Subgroups — C₁₀₉×#32524 #42150518

Order 27904.

720 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×#2512	C₂	1	2	yes	maximal, complemented
2	C₁₀₉×#2093	C₂	1	2	yes	maximal, complemented
3	C₁₀₉×#2067	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#2054	C₂	1	2	yes	maximal
5	C₁₀₉×#1970	C₂	1	1	yes	maximal, characteristic
6	C₁₀₉×#1947	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1942	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#1939	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₀₉×#1925	C₂	1	2	yes	maximal, complemented
10	C₂²×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#126×C₁₀₉	—	2	2	no	—
12	C₂×#107×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×#107×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×#107×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
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×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
22	#397×C₁₀₉	—	2	2	no	—
23	#396×C₁₀₉	—	2	2	no	—
24	#390×C₁₀₉	C₂²	1	2	yes	—
25	#390×C₁₀₉	C₂²	1	2	yes	—
26	#390×C₁₀₉	C₂²	1	2	yes	—
27	#389×C₁₀₉	C₂²	1	2	yes	—
28	#388×C₁₀₉	—	2	2	no	—
29	#384×C₁₀₉	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
32	#340×C₁₀₉	C₂²	1	2	yes	—
33	#340×C₁₀₉	C₂²	1	2	yes	—
34	#336×C₁₀₉	C₂²	1	2	yes	—
35	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
36	C₂×D₈×C₁₀₉	—	2	2	no	—
37	C₂×D₈×C₁₀₉	—	4	1	no	—
38	C₂×D₈×C₁₀₉	—	4	1	no	—
39	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
42	#126×C₁₀₉	—	4	8	no	—
43	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
51	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
54	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
55	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
56	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
57	C₂×C₄²×C₁₀₉	—	2	2	no	—
58	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
59	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
60	#107×C₁₀₉	—	2	2	no	—
61	#107×C₁₀₉	—	4	1	no	—
62	#107×C₁₀₉	—	4	2	no	—
63	#107×C₁₀₉	—	4	1	no	—
64	C₄⋊C₈×C₁₀₉	—	2	4	no	—
65	C₄⋊C₈×C₁₀₉	—	4	1	no	—
66	#102×C₁₀₉	—	2	4	no	—
67	#102×C₁₀₉	—	4	2	no	—
68	#102×C₁₀₉	—	4	2	no	—
69	C₄×C₈×C₁₀₉	—	4	1	no	—
70	#95×C₁₀₉	—	2	2	no	—
71	#95×C₁₀₉	—	2	2	no	—
72	#95×C₁₀₉	D₄	1	2	yes	—
73	C₂⁴×C₁₀₉	—	2	2	no	—
74	C₂×D₄×C₁₀₉	—	2	2	no	—
75	C₂×D₄×C₁₀₉	—	2	2	no	—
76	C₂×D₄×C₁₀₉	—	2	2	no	—
77	C₂×D₄×C₁₀₉	—	2	2	no	—
78	C₂×D₄×C₁₀₉	—	2	2	no	—
79	C₂×D₄×C₁₀₉	—	2	2	no	—
80	C₂²×C₄×C₁₀₉	—	2	2	no	—
81	C₂²×C₄×C₁₀₉	—	2	2	no	—
82	C₂²×C₄×C₁₀₉	—	2	2	no	—
83	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
88	D₈×C₁₀₉	—	4	4	no	—
89	C₂×C₈×C₁₀₉	—	2	1	no	—
90	C₂×C₈×C₁₀₉	—	2	1	no	—
91	C₂×C₈×C₁₀₉	—	2	1	no	—
92	C₂×C₈×C₁₀₉	—	2	1	no	—
93	C₂×C₈×C₁₀₉	—	2	2	no	—
94	C₂×C₈×C₁₀₉	—	2	2	no	—
95	C₂×C₈×C₁₀₉	—	2	2	no	—
96	C₂×C₈×C₁₀₉	#41	1	2	yes	—
97	C₂×C₈×C₁₀₉	#41	1	2	yes	—
98	C₄⋊C₄×C₁₀₉	—	2	2	no	—
99	C₄⋊C₄×C₁₀₉	—	2	2	no	—
100	C₄⋊C₄×C₁₀₉	—	2	2	no	—
101	C₄⋊C₄×C₁₀₉	—	2	2	no	—
102	C₄⋊C₄×C₁₀₉	—	2	2	no	—
103	C₄⋊C₄×C₁₀₉	—	2	2	no	—
104	C₄⋊C₄×C₁₀₉	—	4	1	no	—
105	C₄⋊C₄×C₁₀₉	—	4	4	no	—
106	C₄⋊C₄×C₁₀₉	—	4	4	no	—
107	C₄⋊C₄×C₁₀₉	—	8	2	no	—
108	C₄⋊C₄×C₁₀₉	D₈	1	4	yes	complemented
109	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
110	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
111	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
112	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
113	C₂²⋊C₄×C₁₀₉	—	8	2	no	—
114	C₂²⋊C₄×C₁₀₉	—	8	2	no	—
115	C₄²×C₁₀₉	—	2	2	no	—
116	C₄²×C₁₀₉	—	2	1	no	—
117	C₄²×C₁₀₉	—	2	1	no	—
118	C₄²×C₁₀₉	—	2	2	no	—
119	C₄²×C₁₀₉	—	4	4	no	—
120	C₂³×C₁₀₉	—	4	2	no	—
121	C₂³×C₁₀₉	—	4	2	no	—
122	C₂³×C₁₀₉	—	4	2	no	—
123	C₂³×C₁₀₉	—	4	2	no	—
124	C₂³×C₁₀₉	—	4	2	no	—
125	C₂³×C₁₀₉	—	4	2	no	—
126	C₂³×C₁₀₉	—	4	2	no	—
127	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
128	D₄×C₁₀₉	—	4	2	no	—
129	D₄×C₁₀₉	—	4	2	no	—
130	D₄×C₁₀₉	—	4	2	no	—
131	D₄×C₁₀₉	—	4	2	no	—
132	C₂×C₄×C₁₀₉	—	2	1	no	—
133	C₂×C₄×C₁₀₉	—	2	2	no	—
134	C₂×C₄×C₁₀₉	—	2	1	no	—
135	C₂×C₄×C₁₀₉	—	2	2	no	—
136	C₂×C₄×C₁₀₉	—	2	1	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	2	no	—
141	C₂×C₄×C₁₀₉	—	2	1	no	—
142	C₂×C₄×C₁₀₉	—	4	2	no	—
143	C₂×C₄×C₁₀₉	—	4	2	no	—
144	C₂×C₄×C₁₀₉	—	4	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	2	no	—
149	C₂×C₄×C₁₀₉	—	4	2	no	—
150	C₂×C₄×C₁₀₉	—	4	2	no	—
151	C₂×C₄×C₁₀₉	C₂×D₈	1	4	yes	—
152	C₂×C₄×C₁₀₉	C₂×D₈	1	2	yes	—
153	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
158	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
159	C₈×C₁₀₉	—	2	2	no	—
160	C₈×C₁₀₉	—	2	2	no	—
161	C₈×C₁₀₉	—	4	2	no	—
162	C₂²×C₁₀₉	—	8	2	no	—
163	C₂²×C₁₀₉	—	8	2	no	—
164	C₂²×C₁₀₉	—	8	2	no	—
165	C₂²×C₁₀₉	—	8	2	no	—
166	C₂²×C₁₀₉	—	8	2	no	—
167	C₂²×C₁₀₉	—	8	2	no	—
168	C₂²×C₁₀₉	—	8	2	no	—
169	C₂²×C₁₀₉	#537	1	1	yes	characteristic
170	C₂²×C₁₀₉	#547	1	1	yes	characteristic
171	C₂²×C₁₀₉	#550	1	1	yes	characteristic
172	C₂²×C₁₀₉	C₂²×D₈	1	1	yes	characteristic
173	C₂²×C₁₀₉	#575	1	1	yes	characteristic
174	C₂²×C₁₀₉	#576	1	1	yes	characteristic
175	C₂²×C₁₀₉	#578	1	1	yes	characteristic
176	C₄×C₁₀₉	—	2	4	no	—
177	C₄×C₁₀₉	—	2	2	no	—
178	C₄×C₁₀₉	—	4	2	no	—
179	C₄×C₁₀₉	—	4	1	no	—
180	C₄×C₁₀₉	—	8	2	no	—
181	C₄×C₁₀₉	—	8	2	no	—
182	C₄×C₁₀₉	—	8	2	no	—
183	C₄×C₁₀₉	#538	1	1	yes	characteristic
184	C₄×C₁₀₉	#539	1	1	yes	characteristic
185	C₄×C₁₀₉	#556	1	1	yes	characteristic
186	C₄×C₁₀₉	#563	1	1	yes	characteristic
187	#32524	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂×C₁₀₉	—	16	2	no	—
189	C₂×C₁₀₉	#2727	1	1	yes	characteristic
190	C₂×C₁₀₉	#3245	1	1	yes	characteristic
191	C₂×C₁₀₉	#3262	1	1	yes	characteristic
192	C₂×C₁₀₉	#3334	1	1	yes	characteristic
193	C₂×C₁₀₉	#3355	1	1	yes	characteristic
194	C₂×C₁₀₉	#3391	1	1	yes	characteristic
195	C₂×C₁₀₉	#3402	1	1	yes	characteristic
196	#2512	C₂×C₁₀₉	1	2	yes	complemented
197	#2093	C₂×C₁₀₉	1	2	yes	complemented
198	#2067	C₂×C₁₀₉	1	2	yes	complemented
199	#2054	C₂×C₁₀₉	1	2	yes	—
200	#1970	C₂×C₁₀₉	1	1	yes	characteristic
201	#1947	C₂×C₁₀₉	1	2	yes	complemented
202	#1942	C₂×C₁₀₉	1	1	yes	characteristic, complemented
203	#1939	C₂×C₁₀₉	1	1	yes	characteristic, complemented
204	#1925	C₂×C₁₀₉	1	2	yes	complemented
205	C₁₀₉	#32524	1	1	yes	characteristic, direct product factor, Sylow
206	C₂²×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
207	C₂×#126	—	2	2	no	—
208	C₂×#107	C₂²×C₁₀₉	1	1	yes	characteristic
209	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
210	C₂×#107	C₂²×C₁₀₉	1	1	yes	characteristic
211	C₂×#107	C₂²×C₁₀₉	1	1	yes	characteristic
212	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
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×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
218	#397	—	2	2	no	—
219	#396	—	2	2	no	—
220	#390	C₂²×C₁₀₉	1	2	yes	—
221	#390	C₂²×C₁₀₉	1	2	yes	—
222	#390	C₂²×C₁₀₉	1	2	yes	—
223	#389	C₂²×C₁₀₉	1	2	yes	—
224	#388	—	2	2	no	—
225	#384	C₂²×C₁₀₉	1	2	yes	—
226	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
227	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
228	#340	C₂²×C₁₀₉	1	2	yes	—
229	#340	C₂²×C₁₀₉	1	2	yes	—
230	#336	C₂²×C₁₀₉	1	2	yes	—
231	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
232	C₂×D₈	—	2	2	no	—
233	C₂×D₈	—	4	1	no	—
234	C₂×D₈	—	4	1	no	—
235	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
236	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
237	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
238	#126	—	4	8	no	—
239	C₂×C₄⋊C₄	—	2	2	no	—
240	C₂×C₄⋊C₄	—	2	2	no	—
241	C₂×C₄⋊C₄	—	2	2	no	—
242	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
243	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
244	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
245	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
246	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
247	C₂×C₂²⋊C₄	—	2	2	no	—
248	C₂×C₂²⋊C₄	—	2	2	no	—
249	C₂×C₂²⋊C₄	—	2	2	no	—
250	C₂×C₂²⋊C₄	—	2	2	no	—
251	C₂×C₂²⋊C₄	—	2	2	no	—
252	C₂×C₂²⋊C₄	—	2	2	no	—
253	C₂×C₄²	—	2	2	no	—
254	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
255	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
256	#107	—	2	2	no	—
257	#107	—	4	1	no	—
258	#107	—	4	2	no	—
259	#107	—	4	1	no	—
260	C₄⋊C₈	—	2	4	no	—
261	C₄⋊C₈	—	4	1	no	—
262	#102	—	2	4	no	—
263	#102	—	4	2	no	—
264	#102	—	4	2	no	—
265	C₄×C₈	—	4	1	no	—
266	#95	—	2	2	no	—
267	#95	—	2	2	no	—
268	#95	D₄×C₁₀₉	1	2	yes	—
269	C₂⁴	—	2	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
281	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
282	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
283	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
284	D₈	—	4	4	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	#41×C₁₀₉	1	2	yes	—
293	C₂×C₈	#41×C₁₀₉	1	2	yes	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	4	no	—
302	C₄⋊C₄	—	4	4	no	—
303	C₄⋊C₄	—	8	2	no	—
304	C₄⋊C₄	D₈×C₁₀₉	1	4	yes	complemented
305	C₂²⋊C₄	—	4	4	no	—
306	C₂²⋊C₄	—	4	4	no	—
307	C₂²⋊C₄	—	4	4	no	—
308	C₂²⋊C₄	—	4	4	no	—
309	C₂²⋊C₄	—	8	2	no	—
310	C₂²⋊C₄	—	8	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	2	1	no	—
313	C₄²	—	2	1	no	—
314	C₄²	—	2	2	no	—
315	C₄²	—	4	4	no	—
316	C₂³	—	4	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	4	2	no	—
319	C₂³	—	4	2	no	—
320	C₂³	—	4	2	no	—
321	C₂³	—	4	2	no	—
322	C₂³	—	4	2	no	—
323	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
324	D₄	—	4	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	4	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	C₂×D₈×C₁₀₉	1	4	yes	—
348	C₂×C₄	C₂×D₈×C₁₀₉	1	2	yes	—
349	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
350	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
351	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
352	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
353	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
354	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
355	C₈	—	2	2	no	—
356	C₈	—	2	2	no	—
357	C₈	—	4	2	no	—
358	C₂²	—	8	2	no	—
359	C₂²	—	8	2	no	—
360	C₂²	—	8	2	no	—
361	C₂²	—	8	2	no	—
362	C₂²	—	8	2	no	—
363	C₂²	—	8	2	no	—
364	C₂²	—	8	2	no	—
365	C₂²	#537×C₁₀₉	1	1	yes	characteristic
366	C₂²	#547×C₁₀₉	1	1	yes	characteristic
367	C₂²	#550×C₁₀₉	1	1	yes	characteristic
368	C₂²	C₂²×D₈×C₁₀₉	1	1	yes	characteristic
369	C₂²	#575×C₁₀₉	1	1	yes	characteristic
370	C₂²	#576×C₁₀₉	1	1	yes	characteristic
371	C₂²	#578×C₁₀₉	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	4	no	—
374	C₄	—	4	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	8	2	no	—
377	C₄	—	8	2	no	—
378	C₄	—	8	2	no	—
379	C₄	#538×C₁₀₉	1	1	yes	characteristic
380	C₄	#539×C₁₀₉	1	1	yes	characteristic
381	C₄	#556×C₁₀₉	1	1	yes	characteristic
382	C₄	#563×C₁₀₉	1	1	yes	characteristic
383	C₂	—	16	2	no	—
384	C₂	C₁₀₉×#2727	1	1	yes	characteristic
385	C₂	C₁₀₉×#3245	1	1	yes	characteristic
386	C₂	C₁₀₉×#3262	1	1	yes	characteristic
387	C₂	C₁₀₉×#3334	1	1	yes	characteristic
388	C₂	C₁₀₉×#3355	1	1	yes	characteristic
389	C₂	C₁₀₉×#3391	1	1	yes	characteristic
390	C₂	C₁₀₉×#3402	1	1	yes	characteristic

	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 ◁ C₄×C₄⋊C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₀₉×#2512	C₂	semidirect
2. C₁₀₉×#2093	C₂	semidirect
3. C₁₀₉×#2067	C₂	semidirect
6. C₁₀₉×#1947	C₂	semidirect
7. C₁₀₉×#1942	C₂	semidirect
8. C₁₀₉×#1939	C₂	semidirect
9. C₁₀₉×#1925	C₂	semidirect
108. C₄⋊C₄×C₁₀₉	D₈	semidirect
187. #32524	C₁₀₉	direct (N × H)
196. #2512	C₂×C₁₀₉	semidirect
197. #2093	C₂×C₁₀₉	semidirect
198. #2067	C₂×C₁₀₉	semidirect
201. #1947	C₂×C₁₀₉	semidirect
202. #1942	C₂×C₁₀₉	semidirect
203. #1939	C₂×C₁₀₉	semidirect
204. #1925	C₂×C₁₀₉	semidirect
205. C₁₀₉	#32524	direct (N × H)
304. C₄⋊C₄	D₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#2512	2
C₁₀₉×#2093	2
C₁₀₉×#2067	2
C₁₀₉×#2054	2
C₁₀₉×#1970	2
C₁₀₉×#1947	2
C₁₀₉×#1942	2
C₁₀₉×#1939	2
C₁₀₉×#1925	2
#32524	109

Subgroups — C109×#32524 #42150518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#32524 #42150518