Subgroups — C₁₀₉×#3398 #28360652

Order 13952.

350 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#557×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#556×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#497×C₁₀₉	C₂	1	2	yes	maximal
4	#494×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#464×C₁₀₉	C₂	1	2	yes	maximal
6	#462×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#462×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#461×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#441×C₁₀₉	C₂	1	2	yes	maximal, complemented
11	C₄×SD₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
15	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
20	#126×C₁₀₉	—	2	2	no	—
21	#125×C₁₀₉	—	2	1	no	—
22	#125×C₁₀₉	—	2	2	no	—
23	#124×C₁₀₉	C₂²	1	2	yes	—
24	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
25	C₄×Q₈×C₁₀₉	—	2	1	no	—
26	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
27	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
29	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
30	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
31	#117×C₁₀₉	—	2	1	no	—
32	#106×C₁₀₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
34	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
35	#103×C₁₀₉	C₂²	1	1	yes	characteristic
36	#103×C₁₀₉	C₂²	1	2	yes	—
37	#103×C₁₀₉	C₂²	1	2	yes	—
38	#102×C₁₀₉	C₂²	1	1	yes	characteristic
39	#97×C₁₀₉	C₂²	1	2	yes	—
40	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
43	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
45	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₉	—	2	1	no	—
47	Q₁₆×C₁₀₉	—	2	4	no	—
48	Q₁₆×C₁₀₉	—	4	2	no	—
49	Q₁₆×C₁₀₉	—	4	2	no	—
50	SD₁₆×C₁₀₉	—	2	2	no	—
51	SD₁₆×C₁₀₉	—	4	2	no	—
52	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
54	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₁₀₉	—	2	1	no	—
56	C₄⋊C₄×C₁₀₉	—	2	2	no	—
57	C₄⋊C₄×C₁₀₉	—	2	1	no	—
58	C₄⋊C₄×C₁₀₉	—	2	2	no	—
59	C₄⋊C₄×C₁₀₉	—	2	2	no	—
60	C₄⋊C₄×C₁₀₉	—	2	1	no	—
61	C₄⋊C₄×C₁₀₉	—	2	2	no	—
62	C₄⋊C₄×C₁₀₉	—	2	1	no	—
63	C₄⋊C₄×C₁₀₉	—	2	1	no	—
64	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
66	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
69	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
70	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
71	C₄²×C₁₀₉	—	2	2	no	—
72	C₄²×C₁₀₉	—	2	2	no	—
73	C₄²×C₁₀₉	—	2	1	no	—
74	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
75	C₄²×C₁₀₉	C₂³	1	2	yes	—
76	C₂³×C₁₀₉	—	2	1	no	—
77	Q₈×C₁₀₉	—	2	2	no	—
78	Q₈×C₁₀₉	—	2	1	no	—
79	Q₈×C₁₀₉	—	2	2	no	—
80	Q₈×C₁₀₉	—	2	2	no	—
81	Q₈×C₁₀₉	—	2	2	no	—
82	Q₈×C₁₀₉	—	2	2	no	—
83	Q₈×C₁₀₉	—	2	1	no	—
84	D₄×C₁₀₉	—	2	1	no	—
85	D₄×C₁₀₉	—	2	1	no	—
86	C₂×C₄×C₁₀₉	—	2	1	no	—
87	C₂×C₄×C₁₀₉	—	2	1	no	—
88	C₂×C₄×C₁₀₉	—	2	2	no	—
89	C₂×C₄×C₁₀₉	—	2	1	no	—
90	C₂×C₄×C₁₀₉	—	2	2	no	—
91	C₂×C₄×C₁₀₉	—	4	2	no	—
92	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
97	C₈×C₁₀₉	—	2	2	no	—
98	C₈×C₁₀₉	—	2	1	no	—
99	C₈×C₁₀₉	#41	1	2	yes	—
100	C₂²×C₁₀₉	—	4	1	no	—
101	C₂²×C₁₀₉	—	4	1	no	—
102	C₂²×C₁₀₉	—	4	1	no	—
103	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
104	C₄×C₁₀₉	—	2	2	no	—
105	C₄×C₁₀₉	—	2	2	no	—
106	C₄×C₁₀₉	—	2	1	no	—
107	C₄×C₁₀₉	—	4	2	no	—
108	C₄×C₁₀₉	—	4	1	no	—
109	C₄×C₁₀₉	—	4	2	no	—
110	C₄×C₁₀₉	—	4	1	no	—
111	C₄×C₁₀₉	—	4	1	no	—
112	C₄×C₁₀₉	#137	1	2	yes	—
113	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
114	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×C₁₀₉	—	8	1	no	—
116	C₂×C₁₀₉	#550	1	1	yes	characteristic
117	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
118	C₂×C₁₀₉	#578	1	1	yes	characteristic
119	#3398	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₁₀₉	#3398	1	1	yes	characteristic, direct product factor, Sylow
121	#557	C₂×C₁₀₉	1	1	yes	characteristic, complemented
122	#556	C₂×C₁₀₉	1	1	yes	characteristic
123	#497	C₂×C₁₀₉	1	2	yes	—
124	#494	C₂×C₁₀₉	1	1	yes	characteristic, complemented
125	#464	C₂×C₁₀₉	1	2	yes	—
126	#462	C₂×C₁₀₉	1	2	yes	complemented
127	#462	C₂×C₁₀₉	1	1	yes	characteristic, complemented
128	#461	C₂×C₁₀₉	1	1	yes	characteristic
129	#446	C₂×C₁₀₉	1	1	yes	characteristic, complemented
130	#441	C₂×C₁₀₉	1	2	yes	complemented
131	C₄×SD₁₆	C₂×C₁₀₉	1	1	yes	characteristic
132	C₂×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
133	C₂×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
134	C₂×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
135	C₂×SD₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
136	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
137	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
138	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
139	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
140	#126	—	2	2	no	—
141	#125	—	2	1	no	—
142	#125	—	2	2	no	—
143	#124	C₂²×C₁₀₉	1	2	yes	—
144	C₂²⋊Q₈	—	2	1	no	—
145	C₄×Q₈	—	2	1	no	—
146	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
147	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
148	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
149	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
150	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
151	#117	—	2	1	no	—
152	#106	C₂²×C₁₀₉	1	1	yes	characteristic
153	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
154	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
155	#103	C₂²×C₁₀₉	1	1	yes	characteristic
156	#103	C₂²×C₁₀₉	1	2	yes	—
157	#103	C₂²×C₁₀₉	1	2	yes	—
158	#102	C₂²×C₁₀₉	1	1	yes	characteristic
159	#97	C₂²×C₁₀₉	1	2	yes	—
160	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
161	C₂×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
162	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
163	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
164	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
165	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
166	C₂²×C₄	—	2	1	no	—
167	Q₁₆	—	2	4	no	—
168	Q₁₆	—	4	2	no	—
169	Q₁₆	—	4	2	no	—
170	SD₁₆	—	2	2	no	—
171	SD₁₆	—	4	2	no	—
172	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
174	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
185	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
186	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
187	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	1	no	—
194	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
195	C₄²	C₂³×C₁₀₉	1	2	yes	—
196	C₂³	—	2	1	no	—
197	Q₈	—	2	2	no	—
198	Q₈	—	2	1	no	—
199	Q₈	—	2	2	no	—
200	Q₈	—	2	2	no	—
201	Q₈	—	2	2	no	—
202	Q₈	—	2	2	no	—
203	Q₈	—	2	1	no	—
204	D₄	—	2	1	no	—
205	D₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
213	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
214	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
215	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
216	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	2	1	no	—
219	C₈	#41×C₁₀₉	1	2	yes	—
220	C₂²	—	4	1	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	1	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	1	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	1	no	—
232	C₄	#137×C₁₀₉	1	2	yes	—
233	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
234	C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
235	C₂	—	8	1	no	—
236	C₂	#550×C₁₀₉	1	1	yes	characteristic
237	C₂	C₂×#137×C₁₀₉	1	1	yes	characteristic
238	C₂	#578×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	C₂×D₄×C₁₀₉	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄×Q₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #557×C₁₀₉	C₂	semidirect
4. #494×C₁₀₉	C₂	semidirect
6. #462×C₁₀₉	C₂	semidirect
7. #462×C₁₀₉	C₂	semidirect
9. #446×C₁₀₉	C₂	semidirect
10. #441×C₁₀₉	C₂	semidirect
119. #3398	C₁₀₉	direct (N × H)
120. C₁₀₉	#3398	direct (N × H)
121. #557	C₂×C₁₀₉	semidirect
124. #494	C₂×C₁₀₉	semidirect
126. #462	C₂×C₁₀₉	semidirect
127. #462	C₂×C₁₀₉	semidirect
129. #446	C₂×C₁₀₉	semidirect
130. #441	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#557×C₁₀₉	2
#556×C₁₀₉	2
#497×C₁₀₉	2
#494×C₁₀₉	2
#464×C₁₀₉	2
#462×C₁₀₉	2
#461×C₁₀₉	2
#446×C₁₀₉	2
#441×C₁₀₉	2
C₄×SD₁₆×C₁₀₉	2
#3398	109

Subgroups — C109×#3398 #28360652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3398 #28360652