Subgroups — C₁₀₉×#32648 #42150642

Order 27904.

808 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2538	C₂	1	2	yes	maximal, complemented
2	C₁₀₉×#2093	C₂	1	2	yes	maximal, complemented
3	C₁₀₉×#2083	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#2067	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#1984	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₀₉×#1970	C₂	1	1	yes	maximal, characteristic
7	C₁₀₉×#1942	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×C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
12	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
14	C₂×#107×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×#107×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×#107×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×#102×C₁₀₉	C₂²	1	2	yes	—
20	C₂×#102×C₁₀₉	C₂²	1	2	yes	—
21	C₂×#102×C₁₀₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	#396×C₁₀₉	—	2	2	no	—
24	#396×C₁₀₉	—	2	2	no	—
25	#390×C₁₀₉	C₂²	1	2	yes	—
26	#390×C₁₀₉	C₂²	1	2	yes	—
27	#389×C₁₀₉	C₂²	1	2	yes	—
28	#389×C₁₀₉	C₂²	1	2	yes	—
29	#384×C₁₀₉	C₂²	1	2	yes	—
30	#380×C₁₀₉	—	2	2	no	—
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	C₂²⋊Q₈×C₁₀₉	—	4	8	no	—
45	C₄×Q₈×C₁₀₉	—	4	1	no	—
46	C₄×Q₈×C₁₀₉	—	4	1	no	—
47	C₄×D₄×C₁₀₉	—	4	2	no	—
48	C₄×D₄×C₁₀₉	—	4	2	no	—
49	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
50	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
51	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
52	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
53	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
58	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
63	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
64	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
65	#107×C₁₀₉	—	2	2	no	—
66	#107×C₁₀₉	—	4	2	no	—
67	#107×C₁₀₉	—	4	1	no	—
68	#107×C₁₀₉	—	4	1	no	—
69	C₄⋊C₈×C₁₀₉	—	4	1	no	—
70	#102×C₁₀₉	—	2	4	no	—
71	#102×C₁₀₉	—	4	2	no	—
72	#102×C₁₀₉	—	4	2	no	—
73	C₄×C₈×C₁₀₉	—	4	1	no	—
74	#95×C₁₀₉	—	2	2	no	—
75	#95×C₁₀₉	—	2	2	no	—
76	C₂⁴×C₁₀₉	—	2	2	no	—
77	C₂×Q₈×C₁₀₉	—	4	1	no	—
78	C₂×Q₈×C₁₀₉	—	4	1	no	—
79	C₂×Q₈×C₁₀₉	D₈	1	4	yes	complemented
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	—	2	2	no	—
84	C₂×D₄×C₁₀₉	—	2	2	no	—
85	C₂×D₄×C₁₀₉	C₂×Q₈	1	4	yes	—
86	C₂²×C₄×C₁₀₉	—	2	2	no	—
87	C₂²×C₄×C₁₀₉	—	2	2	no	—
88	C₂²×C₄×C₁₀₉	—	2	2	no	—
89	C₂²×C₄×C₁₀₉	—	4	4	no	—
90	C₂²×C₄×C₁₀₉	—	4	4	no	—
91	C₂²×C₄×C₁₀₉	—	4	4	no	—
92	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
97	D₈×C₁₀₉	—	4	4	no	—
98	C₂×C₈×C₁₀₉	—	2	1	no	—
99	C₂×C₈×C₁₀₉	—	2	1	no	—
100	C₂×C₈×C₁₀₉	—	2	1	no	—
101	C₂×C₈×C₁₀₉	—	2	1	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	C₂×Q₈	1	2	yes	—
106	C₂×C₈×C₁₀₉	#41	1	2	yes	—
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	1	no	—
115	C₄⋊C₄×C₁₀₉	—	4	4	no	—
116	C₄⋊C₄×C₁₀₉	—	4	4	no	—
117	C₄⋊C₄×C₁₀₉	—	4	4	no	—
118	C₄⋊C₄×C₁₀₉	—	8	2	no	—
119	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
120	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
121	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
122	C₂²⋊C₄×C₁₀₉	—	8	2	no	—
123	C₂²⋊C₄×C₁₀₉	—	8	2	no	—
124	C₄²×C₁₀₉	—	2	1	no	—
125	C₄²×C₁₀₉	—	2	1	no	—
126	C₄²×C₁₀₉	—	2	2	no	—
127	C₄²×C₁₀₉	—	2	2	no	—
128	C₂³×C₁₀₉	—	2	4	no	—
129	C₂³×C₁₀₉	—	4	2	no	—
130	C₂³×C₁₀₉	—	4	2	no	—
131	C₂³×C₁₀₉	—	4	2	no	—
132	C₂³×C₁₀₉	—	4	2	no	—
133	C₂³×C₁₀₉	—	4	2	no	—
134	C₂³×C₁₀₉	—	4	2	no	—
135	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
136	Q₈×C₁₀₉	—	4	4	no	—
137	D₄×C₁₀₉	—	2	4	no	—
138	D₄×C₁₀₉	—	2	4	no	—
139	D₄×C₁₀₉	—	4	2	no	—
140	D₄×C₁₀₉	—	4	2	no	—
141	C₂×C₄×C₁₀₉	—	2	2	no	—
142	C₂×C₄×C₁₀₉	—	2	1	no	—
143	C₂×C₄×C₁₀₉	—	2	2	no	—
144	C₂×C₄×C₁₀₉	—	2	1	no	—
145	C₂×C₄×C₁₀₉	—	2	2	no	—
146	C₂×C₄×C₁₀₉	—	2	1	no	—
147	C₂×C₄×C₁₀₉	—	2	1	no	—
148	C₂×C₄×C₁₀₉	—	2	2	no	—
149	C₂×C₄×C₁₀₉	—	2	1	no	—
150	C₂×C₄×C₁₀₉	—	2	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	—	4	2	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	—	8	8	no	—
161	C₂×C₄×C₁₀₉	C₂×D₈	1	2	yes	—
162	C₂×C₄×C₁₀₉	C₂×D₈	1	4	yes	—
163	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
167	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
168	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
169	C₈×C₁₀₉	—	2	2	no	—
170	C₈×C₁₀₉	—	2	2	no	—
171	C₈×C₁₀₉	—	4	2	no	—
172	C₂²×C₁₀₉	—	4	4	no	—
173	C₂²×C₁₀₉	—	4	4	no	—
174	C₂²×C₁₀₉	—	4	4	no	—
175	C₂²×C₁₀₉	—	8	2	no	—
176	C₂²×C₁₀₉	—	8	2	no	—
177	C₂²×C₁₀₉	—	8	2	no	—
178	C₂²×C₁₀₉	—	8	2	no	—
179	C₂²×C₁₀₉	#534	1	1	yes	characteristic
180	C₂²×C₁₀₉	D₄×Q₈	1	1	yes	characteristic
181	C₂²×C₁₀₉	#550	1	1	yes	characteristic
182	C₂²×C₁₀₉	C₂²×D₈	1	1	yes	characteristic
183	C₂²×C₁₀₉	#575	1	1	yes	characteristic
184	C₂²×C₁₀₉	#576	1	1	yes	characteristic
185	C₂²×C₁₀₉	#578	1	1	yes	characteristic
186	C₄×C₁₀₉	—	2	4	no	—
187	C₄×C₁₀₉	—	2	2	no	—
188	C₄×C₁₀₉	—	4	2	no	—
189	C₄×C₁₀₉	—	4	1	no	—
190	C₄×C₁₀₉	—	8	2	no	—
191	C₄×C₁₀₉	—	8	2	no	—
192	C₄×C₁₀₉	—	8	2	no	—
193	C₄×C₁₀₉	#539	1	1	yes	characteristic
194	C₄×C₁₀₉	#543	1	1	yes	characteristic
195	C₄×C₁₀₉	#554	1	1	yes	characteristic
196	C₄×C₁₀₉	#562	1	1	yes	characteristic
197	#32648	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂×C₁₀₉	—	8	4	no	—
199	C₂×C₁₀₉	#2732	1	1	yes	characteristic
200	C₂×C₁₀₉	#3186	1	1	yes	characteristic
201	C₂×C₁₀₉	#3203	1	1	yes	characteristic
202	C₂×C₁₀₉	Q₈×D₈	1	1	yes	characteristic
203	C₂×C₁₀₉	#3389	1	1	yes	characteristic
204	C₂×C₁₀₉	#3391	1	1	yes	characteristic
205	C₂×C₁₀₉	#3402	1	1	yes	characteristic
206	#2538	C₂×C₁₀₉	1	2	yes	complemented
207	#2093	C₂×C₁₀₉	1	2	yes	complemented
208	#2083	C₂×C₁₀₉	1	2	yes	complemented
209	#2067	C₂×C₁₀₉	1	2	yes	complemented
210	#1984	C₂×C₁₀₉	1	1	yes	characteristic, complemented
211	#1970	C₂×C₁₀₉	1	1	yes	characteristic
212	#1942	C₂×C₁₀₉	1	1	yes	characteristic, complemented
213	#1879	C₂×C₁₀₉	1	2	yes	—
214	#1862	C₂×C₁₀₉	1	2	yes	complemented
215	C₁₀₉	#32648	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
217	C₂×C₂²⋊Q₈	—	2	2	no	—
218	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
219	C₂×C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
220	C₂×#107	C₂²×C₁₀₉	1	1	yes	characteristic
221	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
222	C₂×#107	C₂²×C₁₀₉	1	1	yes	characteristic
223	C₂×#107	C₂²×C₁₀₉	1	1	yes	characteristic
224	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
225	C₂×#102	C₂²×C₁₀₉	1	2	yes	—
226	C₂×#102	C₂²×C₁₀₉	1	2	yes	—
227	C₂×#102	C₂²×C₁₀₉	1	2	yes	—
228	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
229	#396	—	2	2	no	—
230	#396	—	2	2	no	—
231	#390	C₂²×C₁₀₉	1	2	yes	—
232	#390	C₂²×C₁₀₉	1	2	yes	—
233	#389	C₂²×C₁₀₉	1	2	yes	—
234	#389	C₂²×C₁₀₉	1	2	yes	—
235	#384	C₂²×C₁₀₉	1	2	yes	—
236	#380	—	2	2	no	—
237	#340	C₂²×C₁₀₉	1	2	yes	—
238	#340	C₂²×C₁₀₉	1	2	yes	—
239	#336	C₂²×C₁₀₉	1	2	yes	—
240	#336	C₂²×C₁₀₉	1	2	yes	—
241	C₂²×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
242	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
243	C₂³×C₄	—	2	2	no	—
244	C₂×D₈	—	4	1	no	—
245	C₂×D₈	—	4	1	no	—
246	C₂×D₈	Q₈×C₁₀₉	1	4	yes	complemented
247	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
248	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
249	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
250	C₂²⋊Q₈	—	4	8	no	—
251	C₄×Q₈	—	4	1	no	—
252	C₄×Q₈	—	4	1	no	—
253	C₄×D₄	—	4	2	no	—
254	C₄×D₄	—	4	2	no	—
255	C₂×C₄⋊C₄	—	2	2	no	—
256	C₂×C₄⋊C₄	—	2	2	no	—
257	C₂×C₄⋊C₄	—	2	2	no	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
260	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
261	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
262	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
263	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
264	C₂×C₂²⋊C₄	—	2	2	no	—
265	C₂×C₂²⋊C₄	—	2	2	no	—
266	C₂×C₂²⋊C₄	—	2	2	no	—
267	C₂×C₂²⋊C₄	—	2	2	no	—
268	C₂×C₂²⋊C₄	—	2	2	no	—
269	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
270	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
271	#107	—	2	2	no	—
272	#107	—	4	2	no	—
273	#107	—	4	1	no	—
274	#107	—	4	1	no	—
275	C₄⋊C₈	—	4	1	no	—
276	#102	—	2	4	no	—
277	#102	—	4	2	no	—
278	#102	—	4	2	no	—
279	C₄×C₈	—	4	1	no	—
280	#95	—	2	2	no	—
281	#95	—	2	2	no	—
282	C₂⁴	—	2	2	no	—
283	C₂×Q₈	—	4	1	no	—
284	C₂×Q₈	—	4	1	no	—
285	C₂×Q₈	D₈×C₁₀₉	1	4	yes	complemented
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	2	no	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	—	2	2	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	C₂×Q₈×C₁₀₉	1	4	yes	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	4	4	no	—
296	C₂²×C₄	—	4	4	no	—
297	C₂²×C₄	—	4	4	no	—
298	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
300	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
301	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
302	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
303	D₈	—	4	4	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	C₂×Q₈×C₁₀₉	1	2	yes	—
312	C₂×C₈	#41×C₁₀₉	1	2	yes	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	4	2	no	—
320	C₄⋊C₄	—	4	1	no	—
321	C₄⋊C₄	—	4	4	no	—
322	C₄⋊C₄	—	4	4	no	—
323	C₄⋊C₄	—	4	4	no	—
324	C₄⋊C₄	—	8	2	no	—
325	C₂²⋊C₄	—	4	4	no	—
326	C₂²⋊C₄	—	4	4	no	—
327	C₂²⋊C₄	—	4	4	no	—
328	C₂²⋊C₄	—	8	2	no	—
329	C₂²⋊C₄	—	8	2	no	—
330	C₄²	—	2	1	no	—
331	C₄²	—	2	1	no	—
332	C₄²	—	2	2	no	—
333	C₄²	—	2	2	no	—
334	C₂³	—	2	4	no	—
335	C₂³	—	4	2	no	—
336	C₂³	—	4	2	no	—
337	C₂³	—	4	2	no	—
338	C₂³	—	4	2	no	—
339	C₂³	—	4	2	no	—
340	C₂³	—	4	2	no	—
341	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
342	Q₈	—	4	4	no	—
343	D₄	—	2	4	no	—
344	D₄	—	2	4	no	—
345	D₄	—	4	2	no	—
346	D₄	—	4	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	8	8	no	—
367	C₂×C₄	C₂×D₈×C₁₀₉	1	2	yes	—
368	C₂×C₄	C₂×D₈×C₁₀₉	1	4	yes	—
369	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
370	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
371	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
372	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
373	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
374	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
375	C₈	—	2	2	no	—
376	C₈	—	2	2	no	—
377	C₈	—	4	2	no	—
378	C₂²	—	4	4	no	—
379	C₂²	—	4	4	no	—
380	C₂²	—	4	4	no	—
381	C₂²	—	8	2	no	—
382	C₂²	—	8	2	no	—
383	C₂²	—	8	2	no	—
384	C₂²	—	8	2	no	—
385	C₂²	#534×C₁₀₉	1	1	yes	characteristic
386	C₂²	D₄×Q₈×C₁₀₉	1	1	yes	characteristic
387	C₂²	#550×C₁₀₉	1	1	yes	characteristic
388	C₂²	C₂²×D₈×C₁₀₉	1	1	yes	characteristic
389	C₂²	#575×C₁₀₉	1	1	yes	characteristic
390	C₂²	#576×C₁₀₉	1	1	yes	characteristic
391	C₂²	#578×C₁₀₉	1	1	yes	characteristic
392	C₄	—	2	4	no	—
393	C₄	—	2	2	no	—
394	C₄	—	4	2	no	—
395	C₄	—	4	1	no	—
396	C₄	—	8	2	no	—
397	C₄	—	8	2	no	—
398	C₄	—	8	2	no	—
399	C₄	#539×C₁₀₉	1	1	yes	characteristic
400	C₄	#543×C₁₀₉	1	1	yes	characteristic
401	C₄	#554×C₁₀₉	1	1	yes	characteristic
402	C₄	#562×C₁₀₉	1	1	yes	characteristic
403	C₂	—	8	4	no	—
404	C₂	C₁₀₉×#2732	1	1	yes	characteristic
405	C₂	C₁₀₉×#3186	1	1	yes	characteristic
406	C₂	C₁₀₉×#3203	1	1	yes	characteristic
407	C₂	Q₈×D₈×C₁₀₉	1	1	yes	characteristic
408	C₂	C₁₀₉×#3389	1	1	yes	characteristic
409	C₂	C₁₀₉×#3391	1	1	yes	characteristic
410	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 ◁ C₂×C₄×Q₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₀₉×#2538	C₂	semidirect
2. C₁₀₉×#2093	C₂	semidirect
3. C₁₀₉×#2083	C₂	semidirect
4. C₁₀₉×#2067	C₂	semidirect
5. C₁₀₉×#1984	C₂	semidirect
7. C₁₀₉×#1942	C₂	semidirect
9. C₁₀₉×#1862	C₂	semidirect
40. C₂×D₈×C₁₀₉	Q₈	semidirect
79. C₂×Q₈×C₁₀₉	D₈	semidirect
197. #32648	C₁₀₉	direct (N × H)
206. #2538	C₂×C₁₀₉	semidirect
207. #2093	C₂×C₁₀₉	semidirect
208. #2083	C₂×C₁₀₉	semidirect
209. #2067	C₂×C₁₀₉	semidirect
210. #1984	C₂×C₁₀₉	semidirect
212. #1942	C₂×C₁₀₉	semidirect
214. #1862	C₂×C₁₀₉	semidirect
215. C₁₀₉	#32648	direct (N × H)
246. C₂×D₈	Q₈×C₁₀₉	semidirect
285. C₂×Q₈	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×#2538	2
C₁₀₉×#2093	2
C₁₀₉×#2083	2
C₁₀₉×#2067	2
C₁₀₉×#1984	2
C₁₀₉×#1970	2
C₁₀₉×#1942	2
C₁₀₉×#1879	2
C₁₀₉×#1862	2
#32648	109

Subgroups — C109×#32648 #42150642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#32648 #42150642