Subgroups — C₁₀₉×#17840 #42135834

Order 27904.

594 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2990	C₂	1	2	yes	maximal
2	C₁₀₉×#2965	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1573	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#1572	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#1571	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#1569	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#1565	C₂	1	2	yes	maximal, complemented
9	C₂×#335×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#41×C₁₀₉	C₂²	1	1	yes	characteristic
11	#446×C₁₀₉	—	2	2	no	—
12	#446×C₁₀₉	C₄	1	1	yes	characteristic
13	#446×C₁₀₉	C₄	1	1	yes	characteristic
14	#436×C₁₀₉	C₄	1	1	yes	characteristic
15	#436×C₁₀₉	C₄	1	1	yes	characteristic
16	#435×C₁₀₉	—	2	2	no	—
17	#435×C₁₀₉	—	2	2	no	—
18	#435×C₁₀₉	C₄	1	2	yes	—
19	#435×C₁₀₉	C₄	1	2	yes	—
20	C₈×D₄×C₁₀₉	—	2	2	no	—
21	#432×C₁₀₉	C₂²	1	2	yes	—
22	#432×C₁₀₉	C₂²	1	2	yes	—
23	#431×C₁₀₉	C₂²	1	2	yes	—
24	#431×C₁₀₉	C₂²	1	2	yes	—
25	#431×C₁₀₉	C₂²	1	1	yes	characteristic
26	#423×C₁₀₉	C₂²	1	2	yes	—
27	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
29	C₄×M₁₆×C₁₀₉	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
31	#335×C₁₀₉	C₂²	1	1	yes	characteristic
32	#335×C₁₀₉	C₂²	1	1	yes	characteristic
33	#335×C₁₀₉	C₂²	1	2	yes	—
34	#335×C₁₀₉	C₂²	1	2	yes	—
35	#335×C₁₀₉	C₂²	1	1	yes	characteristic
36	#335×C₁₀₉	C₂²	1	1	yes	characteristic
37	#335×C₁₀₉	C₂²	1	2	yes	—
38	#334×C₁₀₉	C₂²	1	2	yes	—
39	#334×C₁₀₉	C₂²	1	2	yes	—
40	#334×C₁₀₉	C₂²	1	2	yes	—
41	C₂×#41×C₁₀₉	Q₈	1	1	yes	characteristic
42	C₂×M₁₆×C₁₀₉	—	2	2	no	—
43	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
44	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₁₀₉	—	2	2	no	—
46	C₂²×C₈×C₁₀₉	—	2	2	no	—
47	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
49	C₄×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
50	C₄×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
51	C₄×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
52	C₄×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
53	C₄×D₄×C₁₀₉	D₄	1	2	yes	—
54	C₄×D₄×C₁₀₉	D₄	1	1	yes	characteristic
55	#117×C₁₀₉	Q₈	1	1	yes	characteristic
56	#117×C₁₀₉	Q₈	1	2	yes	—
57	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
59	C₄⋊C₈×C₁₀₉	—	2	2	no	—
60	C₄⋊C₈×C₁₀₉	—	2	2	no	—
61	C₄⋊C₈×C₁₀₉	—	2	2	no	—
62	C₄⋊C₈×C₁₀₉	C₂×C₄	1	2	yes	—
63	C₄⋊C₈×C₁₀₉	C₂×C₄	1	2	yes	—
64	C₄⋊C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
65	C₄⋊C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
66	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
67	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
68	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
69	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
70	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
71	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
72	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
73	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	2	yes	—
74	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	2	yes	—
75	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	2	yes	—
76	#97×C₁₀₉	—	2	2	no	—
77	#97×C₁₀₉	—	2	2	no	—
78	#97×C₁₀₉	C₂×C₄	1	2	yes	—
79	#97×C₁₀₉	C₂×C₄	1	2	yes	—
80	C₄×C₈×C₁₀₉	—	2	2	no	—
81	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₁₀₉	C₂³	1	2	yes	—
85	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
86	#41×C₁₀₉	—	4	2	no	—
87	C₂×Q₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
88	C₂×D₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
89	C₂×D₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
96	M₁₆×C₁₀₉	—	2	4	no	—
97	M₁₆×C₁₀₉	—	4	2	no	—
98	M₁₆×C₁₀₉	—	8	2	no	—
99	C₂×C₈×C₁₀₉	—	2	2	no	—
100	C₂×C₈×C₁₀₉	—	2	2	no	—
101	C₂×C₈×C₁₀₉	—	2	2	no	—
102	C₂×C₈×C₁₀₉	—	2	2	no	—
103	C₂×C₈×C₁₀₉	—	2	2	no	—
104	C₂×C₈×C₁₀₉	—	4	1	no	—
105	C₂×C₈×C₁₀₉	—	8	2	no	—
106	C₂×C₈×C₁₀₉	—	8	2	no	—
107	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
113	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
114	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
115	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
116	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
117	C₂²⋊C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
118	C₂²⋊C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
119	C₂²⋊C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
120	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
122	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
123	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
125	C₄²×C₁₀₉	C₂×Q₈	1	2	yes	—
126	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
127	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
128	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
129	Q₈×C₁₀₉	—	4	1	no	—
130	D₄×C₁₀₉	—	2	4	no	—
131	D₄×C₁₀₉	—	4	1	no	—
132	C₂×C₄×C₁₀₉	—	2	2	no	—
133	C₂×C₄×C₁₀₉	—	2	1	no	—
134	C₂×C₄×C₁₀₉	—	2	2	no	—
135	C₂×C₄×C₁₀₉	—	2	1	no	—
136	C₂×C₄×C₁₀₉	—	2	2	no	—
137	C₂×C₄×C₁₀₉	—	2	1	no	—
138	C₂×C₄×C₁₀₉	—	2	2	no	—
139	C₂×C₄×C₁₀₉	—	2	1	no	—
140	C₂×C₄×C₁₀₉	—	2	2	no	—
141	C₂×C₄×C₁₀₉	—	4	2	no	—
142	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
146	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
147	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
151	C₈×C₁₀₉	—	2	1	no	—
152	C₈×C₁₀₉	—	2	2	no	—
153	C₈×C₁₀₉	—	2	1	no	—
154	C₈×C₁₀₉	—	2	1	no	—
155	C₈×C₁₀₉	—	2	2	no	—
156	C₈×C₁₀₉	—	4	2	no	—
157	C₈×C₁₀₉	—	4	2	no	—
158	C₈×C₁₀₉	—	4	2	no	—
159	C₈×C₁₀₉	—	4	2	no	—
160	C₈×C₁₀₉	#131	1	2	yes	—
161	C₂²×C₁₀₉	—	2	1	no	—
162	C₂²×C₁₀₉	—	2	1	no	—
163	C₂²×C₁₀₉	—	2	2	no	—
164	C₂²×C₁₀₉	—	2	2	no	—
165	C₂²×C₁₀₉	—	2	2	no	—
166	C₂²×C₁₀₉	#430	1	2	yes	—
167	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
168	C₄×C₁₀₉	—	2	1	no	—
169	C₄×C₁₀₉	—	2	2	no	—
170	C₄×C₁₀₉	—	2	1	no	—
171	C₄×C₁₀₉	—	2	2	no	—
172	C₄×C₁₀₉	—	2	1	no	—
173	C₄×C₁₀₉	—	2	1	no	—
174	C₄×C₁₀₉	—	2	2	no	—
175	C₄×C₁₀₉	C₂×#131	1	1	yes	characteristic
176	C₄×C₁₀₉	#568	1	1	yes	characteristic
177	C₄×C₁₀₉	#576	1	1	yes	characteristic
178	C₄×C₁₀₉	#578	1	1	yes	characteristic
179	#17840	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂×C₁₀₉	—	2	2	no	—
181	C₂×C₁₀₉	—	4	2	no	—
182	C₂×C₁₀₉	#2906	1	1	yes	characteristic
183	C₂×C₁₀₉	#2913	1	1	yes	characteristic
184	C₂×C₁₀₉	C₂×#430	1	1	yes	characteristic
185	#2990	C₂×C₁₀₉	1	2	yes	—
186	#2965	C₂×C₁₀₉	1	1	yes	characteristic
187	#1573	C₂×C₁₀₉	1	2	yes	complemented
188	#1572	C₂×C₁₀₉	1	2	yes	complemented
189	#1571	C₂×C₁₀₉	1	2	yes	complemented
190	#1569	C₂×C₁₀₉	1	2	yes	complemented
191	#1568	C₂×C₁₀₉	1	1	yes	characteristic, complemented
192	#1565	C₂×C₁₀₉	1	2	yes	complemented
193	C₂×#335	C₂×C₁₀₉	1	1	yes	characteristic, complemented
194	C₁₀₉	#17840	1	1	yes	characteristic, direct product factor, Sylow
195	C₄×#41	C₂²×C₁₀₉	1	1	yes	characteristic
196	#446	—	2	2	no	—
197	#446	C₄×C₁₀₉	1	1	yes	characteristic
198	#446	C₄×C₁₀₉	1	1	yes	characteristic
199	#436	C₄×C₁₀₉	1	1	yes	characteristic
200	#436	C₄×C₁₀₉	1	1	yes	characteristic
201	#435	—	2	2	no	—
202	#435	—	2	2	no	—
203	#435	C₄×C₁₀₉	1	2	yes	—
204	#435	C₄×C₁₀₉	1	2	yes	—
205	C₈×D₄	—	2	2	no	—
206	#432	C₂²×C₁₀₉	1	2	yes	—
207	#432	C₂²×C₁₀₉	1	2	yes	—
208	#431	C₂²×C₁₀₉	1	2	yes	—
209	#431	C₂²×C₁₀₉	1	2	yes	—
210	#431	C₂²×C₁₀₉	1	1	yes	characteristic
211	#423	C₂²×C₁₀₉	1	2	yes	—
212	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
213	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
214	C₄×M₁₆	C₂²×C₁₀₉	1	2	yes	—
215	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
216	#335	C₂²×C₁₀₉	1	1	yes	characteristic
217	#335	C₂²×C₁₀₉	1	1	yes	characteristic
218	#335	C₂²×C₁₀₉	1	2	yes	—
219	#335	C₂²×C₁₀₉	1	2	yes	—
220	#335	C₂²×C₁₀₉	1	1	yes	characteristic
221	#335	C₂²×C₁₀₉	1	1	yes	characteristic
222	#335	C₂²×C₁₀₉	1	2	yes	—
223	#334	C₂²×C₁₀₉	1	2	yes	—
224	#334	C₂²×C₁₀₉	1	2	yes	—
225	#334	C₂²×C₁₀₉	1	2	yes	—
226	C₂×#41	Q₈×C₁₀₉	1	1	yes	characteristic
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
229	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
230	C₂²×C₈	—	2	2	no	—
231	C₂²×C₈	—	2	2	no	—
232	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
233	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
234	C₄×Q₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
235	C₄×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
236	C₄×D₄	C₂×C₄×C₁₀₉	1	2	yes	—
237	C₄×D₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
238	C₄×D₄	D₄×C₁₀₉	1	2	yes	—
239	C₄×D₄	D₄×C₁₀₉	1	1	yes	characteristic
240	#117	Q₈×C₁₀₉	1	1	yes	characteristic
241	#117	Q₈×C₁₀₉	1	2	yes	—
242	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
243	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	2	2	no	—
247	C₄⋊C₈	C₂×C₄×C₁₀₉	1	2	yes	—
248	C₄⋊C₈	C₂×C₄×C₁₀₉	1	2	yes	—
249	C₄⋊C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
250	C₄⋊C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
251	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
252	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
253	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
254	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
255	C₂²⋊C₈	—	2	2	no	—
256	C₂²⋊C₈	—	2	2	no	—
257	C₂²⋊C₈	—	2	2	no	—
258	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	2	yes	—
259	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	2	yes	—
260	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	2	yes	—
261	#97	—	2	2	no	—
262	#97	—	2	2	no	—
263	#97	C₂×C₄×C₁₀₉	1	2	yes	—
264	#97	C₂×C₄×C₁₀₉	1	2	yes	—
265	C₄×C₈	—	2	2	no	—
266	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
267	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
268	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
269	C₄×C₈	C₂³×C₁₀₉	1	2	yes	—
270	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
271	#41	—	4	2	no	—
272	C₂×Q₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
273	C₂×D₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
274	C₂×D₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
275	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
276	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
277	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
279	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
280	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
281	M₁₆	—	2	4	no	—
282	M₁₆	—	4	2	no	—
283	M₁₆	—	8	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	4	1	no	—
290	C₂×C₈	—	8	2	no	—
291	C₂×C₈	—	8	2	no	—
292	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
293	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
294	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
295	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
296	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
297	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
298	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
299	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
300	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
301	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
302	C₂²⋊C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
303	C₂²⋊C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
304	C₂²⋊C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
305	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
306	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
307	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
308	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
309	C₄²	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
310	C₄²	C₂×Q₈×C₁₀₉	1	2	yes	—
311	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
312	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
313	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
314	Q₈	—	4	1	no	—
315	D₄	—	2	4	no	—
316	D₄	—	4	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
328	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
329	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
330	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
331	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
332	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
333	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
334	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
335	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	2	no	—
338	C₈	—	2	1	no	—
339	C₈	—	2	1	no	—
340	C₈	—	2	2	no	—
341	C₈	—	4	2	no	—
342	C₈	—	4	2	no	—
343	C₈	—	4	2	no	—
344	C₈	—	4	2	no	—
345	C₈	#131×C₁₀₉	1	2	yes	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	2	2	no	—
351	C₂²	#430×C₁₀₉	1	2	yes	—
352	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	2	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	2	no	—
360	C₄	C₂×#131×C₁₀₉	1	1	yes	characteristic
361	C₄	#568×C₁₀₉	1	1	yes	characteristic
362	C₄	#576×C₁₀₉	1	1	yes	characteristic
363	C₄	#578×C₁₀₉	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	4	2	no	—
366	C₂	C₁₀₉×#2906	1	1	yes	characteristic
367	C₂	C₁₀₉×#2913	1	1	yes	characteristic
368	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
3. C₁₀₉×#1573	C₂	semidirect
4. C₁₀₉×#1572	C₂	semidirect
5. C₁₀₉×#1571	C₂	semidirect
6. C₁₀₉×#1569	C₂	semidirect
7. C₁₀₉×#1568	C₂	semidirect
8. C₁₀₉×#1565	C₂	semidirect
9. C₂×#335×C₁₀₉	C₂	semidirect
179. #17840	C₁₀₉	direct (N × H)
187. #1573	C₂×C₁₀₉	semidirect
188. #1572	C₂×C₁₀₉	semidirect
189. #1571	C₂×C₁₀₉	semidirect
190. #1569	C₂×C₁₀₉	semidirect
191. #1568	C₂×C₁₀₉	semidirect
192. #1565	C₂×C₁₀₉	semidirect
193. C₂×#335	C₂×C₁₀₉	semidirect
194. C₁₀₉	#17840	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×#2990	2
C₁₀₉×#2965	2
C₁₀₉×#1573	2
C₁₀₉×#1572	2
C₁₀₉×#1571	2
C₁₀₉×#1569	2
C₁₀₉×#1568	2
C₁₀₉×#1565	2
C₂×#335×C₁₀₉	2
#17840	109

Subgroups — C109×#17840 #42135834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#17840 #42135834