Subgroups — C₁₀₉×#21662 #42139656

Order 27904.

738 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#3135	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2961	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#2932	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1962	C₂	1	1	yes	maximal, characteristic
6	C₁₀₉×#1938	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1937	C₂	1	2	yes	maximal, complemented
8	C₁₀₉×#1838	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₀₉×#1765	C₂	1	2	yes	maximal, complemented
10	C₁₀₉×#1764	C₂	1	2	yes	maximal, complemented
11	C₄×#97×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#131×C₁₀₉	C₂²	1	2	yes	—
13	#533×C₁₀₉	C₂²	1	1	yes	characteristic
14	#532×C₁₀₉	C₂²	1	1	yes	characteristic
15	#491×C₁₀₉	C₄	1	2	yes	complemented
16	#490×C₁₀₉	C₄	1	2	yes	complemented
17	#489×C₁₀₉	C₄	1	4	yes	complemented
18	#446×C₁₀₉	—	2	4	no	—
19	#435×C₁₀₉	—	2	4	no	—
20	#424×C₁₀₉	C₂²	1	2	yes	—
21	#423×C₁₀₉	C₂²	1	1	yes	characteristic
22	#423×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×#104×C₁₀₉	C₂²	1	2	yes	—
24	C₂×#104×C₁₀₉	C₂²	1	2	yes	—
25	#416×C₁₀₉	C₂²	1	2	yes	—
26	#416×C₁₀₉	C₂²	1	2	yes	—
27	#408×C₁₀₉	C₂²	1	2	yes	—
28	#405×C₁₀₉	C₂²	1	2	yes	—
29	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
30	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
32	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
36	#341×C₁₀₉	C₂²	1	2	yes	—
37	#341×C₁₀₉	C₂²	1	2	yes	—
38	#339×C₁₀₉	C₂²	1	2	yes	—
39	#339×C₁₀₉	C₂²	1	2	yes	—
40	C₂×#41×C₁₀₉	C₂³	1	2	yes	—
41	#131×C₁₀₉	—	2	4	no	—
42	#131×C₁₀₉	—	2	4	no	—
43	C₂×M₁₆×C₁₀₉	—	2	2	no	—
44	C₂×M₁₆×C₁₀₉	C₂³	1	2	yes	—
45	C₂×M₁₆×C₁₀₉	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₀₉	—	2	2	no	—
47	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
53	#127×C₁₀₉	C₂×C₄	1	1	yes	characteristic
54	#125×C₁₀₉	C₂×C₄	1	2	yes	—
55	#124×C₁₀₉	C₂×C₄	1	2	yes	—
56	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
57	#121×C₁₀₉	—	2	2	no	—
58	C₄×Q₈×C₁₀₉	—	2	2	no	—
59	C₄×D₄×C₁₀₉	—	2	2	no	—
60	#117×C₁₀₉	C₂³	1	2	yes	—
61	C₂×C₄²×C₁₀₉	—	2	2	no	—
62	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₁₀₉	—	2	4	no	—
66	C₄⋊C₈×C₁₀₉	—	2	2	no	—
67	C₄⋊C₈×C₁₀₉	—	2	2	no	—
68	#104×C₁₀₉	—	2	4	no	—
69	#104×C₁₀₉	—	2	4	no	—
70	#103×C₁₀₉	C₂×C₄	1	4	yes	—
71	#103×C₁₀₉	C₂×C₄	1	4	yes	—
72	#102×C₁₀₉	C₂×C₄	1	4	yes	—
73	#102×C₁₀₉	C₂×C₄	1	4	yes	—
74	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
75	#97×C₁₀₉	—	2	2	no	—
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	#97×C₁₀₉	D₄	1	4	yes	—
81	C₄×C₈×C₁₀₉	—	2	2	no	—
82	C₄×C₈×C₁₀₉	—	2	2	no	—
83	C₄×C₈×C₁₀₉	—	2	2	no	—
84	#41×C₁₀₉	—	2	2	no	—
85	#41×C₁₀₉	—	2	2	no	—
86	#41×C₁₀₉	—	2	2	no	—
87	#41×C₁₀₉	—	2	2	no	—
88	C₂×Q₈×C₁₀₉	—	2	2	no	—
89	C₂×Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
90	C₂×D₄×C₁₀₉	—	2	2	no	—
91	C₂×D₄×C₁₀₉	—	2	2	no	—
92	C₂×D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₁₀₉	—	2	1	no	—
94	C₂²×C₄×C₁₀₉	—	2	1	no	—
95	C₂²×C₄×C₁₀₉	—	2	2	no	—
96	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
97	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
98	M₁₆×C₁₀₉	—	2	2	no	—
99	M₁₆×C₁₀₉	—	2	2	no	—
100	M₁₆×C₁₀₉	—	2	2	no	—
101	M₁₆×C₁₀₉	—	2	2	no	—
102	M₁₆×C₁₀₉	—	4	4	no	—
103	C₂×C₈×C₁₀₉	—	2	1	no	—
104	C₂×C₈×C₁₀₉	—	2	1	no	—
105	C₂×C₈×C₁₀₉	—	2	1	no	—
106	C₂×C₈×C₁₀₉	—	2	1	no	—
107	C₂×C₈×C₁₀₉	—	2	1	no	—
108	C₂×C₈×C₁₀₉	—	2	1	no	—
109	C₂×C₈×C₁₀₉	—	2	2	no	—
110	C₂×C₈×C₁₀₉	—	2	2	no	—
111	C₂×C₈×C₁₀₉	—	4	4	no	—
112	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₁₀₉	#41	1	2	yes	—
117	C₂×C₈×C₁₀₉	#41	1	2	yes	—
118	C₄⋊C₄×C₁₀₉	—	2	2	no	—
119	C₄⋊C₄×C₁₀₉	—	2	4	no	—
120	C₄⋊C₄×C₁₀₉	—	2	2	no	—
121	C₄⋊C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
122	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
123	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
124	C₄²×C₁₀₉	—	2	2	no	—
125	C₄²×C₁₀₉	—	2	1	no	—
126	C₄²×C₁₀₉	—	2	1	no	—
127	C₄²×C₁₀₉	—	4	4	no	—
128	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
131	C₄²×C₁₀₉	#41	1	2	yes	—
132	C₄²×C₁₀₉	#41	1	2	yes	—
133	C₂³×C₁₀₉	—	2	2	no	—
134	C₂³×C₁₀₉	C₂×#41	1	1	yes	characteristic
135	Q₈×C₁₀₉	—	2	2	no	—
136	Q₈×C₁₀₉	—	2	2	no	—
137	Q₈×C₁₀₉	—	8	2	no	—
138	D₄×C₁₀₉	—	2	2	no	—
139	D₄×C₁₀₉	—	2	2	no	—
140	D₄×C₁₀₉	—	4	2	no	—
141	D₄×C₁₀₉	—	4	2	no	—
142	D₄×C₁₀₉	—	8	2	no	—
143	C₂×C₄×C₁₀₉	—	2	4	no	—
144	C₂×C₄×C₁₀₉	—	2	2	no	—
145	C₂×C₄×C₁₀₉	—	2	4	no	—
146	C₂×C₄×C₁₀₉	—	2	2	no	—
147	C₂×C₄×C₁₀₉	—	2	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	C₄×D₄	1	2	yes	—
153	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
154	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
160	C₈×C₁₀₉	—	2	2	no	—
161	C₈×C₁₀₉	—	2	2	no	—
162	C₈×C₁₀₉	—	2	2	no	—
163	C₈×C₁₀₉	—	4	2	no	—
164	C₈×C₁₀₉	—	4	2	no	—
165	C₈×C₁₀₉	#124	1	4	yes	—
166	C₂²×C₁₀₉	—	2	2	no	—
167	C₂²×C₁₀₉	—	4	4	no	—
168	C₂²×C₁₀₉	—	4	2	no	—
169	C₂²×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂²×C₁₀₉	#529	1	1	yes	characteristic
171	C₂²×C₁₀₉	#529	1	1	yes	characteristic
172	C₄×C₁₀₉	—	2	4	no	—
173	C₄×C₁₀₉	—	2	2	no	—
174	C₄×C₁₀₉	—	2	2	no	—
175	C₄×C₁₀₉	—	4	2	no	—
176	C₄×C₁₀₉	—	4	4	no	—
177	C₄×C₁₀₉	—	4	2	no	—
178	C₄×C₁₀₉	C₄×#41	1	1	yes	characteristic
179	C₄×C₁₀₉	C₄×#41	1	1	yes	characteristic
180	C₄×C₁₀₉	C₂×#124	1	1	yes	characteristic
181	C₄×C₁₀₉	#532	1	1	yes	characteristic
182	#21662	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂×C₁₀₉	—	2	1	no	—
184	C₂×C₁₀₉	—	2	1	no	—
185	C₂×C₁₀₉	—	8	2	no	—
186	C₂×C₁₀₉	C₄×#124	1	1	yes	characteristic
187	C₂×C₁₀₉	#2956	1	2	yes	—
188	#3135	C₂×C₁₀₉	1	1	yes	characteristic
189	#2961	C₂×C₁₀₉	1	1	yes	characteristic, complemented
190	#2932	C₂×C₁₀₉	1	1	yes	characteristic
191	#1994	C₂×C₁₀₉	1	1	yes	characteristic, complemented
192	#1962	C₂×C₁₀₉	1	1	yes	characteristic
193	#1938	C₂×C₁₀₉	1	2	yes	complemented
194	#1937	C₂×C₁₀₉	1	2	yes	complemented
195	#1838	C₂×C₁₀₉	1	1	yes	characteristic, complemented
196	#1765	C₂×C₁₀₉	1	2	yes	complemented
197	#1764	C₂×C₁₀₉	1	2	yes	complemented
198	C₄×#97	C₂×C₁₀₉	1	1	yes	characteristic, complemented
199	C₁₀₉	#21662	1	1	yes	characteristic, direct product factor, Sylow
200	C₂×#131	C₂²×C₁₀₉	1	2	yes	—
201	#533	C₂²×C₁₀₉	1	1	yes	characteristic
202	#532	C₂²×C₁₀₉	1	1	yes	characteristic
203	#491	C₄×C₁₀₉	1	2	yes	complemented
204	#490	C₄×C₁₀₉	1	2	yes	complemented
205	#489	C₄×C₁₀₉	1	4	yes	complemented
206	#446	—	2	4	no	—
207	#435	—	2	4	no	—
208	#424	C₂²×C₁₀₉	1	2	yes	—
209	#423	C₂²×C₁₀₉	1	1	yes	characteristic
210	#423	C₂²×C₁₀₉	1	1	yes	characteristic
211	C₂×#104	C₂²×C₁₀₉	1	2	yes	—
212	C₂×#104	C₂²×C₁₀₉	1	2	yes	—
213	#416	C₂²×C₁₀₉	1	2	yes	—
214	#416	C₂²×C₁₀₉	1	2	yes	—
215	#408	C₂²×C₁₀₉	1	2	yes	—
216	#405	C₂²×C₁₀₉	1	2	yes	—
217	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
218	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
219	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
220	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
221	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
222	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
223	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
224	#341	C₂²×C₁₀₉	1	2	yes	—
225	#341	C₂²×C₁₀₉	1	2	yes	—
226	#339	C₂²×C₁₀₉	1	2	yes	—
227	#339	C₂²×C₁₀₉	1	2	yes	—
228	C₂×#41	C₂³×C₁₀₉	1	2	yes	—
229	#131	—	2	4	no	—
230	#131	—	2	4	no	—
231	C₂×M₁₆	—	2	2	no	—
232	C₂×M₁₆	C₂³×C₁₀₉	1	2	yes	—
233	C₂×M₁₆	C₂³×C₁₀₉	1	2	yes	—
234	C₂²×C₈	—	2	2	no	—
235	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
236	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
237	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
238	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
239	C₄⋊Q₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
240	C₄⋊Q₈	C₂×C₄×C₁₀₉	1	2	yes	—
241	#127	C₂×C₄×C₁₀₉	1	1	yes	characteristic
242	#125	C₂×C₄×C₁₀₉	1	2	yes	—
243	#124	C₂×C₄×C₁₀₉	1	2	yes	—
244	C₂²⋊Q₈	—	2	2	no	—
245	#121	—	2	2	no	—
246	C₄×Q₈	—	2	2	no	—
247	C₄×D₄	—	2	2	no	—
248	#117	C₂³×C₁₀₉	1	2	yes	—
249	C₂×C₄²	—	2	2	no	—
250	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
251	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
252	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
253	C₄⋊C₈	—	2	4	no	—
254	C₄⋊C₈	—	2	2	no	—
255	C₄⋊C₈	—	2	2	no	—
256	#104	—	2	4	no	—
257	#104	—	2	4	no	—
258	#103	C₂×C₄×C₁₀₉	1	4	yes	—
259	#103	C₂×C₄×C₁₀₉	1	4	yes	—
260	#102	C₂×C₄×C₁₀₉	1	4	yes	—
261	#102	C₂×C₄×C₁₀₉	1	4	yes	—
262	C₂²⋊C₈	—	2	4	no	—
263	#97	—	2	2	no	—
264	#97	—	2	2	no	—
265	#97	—	2	2	no	—
266	#97	C₂×C₄×C₁₀₉	1	2	yes	—
267	#97	C₂×C₄×C₁₀₉	1	2	yes	—
268	#97	D₄×C₁₀₉	1	4	yes	—
269	C₄×C₈	—	2	2	no	—
270	C₄×C₈	—	2	2	no	—
271	C₄×C₈	—	2	2	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	#41	—	2	2	no	—
275	#41	—	2	2	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	C₂²×C₄×C₁₀₉	1	2	yes	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	C₂²×C₄×C₁₀₉	1	2	yes	—
281	C₂²×C₄	—	2	1	no	—
282	C₂²×C₄	—	2	1	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
285	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	2	2	no	—
290	M₁₆	—	4	4	no	—
291	C₂×C₈	—	2	1	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	1	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	4	4	no	—
300	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
301	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
302	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
303	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
304	C₂×C₈	#41×C₁₀₉	1	2	yes	—
305	C₂×C₈	#41×C₁₀₉	1	2	yes	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	4	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
310	C₂²⋊C₄	—	2	2	no	—
311	C₂²⋊C₄	—	2	4	no	—
312	C₄²	—	2	2	no	—
313	C₄²	—	2	1	no	—
314	C₄²	—	2	1	no	—
315	C₄²	—	4	4	no	—
316	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
317	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
318	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
319	C₄²	#41×C₁₀₉	1	2	yes	—
320	C₄²	#41×C₁₀₉	1	2	yes	—
321	C₂³	—	2	2	no	—
322	C₂³	C₂×#41×C₁₀₉	1	1	yes	characteristic
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	Q₈	—	8	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	2	2	no	—
328	D₄	—	4	2	no	—
329	D₄	—	4	2	no	—
330	D₄	—	8	2	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	4	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
341	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
342	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
343	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
344	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
345	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
346	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
347	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
348	C₈	—	2	2	no	—
349	C₈	—	2	2	no	—
350	C₈	—	2	2	no	—
351	C₈	—	4	2	no	—
352	C₈	—	4	2	no	—
353	C₈	#124×C₁₀₉	1	4	yes	—
354	C₂²	—	2	2	no	—
355	C₂²	—	4	4	no	—
356	C₂²	—	4	2	no	—
357	C₂²	C₂×C₄×D₄×C₁₀₉	1	1	yes	characteristic
358	C₂²	#529×C₁₀₉	1	1	yes	characteristic
359	C₂²	#529×C₁₀₉	1	1	yes	characteristic
360	C₄	—	2	4	no	—
361	C₄	—	2	2	no	—
362	C₄	—	2	2	no	—
363	C₄	—	4	2	no	—
364	C₄	—	4	4	no	—
365	C₄	—	4	2	no	—
366	C₄	C₄×#41×C₁₀₉	1	1	yes	characteristic
367	C₄	C₄×#41×C₁₀₉	1	1	yes	characteristic
368	C₄	C₂×#124×C₁₀₉	1	1	yes	characteristic
369	C₄	#532×C₁₀₉	1	1	yes	characteristic
370	C₂	—	2	1	no	—
371	C₂	—	2	1	no	—
372	C₂	—	8	2	no	—
373	C₂	C₄×#124×C₁₀₉	1	1	yes	characteristic
374	C₂	C₁₀₉×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
socle	C₂²×C₁₀₉	C₂×C₄×D₄
prime radical	#532×C₁₀₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×#97×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₉×#2961	C₂	semidirect
4. C₁₀₉×#1994	C₂	semidirect
6. C₁₀₉×#1938	C₂	semidirect
7. C₁₀₉×#1937	C₂	semidirect
8. C₁₀₉×#1838	C₂	semidirect
9. C₁₀₉×#1765	C₂	semidirect
10. C₁₀₉×#1764	C₂	semidirect
11. C₄×#97×C₁₀₉	C₂	semidirect
15. #491×C₁₀₉	C₄	semidirect
16. #490×C₁₀₉	C₄	semidirect
17. #489×C₁₀₉	C₄	semidirect
182. #21662	C₁₀₉	direct (N × H)
189. #2961	C₂×C₁₀₉	semidirect
191. #1994	C₂×C₁₀₉	semidirect
193. #1938	C₂×C₁₀₉	semidirect
194. #1937	C₂×C₁₀₉	semidirect
195. #1838	C₂×C₁₀₉	semidirect
196. #1765	C₂×C₁₀₉	semidirect
197. #1764	C₂×C₁₀₉	semidirect
198. C₄×#97	C₂×C₁₀₉	semidirect
199. C₁₀₉	#21662	direct (N × H)
203. #491	C₄×C₁₀₉	semidirect
204. #490	C₄×C₁₀₉	semidirect
205. #489	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#3135	2
C₁₀₉×#2961	2
C₁₀₉×#2932	2
C₁₀₉×#1994	2
C₁₀₉×#1962	2
C₁₀₉×#1938	2
C₁₀₉×#1937	2
C₁₀₉×#1838	2
C₁₀₉×#1765	2
C₁₀₉×#1764	2
C₄×#97×C₁₀₉	2
#21662	109

Subgroups — C109×#21662 #42139656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#21662 #42139656