Subgroups — C₁₀₉×#32662 #42150656

Order 27904.

746 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2580	C₂	1	2	yes	maximal, complemented
2	C₁₀₉×#2092	C₂	1	2	yes	maximal, complemented
3	C₁₀₉×#2085	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#2067	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#1985	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₀₉×#1972	C₂	1	1	yes	maximal, characteristic
7	C₁₀₉×#1944	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#1879	C₂	1	2	yes	maximal
9	C₁₀₉×#1862	C₂	1	2	yes	maximal, complemented
10	C₂²×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
13	C₂×#107×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
15	C₂×#106×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×#106×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×#102×C₁₀₉	C₂²	1	2	yes	—
19	C₂×#102×C₁₀₉	C₂²	1	2	yes	—
20	C₂×#102×C₁₀₉	C₂²	1	2	yes	—
21	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
22	#399×C₁₀₉	—	2	2	no	—
23	#393×C₁₀₉	—	2	2	no	—
24	#390×C₁₀₉	C₂²	1	2	yes	—
25	#390×C₁₀₉	C₂²	1	2	yes	—
26	#389×C₁₀₉	C₂²	1	2	yes	—
27	#389×C₁₀₉	C₂²	1	2	yes	—
28	#385×C₁₀₉	—	2	2	no	—
29	#385×C₁₀₉	—	2	2	no	—
30	#382×C₁₀₉	C₂²	1	2	yes	—
31	#340×C₁₀₉	C₂²	1	2	yes	—
32	#340×C₁₀₉	C₂²	1	2	yes	—
33	#336×C₁₀₉	C₂²	1	2	yes	—
34	#336×C₁₀₉	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
36	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
37	C₂³×C₄×C₁₀₉	—	2	2	no	—
38	C₂×D₈×C₁₀₉	—	4	1	no	—
39	C₂×D₈×C₁₀₉	—	4	1	no	—
40	C₂×D₈×C₁₀₉	Q₈	1	4	yes	complemented
41	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₄×Q₈×C₁₀₉	—	4	1	no	—
45	C₄×Q₈×C₁₀₉	—	4	1	no	—
46	C₄×D₄×C₁₀₉	—	4	2	no	—
47	C₄×D₄×C₁₀₉	—	4	2	no	—
48	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
49	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
50	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
55	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
56	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
58	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
60	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
61	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
62	#107×C₁₀₉	—	4	2	no	—
63	#107×C₁₀₉	—	4	1	no	—
64	#106×C₁₀₉	—	2	2	no	—
65	#106×C₁₀₉	—	4	1	no	—
66	C₄⋊C₈×C₁₀₉	—	4	1	no	—
67	#102×C₁₀₉	—	2	4	no	—
68	#102×C₁₀₉	—	4	2	no	—
69	#102×C₁₀₉	—	4	2	no	—
70	#97×C₁₀₉	—	4	1	no	—
71	#95×C₁₀₉	—	2	2	no	—
72	#95×C₁₀₉	—	2	2	no	—
73	#95×C₁₀₉	—	2	2	no	—
74	#95×C₁₀₉	—	2	2	no	—
75	C₂⁴×C₁₀₉	—	2	2	no	—
76	C₂×Q₈×C₁₀₉	—	4	1	no	—
77	C₂×Q₈×C₁₀₉	—	4	2	no	—
78	C₂×D₄×C₁₀₉	—	2	2	no	—
79	C₂×D₄×C₁₀₉	—	2	2	no	—
80	C₂×D₄×C₁₀₉	—	2	2	no	—
81	C₂×D₄×C₁₀₉	—	2	2	no	—
82	C₂×D₄×C₁₀₉	—	2	2	no	—
83	C₂×D₄×C₁₀₉	C₂×Q₈	1	4	yes	—
84	C₂²×C₄×C₁₀₉	—	2	2	no	—
85	C₂²×C₄×C₁₀₉	—	2	2	no	—
86	C₂²×C₄×C₁₀₉	—	2	2	no	—
87	C₂²×C₄×C₁₀₉	—	4	4	no	—
88	C₂²×C₄×C₁₀₉	—	8	2	no	—
89	C₂²×C₄×C₁₀₉	—	8	2	no	—
90	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
95	D₈×C₁₀₉	—	4	4	no	—
96	C₂×C₈×C₁₀₉	—	2	1	no	—
97	C₂×C₈×C₁₀₉	—	2	1	no	—
98	C₂×C₈×C₁₀₉	—	2	1	no	—
99	C₂×C₈×C₁₀₉	—	2	1	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	C₂×Q₈	1	2	yes	—
104	C₂×C₈×C₁₀₉	#41	1	2	yes	—
105	C₄⋊C₄×C₁₀₉	—	2	2	no	—
106	C₄⋊C₄×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	—	4	2	no	—
112	C₄⋊C₄×C₁₀₉	—	4	1	no	—
113	C₄⋊C₄×C₁₀₉	—	8	2	no	—
114	C₄⋊C₄×C₁₀₉	—	8	2	no	—
115	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
116	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
117	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
118	C₂²⋊C₄×C₁₀₉	—	8	2	no	—
119	C₂²⋊C₄×C₁₀₉	—	8	2	no	—
120	C₄²×C₁₀₉	—	2	1	no	—
121	C₄²×C₁₀₉	—	2	1	no	—
122	C₄²×C₁₀₉	—	2	2	no	—
123	C₄²×C₁₀₉	—	2	2	no	—
124	C₂³×C₁₀₉	—	2	4	no	—
125	C₂³×C₁₀₉	—	4	2	no	—
126	C₂³×C₁₀₉	—	4	2	no	—
127	C₂³×C₁₀₉	—	4	2	no	—
128	C₂³×C₁₀₉	—	4	2	no	—
129	C₂³×C₁₀₉	—	4	2	no	—
130	C₂³×C₁₀₉	—	4	2	no	—
131	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
132	Q₈×C₁₀₉	—	4	4	no	—
133	D₄×C₁₀₉	—	2	4	no	—
134	D₄×C₁₀₉	—	2	4	no	—
135	D₄×C₁₀₉	—	4	2	no	—
136	D₄×C₁₀₉	—	4	2	no	—
137	C₂×C₄×C₁₀₉	—	2	2	no	—
138	C₂×C₄×C₁₀₉	—	2	2	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	—	2	2	no	—
146	C₂×C₄×C₁₀₉	—	2	2	no	—
147	C₂×C₄×C₁₀₉	—	4	2	no	—
148	C₂×C₄×C₁₀₉	—	4	2	no	—
149	C₂×C₄×C₁₀₉	—	4	2	no	—
150	C₂×C₄×C₁₀₉	—	4	2	no	—
151	C₂×C₄×C₁₀₉	—	4	2	no	—
152	C₂×C₄×C₁₀₉	—	4	2	no	—
153	C₂×C₄×C₁₀₉	—	4	2	no	—
154	C₂×C₄×C₁₀₉	—	4	2	no	—
155	C₂×C₄×C₁₀₉	—	4	2	no	—
156	C₂×C₄×C₁₀₉	—	8	8	no	—
157	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
158	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
163	C₈×C₁₀₉	—	2	2	no	—
164	C₈×C₁₀₉	—	2	2	no	—
165	C₈×C₁₀₉	—	4	2	no	—
166	C₂²×C₁₀₉	—	4	4	no	—
167	C₂²×C₁₀₉	—	4	4	no	—
168	C₂²×C₁₀₉	—	4	4	no	—
169	C₂²×C₁₀₉	—	8	2	no	—
170	C₂²×C₁₀₉	—	8	2	no	—
171	C₂²×C₁₀₉	—	8	2	no	—
172	C₂²×C₁₀₉	—	8	2	no	—
173	C₂²×C₁₀₉	#534	1	1	yes	characteristic
174	C₂²×C₁₀₉	D₄×Q₈	1	1	yes	characteristic
175	C₂²×C₁₀₉	#550	1	1	yes	characteristic
176	C₂²×C₁₀₉	#575	1	2	yes	—
177	C₂²×C₁₀₉	#576	1	2	yes	—
178	C₄×C₁₀₉	—	4	2	no	—
179	C₄×C₁₀₉	—	4	1	no	—
180	C₄×C₁₀₉	—	4	1	no	—
181	C₄×C₁₀₉	—	4	2	no	—
182	C₄×C₁₀₉	—	8	2	no	—
183	C₄×C₁₀₉	—	8	2	no	—
184	C₄×C₁₀₉	—	8	2	no	—
185	C₄×C₁₀₉	#539	1	1	yes	characteristic
186	C₄×C₁₀₉	#543	1	1	yes	characteristic
187	C₄×C₁₀₉	#554	1	1	yes	characteristic
188	C₄×C₁₀₉	#562	1	1	yes	characteristic
189	#32662	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂×C₁₀₉	—	8	4	no	—
191	C₂×C₁₀₉	#2732	1	1	yes	characteristic
192	C₂×C₁₀₉	#3197	1	1	yes	characteristic
193	C₂×C₁₀₉	#3199	1	1	yes	characteristic
194	C₂×C₁₀₉	#3389	1	2	yes	—
195	C₂×C₁₀₉	#3403	1	2	yes	—
196	#2580	C₂×C₁₀₉	1	2	yes	complemented
197	#2092	C₂×C₁₀₉	1	2	yes	complemented
198	#2085	C₂×C₁₀₉	1	2	yes	complemented
199	#2067	C₂×C₁₀₉	1	2	yes	complemented
200	#1985	C₂×C₁₀₉	1	1	yes	characteristic, complemented
201	#1972	C₂×C₁₀₉	1	1	yes	characteristic
202	#1944	C₂×C₁₀₉	1	1	yes	characteristic, complemented
203	#1879	C₂×C₁₀₉	1	2	yes	—
204	#1862	C₂×C₁₀₉	1	2	yes	complemented
205	C₁₀₉	#32662	1	1	yes	characteristic, direct product factor, Sylow
206	C₂²×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
207	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
208	C₂×C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
209	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
210	C₂×#107	C₂²×C₁₀₉	1	1	yes	characteristic
211	C₂×#106	C₂²×C₁₀₉	1	1	yes	characteristic
212	C₂×#106	C₂²×C₁₀₉	1	1	yes	characteristic
213	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
214	C₂×#102	C₂²×C₁₀₉	1	2	yes	—
215	C₂×#102	C₂²×C₁₀₉	1	2	yes	—
216	C₂×#102	C₂²×C₁₀₉	1	2	yes	—
217	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
218	#399	—	2	2	no	—
219	#393	—	2	2	no	—
220	#390	C₂²×C₁₀₉	1	2	yes	—
221	#390	C₂²×C₁₀₉	1	2	yes	—
222	#389	C₂²×C₁₀₉	1	2	yes	—
223	#389	C₂²×C₁₀₉	1	2	yes	—
224	#385	—	2	2	no	—
225	#385	—	2	2	no	—
226	#382	C₂²×C₁₀₉	1	2	yes	—
227	#340	C₂²×C₁₀₉	1	2	yes	—
228	#340	C₂²×C₁₀₉	1	2	yes	—
229	#336	C₂²×C₁₀₉	1	2	yes	—
230	#336	C₂²×C₁₀₉	1	2	yes	—
231	C₂²×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
232	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
233	C₂³×C₄	—	2	2	no	—
234	C₂×D₈	—	4	1	no	—
235	C₂×D₈	—	4	1	no	—
236	C₂×D₈	Q₈×C₁₀₉	1	4	yes	complemented
237	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
238	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
239	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
240	C₄×Q₈	—	4	1	no	—
241	C₄×Q₈	—	4	1	no	—
242	C₄×D₄	—	4	2	no	—
243	C₄×D₄	—	4	2	no	—
244	C₂×C₄⋊C₄	—	2	2	no	—
245	C₂×C₄⋊C₄	—	2	2	no	—
246	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
247	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
248	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
249	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
250	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
251	C₂×C₂²⋊C₄	—	2	2	no	—
252	C₂×C₂²⋊C₄	—	2	2	no	—
253	C₂×C₂²⋊C₄	—	2	2	no	—
254	C₂×C₂²⋊C₄	—	2	2	no	—
255	C₂×C₂²⋊C₄	—	2	2	no	—
256	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
257	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
258	#107	—	4	2	no	—
259	#107	—	4	1	no	—
260	#106	—	2	2	no	—
261	#106	—	4	1	no	—
262	C₄⋊C₈	—	4	1	no	—
263	#102	—	2	4	no	—
264	#102	—	4	2	no	—
265	#102	—	4	2	no	—
266	#97	—	4	1	no	—
267	#95	—	2	2	no	—
268	#95	—	2	2	no	—
269	#95	—	2	2	no	—
270	#95	—	2	2	no	—
271	C₂⁴	—	2	2	no	—
272	C₂×Q₈	—	4	1	no	—
273	C₂×Q₈	—	4	2	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	2	2	no	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	C₂×Q₈×C₁₀₉	1	4	yes	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	4	4	no	—
284	C₂²×C₄	—	8	2	no	—
285	C₂²×C₄	—	8	2	no	—
286	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
287	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
289	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
290	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
291	D₈	—	4	4	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	C₂×Q₈×C₁₀₉	1	2	yes	—
300	C₂×C₈	#41×C₁₀₉	1	2	yes	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	4	2	no	—
308	C₄⋊C₄	—	4	1	no	—
309	C₄⋊C₄	—	8	2	no	—
310	C₄⋊C₄	—	8	2	no	—
311	C₂²⋊C₄	—	4	4	no	—
312	C₂²⋊C₄	—	4	4	no	—
313	C₂²⋊C₄	—	4	4	no	—
314	C₂²⋊C₄	—	8	2	no	—
315	C₂²⋊C₄	—	8	2	no	—
316	C₄²	—	2	1	no	—
317	C₄²	—	2	1	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	2	2	no	—
320	C₂³	—	2	4	no	—
321	C₂³	—	4	2	no	—
322	C₂³	—	4	2	no	—
323	C₂³	—	4	2	no	—
324	C₂³	—	4	2	no	—
325	C₂³	—	4	2	no	—
326	C₂³	—	4	2	no	—
327	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
328	Q₈	—	4	4	no	—
329	D₄	—	2	4	no	—
330	D₄	—	2	4	no	—
331	D₄	—	4	2	no	—
332	D₄	—	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	2	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	—	2	2	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	8	8	no	—
353	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
354	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
355	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
356	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
357	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
358	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
359	C₈	—	2	2	no	—
360	C₈	—	2	2	no	—
361	C₈	—	4	2	no	—
362	C₂²	—	4	4	no	—
363	C₂²	—	4	4	no	—
364	C₂²	—	4	4	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	2	no	—
368	C₂²	—	8	2	no	—
369	C₂²	#534×C₁₀₉	1	1	yes	characteristic
370	C₂²	D₄×Q₈×C₁₀₉	1	1	yes	characteristic
371	C₂²	#550×C₁₀₉	1	1	yes	characteristic
372	C₂²	#575×C₁₀₉	1	2	yes	—
373	C₂²	#576×C₁₀₉	1	2	yes	—
374	C₄	—	4	2	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	4	1	no	—
378	C₄	—	8	2	no	—
379	C₄	—	8	2	no	—
380	C₄	—	8	2	no	—
381	C₄	#539×C₁₀₉	1	1	yes	characteristic
382	C₄	#543×C₁₀₉	1	1	yes	characteristic
383	C₄	#554×C₁₀₉	1	1	yes	characteristic
384	C₄	#562×C₁₀₉	1	1	yes	characteristic
385	C₂	—	8	4	no	—
386	C₂	C₁₀₉×#2732	1	1	yes	characteristic
387	C₂	C₁₀₉×#3197	1	1	yes	characteristic
388	C₂	C₁₀₉×#3199	1	1	yes	characteristic
389	C₂	C₁₀₉×#3389	1	2	yes	—
390	C₂	C₁₀₉×#3403	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
prime radical	C₂²×D₈×C₁₀₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄×Q₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₀₉×#2580	C₂	semidirect
2. C₁₀₉×#2092	C₂	semidirect
3. C₁₀₉×#2085	C₂	semidirect
4. C₁₀₉×#2067	C₂	semidirect
5. C₁₀₉×#1985	C₂	semidirect
7. C₁₀₉×#1944	C₂	semidirect
9. C₁₀₉×#1862	C₂	semidirect
40. C₂×D₈×C₁₀₉	Q₈	semidirect
189. #32662	C₁₀₉	direct (N × H)
196. #2580	C₂×C₁₀₉	semidirect
197. #2092	C₂×C₁₀₉	semidirect
198. #2085	C₂×C₁₀₉	semidirect
199. #2067	C₂×C₁₀₉	semidirect
200. #1985	C₂×C₁₀₉	semidirect
202. #1944	C₂×C₁₀₉	semidirect
204. #1862	C₂×C₁₀₉	semidirect
205. C₁₀₉	#32662	direct (N × H)
236. C₂×D₈	Q₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#2580	2
C₁₀₉×#2092	2
C₁₀₉×#2085	2
C₁₀₉×#2067	2
C₁₀₉×#1985	2
C₁₀₉×#1972	2
C₁₀₉×#1944	2
C₁₀₉×#1879	2
C₁₀₉×#1862	2
#32662	109

Subgroups — C109×#32662 #42150656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#32662 #42150656