Subgroups — C₁₀₉×#17839 #42135833

Order 27904.

592 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2991	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2990	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2965	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1574	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1574	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₀₉×#1572	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1571	C₂	1	2	yes	maximal, complemented
8	C₁₀₉×#1569	C₂	1	2	yes	maximal, complemented
9	C₁₀₉×#1567	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₀₉×#1565	C₂	1	2	yes	maximal, complemented
11	C₁₀₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#41×C₁₀₉	C₂²	1	1	yes	characteristic
13	#446×C₁₀₉	—	2	1	no	—
14	#446×C₁₀₉	C₄	1	1	yes	characteristic
15	#446×C₁₀₉	C₄	1	1	yes	characteristic
16	C₈×Q₈×C₁₀₉	—	2	1	no	—
17	#436×C₁₀₉	—	2	1	no	—
18	#436×C₁₀₉	—	2	2	no	—
19	#436×C₁₀₉	C₄	1	1	yes	characteristic
20	#436×C₁₀₉	C₄	1	1	yes	characteristic
21	#435×C₁₀₉	—	2	2	no	—
22	#435×C₁₀₉	C₄	1	2	yes	—
23	#435×C₁₀₉	C₄	1	2	yes	—
24	C₈×D₄×C₁₀₉	—	2	1	no	—
25	#432×C₁₀₉	C₂²	1	2	yes	—
26	#431×C₁₀₉	C₂²	1	1	yes	characteristic
27	#431×C₁₀₉	C₂²	1	1	yes	characteristic
28	#431×C₁₀₉	C₂²	1	2	yes	—
29	#431×C₁₀₉	C₂²	1	2	yes	—
30	#431×C₁₀₉	C₂²	1	1	yes	characteristic
31	#423×C₁₀₉	C₂²	1	1	yes	characteristic
32	#423×C₁₀₉	C₂²	1	1	yes	characteristic
33	#423×C₁₀₉	C₂²	1	2	yes	—
34	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
35	C₄×M₁₆×C₁₀₉	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
37	#335×C₁₀₉	C₂²	1	1	yes	characteristic
38	#335×C₁₀₉	C₂²	1	1	yes	characteristic
39	#335×C₁₀₉	C₂²	1	2	yes	—
40	#335×C₁₀₉	C₂²	1	2	yes	—
41	#335×C₁₀₉	C₂²	1	2	yes	—
42	#335×C₁₀₉	C₂²	1	2	yes	—
43	#334×C₁₀₉	C₂²	1	1	yes	characteristic
44	#334×C₁₀₉	C₂²	1	1	yes	characteristic
45	#334×C₁₀₉	C₂²	1	2	yes	—
46	#334×C₁₀₉	C₂²	1	2	yes	—
47	C₂×#41×C₁₀₉	Q₈	1	1	yes	characteristic
48	C₂×M₁₆×C₁₀₉	—	2	1	no	—
49	C₂×M₁₆×C₁₀₉	—	2	2	no	—
50	C₂×M₁₆×C₁₀₉	—	2	1	no	—
51	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
52	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₁₀₉	—	2	2	no	—
54	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
56	C₄×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
57	C₄×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
58	C₄×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
59	C₄×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
60	C₄×D₄×C₁₀₉	D₄	1	1	yes	characteristic
61	C₄×D₄×C₁₀₉	D₄	1	2	yes	—
62	#117×C₁₀₉	Q₈	1	1	yes	characteristic
63	#117×C₁₀₉	Q₈	1	2	yes	—
64	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
66	C₄⋊C₈×C₁₀₉	—	2	2	no	—
67	C₄⋊C₈×C₁₀₉	—	2	1	no	—
68	C₄⋊C₈×C₁₀₉	—	2	1	no	—
69	C₄⋊C₈×C₁₀₉	—	2	2	no	—
70	C₄⋊C₈×C₁₀₉	C₂×C₄	1	2	yes	—
71	C₄⋊C₈×C₁₀₉	C₂×C₄	1	2	yes	—
72	C₄⋊C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
73	C₄⋊C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
74	C₄⋊C₈×C₁₀₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₁₀₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
78	C₄⋊C₈×C₁₀₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
80	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
81	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
82	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
83	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
84	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	2	yes	—
85	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	2	yes	—
86	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	2	yes	—
87	#97×C₁₀₉	—	2	2	no	—
88	#97×C₁₀₉	C₂×C₄	1	2	yes	—
89	#97×C₁₀₉	C₂×C₄	1	2	yes	—
90	C₄×C₈×C₁₀₉	—	2	1	no	—
91	C₄×C₈×C₁₀₉	—	2	1	no	—
92	C₄×C₈×C₁₀₉	—	2	2	no	—
93	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
94	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
95	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
96	C₄×C₈×C₁₀₉	C₂³	1	2	yes	—
97	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	#41×C₁₀₉	—	4	2	no	—
99	C₂×Q₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
100	C₂×D₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
101	C₂×D₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
102	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
107	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
108	M₁₆×C₁₀₉	—	2	4	no	—
109	M₁₆×C₁₀₉	—	4	2	no	—
110	M₁₆×C₁₀₉	—	8	1	no	—
111	M₁₆×C₁₀₉	—	8	2	no	—
112	M₁₆×C₁₀₉	—	8	1	no	—
113	C₂×C₈×C₁₀₉	—	2	2	no	—
114	C₂×C₈×C₁₀₉	—	2	1	no	—
115	C₂×C₈×C₁₀₉	—	2	1	no	—
116	C₂×C₈×C₁₀₉	—	2	1	no	—
117	C₂×C₈×C₁₀₉	—	2	1	no	—
118	C₂×C₈×C₁₀₉	—	2	2	no	—
119	C₂×C₈×C₁₀₉	—	2	2	no	—
120	C₂×C₈×C₁₀₉	—	4	1	no	—
121	C₂×C₈×C₁₀₉	—	8	2	no	—
122	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
127	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
128	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
129	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
130	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
131	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
132	C₂²⋊C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
133	C₂²⋊C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
134	C₂²⋊C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
135	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
136	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
137	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
139	C₄²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
140	C₄²×C₁₀₉	C₂×Q₈	1	2	yes	—
141	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
142	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
144	Q₈×C₁₀₉	—	2	2	no	—
145	D₄×C₁₀₉	—	2	2	no	—
146	D₄×C₁₀₉	—	4	2	no	—
147	C₂×C₄×C₁₀₉	—	2	1	no	—
148	C₂×C₄×C₁₀₉	—	2	2	no	—
149	C₂×C₄×C₁₀₉	—	2	1	no	—
150	C₂×C₄×C₁₀₉	—	2	2	no	—
151	C₂×C₄×C₁₀₉	—	2	2	no	—
152	C₂×C₄×C₁₀₉	—	2	1	no	—
153	C₂×C₄×C₁₀₉	—	2	2	no	—
154	C₂×C₄×C₁₀₉	—	2	1	no	—
155	C₂×C₄×C₁₀₉	—	2	2	no	—
156	C₂×C₄×C₁₀₉	—	4	2	no	—
157	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
158	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
160	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
162	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
164	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
166	C₈×C₁₀₉	—	2	1	no	—
167	C₈×C₁₀₉	—	2	2	no	—
168	C₈×C₁₀₉	—	2	1	no	—
169	C₈×C₁₀₉	—	2	1	no	—
170	C₈×C₁₀₉	—	2	2	no	—
171	C₈×C₁₀₉	—	4	1	no	—
172	C₈×C₁₀₉	—	4	2	no	—
173	C₈×C₁₀₉	—	4	1	no	—
174	C₈×C₁₀₉	—	4	2	no	—
175	C₈×C₁₀₉	—	4	1	no	—
176	C₈×C₁₀₉	—	4	1	no	—
177	C₈×C₁₀₉	#131	1	2	yes	—
178	C₂²×C₁₀₉	—	2	1	no	—
179	C₂²×C₁₀₉	—	2	2	no	—
180	C₂²×C₁₀₉	—	2	2	no	—
181	C₂²×C₁₀₉	—	2	1	no	—
182	C₂²×C₁₀₉	—	2	1	no	—
183	C₂²×C₁₀₉	—	2	2	no	—
184	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
185	C₄×C₁₀₉	—	2	1	no	—
186	C₄×C₁₀₉	—	2	2	no	—
187	C₄×C₁₀₉	—	2	2	no	—
188	C₄×C₁₀₉	—	2	1	no	—
189	C₄×C₁₀₉	—	2	1	no	—
190	C₄×C₁₀₉	—	2	2	no	—
191	C₄×C₁₀₉	#430	1	2	yes	—
192	C₄×C₁₀₉	C₂×#131	1	1	yes	characteristic
193	C₄×C₁₀₉	#568	1	1	yes	characteristic
194	C₄×C₁₀₉	#576	1	1	yes	characteristic
195	C₄×C₁₀₉	#578	1	1	yes	characteristic
196	#17839	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂×C₁₀₉	—	4	1	no	—
198	C₂×C₁₀₉	—	4	2	no	—
199	C₂×C₁₀₉	#2906	1	1	yes	characteristic
200	C₂×C₁₀₉	#2913	1	1	yes	characteristic
201	C₂×C₁₀₉	C₂×#430	1	1	yes	characteristic
202	#2991	C₂×C₁₀₉	1	1	yes	characteristic
203	#2990	C₂×C₁₀₉	1	1	yes	characteristic
204	#2965	C₂×C₁₀₉	1	1	yes	characteristic
205	#1574	C₂×C₁₀₉	1	1	yes	characteristic, complemented
206	#1574	C₂×C₁₀₉	1	1	yes	characteristic, complemented
207	#1572	C₂×C₁₀₉	1	2	yes	complemented
208	#1571	C₂×C₁₀₉	1	2	yes	complemented
209	#1569	C₂×C₁₀₉	1	2	yes	complemented
210	#1567	C₂×C₁₀₉	1	1	yes	characteristic, complemented
211	#1565	C₂×C₁₀₉	1	2	yes	complemented
212	#1564	C₂×C₁₀₉	1	1	yes	characteristic, complemented
213	C₁₀₉	#17839	1	1	yes	characteristic, direct product factor, Sylow
214	C₄×#41	C₂²×C₁₀₉	1	1	yes	characteristic
215	#446	—	2	1	no	—
216	#446	C₄×C₁₀₉	1	1	yes	characteristic
217	#446	C₄×C₁₀₉	1	1	yes	characteristic
218	C₈×Q₈	—	2	1	no	—
219	#436	—	2	1	no	—
220	#436	—	2	2	no	—
221	#436	C₄×C₁₀₉	1	1	yes	characteristic
222	#436	C₄×C₁₀₉	1	1	yes	characteristic
223	#435	—	2	2	no	—
224	#435	C₄×C₁₀₉	1	2	yes	—
225	#435	C₄×C₁₀₉	1	2	yes	—
226	C₈×D₄	—	2	1	no	—
227	#432	C₂²×C₁₀₉	1	2	yes	—
228	#431	C₂²×C₁₀₉	1	1	yes	characteristic
229	#431	C₂²×C₁₀₉	1	1	yes	characteristic
230	#431	C₂²×C₁₀₉	1	2	yes	—
231	#431	C₂²×C₁₀₉	1	2	yes	—
232	#431	C₂²×C₁₀₉	1	1	yes	characteristic
233	#423	C₂²×C₁₀₉	1	1	yes	characteristic
234	#423	C₂²×C₁₀₉	1	1	yes	characteristic
235	#423	C₂²×C₁₀₉	1	2	yes	—
236	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
237	C₄×M₁₆	C₂²×C₁₀₉	1	2	yes	—
238	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
239	#335	C₂²×C₁₀₉	1	1	yes	characteristic
240	#335	C₂²×C₁₀₉	1	1	yes	characteristic
241	#335	C₂²×C₁₀₉	1	2	yes	—
242	#335	C₂²×C₁₀₉	1	2	yes	—
243	#335	C₂²×C₁₀₉	1	2	yes	—
244	#335	C₂²×C₁₀₉	1	2	yes	—
245	#334	C₂²×C₁₀₉	1	1	yes	characteristic
246	#334	C₂²×C₁₀₉	1	1	yes	characteristic
247	#334	C₂²×C₁₀₉	1	2	yes	—
248	#334	C₂²×C₁₀₉	1	2	yes	—
249	C₂×#41	Q₈×C₁₀₉	1	1	yes	characteristic
250	C₂×M₁₆	—	2	1	no	—
251	C₂×M₁₆	—	2	2	no	—
252	C₂×M₁₆	—	2	1	no	—
253	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
254	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
255	C₂²×C₈	—	2	2	no	—
256	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
257	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
258	C₄×Q₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
259	C₄×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
260	C₄×D₄	C₂×C₄×C₁₀₉	1	2	yes	—
261	C₄×D₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
262	C₄×D₄	D₄×C₁₀₉	1	1	yes	characteristic
263	C₄×D₄	D₄×C₁₀₉	1	2	yes	—
264	#117	Q₈×C₁₀₉	1	1	yes	characteristic
265	#117	Q₈×C₁₀₉	1	2	yes	—
266	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
267	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
268	C₄⋊C₈	—	2	2	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	2	1	no	—
271	C₄⋊C₈	—	2	2	no	—
272	C₄⋊C₈	C₂×C₄×C₁₀₉	1	2	yes	—
273	C₄⋊C₈	C₂×C₄×C₁₀₉	1	2	yes	—
274	C₄⋊C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
275	C₄⋊C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
276	C₄⋊C₈	C₂³×C₁₀₉	1	1	yes	characteristic
277	C₄⋊C₈	C₂³×C₁₀₉	1	1	yes	characteristic
278	C₄⋊C₈	C₂³×C₁₀₉	1	1	yes	characteristic
279	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
280	C₄⋊C₈	C₂³×C₁₀₉	1	1	yes	characteristic
281	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
282	C₂²⋊C₈	—	2	1	no	—
283	C₂²⋊C₈	—	2	2	no	—
284	C₂²⋊C₈	—	2	1	no	—
285	C₂²⋊C₈	—	2	2	no	—
286	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	2	yes	—
287	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	2	yes	—
288	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	2	yes	—
289	#97	—	2	2	no	—
290	#97	C₂×C₄×C₁₀₉	1	2	yes	—
291	#97	C₂×C₄×C₁₀₉	1	2	yes	—
292	C₄×C₈	—	2	1	no	—
293	C₄×C₈	—	2	1	no	—
294	C₄×C₈	—	2	2	no	—
295	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
296	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
297	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
298	C₄×C₈	C₂³×C₁₀₉	1	2	yes	—
299	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
300	#41	—	4	2	no	—
301	C₂×Q₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
302	C₂×D₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
303	C₂×D₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
304	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
305	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
306	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
307	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
308	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
309	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
310	M₁₆	—	2	4	no	—
311	M₁₆	—	4	2	no	—
312	M₁₆	—	8	1	no	—
313	M₁₆	—	8	2	no	—
314	M₁₆	—	8	1	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	4	1	no	—
323	C₂×C₈	—	8	2	no	—
324	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
325	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
326	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
327	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
328	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
329	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
330	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
331	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
332	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
333	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
334	C₂²⋊C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
335	C₂²⋊C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
336	C₂²⋊C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
337	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
338	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
339	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
340	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
341	C₄²	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
342	C₄²	C₂×Q₈×C₁₀₉	1	2	yes	—
343	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
344	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
345	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
346	Q₈	—	2	2	no	—
347	D₄	—	2	2	no	—
348	D₄	—	4	2	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
360	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
361	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
362	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
363	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
364	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
365	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
366	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
367	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
368	C₈	—	2	1	no	—
369	C₈	—	2	2	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	2	no	—
373	C₈	—	4	1	no	—
374	C₈	—	4	2	no	—
375	C₈	—	4	1	no	—
376	C₈	—	4	2	no	—
377	C₈	—	4	1	no	—
378	C₈	—	4	1	no	—
379	C₈	#131×C₁₀₉	1	2	yes	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	2	no	—
382	C₂²	—	2	2	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	2	no	—
386	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
387	C₄	—	2	1	no	—
388	C₄	—	2	2	no	—
389	C₄	—	2	2	no	—
390	C₄	—	2	1	no	—
391	C₄	—	2	1	no	—
392	C₄	—	2	2	no	—
393	C₄	#430×C₁₀₉	1	2	yes	—
394	C₄	C₂×#131×C₁₀₉	1	1	yes	characteristic
395	C₄	#568×C₁₀₉	1	1	yes	characteristic
396	C₄	#576×C₁₀₉	1	1	yes	characteristic
397	C₄	#578×C₁₀₉	1	1	yes	characteristic
398	C₂	—	4	1	no	—
399	C₂	—	4	2	no	—
400	C₂	C₁₀₉×#2906	1	1	yes	characteristic
401	C₂	C₁₀₉×#2913	1	1	yes	characteristic
402	C₂	C₂×#430×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₄²	C₂⁴×C₁₀₉
socle	C₂²×C₁₀₉	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₁₀₉	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₄×#41×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₀₉×#1574	C₂	semidirect
5. C₁₀₉×#1574	C₂	semidirect
6. C₁₀₉×#1572	C₂	semidirect
7. C₁₀₉×#1571	C₂	semidirect
8. C₁₀₉×#1569	C₂	semidirect
9. C₁₀₉×#1567	C₂	semidirect
10. C₁₀₉×#1565	C₂	semidirect
11. C₁₀₉×#1564	C₂	semidirect
196. #17839	C₁₀₉	direct (N × H)
205. #1574	C₂×C₁₀₉	semidirect
206. #1574	C₂×C₁₀₉	semidirect
207. #1572	C₂×C₁₀₉	semidirect
208. #1571	C₂×C₁₀₉	semidirect
209. #1569	C₂×C₁₀₉	semidirect
210. #1567	C₂×C₁₀₉	semidirect
211. #1565	C₂×C₁₀₉	semidirect
212. #1564	C₂×C₁₀₉	semidirect
213. C₁₀₉	#17839	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×#2991	2
C₁₀₉×#2990	2
C₁₀₉×#2965	2
C₁₀₉×#1574	2
C₁₀₉×#1572	2
C₁₀₉×#1571	2
C₁₀₉×#1569	2
C₁₀₉×#1567	2
C₁₀₉×#1565	2
C₁₀₉×#1564	2
#17839	109

Subgroups — C109×#17839 #42135833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#17839 #42135833