Subgroups — C₁₀₉×#17342 #42135336

Order 27904.

688 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2965	C₂	1	2	yes	maximal
2	C₁₀₉×#2922	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1455	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#1454	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#1453	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#1453	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₀₉×#1452	C₂	1	2	yes	maximal, complemented
8	C₁₀₉×#1451	C₂	1	2	yes	maximal, complemented
9	C₁₀₉×#1451	C₂	1	2	yes	maximal, complemented
10	C₄×#41×C₁₀₉	C₂²	1	1	yes	characteristic
11	#446×C₁₀₉	C₄	1	4	yes	—
12	#436×C₁₀₉	C₄	1	4	yes	—
13	#435×C₁₀₉	C₄	1	4	yes	—
14	#435×C₁₀₉	C₄	1	4	yes	—
15	#433×C₁₀₉	C₄	1	4	yes	—
16	#431×C₁₀₉	C₂²	1	2	yes	—
17	#431×C₁₀₉	C₂²	1	2	yes	—
18	#431×C₁₀₉	C₂²	1	2	yes	—
19	#431×C₁₀₉	C₂²	1	2	yes	—
20	#431×C₁₀₉	C₂²	1	2	yes	—
21	#424×C₁₀₉	C₄	1	2	yes	—
22	#408×C₁₀₉	C₄	1	2	yes	—
23	C₄×M₁₆×C₁₀₉	C₂²	1	2	yes	—
24	C₄×M₁₆×C₁₀₉	C₂²	1	2	yes	—
25	C₄×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
28	#322×C₁₀₉	C₂²	1	2	yes	—
29	#322×C₁₀₉	C₂²	1	2	yes	—
30	#322×C₁₀₉	C₂²	1	1	yes	characteristic
31	#322×C₁₀₉	C₂²	1	2	yes	—
32	#322×C₁₀₉	C₂²	1	2	yes	—
33	#322×C₁₀₉	C₂²	1	2	yes	—
34	#322×C₁₀₉	C₂²	1	1	yes	characteristic
35	#322×C₁₀₉	C₂²	1	2	yes	—
36	C₈²×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₈²×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₂×#41×C₁₀₉	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
40	C₂×M₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
41	C₂×M₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
44	C₄×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
45	C₄×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
47	C₄×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
48	#117×C₁₀₉	C₂×C₄	1	1	yes	characteristic
49	#117×C₁₀₉	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
52	C₄⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
53	C₄⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
54	C₄⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
55	C₄⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
56	C₄⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
57	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
58	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
59	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
60	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
61	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
62	#97×C₁₀₉	C₂×C₄	1	2	yes	—
63	#97×C₁₀₉	C₂×C₄	1	4	yes	—
64	#97×C₁₀₉	C₂×C₄	1	4	yes	—
65	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
67	C₄×C₈×C₁₀₉	C₂³	1	2	yes	—
68	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₀₉	C₂³	1	2	yes	—
71	C₄×C₈×C₁₀₉	C₂³	1	2	yes	—
72	C₄×C₈×C₁₀₉	C₂³	1	2	yes	—
73	C₄×C₈×C₁₀₉	C₂³	1	2	yes	—
74	#41×C₁₀₉	—	4	2	no	—
75	C₂×Q₈×C₁₀₉	C₄²	1	1	yes	characteristic
76	C₂×D₄×C₁₀₉	C₄²	1	1	yes	characteristic
77	C₂×D₄×C₁₀₉	C₄²	1	2	yes	—
78	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
83	M₁₆×C₁₀₉	—	4	4	no	—
84	M₁₆×C₁₀₉	—	4	4	no	—
85	M₁₆×C₁₀₉	—	4	2	no	—
86	C₂×C₈×C₁₀₉	—	4	2	no	—
87	C₂×C₈×C₁₀₉	—	4	4	no	—
88	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
95	C₄⋊C₄×C₁₀₉	C₄²	1	2	yes	—
96	C₄⋊C₄×C₁₀₉	C₄²	1	2	yes	—
97	C₄⋊C₄×C₁₀₉	C₄²	1	2	yes	—
98	C₂²⋊C₄×C₁₀₉	C₄²	1	1	yes	characteristic
99	C₂²⋊C₄×C₁₀₉	C₄²	1	1	yes	characteristic
100	C₂²⋊C₄×C₁₀₉	C₄²	1	2	yes	—
101	C₂²⋊C₄×C₁₀₉	C₄²	1	2	yes	—
102	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
104	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
105	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
106	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
107	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
108	C₂³×C₁₀₉	C₂×C₄²	1	2	yes	—
109	C₂³×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
110	Q₈×C₁₀₉	—	4	1	no	—
111	D₄×C₁₀₉	—	4	1	no	—
112	D₄×C₁₀₉	—	4	2	no	—
113	C₂×C₄×C₁₀₉	—	2	2	no	—
114	C₂×C₄×C₁₀₉	—	2	4	no	—
115	C₂×C₄×C₁₀₉	—	2	4	no	—
116	C₂×C₄×C₁₀₉	—	2	4	no	—
117	C₂×C₄×C₁₀₉	—	2	4	no	—
118	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
125	C₈×C₁₀₉	—	2	4	no	—
126	C₈×C₁₀₉	—	2	4	no	—
127	C₈×C₁₀₉	—	2	4	no	—
128	C₈×C₁₀₉	—	2	2	no	—
129	C₈×C₁₀₉	—	2	2	no	—
130	C₈×C₁₀₉	—	2	4	no	—
131	C₈×C₁₀₉	—	2	4	no	—
132	C₂²×C₁₀₉	—	2	1	no	—
133	C₂²×C₁₀₉	—	2	2	no	—
134	C₂²×C₁₀₉	—	2	2	no	—
135	C₂²×C₁₀₉	—	2	2	no	—
136	C₂²×C₁₀₉	—	2	2	no	—
137	C₂²×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
138	C₄×C₁₀₉	—	2	1	no	—
139	C₄×C₁₀₉	—	2	2	no	—
140	C₄×C₁₀₉	—	2	2	no	—
141	C₄×C₁₀₉	—	2	2	no	—
142	C₄×C₁₀₉	—	2	2	no	—
143	C₄×C₁₀₉	#568	1	2	yes	—
144	C₄×C₁₀₉	#568	1	4	yes	—
145	#17342	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂×C₁₀₉	—	4	1	no	—
147	C₂×C₁₀₉	—	4	2	no	—
148	C₂×C₁₀₉	#2873	1	1	yes	characteristic
149	C₂×C₁₀₉	#2875	1	2	yes	—
150	#2965	C₂×C₁₀₉	1	2	yes	—
151	#2922	C₂×C₁₀₉	1	1	yes	characteristic
152	#1455	C₂×C₁₀₉	1	1	yes	characteristic, complemented
153	#1454	C₂×C₁₀₉	1	2	yes	complemented
154	#1453	C₂×C₁₀₉	1	2	yes	complemented
155	#1453	C₂×C₁₀₉	1	1	yes	characteristic, complemented
156	#1452	C₂×C₁₀₉	1	2	yes	complemented
157	#1451	C₂×C₁₀₉	1	2	yes	complemented
158	#1451	C₂×C₁₀₉	1	2	yes	complemented
159	C₁₀₉	#17342	1	1	yes	characteristic, direct product factor, Sylow
160	C₄×#41	C₂²×C₁₀₉	1	1	yes	characteristic
161	#446	C₄×C₁₀₉	1	4	yes	—
162	#436	C₄×C₁₀₉	1	4	yes	—
163	#435	C₄×C₁₀₉	1	4	yes	—
164	#435	C₄×C₁₀₉	1	4	yes	—
165	#433	C₄×C₁₀₉	1	4	yes	—
166	#431	C₂²×C₁₀₉	1	2	yes	—
167	#431	C₂²×C₁₀₉	1	2	yes	—
168	#431	C₂²×C₁₀₉	1	2	yes	—
169	#431	C₂²×C₁₀₉	1	2	yes	—
170	#431	C₂²×C₁₀₉	1	2	yes	—
171	#424	C₄×C₁₀₉	1	2	yes	—
172	#408	C₄×C₁₀₉	1	2	yes	—
173	C₄×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
174	C₄×M₁₆	C₂²×C₁₀₉	1	2	yes	—
175	C₄×M₁₆	C₂²×C₁₀₉	1	2	yes	—
176	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
177	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
178	#322	C₂²×C₁₀₉	1	2	yes	—
179	#322	C₂²×C₁₀₉	1	2	yes	—
180	#322	C₂²×C₁₀₉	1	1	yes	characteristic
181	#322	C₂²×C₁₀₉	1	2	yes	—
182	#322	C₂²×C₁₀₉	1	2	yes	—
183	#322	C₂²×C₁₀₉	1	2	yes	—
184	#322	C₂²×C₁₀₉	1	1	yes	characteristic
185	#322	C₂²×C₁₀₉	1	2	yes	—
186	C₈²	C₂²×C₁₀₉	1	1	yes	characteristic
187	C₈²	C₂²×C₁₀₉	1	1	yes	characteristic
188	C₂×#41	C₂×C₄×C₁₀₉	1	1	yes	characteristic
189	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
190	C₂×M₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
191	C₂×M₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
192	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
193	C₂²×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
194	C₄×Q₈	C₂×C₄×C₁₀₉	1	2	yes	—
195	C₄×D₄	C₂×C₄×C₁₀₉	1	2	yes	—
196	C₄×D₄	C₂×C₄×C₁₀₉	1	2	yes	—
197	C₄×D₄	C₂×C₄×C₁₀₉	1	2	yes	—
198	#117	C₂×C₄×C₁₀₉	1	1	yes	characteristic
199	#117	C₂×C₄×C₁₀₉	1	2	yes	—
200	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
202	C₄⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
203	C₄⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
204	C₄⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
205	C₄⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
206	C₄⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
207	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
208	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
209	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
210	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
211	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
212	#97	C₂×C₄×C₁₀₉	1	2	yes	—
213	#97	C₂×C₄×C₁₀₉	1	4	yes	—
214	#97	C₂×C₄×C₁₀₉	1	4	yes	—
215	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
216	C₄×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
217	C₄×C₈	C₂³×C₁₀₉	1	2	yes	—
218	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
219	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
220	C₄×C₈	C₂³×C₁₀₉	1	2	yes	—
221	C₄×C₈	C₂³×C₁₀₉	1	2	yes	—
222	C₄×C₈	C₂³×C₁₀₉	1	2	yes	—
223	C₄×C₈	C₂³×C₁₀₉	1	2	yes	—
224	#41	—	4	2	no	—
225	C₂×Q₈	C₄²×C₁₀₉	1	1	yes	characteristic
226	C₂×D₄	C₄²×C₁₀₉	1	1	yes	characteristic
227	C₂×D₄	C₄²×C₁₀₉	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
230	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
231	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
232	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
233	M₁₆	—	4	4	no	—
234	M₁₆	—	4	4	no	—
235	M₁₆	—	4	2	no	—
236	C₂×C₈	—	4	4	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
239	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
240	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
241	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
242	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
243	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
244	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
245	C₄⋊C₄	C₄²×C₁₀₉	1	2	yes	—
246	C₄⋊C₄	C₄²×C₁₀₉	1	2	yes	—
247	C₄⋊C₄	C₄²×C₁₀₉	1	2	yes	—
248	C₂²⋊C₄	C₄²×C₁₀₉	1	1	yes	characteristic
249	C₂²⋊C₄	C₄²×C₁₀₉	1	1	yes	characteristic
250	C₂²⋊C₄	C₄²×C₁₀₉	1	2	yes	—
251	C₂²⋊C₄	C₄²×C₁₀₉	1	2	yes	—
252	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
253	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
254	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
255	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
256	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
257	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
258	C₂³	C₂×C₄²×C₁₀₉	1	2	yes	—
259	C₂³	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
260	Q₈	—	4	1	no	—
261	D₄	—	4	1	no	—
262	D₄	—	4	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
269	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
270	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
271	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
272	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
273	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
274	C₂×C₄	C₂³×C₄×C₁₀₉	1	2	yes	—
275	C₈	—	2	4	no	—
276	C₈	—	2	4	no	—
277	C₈	—	2	4	no	—
278	C₈	—	2	2	no	—
279	C₈	—	2	2	no	—
280	C₈	—	2	4	no	—
281	C₈	—	2	4	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	C₂²×C₄²×C₁₀₉	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	#568×C₁₀₉	1	2	yes	—
294	C₄	#568×C₁₀₉	1	4	yes	—
295	C₂	—	4	1	no	—
296	C₂	—	4	2	no	—
297	C₂	C₁₀₉×#2873	1	1	yes	characteristic
298	C₂	C₁₀₉×#2875	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₀₉	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₀₉
Frattini, cosocle	C₄²	C₂⁴×C₁₀₉
socle	C₂²×C₁₀₉	C₂²×C₄²
prime radical	C₂×#41×C₁₀₉	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1455	C₂	semidirect
4. C₁₀₉×#1454	C₂	semidirect
5. C₁₀₉×#1453	C₂	semidirect
6. C₁₀₉×#1453	C₂	semidirect
7. C₁₀₉×#1452	C₂	semidirect
8. C₁₀₉×#1451	C₂	semidirect
9. C₁₀₉×#1451	C₂	semidirect
145. #17342	C₁₀₉	direct (N × H)
152. #1455	C₂×C₁₀₉	semidirect
153. #1454	C₂×C₁₀₉	semidirect
154. #1453	C₂×C₁₀₉	semidirect
155. #1453	C₂×C₁₀₉	semidirect
156. #1452	C₂×C₁₀₉	semidirect
157. #1451	C₂×C₁₀₉	semidirect
158. #1451	C₂×C₁₀₉	semidirect
159. C₁₀₉	#17342	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×#2965	2
C₁₀₉×#2922	2
C₁₀₉×#1455	2
C₁₀₉×#1454	2
C₁₀₉×#1453	2
C₁₀₉×#1452	2
C₁₀₉×#1451	2
#17342	109

Subgroups — C109×#17342 #42135336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#17342 #42135336