Subgroups — C₁₀₉×#32784 #42150778

Order 27904.

644 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2629	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2621	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2096	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#2095	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#2085	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#2043	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#2036	C₂	1	2	yes	maximal
8	C₁₀₉×#2035	C₂	1	2	yes	maximal
9	C₁₀₉×#2026	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₀₉×#2026	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₀₉×#1818	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×#125×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×#117×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
18	C₂×#106×C₁₀₉	C₂²	1	2	yes	—
19	C₂×#103×C₁₀₉	C₂²	1	2	yes	—
20	C₂×#103×C₁₀₉	C₂²	1	2	yes	—
21	C₂×#103×C₁₀₉	C₂²	1	2	yes	—
22	C₂×#103×C₁₀₉	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
24	#398×C₁₀₉	—	2	2	no	—
25	#398×C₁₀₉	—	2	2	no	—
26	#397×C₁₀₉	—	2	2	no	—
27	#396×C₁₀₉	—	2	2	no	—
28	#391×C₁₀₉	C₂²	1	1	yes	characteristic
29	#391×C₁₀₉	C₂²	1	1	yes	characteristic
30	#389×C₁₀₉	C₂²	1	2	yes	—
31	#384×C₁₀₉	C₂²	1	1	yes	characteristic
32	#382×C₁₀₉	C₂²	1	1	yes	characteristic
33	#380×C₁₀₉	C₂²	1	2	yes	—
34	#340×C₁₀₉	C₂²	1	2	yes	—
35	#340×C₁₀₉	C₂²	1	2	yes	—
36	#336×C₁₀₉	C₂²	1	1	yes	characteristic
37	#336×C₁₀₉	C₂²	1	1	yes	characteristic
38	#336×C₁₀₉	C₂²	1	2	yes	—
39	C₂²×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₂²×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
43	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₁₀₉	—	4	2	no	—
45	#125×C₁₀₉	—	4	2	no	—
46	C₂²⋊Q₈×C₁₀₉	—	4	2	no	—
47	C₂²⋊Q₈×C₁₀₉	—	4	2	no	—
48	#117×C₁₀₉	—	4	2	no	—
49	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
50	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
51	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
52	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
53	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
54	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
55	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
60	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
61	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
62	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
64	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
66	#107×C₁₀₉	—	4	2	no	—
67	#106×C₁₀₉	—	4	2	no	—
68	#103×C₁₀₉	—	4	2	no	—
69	#103×C₁₀₉	—	4	2	no	—
70	#103×C₁₀₉	—	4	2	no	—
71	#103×C₁₀₉	—	4	2	no	—
72	C₂²⋊C₈×C₁₀₉	—	4	2	no	—
73	#95×C₁₀₉	—	2	2	no	—
74	#95×C₁₀₉	—	2	2	no	—
75	#95×C₁₀₉	—	2	2	no	—
76	#95×C₁₀₉	—	2	2	no	—
77	#95×C₁₀₉	—	2	2	no	—
78	C₂⁴×C₁₀₉	C₂×D₄	1	1	yes	characteristic
79	C₂×Q₈×C₁₀₉	—	2	2	no	—
80	C₂×Q₈×C₁₀₉	—	2	2	no	—
81	C₂×Q₈×C₁₀₉	—	2	2	no	—
82	C₂×Q₈×C₁₀₉	—	2	2	no	—
83	C₂×Q₈×C₁₀₉	—	2	1	no	—
84	C₂×Q₈×C₁₀₉	—	2	1	no	—
85	C₂×Q₈×C₁₀₉	—	2	1	no	—
86	C₂×Q₈×C₁₀₉	—	2	1	no	—
87	C₂²×C₄×C₁₀₉	—	2	2	no	—
88	C₂²×C₄×C₁₀₉	—	2	2	no	—
89	C₂²×C₄×C₁₀₉	—	2	1	no	—
90	C₂²×C₄×C₁₀₉	—	2	1	no	—
91	C₂²×C₄×C₁₀₉	—	2	1	no	—
92	C₂²×C₄×C₁₀₉	—	2	1	no	—
93	C₂²×C₄×C₁₀₉	—	2	2	no	—
94	C₂²×C₄×C₁₀₉	—	2	2	no	—
95	C₂²×C₄×C₁₀₉	—	2	1	no	—
96	C₂²×C₄×C₁₀₉	—	2	1	no	—
97	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
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	—	2	2	no	—
105	C₂×C₈×C₁₀₉	—	2	2	no	—
106	C₂×C₈×C₁₀₉	—	2	2	no	—
107	C₄⋊C₄×C₁₀₉	—	2	2	no	—
108	C₄⋊C₄×C₁₀₉	—	2	2	no	—
109	C₄⋊C₄×C₁₀₉	—	2	2	no	—
110	C₄⋊C₄×C₁₀₉	—	2	2	no	—
111	C₄⋊C₄×C₁₀₉	—	2	2	no	—
112	C₄⋊C₄×C₁₀₉	—	2	2	no	—
113	C₄⋊C₄×C₁₀₉	—	4	2	no	—
114	C₄⋊C₄×C₁₀₉	—	4	2	no	—
115	C₄⋊C₄×C₁₀₉	—	4	2	no	—
116	C₄⋊C₄×C₁₀₉	—	4	4	no	—
117	C₄⋊C₄×C₁₀₉	—	4	4	no	—
118	C₄⋊C₄×C₁₀₉	—	8	2	no	—
119	C₄⋊C₄×C₁₀₉	—	8	2	no	—
120	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
121	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
122	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
123	C₂²⋊C₄×C₁₀₉	—	8	2	no	—
124	C₄²×C₁₀₉	—	2	2	no	—
125	C₄²×C₁₀₉	—	2	2	no	—
126	C₂³×C₁₀₉	—	2	1	no	—
127	C₂³×C₁₀₉	—	2	1	no	—
128	C₂³×C₁₀₉	—	2	1	no	—
129	C₂³×C₁₀₉	—	2	2	no	—
130	C₂³×C₁₀₉	—	2	2	no	—
131	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
132	Q₈×C₁₀₉	—	4	2	no	—
133	Q₈×C₁₀₉	—	4	2	no	—
134	Q₈×C₁₀₉	—	4	2	no	—
135	Q₈×C₁₀₉	—	4	2	no	—
136	C₂×C₄×C₁₀₉	—	2	2	no	—
137	C₂×C₄×C₁₀₉	—	2	2	no	—
138	C₂×C₄×C₁₀₉	—	2	1	no	—
139	C₂×C₄×C₁₀₉	—	2	1	no	—
140	C₂×C₄×C₁₀₉	—	2	1	no	—
141	C₂×C₄×C₁₀₉	—	2	2	no	—
142	C₂×C₄×C₁₀₉	—	2	2	no	—
143	C₂×C₄×C₁₀₉	—	2	2	no	—
144	C₂×C₄×C₁₀₉	—	2	2	no	—
145	C₂×C₄×C₁₀₉	—	4	2	no	—
146	C₂×C₄×C₁₀₉	—	4	2	no	—
147	C₂×C₄×C₁₀₉	—	4	2	no	—
148	C₂×C₄×C₁₀₉	—	4	1	no	—
149	C₂×C₄×C₁₀₉	—	4	1	no	—
150	C₂×C₄×C₁₀₉	—	4	1	no	—
151	C₂×C₄×C₁₀₉	—	4	1	no	—
152	C₂×C₄×C₁₀₉	—	4	1	no	—
153	C₂×C₄×C₁₀₉	—	4	1	no	—
154	C₂×C₄×C₁₀₉	—	4	2	no	—
155	C₂×C₄×C₁₀₉	—	4	1	no	—
156	C₂×C₄×C₁₀₉	—	4	1	no	—
157	C₂×C₄×C₁₀₉	—	4	2	no	—
158	C₂×C₄×C₁₀₉	—	4	2	no	—
159	C₂×C₄×C₁₀₉	—	4	2	no	—
160	C₂×C₄×C₁₀₉	—	4	2	no	—
161	C₂×C₄×C₁₀₉	—	4	2	no	—
162	C₂×C₄×C₁₀₉	—	4	2	no	—
163	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
164	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
165	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
166	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
167	C₈×C₁₀₉	—	4	2	no	—
168	C₈×C₁₀₉	—	4	2	no	—
169	C₂²×C₁₀₉	—	4	1	no	—
170	C₂²×C₁₀₉	—	4	2	no	—
171	C₂²×C₁₀₉	—	4	1	no	—
172	C₂²×C₁₀₉	—	4	1	no	—
173	C₂²×C₁₀₉	—	4	2	no	—
174	C₂²×C₁₀₉	#534	1	1	yes	characteristic
175	C₂²×C₁₀₉	#535	1	1	yes	characteristic
176	C₂²×C₁₀₉	#536	1	1	yes	characteristic
177	C₂²×C₁₀₉	#577	1	2	yes	—
178	C₂²×C₁₀₉	#578	1	2	yes	—
179	C₄×C₁₀₉	—	4	1	no	—
180	C₄×C₁₀₉	—	4	2	no	—
181	C₄×C₁₀₉	—	4	2	no	—
182	C₄×C₁₀₉	—	8	1	no	—
183	C₄×C₁₀₉	—	8	1	no	—
184	C₄×C₁₀₉	—	8	2	no	—
185	C₄×C₁₀₉	—	8	1	no	—
186	C₄×C₁₀₉	—	8	2	no	—
187	C₄×C₁₀₉	—	8	1	no	—
188	C₄×C₁₀₉	#562	1	2	yes	—
189	C₄×C₁₀₉	#562	1	2	yes	—
190	#32784	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂×C₁₀₉	—	8	1	no	—
192	C₂×C₁₀₉	#2816	1	1	yes	characteristic
193	C₂×C₁₀₉	#3201	1	2	yes	—
194	C₂×C₁₀₉	#3221	1	1	yes	characteristic
195	C₂×C₁₀₉	#3223	1	1	yes	characteristic
196	C₂×C₁₀₉	#3242	1	2	yes	—
197	#2629	C₂×C₁₀₉	1	1	yes	characteristic
198	#2621	C₂×C₁₀₉	1	1	yes	characteristic
199	#2096	C₂×C₁₀₉	1	1	yes	characteristic, complemented
200	#2095	C₂×C₁₀₉	1	1	yes	characteristic, complemented
201	#2085	C₂×C₁₀₉	1	2	yes	complemented
202	#2043	C₂×C₁₀₉	1	2	yes	complemented
203	#2036	C₂×C₁₀₉	1	2	yes	—
204	#2035	C₂×C₁₀₉	1	2	yes	—
205	#2026	C₂×C₁₀₉	1	1	yes	characteristic, complemented
206	#2026	C₂×C₁₀₉	1	1	yes	characteristic, complemented
207	#1818	C₂×C₁₀₉	1	1	yes	characteristic
208	C₁₀₉	#32784	1	1	yes	characteristic, direct product factor, Sylow
209	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
210	C₂×#125	C₂²×C₁₀₉	1	1	yes	characteristic
211	C₂×C₂²⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
212	C₂×C₂²⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
213	C₂×#117	C₂²×C₁₀₉	1	1	yes	characteristic
214	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
215	C₂×#106	C₂²×C₁₀₉	1	2	yes	—
216	C₂×#103	C₂²×C₁₀₉	1	2	yes	—
217	C₂×#103	C₂²×C₁₀₉	1	2	yes	—
218	C₂×#103	C₂²×C₁₀₉	1	2	yes	—
219	C₂×#103	C₂²×C₁₀₉	1	2	yes	—
220	C₂×C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
221	#398	—	2	2	no	—
222	#398	—	2	2	no	—
223	#397	—	2	2	no	—
224	#396	—	2	2	no	—
225	#391	C₂²×C₁₀₉	1	1	yes	characteristic
226	#391	C₂²×C₁₀₉	1	1	yes	characteristic
227	#389	C₂²×C₁₀₉	1	2	yes	—
228	#384	C₂²×C₁₀₉	1	1	yes	characteristic
229	#382	C₂²×C₁₀₉	1	1	yes	characteristic
230	#380	C₂²×C₁₀₉	1	2	yes	—
231	#340	C₂²×C₁₀₉	1	2	yes	—
232	#340	C₂²×C₁₀₉	1	2	yes	—
233	#336	C₂²×C₁₀₉	1	2	yes	—
234	#336	C₂²×C₁₀₉	1	1	yes	characteristic
235	#336	C₂²×C₁₀₉	1	1	yes	characteristic
236	C₂²×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
237	C₂²×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
238	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
239	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
240	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
241	C₄⋊Q₈	—	4	2	no	—
242	#125	—	4	2	no	—
243	C₂²⋊Q₈	—	4	2	no	—
244	C₂²⋊Q₈	—	4	2	no	—
245	#117	—	4	2	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄⋊C₄	—	2	2	no	—
249	C₂×C₄⋊C₄	—	2	2	no	—
250	C₂×C₄⋊C₄	—	2	2	no	—
251	C₂×C₄⋊C₄	—	2	2	no	—
252	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
253	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
254	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
255	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
256	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
257	C₂×C₂²⋊C₄	—	2	1	no	—
258	C₂×C₂²⋊C₄	—	2	1	no	—
259	C₂×C₂²⋊C₄	—	2	2	no	—
260	C₂×C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
261	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
262	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
263	#107	—	4	2	no	—
264	#106	—	4	2	no	—
265	#103	—	4	2	no	—
266	#103	—	4	2	no	—
267	#103	—	4	2	no	—
268	#103	—	4	2	no	—
269	C₂²⋊C₈	—	4	2	no	—
270	#95	—	2	2	no	—
271	#95	—	2	2	no	—
272	#95	—	2	2	no	—
273	#95	—	2	2	no	—
274	#95	—	2	2	no	—
275	C₂⁴	C₂×D₄×C₁₀₉	1	1	yes	characteristic
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	2	1	no	—
281	C₂×Q₈	—	2	1	no	—
282	C₂×Q₈	—	2	1	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	1	no	—
293	C₂²×C₄	—	2	1	no	—
294	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
295	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
296	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
297	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	4	2	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₄⋊C₄	—	4	2	no	—
313	C₄⋊C₄	—	4	4	no	—
314	C₄⋊C₄	—	4	4	no	—
315	C₄⋊C₄	—	8	2	no	—
316	C₄⋊C₄	—	8	2	no	—
317	C₂²⋊C₄	—	4	2	no	—
318	C₂²⋊C₄	—	4	2	no	—
319	C₂²⋊C₄	—	4	2	no	—
320	C₂²⋊C₄	—	8	2	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	2	2	no	—
323	C₂³	—	2	1	no	—
324	C₂³	—	2	1	no	—
325	C₂³	—	2	1	no	—
326	C₂³	—	2	2	no	—
327	C₂³	—	2	2	no	—
328	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
329	Q₈	—	4	2	no	—
330	Q₈	—	4	2	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	4	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	1	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
361	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
362	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
363	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
364	C₈	—	4	2	no	—
365	C₈	—	4	2	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	4	2	no	—
370	C₂²	—	4	1	no	—
371	C₂²	#534×C₁₀₉	1	1	yes	characteristic
372	C₂²	#535×C₁₀₉	1	1	yes	characteristic
373	C₂²	#536×C₁₀₉	1	1	yes	characteristic
374	C₂²	#577×C₁₀₉	1	2	yes	—
375	C₂²	#578×C₁₀₉	1	2	yes	—
376	C₄	—	4	2	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	1	no	—
379	C₄	—	8	2	no	—
380	C₄	—	8	1	no	—
381	C₄	—	8	2	no	—
382	C₄	—	8	1	no	—
383	C₄	—	8	1	no	—
384	C₄	—	8	1	no	—
385	C₄	#562×C₁₀₉	1	2	yes	—
386	C₄	#562×C₁₀₉	1	2	yes	—
387	C₂	—	8	1	no	—
388	C₂	C₁₀₉×#2816	1	1	yes	characteristic
389	C₂	C₁₀₉×#3201	1	2	yes	—
390	C₂	C₁₀₉×#3221	1	1	yes	characteristic
391	C₂	C₁₀₉×#3223	1	1	yes	characteristic
392	C₂	C₁₀₉×#3242	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
prime radical	C₂⁴×C₁₀₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×#117×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#2096	C₂	semidirect
4. C₁₀₉×#2095	C₂	semidirect
5. C₁₀₉×#2085	C₂	semidirect
6. C₁₀₉×#2043	C₂	semidirect
9. C₁₀₉×#2026	C₂	semidirect
10. C₁₀₉×#2026	C₂	semidirect
190. #32784	C₁₀₉	direct (N × H)
199. #2096	C₂×C₁₀₉	semidirect
200. #2095	C₂×C₁₀₉	semidirect
201. #2085	C₂×C₁₀₉	semidirect
202. #2043	C₂×C₁₀₉	semidirect
205. #2026	C₂×C₁₀₉	semidirect
206. #2026	C₂×C₁₀₉	semidirect
208. C₁₀₉	#32784	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×#2629	2
C₁₀₉×#2621	2
C₁₀₉×#2096	2
C₁₀₉×#2095	2
C₁₀₉×#2085	2
C₁₀₉×#2043	2
C₁₀₉×#2036	2
C₁₀₉×#2035	2
C₁₀₉×#2026	2
C₁₀₉×#1818	2
#32784	109

Subgroups — C109×#32784 #42150778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#32784 #42150778