Subgroups — C₁₀₉×#3045 #28360299

Order 13952.

504 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×AGL(1,ℤ/8)×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#135×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#535×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#520×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#464×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#463×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#460×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	#459×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	#424×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#417×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#102×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
15	AGL(1,ℤ/8)×C₁₀₉	—	2	4	no	—
16	#135×C₁₀₉	—	2	4	no	—
17	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
18	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
19	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
20	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
21	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
22	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
23	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
25	#127×C₁₀₉	C₂²	1	2	yes	—
26	#124×C₁₀₉	C₂²	1	2	yes	—
27	#121×C₁₀₉	—	2	1	no	—
28	#121×C₁₀₉	—	2	1	no	—
29	#120×C₁₀₉	—	2	2	no	—
30	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	complemented
31	C₄×D₄×C₁₀₉	—	2	2	no	—
32	C₄×D₄×C₁₀₉	C₂²	1	2	yes	complemented
33	#117×C₁₀₉	—	2	1	no	—
34	#117×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	complemented
38	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	complemented
39	#103×C₁₀₉	C₂²	1	2	yes	complemented
40	#102×C₁₀₉	C₂²	1	2	yes	—
41	#102×C₁₀₉	C₂²	1	2	yes	—
42	#102×C₁₀₉	C₂²	1	2	yes	complemented
43	C₂⁴×C₁₀₉	—	2	1	no	—
44	#41×C₁₀₉	—	2	2	no	—
45	#41×C₁₀₉	—	2	2	no	—
46	#41×C₁₀₉	—	2	2	no	—
47	#41×C₁₀₉	D₄	1	4	yes	complemented
48	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₁₀₉	—	2	1	no	—
51	C₂×D₄×C₁₀₉	—	2	1	no	—
52	C₂×D₄×C₁₀₉	—	2	2	no	—
53	C₂×D₄×C₁₀₉	—	2	1	no	—
54	C₂×D₄×C₁₀₉	—	2	2	no	—
55	C₂×D₄×C₁₀₉	—	2	2	no	—
56	C₂×D₄×C₁₀₉	—	2	2	no	—
57	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
58	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
59	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	—	2	1	no	—
61	C₂²×C₄×C₁₀₉	—	2	1	no	—
62	C₂²×C₄×C₁₀₉	—	2	1	no	—
63	C₂²×C₄×C₁₀₉	—	2	2	no	—
64	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
65	Q₁₆×C₁₀₉	—	2	2	no	—
66	SD₁₆×C₁₀₉	—	2	4	no	—
67	SD₁₆×C₁₀₉	—	2	2	no	—
68	SD₁₆×C₁₀₉	—	2	2	no	—
69	D₈×C₁₀₉	—	2	4	no	—
70	D₈×C₁₀₉	—	2	2	no	—
71	M₁₆×C₁₀₉	—	2	2	no	—
72	C₂×C₈×C₁₀₉	—	2	2	no	—
73	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
76	C₄⋊C₄×C₁₀₉	—	2	1	no	—
77	C₄⋊C₄×C₁₀₉	—	2	2	no	—
78	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
79	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
80	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
81	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
82	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
83	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
84	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
85	C₄²×C₁₀₉	—	2	2	no	—
86	C₄²×C₁₀₉	C₂³	1	2	yes	—
87	C₂³×C₁₀₉	—	2	1	no	—
88	C₂³×C₁₀₉	—	2	2	no	—
89	C₂³×C₁₀₉	—	2	1	no	—
90	C₂³×C₁₀₉	—	4	2	no	—
91	C₂³×C₁₀₉	—	8	1	no	—
92	C₂³×C₁₀₉	—	8	1	no	—
93	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
94	Q₈×C₁₀₉	—	2	1	no	—
95	Q₈×C₁₀₉	—	2	1	no	—
96	Q₈×C₁₀₉	—	2	1	no	—
97	Q₈×C₁₀₉	C₂×D₄	1	2	yes	—
98	D₄×C₁₀₉	—	2	4	no	—
99	D₄×C₁₀₉	—	2	2	no	—
100	D₄×C₁₀₉	—	2	1	no	—
101	D₄×C₁₀₉	—	2	1	no	—
102	D₄×C₁₀₉	—	2	2	no	—
103	D₄×C₁₀₉	—	2	1	no	—
104	D₄×C₁₀₉	—	4	2	no	—
105	D₄×C₁₀₉	—	4	2	no	—
106	D₄×C₁₀₉	—	8	2	no	—
107	D₄×C₁₀₉	—	8	2	no	—
108	D₄×C₁₀₉	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₀₉	—	2	1	no	—
110	C₂×C₄×C₁₀₉	—	2	4	no	—
111	C₂×C₄×C₁₀₉	—	2	2	no	—
112	C₂×C₄×C₁₀₉	—	2	1	no	—
113	C₂×C₄×C₁₀₉	—	4	2	no	—
114	C₂×C₄×C₁₀₉	—	8	2	no	—
115	C₂×C₄×C₁₀₉	—	8	1	no	—
116	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
120	C₂×C₄×C₁₀₉	#41	1	2	yes	—
121	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
122	C₈×C₁₀₉	—	2	1	no	—
123	C₈×C₁₀₉	—	2	1	no	—
124	C₈×C₁₀₉	—	2	2	no	—
125	C₂²×C₁₀₉	—	2	1	no	—
126	C₂²×C₁₀₉	—	2	1	no	—
127	C₂²×C₁₀₉	—	2	2	no	—
128	C₂²×C₁₀₉	—	4	1	no	—
129	C₂²×C₁₀₉	—	4	1	no	—
130	C₂²×C₁₀₉	—	4	1	no	—
131	C₂²×C₁₀₉	—	4	2	no	—
132	C₂²×C₁₀₉	—	4	1	no	—
133	C₂²×C₁₀₉	—	4	2	no	—
134	C₂²×C₁₀₉	—	4	1	no	—
135	C₂²×C₁₀₉	—	4	2	no	—
136	C₂²×C₁₀₉	—	8	4	no	—
137	C₂²×C₁₀₉	#121	1	2	yes	—
138	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₁₀₉	—	2	2	no	—
140	C₄×C₁₀₉	—	2	2	no	—
141	C₄×C₁₀₉	—	2	2	no	—
142	C₄×C₁₀₉	—	4	1	no	—
143	C₄×C₁₀₉	—	4	2	no	—
144	C₄×C₁₀₉	#121	1	2	yes	—
145	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
146	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
147	C₂×C₁₀₉	—	2	2	no	—
148	C₂×C₁₀₉	—	4	2	no	—
149	C₂×C₁₀₉	—	8	1	no	—
150	C₂×C₁₀₉	—	8	2	no	—
151	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
152	C₂×C₁₀₉	#576	1	1	yes	characteristic
153	C₂×C₁₀₉	#577	1	1	yes	characteristic
154	#3045	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₁₀₉	#3045	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×AGL(1,ℤ/8)	C₂×C₁₀₉	1	1	yes	characteristic, complemented
157	C₂×#135	C₂×C₁₀₉	1	1	yes	characteristic, complemented
158	#535	C₂×C₁₀₉	1	1	yes	characteristic, complemented
159	#520	C₂×C₁₀₉	1	1	yes	characteristic, complemented
160	#464	C₂×C₁₀₉	1	2	yes	complemented
161	#463	C₂×C₁₀₉	1	2	yes	complemented
162	#460	C₂×C₁₀₉	1	2	yes	complemented
163	#459	C₂×C₁₀₉	1	2	yes	complemented
164	#424	C₂×C₁₀₉	1	1	yes	characteristic, complemented
165	#417	C₂×C₁₀₉	1	1	yes	characteristic, complemented
166	C₂×#102	C₂×C₁₀₉	1	1	yes	characteristic, complemented
167	C₂×#41	C₂²×C₁₀₉	1	1	yes	characteristic
168	C₂×#41	C₂²×C₁₀₉	1	1	yes	characteristic
169	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
170	AGL(1,ℤ/8)	—	2	4	no	—
171	#135	—	2	4	no	—
172	C₂×Q₁₆	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
173	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
174	C₂×SD₁₆	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
175	C₂×SD₁₆	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
176	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
177	C₂×D₈	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
178	C₂×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
179	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
180	#127	C₂²×C₁₀₉	1	2	yes	—
181	#124	C₂²×C₁₀₉	1	2	yes	—
182	#121	—	2	1	no	—
183	#121	—	2	1	no	—
184	#120	—	2	2	no	—
185	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	complemented
186	C₄×D₄	—	2	2	no	—
187	C₄×D₄	C₂²×C₁₀₉	1	2	yes	complemented
188	#117	—	2	1	no	—
189	#117	C₂²×C₁₀₉	1	1	yes	characteristic
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
192	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	complemented
193	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	complemented
194	#103	C₂²×C₁₀₉	1	2	yes	complemented
195	#102	C₂²×C₁₀₉	1	2	yes	—
196	#102	C₂²×C₁₀₉	1	2	yes	—
197	#102	C₂²×C₁₀₉	1	2	yes	complemented
198	C₂⁴	—	2	1	no	—
199	#41	—	2	2	no	—
200	#41	—	2	2	no	—
201	#41	—	2	2	no	—
202	#41	D₄×C₁₀₉	1	4	yes	complemented
203	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
204	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
213	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
214	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
220	Q₁₆	—	2	2	no	—
221	SD₁₆	—	2	4	no	—
222	SD₁₆	—	2	2	no	—
223	SD₁₆	—	2	2	no	—
224	D₈	—	2	4	no	—
225	D₈	—	2	2	no	—
226	M₁₆	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
229	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
230	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
234	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
235	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
240	C₄²	—	2	2	no	—
241	C₄²	C₂³×C₁₀₉	1	2	yes	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	8	1	no	—
247	C₂³	—	8	1	no	—
248	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
249	Q₈	—	2	1	no	—
250	Q₈	—	2	1	no	—
251	Q₈	—	2	1	no	—
252	Q₈	C₂×D₄×C₁₀₉	1	2	yes	—
253	D₄	—	2	4	no	—
254	D₄	—	2	2	no	—
255	D₄	—	2	1	no	—
256	D₄	—	2	1	no	—
257	D₄	—	2	2	no	—
258	D₄	—	2	1	no	—
259	D₄	—	4	2	no	—
260	D₄	—	4	2	no	—
261	D₄	—	8	2	no	—
262	D₄	—	8	2	no	—
263	D₄	C₂×D₄×C₁₀₉	1	2	yes	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
272	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
273	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
274	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
275	C₂×C₄	#41×C₁₀₉	1	2	yes	—
276	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	2	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	8	4	no	—
292	C₂²	#121×C₁₀₉	1	2	yes	—
293	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	1	no	—
298	C₄	—	4	2	no	—
299	C₄	#121×C₁₀₉	1	2	yes	—
300	C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
301	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	4	2	no	—
304	C₂	—	8	2	no	—
305	C₂	—	8	1	no	—
306	C₂	C₂×#121×C₁₀₉	1	1	yes	characteristic
307	C₂	#576×C₁₀₉	1	1	yes	characteristic
308	C₂	#577×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ #117×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×AGL(1,ℤ/8)×C₁₀₉	C₂	semidirect
2. C₂×#135×C₁₀₉	C₂	semidirect
3. #535×C₁₀₉	C₂	semidirect
4. #520×C₁₀₉	C₂	semidirect
5. #464×C₁₀₉	C₂	semidirect
6. #463×C₁₀₉	C₂	semidirect
7. #460×C₁₀₉	C₂	semidirect
8. #459×C₁₀₉	C₂	semidirect
9. #424×C₁₀₉	C₂	semidirect
10. #417×C₁₀₉	C₂	semidirect
11. C₂×#102×C₁₀₉	C₂	semidirect
17. C₂×Q₁₆×C₁₀₉	C₂²	semidirect
19. C₂×SD₁₆×C₁₀₉	C₂²	semidirect
20. C₂×SD₁₆×C₁₀₉	C₂²	semidirect
22. C₂×D₈×C₁₀₉	C₂²	semidirect
30. C₄×Q₈×C₁₀₉	C₂²	semidirect
32. C₄×D₄×C₁₀₉	C₂²	semidirect
37. C₄⋊C₈×C₁₀₉	C₂²	semidirect
38. C₄⋊C₈×C₁₀₉	C₂²	semidirect
39. #103×C₁₀₉	C₂²	semidirect
42. #102×C₁₀₉	C₂²	semidirect
47. #41×C₁₀₉	D₄	semidirect
154. #3045	C₁₀₉	direct (N × H)
155. C₁₀₉	#3045	direct (N × H)
156. C₂×AGL(1,ℤ/8)	C₂×C₁₀₉	semidirect
157. C₂×#135	C₂×C₁₀₉	semidirect
158. #535	C₂×C₁₀₉	semidirect
159. #520	C₂×C₁₀₉	semidirect
160. #464	C₂×C₁₀₉	semidirect
161. #463	C₂×C₁₀₉	semidirect
162. #460	C₂×C₁₀₉	semidirect
163. #459	C₂×C₁₀₉	semidirect
164. #424	C₂×C₁₀₉	semidirect
165. #417	C₂×C₁₀₉	semidirect
166. C₂×#102	C₂×C₁₀₉	semidirect
172. C₂×Q₁₆	C₂²×C₁₀₉	semidirect
174. C₂×SD₁₆	C₂²×C₁₀₉	semidirect
175. C₂×SD₁₆	C₂²×C₁₀₉	semidirect
177. C₂×D₈	C₂²×C₁₀₉	semidirect
185. C₄×Q₈	C₂²×C₁₀₉	semidirect
187. C₄×D₄	C₂²×C₁₀₉	semidirect
192. C₄⋊C₈	C₂²×C₁₀₉	semidirect
193. C₄⋊C₈	C₂²×C₁₀₉	semidirect
194. #103	C₂²×C₁₀₉	semidirect
197. #102	C₂²×C₁₀₉	semidirect
202. #41	D₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×AGL(1,ℤ/8)×C₁₀₉	2
C₂×#135×C₁₀₉	2
#535×C₁₀₉	2
#520×C₁₀₉	2
#464×C₁₀₉	2
#463×C₁₀₉	2
#460×C₁₀₉	2
#459×C₁₀₉	2
#424×C₁₀₉	2
#417×C₁₀₉	2
C₂×#102×C₁₀₉	2
#3045	109

Subgroups — C109×#3045 #28360299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3045 #28360299