Subgroups — C₁₀₉×#3354 #28360608

Order 13952.

386 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#555×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#547×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#498×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#482×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#481×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#476×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#474×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#469×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₄×D₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#436×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#428×C₁₀₉	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₁₀₉	—	2	1	no	—
13	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	#124×C₁₀₉	—	2	2	no	—
18	#123×C₁₀₉	—	2	2	no	—
19	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
20	#121×C₁₀₉	—	2	1	no	—
21	#121×C₁₀₉	C₂²	1	2	yes	—
22	#121×C₁₀₉	C₂²	1	2	yes	—
23	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
26	#117×C₁₀₉	—	2	1	no	—
27	#117×C₁₀₉	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
29	#107×C₁₀₉	C₂²	1	1	yes	characteristic
30	#107×C₁₀₉	C₂²	1	2	yes	—
31	#106×C₁₀₉	C₂²	1	2	yes	—
32	#106×C₁₀₉	C₂²	1	2	yes	—
33	#106×C₁₀₉	C₂²	1	2	yes	—
34	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
35	#102×C₁₀₉	C₂²	1	1	yes	characteristic
36	#102×C₁₀₉	C₂²	1	1	yes	characteristic
37	#102×C₁₀₉	C₂²	1	2	yes	—
38	#102×C₁₀₉	C₂²	1	2	yes	—
39	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
40	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	#41×C₁₀₉	—	4	4	no	—
42	C₂×Q₈×C₁₀₉	—	2	1	no	—
43	C₂×Q₈×C₁₀₉	—	2	1	no	—
44	C₂×D₄×C₁₀₉	—	2	2	no	—
45	C₂×D₄×C₁₀₉	—	2	2	no	—
46	C₂×D₄×C₁₀₉	D₄	1	1	yes	characteristic
47	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	—	2	2	no	—
50	C₂²×C₄×C₁₀₉	—	2	1	no	—
51	C₂²×C₄×C₁₀₉	—	2	1	no	—
52	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
53	D₈×C₁₀₉	—	2	2	no	—
54	M₁₆×C₁₀₉	—	2	4	no	—
55	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
57	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₁₀₉	—	2	2	no	—
59	C₄⋊C₄×C₁₀₉	—	2	1	no	—
60	C₄⋊C₄×C₁₀₉	—	2	1	no	—
61	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
63	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
65	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
66	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
67	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
68	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
69	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
70	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
71	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
72	C₄²×C₁₀₉	—	2	2	no	—
73	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₂³×C₁₀₉	—	2	1	no	—
75	C₂³×C₁₀₉	—	2	1	no	—
76	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
77	Q₈×C₁₀₉	—	4	2	no	—
78	Q₈×C₁₀₉	—	8	1	no	—
79	D₄×C₁₀₉	—	2	2	no	—
80	D₄×C₁₀₉	—	2	1	no	—
81	D₄×C₁₀₉	—	2	1	no	—
82	D₄×C₁₀₉	—	2	1	no	—
83	D₄×C₁₀₉	—	2	1	no	—
84	D₄×C₁₀₉	—	4	4	no	—
85	D₄×C₁₀₉	—	8	2	no	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	2	2	no	—
88	C₂×C₄×C₁₀₉	—	2	1	no	—
89	C₂×C₄×C₁₀₉	—	2	2	no	—
90	C₂×C₄×C₁₀₉	—	2	1	no	—
91	C₂×C₄×C₁₀₉	—	2	2	no	—
92	C₂×C₄×C₁₀₉	—	4	4	no	—
93	C₂×C₄×C₁₀₉	—	4	2	no	—
94	C₂×C₄×C₁₀₉	—	4	2	no	—
95	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
99	C₈×C₁₀₉	—	2	2	no	—
100	C₈×C₁₀₉	—	2	1	no	—
101	C₈×C₁₀₉	#41	1	2	yes	complemented
102	C₂²×C₁₀₉	—	2	2	no	—
103	C₂²×C₁₀₉	—	2	2	no	—
104	C₂²×C₁₀₉	—	2	2	no	—
105	C₂²×C₁₀₉	—	4	1	no	—
106	C₂²×C₁₀₉	—	4	1	no	—
107	C₂²×C₁₀₉	—	4	1	no	—
108	C₂²×C₁₀₉	—	4	1	no	—
109	C₂²×C₁₀₉	—	4	1	no	—
110	C₂²×C₁₀₉	—	4	1	no	—
111	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
112	C₄×C₁₀₉	—	2	2	no	—
113	C₄×C₁₀₉	—	2	1	no	—
114	C₄×C₁₀₉	—	4	1	no	—
115	C₄×C₁₀₉	—	4	1	no	—
116	C₄×C₁₀₉	—	4	2	no	—
117	C₄×C₁₀₉	—	4	2	no	—
118	C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
119	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
120	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
121	C₂×C₁₀₉	—	4	2	no	—
122	C₂×C₁₀₉	—	8	1	no	—
123	C₂×C₁₀₉	—	8	1	no	—
124	C₂×C₁₀₉	#547	1	1	yes	characteristic
125	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₁₀₉	#577	1	1	yes	characteristic
127	#3354	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₁₀₉	#3354	1	1	yes	characteristic, direct product factor, Sylow
129	#555	C₂×C₁₀₉	1	1	yes	characteristic, complemented
130	#547	C₂×C₁₀₉	1	1	yes	characteristic, complemented
131	#498	C₂×C₁₀₉	1	1	yes	characteristic, complemented
132	#482	C₂×C₁₀₉	1	2	yes	complemented
133	#481	C₂×C₁₀₉	1	2	yes	complemented
134	#476	C₂×C₁₀₉	1	1	yes	characteristic, complemented
135	#474	C₂×C₁₀₉	1	1	yes	characteristic, complemented
136	#469	C₂×C₁₀₉	1	2	yes	complemented
137	C₄×D₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
138	#436	C₂×C₁₀₉	1	1	yes	characteristic, complemented
139	#428	C₂×C₁₀₉	1	2	yes	complemented
140	C₂×#41	—	2	1	no	—
141	C₂×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
142	C₂×M₁₆	C₂²×C₁₀₉	1	2	yes	—
143	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
144	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
145	#124	—	2	2	no	—
146	#123	—	2	2	no	—
147	C₂²⋊Q₈	—	2	1	no	—
148	#121	—	2	1	no	—
149	#121	C₂²×C₁₀₉	1	2	yes	—
150	#121	C₂²×C₁₀₉	1	2	yes	—
151	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
152	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
153	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
154	#117	—	2	1	no	—
155	#117	C₂²×C₁₀₉	1	2	yes	—
156	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
157	#107	C₂²×C₁₀₉	1	1	yes	characteristic
158	#107	C₂²×C₁₀₉	1	2	yes	—
159	#106	C₂²×C₁₀₉	1	2	yes	—
160	#106	C₂²×C₁₀₉	1	2	yes	—
161	#106	C₂²×C₁₀₉	1	2	yes	—
162	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
163	#102	C₂²×C₁₀₉	1	1	yes	characteristic
164	#102	C₂²×C₁₀₉	1	1	yes	characteristic
165	#102	C₂²×C₁₀₉	1	2	yes	—
166	#102	C₂²×C₁₀₉	1	2	yes	—
167	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
168	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
169	#41	—	4	4	no	—
170	C₂×Q₈	—	2	1	no	—
171	C₂×Q₈	—	2	1	no	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	D₄×C₁₀₉	1	1	yes	characteristic
175	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
176	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
181	D₈	—	2	2	no	—
182	M₁₆	—	2	4	no	—
183	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
184	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
185	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
190	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
191	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
192	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
193	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	—	2	1	no	—
197	C₂²⋊C₄	—	2	1	no	—
198	C₂²⋊C₄	—	2	2	no	—
199	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
200	C₄²	—	2	2	no	—
201	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
205	Q₈	—	4	2	no	—
206	Q₈	—	8	1	no	—
207	D₄	—	2	2	no	—
208	D₄	—	2	1	no	—
209	D₄	—	2	1	no	—
210	D₄	—	2	1	no	—
211	D₄	—	2	1	no	—
212	D₄	—	4	4	no	—
213	D₄	—	8	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
224	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
225	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
226	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
227	C₈	—	2	2	no	—
228	C₈	—	2	1	no	—
229	C₈	#41×C₁₀₉	1	2	yes	complemented
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	1	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	1	no	—
239	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	1	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	2	no	—
246	C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
247	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
248	C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
249	C₂	—	4	2	no	—
250	C₂	—	8	1	no	—
251	C₂	—	8	1	no	—
252	C₂	#547×C₁₀₉	1	1	yes	characteristic
253	C₂	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
254	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 ◁ C₄×D₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #555×C₁₀₉	C₂	semidirect
2. #547×C₁₀₉	C₂	semidirect
3. #498×C₁₀₉	C₂	semidirect
4. #482×C₁₀₉	C₂	semidirect
5. #481×C₁₀₉	C₂	semidirect
6. #476×C₁₀₉	C₂	semidirect
7. #474×C₁₀₉	C₂	semidirect
8. #469×C₁₀₉	C₂	semidirect
9. C₄×D₈×C₁₀₉	C₂	semidirect
10. #436×C₁₀₉	C₂	semidirect
11. #428×C₁₀₉	C₂	semidirect
101. C₈×C₁₀₉	#41	semidirect
127. #3354	C₁₀₉	direct (N × H)
128. C₁₀₉	#3354	direct (N × H)
129. #555	C₂×C₁₀₉	semidirect
130. #547	C₂×C₁₀₉	semidirect
131. #498	C₂×C₁₀₉	semidirect
132. #482	C₂×C₁₀₉	semidirect
133. #481	C₂×C₁₀₉	semidirect
134. #476	C₂×C₁₀₉	semidirect
135. #474	C₂×C₁₀₉	semidirect
136. #469	C₂×C₁₀₉	semidirect
137. C₄×D₈	C₂×C₁₀₉	semidirect
138. #436	C₂×C₁₀₉	semidirect
139. #428	C₂×C₁₀₉	semidirect
229. C₈	#41×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#555×C₁₀₉	2
#547×C₁₀₉	2
#498×C₁₀₉	2
#482×C₁₀₉	2
#481×C₁₀₉	2
#476×C₁₀₉	2
#474×C₁₀₉	2
#469×C₁₀₉	2
C₄×D₈×C₁₀₉	2
#436×C₁₀₉	2
#428×C₁₀₉	2
#3354	109

Subgroups — C109×#3354 #28360608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3354 #28360608