Subgroups — C₁₀₉×#23667 #42141661

Order 27904.

874 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#424×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2317	C₂	1	2	yes	maximal
4	C₁₀₉×#2087	C₂	1	4	yes	maximal, complemented
5	C₁₀₉×#2086	C₂	1	4	yes	maximal, complemented
6	C₁₀₉×#1736	C₂	1	2	yes	maximal
7	C₁₀₉×#1735	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×#117×C₁₀₉	C₂²	1	2	yes	—
11	C₂×#117×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
13	C₂×#107×C₁₀₉	—	2	4	no	—
14	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
15	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
16	#424×C₁₀₉	—	2	4	no	—
17	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
19	#389×C₁₀₉	C₂²	1	8	yes	—
20	#385×C₁₀₉	—	2	4	no	—
21	#384×C₁₀₉	—	2	2	no	—
22	#382×C₁₀₉	—	2	2	no	—
23	#340×C₁₀₉	C₂²	1	2	yes	—
24	#340×C₁₀₉	C₂²	1	2	yes	—
25	#340×C₁₀₉	C₂²	1	8	yes	—
26	C₂³×C₄×C₁₀₉	—	2	2	no	—
27	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₁₀₉	—	2	2	no	—
29	C₂×M₁₆×C₁₀₉	—	2	4	no	—
30	C₂²×C₈×C₁₀₉	—	2	4	no	—
31	C₂²×C₈×C₁₀₉	D₄	1	4	yes	—
32	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
35	#117×C₁₀₉	—	2	2	no	—
36	#117×C₁₀₉	—	2	4	no	—
37	#117×C₁₀₉	—	2	2	no	—
38	#117×C₁₀₉	Q₈	1	8	yes	—
39	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
41	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₀₉	—	2	8	no	—
43	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
47	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
49	C₂×C₄²×C₁₀₉	—	2	2	no	—
50	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
51	#107×C₁₀₉	—	2	4	no	—
52	#107×C₁₀₉	—	2	4	no	—
53	C₄⋊C₈×C₁₀₉	—	2	4	no	—
54	C₄⋊C₈×C₁₀₉	—	2	4	no	—
55	#95×C₁₀₉	—	2	8	no	—
56	C₂⁴×C₁₀₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₉	—	2	4	no	—
58	C₂²×C₄×C₁₀₉	—	2	4	no	—
59	C₂²×C₄×C₁₀₉	—	4	4	no	—
60	C₂²×C₄×C₁₀₉	—	8	4	no	—
61	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
67	C₂²×C₄×C₁₀₉	#41	1	4	yes	—
68	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₁₀₉	—	4	4	no	—
70	C₂×C₈×C₁₀₉	—	2	8	no	—
71	C₂×C₈×C₁₀₉	—	2	2	no	—
72	C₂×C₈×C₁₀₉	—	2	2	no	—
73	C₂×C₈×C₁₀₉	—	2	2	no	—
74	C₂×C₈×C₁₀₉	—	2	4	no	—
75	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
77	C₄⋊C₄×C₁₀₉	—	2	2	no	—
78	C₄⋊C₄×C₁₀₉	—	2	2	no	—
79	C₄⋊C₄×C₁₀₉	—	2	4	no	—
80	C₄⋊C₄×C₁₀₉	—	2	4	no	—
81	C₄⋊C₄×C₁₀₉	—	2	4	no	—
82	C₄⋊C₄×C₁₀₉	—	4	4	no	—
83	C₄⋊C₄×C₁₀₉	—	8	8	no	—
84	C₄⋊C₄×C₁₀₉	C₂×Q₈	1	8	yes	—
85	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
86	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
87	C₄²×C₁₀₉	—	2	8	no	—
88	C₄²×C₁₀₉	—	2	2	no	—
89	C₄²×C₁₀₉	—	2	2	no	—
90	C₂³×C₁₀₉	—	2	2	no	—
91	C₂³×C₁₀₉	—	2	2	no	—
92	C₂³×C₁₀₉	#121	1	2	yes	—
93	C₂³×C₁₀₉	C₂²⋊Q₈	1	4	yes	—
94	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	—	2	2	no	—
96	C₂×C₄×C₁₀₉	—	2	2	no	—
97	C₂×C₄×C₁₀₉	—	2	8	no	—
98	C₂×C₄×C₁₀₉	—	2	4	no	—
99	C₂×C₄×C₁₀₉	—	2	4	no	—
100	C₂×C₄×C₁₀₉	—	4	4	no	—
101	C₂×C₄×C₁₀₉	—	4	4	no	—
102	C₂×C₄×C₁₀₉	—	4	4	no	—
103	C₂×C₄×C₁₀₉	—	4	8	no	—
104	C₂×C₄×C₁₀₉	—	8	8	no	—
105	C₂×C₄×C₁₀₉	#121	1	2	yes	—
106	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	4	yes	—
107	C₂×C₄×C₁₀₉	#123	1	4	yes	—
108	C₂×C₄×C₁₀₉	#125	1	8	yes	—
109	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	4	yes	—
110	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₀₉	C₂²×Q₈	1	2	yes	—
112	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
113	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
114	C₈×C₁₀₉	—	2	2	no	—
115	C₈×C₁₀₉	—	2	2	no	—
116	C₈×C₁₀₉	—	4	2	no	—
117	C₂²×C₁₀₉	—	2	8	no	—
118	C₂²×C₁₀₉	—	2	2	no	—
119	C₂²×C₁₀₉	—	2	2	no	—
120	C₂²×C₁₀₉	#398	1	4	yes	—
121	C₂²×C₁₀₉	C₂×#121	1	1	yes	characteristic
122	C₂²×C₁₀₉	C₂×C₂²⋊Q₈	1	2	yes	—
123	C₂²×C₁₀₉	#576	1	2	yes	—
124	C₂²×C₁₀₉	#578	1	2	yes	—
125	C₄×C₁₀₉	—	4	8	no	—
126	C₄×C₁₀₉	—	4	2	no	—
127	C₄×C₁₀₉	—	4	2	no	—
128	C₄×C₁₀₉	—	8	4	no	—
129	C₄×C₁₀₉	#398	1	4	yes	—
130	C₄×C₁₀₉	C₂×#123	1	1	yes	characteristic
131	C₄×C₁₀₉	C₂×#125	1	2	yes	—
132	C₄×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
133	#23667	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂×C₁₀₉	—	2	4	no	—
135	C₂×C₁₀₉	C₂×#398	1	1	yes	characteristic
136	C₂×C₁₀₉	#3058	1	1	yes	characteristic
137	C₂×C₁₀₉	#3059	1	1	yes	characteristic
138	C₂×C₁₀₉	#3083	1	4	yes	—
139	C₂²×#107	C₂×C₁₀₉	1	1	yes	characteristic
140	C₂×#424	C₂×C₁₀₉	1	1	yes	characteristic
141	#2317	C₂×C₁₀₉	1	2	yes	—
142	#2087	C₂×C₁₀₉	1	4	yes	complemented
143	#2086	C₂×C₁₀₉	1	4	yes	complemented
144	#1736	C₂×C₁₀₉	1	2	yes	—
145	#1735	C₂×C₁₀₉	1	1	yes	characteristic
146	C₁₀₉	#23667	1	1	yes	characteristic, direct product factor, Sylow
147	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
148	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
149	C₂×#117	C₂²×C₁₀₉	1	2	yes	—
150	C₂×#117	C₂²×C₁₀₉	1	1	yes	characteristic
151	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
152	C₂×#107	—	2	4	no	—
153	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
154	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
155	#424	—	2	4	no	—
156	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
157	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
158	#389	C₂²×C₁₀₉	1	8	yes	—
159	#385	—	2	4	no	—
160	#384	—	2	2	no	—
161	#382	—	2	2	no	—
162	#340	C₂²×C₁₀₉	1	2	yes	—
163	#340	C₂²×C₁₀₉	1	8	yes	—
164	#340	C₂²×C₁₀₉	1	2	yes	—
165	C₂³×C₄	—	2	2	no	—
166	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	4	no	—
169	C₂²×C₈	—	2	4	no	—
170	C₂²×C₈	D₄×C₁₀₉	1	4	yes	—
171	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
172	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
174	#117	—	2	2	no	—
175	#117	—	2	4	no	—
176	#117	—	2	2	no	—
177	#117	Q₈×C₁₀₉	1	8	yes	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	8	no	—
182	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
183	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
185	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
190	#107	—	2	4	no	—
191	#107	—	2	4	no	—
192	C₄⋊C₈	—	2	4	no	—
193	C₄⋊C₈	—	2	4	no	—
194	#95	—	2	8	no	—
195	C₂⁴	C₂×D₄×C₁₀₉	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	—	8	4	no	—
200	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
201	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
202	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
204	C₂²×C₄	C₂×Q₈×C₁₀₉	1	4	yes	—
205	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
206	C₂²×C₄	#41×C₁₀₉	1	4	yes	—
207	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
208	M₁₆	—	4	4	no	—
209	C₂×C₈	—	2	8	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
215	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	8	8	no	—
223	C₄⋊C₄	C₂×Q₈×C₁₀₉	1	8	yes	—
224	C₂²⋊C₄	—	2	4	no	—
225	C₂²⋊C₄	—	2	8	no	—
226	C₄²	—	2	8	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	#121×C₁₀₉	1	2	yes	—
232	C₂³	C₂²⋊Q₈×C₁₀₉	1	4	yes	—
233	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	8	8	no	—
244	C₂×C₄	#121×C₁₀₉	1	2	yes	—
245	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	4	yes	—
246	C₂×C₄	#123×C₁₀₉	1	4	yes	—
247	C₂×C₄	#125×C₁₀₉	1	8	yes	—
248	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	4	yes	—
249	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
250	C₂×C₄	C₂²×Q₈×C₁₀₉	1	2	yes	—
251	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
252	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	2	2	no	—
255	C₈	—	4	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	8	no	—
259	C₂²	#398×C₁₀₉	1	4	yes	—
260	C₂²	C₂×#121×C₁₀₉	1	1	yes	characteristic
261	C₂²	C₂×C₂²⋊Q₈×C₁₀₉	1	2	yes	—
262	C₂²	#576×C₁₀₉	1	2	yes	—
263	C₂²	#578×C₁₀₉	1	2	yes	—
264	C₄	—	4	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	8	no	—
267	C₄	—	8	4	no	—
268	C₄	#398×C₁₀₉	1	4	yes	—
269	C₄	C₂×#123×C₁₀₉	1	1	yes	characteristic
270	C₄	C₂×#125×C₁₀₉	1	2	yes	—
271	C₄	C₂×C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
272	C₂	—	2	4	no	—
273	C₂	C₂×#398×C₁₀₉	1	1	yes	characteristic
274	C₂	C₁₀₉×#3058	1	1	yes	characteristic
275	C₂	C₁₀₉×#3059	1	1	yes	characteristic
276	C₂	C₁₀₉×#3083	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
prime radical	C₂⁴×C₁₀₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×#117×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₀₉×#2087	C₂	semidirect
5. C₁₀₉×#2086	C₂	semidirect
133. #23667	C₁₀₉	direct (N × H)
142. #2087	C₂×C₁₀₉	semidirect
143. #2086	C₂×C₁₀₉	semidirect
146. C₁₀₉	#23667	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#107×C₁₀₉	2
C₂×#424×C₁₀₉	2
C₁₀₉×#2317	2
C₁₀₉×#2087	2
C₁₀₉×#2086	2
C₁₀₉×#1736	2
C₁₀₉×#1735	2
#23667	109

Subgroups — C109×#23667 #42141661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#23667 #42141661