1	2	yes	—
403	C₂²⋊C₄	—	4	4	no	—
404	C₂²⋊C₄	—	4	2	no	—
405	C₂²⋊C₄	—	4	4	no	—
406	C₄²	—	2	4	no	—
407	C₄²	—	2	2	no	—
408	C₄²	—	2	2	no	—
409	C₄²	—	4	4	no	—
410	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
411	C₄²	#41×C₁₀₉	1	2	yes	—
412	C₂³	—	2	2	no	—
413	C₂³	—	4	1	no	—
414	C₂³	—	4	1	no	—
415	C₂³	—	4	1	no	—
416	C₂³	—	4	1	no	—
417	C₂³	—	4	1	no	—
418	C₂³	—	4	1	no	—
419	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
420	Q₈	—	2	2	no	—
421	Q₈	—	2	2	no	—
422	Q₈	—	2	2	no	—
423	Q₈	—	8	4	no	—
424	Q₈	C₂²⋊Q₈×C₁₀₉	1	4	yes	complemented
425	D₄	—	2	2	no	—
426	D₄	—	2	2	no	—
427	D₄	—	4	1	no	—
428	D₄	—	4	1	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	1	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	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	2	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	8	4	no	—
452	C₂×C₄	#121×C₁₀₉	1	2	yes	—
453	C₂×C₄	#121×C₁₀₉	1	2	yes	—
454	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	4	yes	—
455	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
456	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
457	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
458	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
459	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
460	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
461	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
462	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
463	C₈	—	2	2	no	—
464	C₈	—	4	1	no	—
465	C₈	—	4	2	no	—
466	C₂²	—	4	2	no	—
467	C₂²	—	4	2	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	8	1	no	—
470	C₂²	—	8	1	no	—
471	C₂²	—	8	1	no	—
472	C₂²	—	8	1	no	—
473	C₂²	C₂×#121×C₁₀₉	1	1	yes	characteristic
474	C₂²	#546×C₁₀₉	1	1	yes	characteristic
475	C₂²	D₄×Q₈×C₁₀₉	1	1	yes	characteristic
476	C₂²	C₂²×SD₁₆×C₁₀₉	1	1	yes	characteristic
477	C₂²	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
478	C₂²	#577×C₁₀₉	1	1	yes	characteristic
479	C₂²	#578×C₁₀₉	1	1	yes	characteristic
480	C₄	—	2	4	no	—
481	C₄	—	2	2	no	—
482	C₄	—	2	4	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	2	no	—
485	C₄	—	8	1	no	—
486	C₄	—	8	2	no	—
487	C₄	—	8	2	no	—
488	C₄	#460×C₁₀₉	1	4	yes	—
489	C₄	C₂×C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
490	C₄	#537×C₁₀₉	1	1	yes	characteristic
491	C₄	#552×C₁₀₉	1	1	yes	characteristic
492	C₄	#554×C₁₀₉	1	1	yes	characteristic
493	C₂	—	8	2	no	—
494	C₂	C₁₀₉×#2417	1	1	yes	characteristic
495	C₂	C₂×#460×C₁₀₉	1	1	yes	characteristic
496	C₂	C₁₀₉×#3046	1	1	yes	characteristic
497	C₂	C₁₀₉×#3298	1	1	yes	characteristic
498	C₂	C₁₀₉×#3332	1	1	yes	characteristic
499	C₂	Q₈×SD₁₆×C₁₀₉	1	1	yes	characteristic
500	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
1. C₂×#475×C₁₀₉	C₂	semidirect
4. Q₈×C₄⋊C₄×C₁₀₉	C₂	semidirect
5. C₁₀₉×#2084	C₂	semidirect
7. C₁₀₉×#1990	C₂	semidirect
8. C₁₀₉×#1939	C₂	semidirect
10. C₁₀₉×#1803	C₂	semidirect
49. C₂×SD₁₆×C₁₀₉	Q₈	semidirect
150. C₄⋊C₄×C₁₀₉	SD₁₆	semidirect
173. Q₈×C₁₀₉	C₂²⋊Q₈	semidirect
242. #24813	C₁₀₉	direct (N × H)
251. C₂×#475	C₂×C₁₀₉	semidirect
254. Q₈×C₄⋊C₄	C₂×C₁₀₉	semidirect
255. #2084	C₂×C₁₀₉	semidirect
257. #1990	C₂×C₁₀₉	semidirect
258. #1939	C₂×C₁₀₉	semidirect
260. #1803	C₂×C₁₀₉	semidirect
262. C₁₀₉	#24813	direct (N × H)
300. C₂×SD₁₆	Q₈×C₁₀₉	semidirect
401. C₄⋊C₄	SD₁₆×C₁₀₉	semidirect
424. Q₈	C₂²⋊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×#475×C₁₀₉	2
C₂×C₄×SD₁₆×C₁₀₉	2
C₁₀₉×#2542	2
Q₈×C₄⋊C₄×C₁₀₉	2
C₁₀₉×#2084	2
C₁₀₉×#2065	2
C₁₀₉×#1990	2
C₁₀₉×#1939	2
C₁₀₉×#1880	2
C₁₀₉×#1803	2
C₁₀₉×#1802	2
#24813	109

Subgroups — C109×#24813 #42142807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#24813 #42142807