Subgroups — C₁₀₉×#34172 #42152166

Order 27904.

5430 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#518×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₁₀₉×#2339	C₂	1	16	yes	maximal, complemented
4	C₁₀₉×#2310	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#2308	C₂	1	4	yes	maximal, complemented
6	C₁₀₉×#2280	C₂	1	2	yes	maximal, complemented
7	C₂×#381×C₁₀₉	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₁₀₉	—	2	16	no	—
9	C₂²×#41×C₁₀₉	C₄	1	16	yes	complemented
10	C₂²×#41×C₁₀₉	C₂²	1	8	yes	—
11	C₂³×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₁₀₉	C₂²	1	2	yes	—
13	C₂³×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₁₀₉	C₂²	1	2	yes	—
15	C₂⁴×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	#518×C₁₀₉	—	2	16	no	—
17	C₂×C₄×D₄×C₁₀₉	C₂²	1	16	yes	complemented
18	C₂×#117×C₁₀₉	C₂²	1	4	yes	—
19	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	8	yes	—
20	#391×C₁₀₉	C₂²	1	16	yes	complemented
21	#386×C₁₀₉	C₂²	1	32	yes	complemented
22	#381×C₁₀₉	C₂²	1	4	yes	—
23	#381×C₁₀₉	C₂²	1	8	yes	—
24	#381×C₁₀₉	C₂²	1	16	yes	complemented
25	#380×C₁₀₉	C₂²	1	8	yes	complemented
26	#379×C₁₀₉	C₂²	1	4	yes	—
27	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	8	yes	complemented
28	C₂×#95×C₁₀₉	C₂²	1	16	yes	complemented
29	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
30	C₂⁵×C₁₀₉	C₂³	1	2	yes	—
31	C₂×#41×C₁₀₉	—	2	128	no	—
32	C₂×#41×C₁₀₉	—	2	64	no	—
33	C₂×#41×C₁₀₉	C₂×C₄	1	64	yes	complemented
34	C₂²×Q₈×C₁₀₉	—	2	8	no	—
35	C₂²×Q₈×C₁₀₉	C₂×C₄	1	8	yes	complemented
36	C₂²×Q₈×C₁₀₉	C₂³	1	4	yes	—
37	C₂²×D₄×C₁₀₉	—	2	16	no	—
38	C₂²×D₄×C₁₀₉	C₂×C₄	1	8	yes	—
39	C₂²×D₄×C₁₀₉	C₂×C₄	1	16	yes	complemented
40	C₂²×D₄×C₁₀₉	D₄	1	16	yes	—
41	C₂²×D₄×C₁₀₉	C₂³	1	8	yes	—
42	C₂²×D₄×C₁₀₉	C₂³	1	4	yes	—
43	C₂³×C₄×C₁₀₉	—	2	16	no	—
44	C₂³×C₄×C₁₀₉	—	2	8	no	—
45	C₂³×C₄×C₁₀₉	C₂×C₄	1	16	yes	complemented
46	C₂³×C₄×C₁₀₉	C₂×C₄	1	8	yes	complemented
47	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
49	C₂³×C₄×C₁₀₉	C₂³	1	8	yes	—
50	C₂³×C₄×C₁₀₉	C₂³	1	4	yes	—
51	C₂³×C₄×C₁₀₉	C₂³	1	16	yes	—
52	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₄×D₄×C₁₀₉	—	2	64	no	—
54	#117×C₁₀₉	—	2	16	no	—
55	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
56	C₂×C₂²⋊C₄×C₁₀₉	—	2	32	no	—
57	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
58	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	16	yes	—
59	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	16	yes	—
60	C₂×C₄²×C₁₀₉	C₂³	1	8	yes	—
61	#95×C₁₀₉	C₂³	1	16	yes	—
62	#95×C₁₀₉	C₂³	1	32	yes	—
63	C₂⁴×C₁₀₉	—	2	4	no	—
64	C₂⁴×C₁₀₉	—	4	8	no	—
65	C₂⁴×C₁₀₉	—	4	4	no	—
66	C₂⁴×C₁₀₉	C₂²×C₄	1	4	yes	—
67	C₂⁴×C₁₀₉	C₂²×C₄	1	8	yes	—
68	C₂⁴×C₁₀₉	C₂×D₄	1	16	yes	—
69	C₂⁴×C₁₀₉	C₂⁴	1	2	yes	—
70	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
71	C₂⁴×C₁₀₉	C₂⁴	1	4	yes	—
72	#41×C₁₀₉	—	2	256	no	—
73	C₂×Q₈×C₁₀₉	—	2	32	no	—
74	C₂×Q₈×C₁₀₉	—	2	16	no	—
75	C₂×Q₈×C₁₀₉	C₂²×C₄	1	16	yes	—
76	C₂×D₄×C₁₀₉	—	2	64	no	—
77	C₂×D₄×C₁₀₉	—	2	32	no	—
78	C₂×D₄×C₁₀₉	C₂²⋊C₄	1	64	yes	complemented
79	C₂×D₄×C₁₀₉	C₂²×C₄	1	32	yes	—
80	C₂×D₄×C₁₀₉	C₂²×C₄	1	16	yes	—
81	C₂×D₄×C₁₀₉	C₂×D₄	1	32	yes	—
82	C₂²×C₄×C₁₀₉	—	2	64	no	—
83	C₂²×C₄×C₁₀₉	—	2	4	no	—
84	C₂²×C₄×C₁₀₉	—	2	8	no	—
85	C₂²×C₄×C₁₀₉	—	2	32	no	—
86	C₂²×C₄×C₁₀₉	—	2	32	no	—
87	C₂²×C₄×C₁₀₉	—	2	16	no	—
88	C₂²×C₄×C₁₀₉	—	2	32	no	—
89	C₂²×C₄×C₁₀₉	—	4	32	no	—
90	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
91	C₂²×C₄×C₁₀₉	C₂²×C₄	1	8	yes	—
92	C₂²×C₄×C₁₀₉	C₂²×C₄	1	32	yes	—
93	C₂²×C₄×C₁₀₉	C₂²×C₄	1	16	yes	—
94	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
95	C₂²×C₄×C₁₀₉	C₂×D₄	1	8	yes	—
96	C₂²×C₄×C₁₀₉	C₂⁴	1	16	yes	—
97	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
98	C₂²×C₄×C₁₀₉	C₂⁴	1	4	yes	—
99	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
100	C₄⋊C₄×C₁₀₉	—	2	16	no	—
101	C₂²⋊C₄×C₁₀₉	—	2	16	no	—
102	C₂²⋊C₄×C₁₀₉	—	2	32	no	—
103	C₂²⋊C₄×C₁₀₉	—	4	16	no	—
104	C₄²×C₁₀₉	—	2	16	no	—
105	C₂³×C₁₀₉	—	2	16	no	—
106	C₂³×C₁₀₉	—	2	4	no	—
107	C₂³×C₁₀₉	—	2	8	no	—
108	C₂³×C₁₀₉	—	2	4	no	—
109	C₂³×C₁₀₉	—	2	8	no	—
110	C₂³×C₁₀₉	—	4	16	no	—
111	C₂³×C₁₀₉	—	4	8	no	—
112	C₂³×C₁₀₉	—	4	32	no	—
113	C₂³×C₁₀₉	—	4	16	no	—
114	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	32	yes	—
115	C₂³×C₁₀₉	C₂³×C₄	1	4	yes	—
116	C₂³×C₁₀₉	C₂³×C₄	1	2	yes	—
117	C₂³×C₁₀₉	C₂³×C₄	1	8	yes	—
118	C₂³×C₁₀₉	C₂²×D₄	1	4	yes	—
119	C₂³×C₁₀₉	C₂²×D₄	1	16	yes	—
120	C₂³×C₁₀₉	C₂⁵	1	1	yes	characteristic
121	Q₈×C₁₀₉	—	2	32	no	—
122	D₄×C₁₀₉	—	2	64	no	—
123	D₄×C₁₀₉	C₂×C₂²⋊C₄	1	64	yes	—
124	C₂×C₄×C₁₀₉	—	2	8	no	—
125	C₂×C₄×C₁₀₉	—	2	16	no	—
126	C₂×C₄×C₁₀₉	—	2	64	no	—
127	C₂×C₄×C₁₀₉	—	2	4	no	—
128	C₂×C₄×C₁₀₉	—	2	8	no	—
129	C₂×C₄×C₁₀₉	—	2	32	no	—
130	C₂×C₄×C₁₀₉	—	2	16	no	—
131	C₂×C₄×C₁₀₉	—	2	32	no	—
132	C₂×C₄×C₁₀₉	—	8	32	no	—
133	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	16	yes	—
134	C₂×C₄×C₁₀₉	C₂³×C₄	1	4	yes	—
135	C₂×C₄×C₁₀₉	C₂³×C₄	1	8	yes	—
136	C₂×C₄×C₁₀₉	C₂³×C₄	1	4	yes	—
137	C₂×C₄×C₁₀₉	C₂²×D₄	1	8	yes	—
138	C₂²×C₁₀₉	—	2	4	no	—
139	C₂²×C₁₀₉	—	2	16	no	—
140	C₂²×C₁₀₉	—	2	32	no	—
141	C₂²×C₁₀₉	—	2	8	no	—
142	C₂²×C₁₀₉	—	2	4	no	—
143	C₂²×C₁₀₉	—	2	8	no	—
144	C₂²×C₁₀₉	—	4	32	no	—
145	C₂²×C₁₀₉	—	4	16	no	—
146	C₂²×C₁₀₉	—	4	8	no	—
147	C₂²×C₁₀₉	C₂²×C₂²⋊C₄	1	4	yes	—
148	C₂²×C₁₀₉	C₂²×C₂²⋊C₄	1	16	yes	—
149	C₂²×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
150	C₂²×C₁₀₉	C₂³×D₄	1	2	yes	—
151	C₂²×C₁₀₉	#585	1	4	yes	—
152	C₄×C₁₀₉	—	2	4	no	—
153	C₄×C₁₀₉	—	2	8	no	—
154	C₄×C₁₀₉	—	4	16	no	—
155	C₄×C₁₀₉	C₂²×C₂²⋊C₄	1	8	yes	—
156	#34172	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂×C₁₀₉	—	2	4	no	—
158	C₂×C₁₀₉	—	2	16	no	—
159	C₂×C₁₀₉	—	4	4	no	—
160	C₂×C₁₀₉	C₂³×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₁₀₉	#3428	1	2	yes	—
162	C₂×C₁₀₉	#3485	1	4	yes	—
163	C₂³×#41	C₂×C₁₀₉	1	1	yes	characteristic
164	C₂×#518	C₂×C₁₀₉	1	2	yes	complemented
165	#2339	C₂×C₁₀₉	1	16	yes	complemented
166	#2310	C₂×C₁₀₉	1	2	yes	complemented
167	#2308	C₂×C₁₀₉	1	4	yes	complemented
168	#2280	C₂×C₁₀₉	1	2	yes	complemented
169	C₂×#381	C₂×C₁₀₉	1	4	yes	complemented
170	C₁₀₉	#34172	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²×#41	—	2	16	no	—
172	C₂²×#41	C₄×C₁₀₉	1	16	yes	complemented
173	C₂²×#41	C₂²×C₁₀₉	1	8	yes	—
174	C₂³×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
175	C₂³×D₄	C₂²×C₁₀₉	1	2	yes	—
176	C₂³×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
177	C₂⁴×C₄	C₂²×C₁₀₉	1	2	yes	—
178	C₂⁴×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
179	#518	—	2	16	no	—
180	C₂×C₄×D₄	C₂²×C₁₀₉	1	16	yes	complemented
181	C₂×#117	C₂²×C₁₀₉	1	4	yes	—
182	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	8	yes	—
183	#391	C₂²×C₁₀₉	1	16	yes	complemented
184	#386	C₂²×C₁₀₉	1	32	yes	complemented
185	#381	C₂²×C₁₀₉	1	8	yes	—
186	#381	C₂²×C₁₀₉	1	4	yes	—
187	#381	C₂²×C₁₀₉	1	16	yes	complemented
188	#380	C₂²×C₁₀₉	1	8	yes	complemented
189	#379	C₂²×C₁₀₉	1	4	yes	—
190	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	8	yes	complemented
191	C₂×#95	C₂²×C₁₀₉	1	16	yes	complemented
192	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
193	C₂⁵	C₂³×C₁₀₉	1	2	yes	—
194	C₂×#41	—	2	128	no	—
195	C₂×#41	—	2	64	no	—
196	C₂×#41	C₂×C₄×C₁₀₉	1	64	yes	complemented
197	C₂²×Q₈	—	2	8	no	—
198	C₂²×Q₈	C₂×C₄×C₁₀₉	1	8	yes	complemented
199	C₂²×Q₈	C₂³×C₁₀₉	1	4	yes	—
200	C₂²×D₄	—	2	16	no	—
201	C₂²×D₄	C₂×C₄×C₁₀₉	1	8	yes	—
202	C₂²×D₄	C₂×C₄×C₁₀₉	1	16	yes	complemented
203	C₂²×D₄	D₄×C₁₀₉	1	16	yes	—
204	C₂²×D₄	C₂³×C₁₀₉	1	8	yes	—
205	C₂²×D₄	C₂³×C₁₀₉	1	4	yes	—
206	C₂³×C₄	—	2	16	no	—
207	C₂³×C₄	—	2	8	no	—
208	C₂³×C₄	C₂×C₄×C₁₀₉	1	16	yes	complemented
209	C₂³×C₄	C₂×C₄×C₁₀₉	1	8	yes	complemented
210	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
211	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
212	C₂³×C₄	C₂³×C₁₀₉	1	8	yes	—
213	C₂³×C₄	C₂³×C₁₀₉	1	4	yes	—
214	C₂³×C₄	C₂³×C₁₀₉	1	16	yes	—
215	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
216	C₄×D₄	—	2	64	no	—
217	#117	—	2	16	no	—
218	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	8	yes	—
219	C₂×C₂²⋊C₄	—	2	32	no	—
220	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
221	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	16	yes	—
222	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	16	yes	—
223	C₂×C₄²	C₂³×C₁₀₉	1	8	yes	—
224	#95	C₂³×C₁₀₉	1	16	yes	—
225	#95	C₂³×C₁₀₉	1	32	yes	—
226	C₂⁴	—	2	4	no	—
227	C₂⁴	—	4	8	no	—
228	C₂⁴	—	4	4	no	—
229	C₂⁴	C₂²×C₄×C₁₀₉	1	4	yes	—
230	C₂⁴	C₂²×C₄×C₁₀₉	1	8	yes	—
231	C₂⁴	C₂×D₄×C₁₀₉	1	16	yes	—
232	C₂⁴	C₂⁴×C₁₀₉	1	2	yes	—
233	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
234	C₂⁴	C₂⁴×C₁₀₉	1	4	yes	—
235	#41	—	2	256	no	—
236	C₂×Q₈	—	2	32	no	—
237	C₂×Q₈	—	2	16	no	—
238	C₂×Q₈	C₂²×C₄×C₁₀₉	1	16	yes	—
239	C₂×D₄	—	2	64	no	—
240	C₂×D₄	—	2	32	no	—
241	C₂×D₄	C₂²⋊C₄×C₁₀₉	1	64	yes	complemented
242	C₂×D₄	C₂²×C₄×C₁₀₉	1	32	yes	—
243	C₂×D₄	C₂²×C₄×C₁₀₉	1	16	yes	—
244	C₂×D₄	C₂×D₄×C₁₀₉	1	32	yes	—
245	C₂²×C₄	—	2	64	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	8	no	—
248	C₂²×C₄	—	2	32	no	—
249	C₂²×C₄	—	2	32	no	—
250	C₂²×C₄	—	2	16	no	—
251	C₂²×C₄	—	2	32	no	—
252	C₂²×C₄	—	4	32	no	—
253	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
254	C₂²×C₄	C₂²×C₄×C₁₀₉	1	8	yes	—
255	C₂²×C₄	C₂²×C₄×C₁₀₉	1	32	yes	—
256	C₂²×C₄	C₂²×C₄×C₁₀₉	1	16	yes	—
257	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
258	C₂²×C₄	C₂×D₄×C₁₀₉	1	8	yes	—
259	C₂²×C₄	C₂⁴×C₁₀₉	1	16	yes	—
260	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
261	C₂²×C₄	C₂⁴×C₁₀₉	1	4	yes	—
262	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
263	C₄⋊C₄	—	2	16	no	—
264	C₂²⋊C₄	—	2	16	no	—
265	C₂²⋊C₄	—	2	32	no	—
266	C₂²⋊C₄	—	4	16	no	—
267	C₄²	—	2	16	no	—
268	C₂³	—	2	16	no	—
269	C₂³	—	2	4	no	—
270	C₂³	—	2	8	no	—
271	C₂³	—	2	4	no	—
272	C₂³	—	2	8	no	—
273	C₂³	—	4	16	no	—
274	C₂³	—	4	8	no	—
275	C₂³	—	4	32	no	—
276	C₂³	—	4	16	no	—
277	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	32	yes	—
278	C₂³	C₂³×C₄×C₁₀₉	1	4	yes	—
279	C₂³	C₂³×C₄×C₁₀₉	1	2	yes	—
280	C₂³	C₂³×C₄×C₁₀₉	1	8	yes	—
281	C₂³	C₂²×D₄×C₁₀₉	1	4	yes	—
282	C₂³	C₂²×D₄×C₁₀₉	1	16	yes	—
283	C₂³	C₂⁵×C₁₀₉	1	1	yes	characteristic
284	Q₈	—	2	32	no	—
285	D₄	—	2	64	no	—
286	D₄	C₂×C₂²⋊C₄×C₁₀₉	1	64	yes	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	2	16	no	—
289	C₂×C₄	—	2	64	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	8	no	—
292	C₂×C₄	—	2	32	no	—
293	C₂×C₄	—	2	16	no	—
294	C₂×C₄	—	2	32	no	—
295	C₂×C₄	—	8	32	no	—
296	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	16	yes	—
297	C₂×C₄	C₂³×C₄×C₁₀₉	1	4	yes	—
298	C₂×C₄	C₂³×C₄×C₁₀₉	1	8	yes	—
299	C₂×C₄	C₂³×C₄×C₁₀₉	1	4	yes	—
300	C₂×C₄	C₂²×D₄×C₁₀₉	1	8	yes	—
301	C₂²	—	2	32	no	—
302	C₂²	—	2	8	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	2	16	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	8	no	—
307	C₂²	—	4	32	no	—
308	C₂²	—	4	16	no	—
309	C₂²	—	4	8	no	—
310	C₂²	C₂²×C₂²⋊C₄×C₁₀₉	1	4	yes	—
311	C₂²	C₂²×C₂²⋊C₄×C₁₀₉	1	16	yes	—
312	C₂²	C₂⁴×C₄×C₁₀₉	1	1	yes	characteristic
313	C₂²	C₂³×D₄×C₁₀₉	1	2	yes	—
314	C₂²	#585×C₁₀₉	1	4	yes	—
315	C₄	—	2	4	no	—
316	C₄	—	2	8	no	—
317	C₄	—	4	16	no	—
318	C₄	C₂²×C₂²⋊C₄×C₁₀₉	1	8	yes	—
319	C₂	—	2	4	no	—
320	C₂	—	2	16	no	—
321	C₂	—	4	4	no	—
322	C₂	C₂³×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
323	C₂	C₁₀₉×#3428	1	2	yes	—
324	C₂	C₁₀₉×#3485	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₁₀₉
Frattini, cosocle	C₂³	C₂⁵×C₁₀₉
prime radical	C₂³×#41×C₁₀₉	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#518×C₁₀₉	C₂	semidirect
3. C₁₀₉×#2339	C₂	semidirect
4. C₁₀₉×#2310	C₂	semidirect
5. C₁₀₉×#2308	C₂	semidirect
6. C₁₀₉×#2280	C₂	semidirect
7. C₂×#381×C₁₀₉	C₂	semidirect
9. C₂²×#41×C₁₀₉	C₄	semidirect
17. C₂×C₄×D₄×C₁₀₉	C₂²	semidirect
20. #391×C₁₀₉	C₂²	semidirect
21. #386×C₁₀₉	C₂²	semidirect
24. #381×C₁₀₉	C₂²	semidirect
25. #380×C₁₀₉	C₂²	semidirect
27. C₄×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
28. C₂×#95×C₁₀₉	C₂²	semidirect
33. C₂×#41×C₁₀₉	C₂×C₄	semidirect
35. C₂²×Q₈×C₁₀₉	C₂×C₄	semidirect
39. C₂²×D₄×C₁₀₉	C₂×C₄	semidirect
45. C₂³×C₄×C₁₀₉	C₂×C₄	semidirect
46. C₂³×C₄×C₁₀₉	C₂×C₄	semidirect
78. C₂×D₄×C₁₀₉	C₂²⋊C₄	semidirect
156. #34172	C₁₀₉	direct (N × H)
164. C₂×#518	C₂×C₁₀₉	semidirect
165. #2339	C₂×C₁₀₉	semidirect
166. #2310	C₂×C₁₀₉	semidirect
167. #2308	C₂×C₁₀₉	semidirect
168. #2280	C₂×C₁₀₉	semidirect
169. C₂×#381	C₂×C₁₀₉	semidirect
170. C₁₀₉	#34172	direct (N × H)
172. C₂²×#41	C₄×C₁₀₉	semidirect
180. C₂×C₄×D₄	C₂²×C₁₀₉	semidirect
183. #391	C₂²×C₁₀₉	semidirect
184. #386	C₂²×C₁₀₉	semidirect
187. #381	C₂²×C₁₀₉	semidirect
188. #380	C₂²×C₁₀₉	semidirect
190. C₄×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
191. C₂×#95	C₂²×C₁₀₉	semidirect
196. C₂×#41	C₂×C₄×C₁₀₉	semidirect
198. C₂²×Q₈	C₂×C₄×C₁₀₉	semidirect
202. C₂²×D₄	C₂×C₄×C₁₀₉	semidirect
208. C₂³×C₄	C₂×C₄×C₁₀₉	semidirect
209. C₂³×C₄	C₂×C₄×C₁₀₉	semidirect
241. C₂×D₄	C₂²⋊C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#41×C₁₀₉	2
C₂×#518×C₁₀₉	2
C₁₀₉×#2339	2
C₁₀₉×#2310	2
C₁₀₉×#2308	2
C₁₀₉×#2280	2
C₂×#381×C₁₀₉	2
#34172	109

Subgroups — C109×#34172 #42152166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#34172 #42152166