Subgroups — C₁₀₉×#3204 #28360458

Order 13952.

402 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#537×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	#483×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	#481×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#470×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#469×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#465×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#452×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	#449×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	#408×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₁₀₉	C₂²	1	2	yes	—
11	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
18	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
20	#121×C₁₀₉	—	2	2	no	—
21	#121×C₁₀₉	—	2	2	no	—
22	#121×C₁₀₉	—	2	2	no	—
23	#121×C₁₀₉	C₂²	1	2	yes	—
24	#121×C₁₀₉	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
26	#106×C₁₀₉	C₂²	1	2	yes	—
27	#106×C₁₀₉	C₂²	1	1	yes	characteristic
28	#106×C₁₀₉	C₂²	1	1	yes	characteristic
29	#103×C₁₀₉	C₂²	1	2	yes	—
30	#103×C₁₀₉	C₂²	1	2	yes	—
31	#102×C₁₀₉	C₂²	1	2	yes	—
32	#102×C₁₀₉	C₂²	1	2	yes	—
33	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
34	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
35	#41×C₁₀₉	—	4	1	no	—
36	#41×C₁₀₉	—	4	1	no	—
37	#41×C₁₀₉	—	4	2	no	—
38	#41×C₁₀₉	—	4	2	no	—
39	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
40	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
41	C₂×D₄×C₁₀₉	—	2	2	no	—
42	C₂×D₄×C₁₀₉	—	2	2	no	—
43	C₂×D₄×C₁₀₉	—	2	2	no	—
44	C₂×D₄×C₁₀₉	D₄	1	1	yes	characteristic
45	C₂×D₄×C₁₀₉	D₄	1	2	yes	—
46	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₀₉	—	2	2	no	—
48	C₂²×C₄×C₁₀₉	—	2	2	no	—
49	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
51	Q₁₆×C₁₀₉	—	4	1	no	—
52	SD₁₆×C₁₀₉	—	4	2	no	—
53	D₈×C₁₀₉	—	4	1	no	—
54	M₁₆×C₁₀₉	—	4	1	no	—
55	C₂×C₈×C₁₀₉	—	4	1	no	—
56	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
58	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₁₀₉	—	2	2	no	—
60	C₄⋊C₄×C₁₀₉	—	2	2	no	—
61	C₄⋊C₄×C₁₀₉	—	2	2	no	—
62	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
63	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
64	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
65	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
66	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
67	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
68	C₂³×C₁₀₉	—	2	2	no	—
69	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
70	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
71	Q₈×C₁₀₉	—	2	2	no	—
72	Q₈×C₁₀₉	—	2	2	no	—
73	Q₈×C₁₀₉	—	4	1	no	—
74	D₄×C₁₀₉	—	2	2	no	—
75	D₄×C₁₀₉	—	2	2	no	—
76	D₄×C₁₀₉	—	4	2	no	—
77	D₄×C₁₀₉	—	4	1	no	—
78	D₄×C₁₀₉	—	4	4	no	—
79	D₄×C₁₀₉	—	8	2	no	—
80	D₄×C₁₀₉	—	8	2	no	—
81	C₂×C₄×C₁₀₉	—	2	1	no	—
82	C₂×C₄×C₁₀₉	—	2	1	no	—
83	C₂×C₄×C₁₀₉	—	2	2	no	—
84	C₂×C₄×C₁₀₉	—	2	2	no	—
85	C₂×C₄×C₁₀₉	—	2	2	no	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	2	2	no	—
88	C₂×C₄×C₁₀₉	—	4	4	no	—
89	C₂×C₄×C₁₀₉	—	8	2	no	—
90	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
93	C₈×C₁₀₉	—	2	2	no	—
94	C₈×C₁₀₉	—	2	1	no	—
95	C₈×C₁₀₉	—	2	1	no	—
96	C₂²×C₁₀₉	—	2	2	no	—
97	C₂²×C₁₀₉	—	2	1	no	—
98	C₂²×C₁₀₉	—	2	2	no	—
99	C₂²×C₁₀₉	—	2	2	no	—
100	C₂²×C₁₀₉	—	2	2	no	—
101	C₂²×C₁₀₉	—	4	2	no	—
102	C₂²×C₁₀₉	—	4	2	no	—
103	C₂²×C₁₀₉	—	4	2	no	—
104	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₁₀₉	—	2	2	no	—
106	C₄×C₁₀₉	—	2	1	no	—
107	C₄×C₁₀₉	—	4	2	no	—
108	C₄×C₁₀₉	—	4	2	no	—
109	C₄×C₁₀₉	—	4	2	no	—
110	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
111	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂×C₁₀₉	—	4	2	no	—
113	C₂×C₁₀₉	—	4	1	no	—
114	C₂×C₁₀₉	—	8	2	no	—
115	C₂×C₁₀₉	#534	1	1	yes	characteristic
116	C₂×C₁₀₉	#577	1	2	yes	—
117	#3204	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₁₀₉	#3204	1	1	yes	characteristic, direct product factor, Sylow
119	#537	C₂×C₁₀₉	1	2	yes	complemented
120	#483	C₂×C₁₀₉	1	2	yes	complemented
121	#481	C₂×C₁₀₉	1	2	yes	complemented
122	#470	C₂×C₁₀₉	1	1	yes	characteristic, complemented
123	#469	C₂×C₁₀₉	1	1	yes	characteristic, complemented
124	#465	C₂×C₁₀₉	1	2	yes	complemented
125	#452	C₂×C₁₀₉	1	2	yes	complemented
126	#449	C₂×C₁₀₉	1	2	yes	complemented
127	#408	C₂×C₁₀₉	1	1	yes	characteristic, complemented
128	C₂×#41	C₂²×C₁₀₉	1	2	yes	—
129	C₂×#41	C₂²×C₁₀₉	1	1	yes	characteristic
130	C₂×Q₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
131	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
132	C₂×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
133	C₂×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
134	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
135	C₂²⋊Q₈	—	2	2	no	—
136	C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
137	C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
138	#121	—	2	2	no	—
139	#121	—	2	2	no	—
140	#121	—	2	2	no	—
141	#121	C₂²×C₁₀₉	1	2	yes	—
142	#121	C₂²×C₁₀₉	1	2	yes	—
143	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
144	#106	C₂²×C₁₀₉	1	2	yes	—
145	#106	C₂²×C₁₀₉	1	1	yes	characteristic
146	#106	C₂²×C₁₀₉	1	1	yes	characteristic
147	#103	C₂²×C₁₀₉	1	2	yes	—
148	#103	C₂²×C₁₀₉	1	2	yes	—
149	#102	C₂²×C₁₀₉	1	2	yes	—
150	#102	C₂²×C₁₀₉	1	2	yes	—
151	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
152	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
153	#41	—	4	1	no	—
154	#41	—	4	1	no	—
155	#41	—	4	2	no	—
156	#41	—	4	2	no	—
157	C₂×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
158	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	D₄×C₁₀₉	1	1	yes	characteristic
163	C₂×D₄	D₄×C₁₀₉	1	2	yes	—
164	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
168	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
169	Q₁₆	—	4	1	no	—
170	SD₁₆	—	4	2	no	—
171	D₈	—	4	1	no	—
172	M₁₆	—	4	1	no	—
173	C₂×C₈	—	4	1	no	—
174	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
175	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
176	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
181	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
188	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
189	Q₈	—	2	2	no	—
190	Q₈	—	2	2	no	—
191	Q₈	—	4	1	no	—
192	D₄	—	2	2	no	—
193	D₄	—	2	2	no	—
194	D₄	—	4	2	no	—
195	D₄	—	4	1	no	—
196	D₄	—	4	4	no	—
197	D₄	—	8	2	no	—
198	D₄	—	8	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	8	2	no	—
208	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
209	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
210	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	1	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	2	no	—
228	C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
229	C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
230	C₂	—	4	2	no	—
231	C₂	—	4	1	no	—
232	C₂	—	8	2	no	—
233	C₂	#534×C₁₀₉	1	1	yes	characteristic
234	C₂	#577×C₁₀₉	1	2	yes	—

	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 ◁ C₂×#41×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #537×C₁₀₉	C₂	semidirect
2. #483×C₁₀₉	C₂	semidirect
3. #481×C₁₀₉	C₂	semidirect
4. #470×C₁₀₉	C₂	semidirect
5. #469×C₁₀₉	C₂	semidirect
6. #465×C₁₀₉	C₂	semidirect
7. #452×C₁₀₉	C₂	semidirect
8. #449×C₁₀₉	C₂	semidirect
9. #408×C₁₀₉	C₂	semidirect
117. #3204	C₁₀₉	direct (N × H)
118. C₁₀₉	#3204	direct (N × H)
119. #537	C₂×C₁₀₉	semidirect
120. #483	C₂×C₁₀₉	semidirect
121. #481	C₂×C₁₀₉	semidirect
122. #470	C₂×C₁₀₉	semidirect
123. #469	C₂×C₁₀₉	semidirect
124. #465	C₂×C₁₀₉	semidirect
125. #452	C₂×C₁₀₉	semidirect
126. #449	C₂×C₁₀₉	semidirect
127. #408	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#537×C₁₀₉	2
#483×C₁₀₉	2
#481×C₁₀₉	2
#470×C₁₀₉	2
#469×C₁₀₉	2
#465×C₁₀₉	2
#452×C₁₀₉	2
#449×C₁₀₉	2
#408×C₁₀₉	2
#3204	109

Subgroups — C109×#3204 #28360458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3204 #28360458