Subgroups — #135×#89137 #28379782

Order 13952.

500 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#89137	C₂	1	2	yes	maximal, complemented
2	Q₁₆×#89137	C₂	1	2	yes	maximal, complemented
3	SD₁₆×#89137	C₂	1	4	yes	maximal, complemented
4	D₈×#89137	C₂	1	2	yes	maximal, complemented
5	#20296036	C₂	1	1	yes	maximal, characteristic, complemented
6	#20296016	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×#89137	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂×D₁₀₉⋊C₈	C₂²	1	1	yes	characteristic
10	Q₈×#89137	C₂²	1	4	yes	—
11	D₄×#89137	—	2	4	no	—
12	D₄×#89137	C₂²	1	4	yes	—
13	#16812778	—	2	2	no	—
14	#16812778	C₂²	1	2	yes	—
15	C₂×C₄×#89137	—	2	2	no	—
16	C₂×C₄×#89137	C₂²	1	1	yes	characteristic
17	#16812768	C₂²	1	2	yes	—
18	#16812767	C₂²	1	2	yes	—
19	#16812762	C₂²	1	4	yes	—
20	#16812761	C₂²	1	4	yes	—
21	C₈×#89137	C₂²	1	2	yes	—
22	C₈×#89137	C₂²	1	2	yes	—
23	#41×D₁₀₉	C₂²	1	2	yes	—
24	#16812722	C₄	1	1	yes	characteristic, complemented
25	#16812721	C₄	1	1	yes	characteristic, complemented
26	#16812720	C₄	1	2	yes	complemented
27	#16812719	C₄	1	2	yes	complemented
28	#16812718	C₄	1	1	yes	characteristic, complemented
29	Q₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
30	SD₁₆×D₁₀₉	C₂²	1	2	yes	—
31	D₈×D₁₀₉	C₂²	1	1	yes	characteristic
32	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
33	#135×C₁₀₉	C₄	1	1	yes	characteristic, complemented
34	C₂²×#89137	—	2	4	no	—
35	C₂²×#89137	D₄	1	2	yes	—
36	D₁₀₉⋊C₈	—	2	1	no	—
37	D₁₀₉⋊C₈	C₂³	1	2	yes	—
38	C₂×#11724786	—	2	1	no	—
39	C₄⋊#89137	—	2	4	no	—
40	C₄⋊#89137	C₂³	1	4	yes	—
41	C₂²⋊#89137	—	2	2	no	—
42	C₂²⋊#89137	—	2	4	no	—
43	C₄×#89137	—	2	4	no	—
44	C₄×#89137	—	2	2	no	—
45	C₄×#89137	D₄	1	1	yes	characteristic
46	C₄×#89137	D₄	1	1	yes	characteristic
47	C₄×#89137	C₂³	1	2	yes	—
48	#13182030	C₂×C₄	1	2	yes	complemented
49	#13182029	—	2	2	no	—
50	#13182029	C₂×C₄	1	2	yes	complemented
51	#13182028	—	2	2	no	—
52	#13182028	C₂×C₄	1	2	yes	complemented
53	Q₈×D₁₀₉	C₂³	1	2	yes	—
54	D₄×D₁₀₉	—	2	2	no	—
55	D₄×D₁₀₉	C₂³	1	2	yes	—
56	C₂×C₄×D₁₀₉	—	2	2	no	—
57	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
58	Dic₄₃₆	C₂×C₄	1	1	yes	characteristic, complemented
59	#13182019	C₂×C₄	1	2	yes	complemented
60	#13182018	C₂×C₄	1	2	yes	complemented
61	#13182017	C₂×C₄	1	2	yes	complemented
62	#13182016	C₂×C₄	1	2	yes	complemented
63	D₈₇₂	C₂×C₄	1	1	yes	characteristic, complemented
64	#13182014	C₂×C₄	1	2	yes	complemented
65	C₂×#11724784	C₂×C₄	1	1	yes	characteristic, complemented
66	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
67	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
68	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
69	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
70	#41×C₁₀₉	C₂×C₄	1	2	yes	complemented
71	Q₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
72	SD₁₆×C₁₀₉	C₂×C₄	1	2	yes	complemented
73	D₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
74	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
75	C₂²×D₁₀₉	—	2	2	no	—
76	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
77	C₂×#89137	—	2	4	no	—
78	C₂×#89137	—	2	4	no	—
79	C₂×#89137	—	4	8	no	—
80	C₂×#89137	C₂×D₄	1	2	yes	—
81	C₂×#89137	C₂×D₄	1	2	yes	—
82	Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
83	D₄×C₁₀₉	—	2	2	no	—
84	D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₀₉	—	2	2	no	—
86	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²⋊D₁₀₉	—	2	2	no	—
88	C₂²⋊D₁₀₉	—	2	2	no	—
89	C₂²⋊D₁₀₉	—	2	2	no	—
90	C₂×Dic₁₀₉	—	2	2	no	—
91	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
92	D₄₃₆	—	2	2	no	—
93	D₄₃₆	C₂²×C₄	1	2	yes	—
94	C₄×D₁₀₉	—	2	2	no	—
95	C₄×D₁₀₉	—	2	2	no	—
96	C₄×D₁₀₉	—	2	2	no	—
97	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
98	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
99	C₄×D₁₀₉	#41	1	1	yes	characteristic
100	C₄×D₁₀₉	C₂⁴	1	1	yes	characteristic
101	Dic₂₁₈	—	2	2	no	—
102	Dic₂₁₈	C₂²×C₄	1	2	yes	—
103	#11724786	—	2	2	no	—
104	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
105	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
106	#11724784	C₂²×C₄	1	1	yes	characteristic
107	#11724784	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₁₀₉	—	2	2	no	—
109	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
110	C₂×D₁₀₉	—	2	2	no	—
111	C₂×D₁₀₉	—	2	2	no	—
112	C₂×D₁₀₉	—	4	4	no	—
113	C₂×D₁₀₉	C₄×D₄	1	1	yes	characteristic
114	C₂×D₁₀₉	C₂²×D₄	1	1	yes	characteristic
115	#89137	—	2	2	no	—
116	#89137	—	4	4	no	—
117	#89137	#135	1	4	yes	direct product factor
118	C₄×C₁₀₉	—	2	2	no	—
119	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
120	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
121	Dic₁₀₉	—	2	2	no	—
122	Dic₁₀₉	C₄×D₄	1	1	yes	characteristic
123	Dic₁₀₉	C₂×#41	1	1	yes	characteristic
124	C₂×C₁₀₉	—	2	1	no	—
125	C₂×C₁₀₉	—	4	2	no	—
126	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
127	D₁₀₉	—	2	1	no	—
128	D₁₀₉	—	4	2	no	—
129	D₁₀₉	C₂×#135	1	2	yes	—
130	C₄×#135	—	109	1	no	maximal, Sylow
131	C₁₀₉	C₄×#135	1	1	yes	characteristic, complemented, Sylow
132	C₂×#135	—	109	1	no	—
133	C₄×#41	—	109	2	no	—
134	C₄×Q₁₆	—	109	2	no	—
135	C₄×SD₁₆	—	109	4	no	—
136	C₄×D₈	—	109	2	no	—
137	#427	—	109	1	no	—
138	#416	—	109	2	no	—
139	C₂×C₄×C₈	—	109	1	no	—
140	C₂×#41	—	109	2	no	—
141	#135	—	109	2	no	—
142	#135	—	109	1	no	—
143	#135	—	109	2	no	—
144	#135	—	109	1	no	—
145	#135	—	109	1	no	—
146	#135	#89137	1	1	yes	characteristic, direct product factor
147	C₂×Q₁₆	—	109	1	no	—
148	C₂×SD₁₆	—	109	2	no	—
149	C₂×D₈	—	109	1	no	—
150	C₂²×C₈	—	109	1	no	—
151	C₂²×C₈	—	109	1	no	—
152	C₄×Q₈	—	109	4	no	—
153	C₄×D₄	—	109	4	no	—
154	C₄×D₄	—	218	4	no	—
155	#117	—	109	2	no	—
156	#117	—	218	2	no	—
157	C₂×C₄²	—	109	1	no	—
158	C₂×C₄²	—	218	2	no	—
159	#107	—	109	2	no	—
160	#106	—	109	2	no	—
161	#103	—	109	4	no	—
162	#102	—	109	4	no	—
163	C₄×C₈	—	109	2	no	—
164	C₄×C₈	—	109	2	no	—
165	#41	—	109	2	no	—
166	#41	—	109	2	no	—
167	#41	—	109	2	no	—
168	#41	—	218	2	no	—
169	#41	—	218	2	no	—
170	#41	C₂×#89137	1	2	yes	complemented
171	C₂×Q₈	—	109	2	no	—
172	C₂×D₄	—	109	2	no	—
173	C₂×D₄	—	218	2	no	—
174	C₂²×C₄	—	109	1	no	—
175	C₂²×C₄	—	109	2	no	—
176	C₂²×C₄	—	218	2	no	—
177	C₂²×C₄	—	218	4	no	—
178	Q₁₆	—	109	2	no	—
179	Q₁₆	—	109	1	no	—
180	Q₁₆	C₂×#89137	1	1	yes	characteristic, complemented
181	SD₁₆	—	109	2	no	—
182	SD₁₆	—	109	2	no	—
183	SD₁₆	—	109	2	no	—
184	SD₁₆	C₂×#89137	1	2	yes	complemented
185	D₈	—	109	2	no	—
186	D₈	—	109	1	no	—
187	D₈	C₂×#89137	1	1	yes	characteristic, complemented
188	C₂×C₈	—	109	1	no	—
189	C₂×C₈	—	109	1	no	—
190	C₂×C₈	—	109	1	no	—
191	C₂×C₈	—	109	1	no	—
192	C₂×C₈	—	109	1	no	—
193	C₂×C₈	—	109	2	no	—
194	C₂×C₈	—	218	1	no	—
195	C₂×C₈	—	218	1	no	—
196	C₂×C₈	C₂×#89137	1	1	yes	characteristic, complemented
197	C₄⋊C₄	—	109	4	no	—
198	C₄⋊C₄	—	218	4	no	—
199	C₂²⋊C₄	—	218	4	no	—
200	C₂²⋊C₄	—	218	2	no	—
201	C₄²	—	109	1	no	—
202	C₄²	—	109	1	no	—
203	C₄²	—	109	2	no	—
204	C₄²	—	218	2	no	—
205	C₄²	—	218	4	no	—
206	C₂³	—	109	1	no	—
207	C₂³	—	218	2	no	—
208	Q₈	—	109	2	no	—
209	Q₈	—	218	2	no	—
210	Q₈	C₂²×#89137	1	2	yes	—
211	D₄	—	2	2	no	—
212	D₄	—	109	2	no	—
213	D₄	—	218	2	no	—
214	D₄	—	218	2	no	—
215	D₄	—	218	2	no	—
216	D₄	—	218	2	no	—
217	D₄	C₂²×#89137	1	2	yes	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	109	2	no	—
220	C₂×C₄	—	109	1	no	—
221	C₂×C₄	—	109	1	no	—
222	C₂×C₄	—	109	1	no	—
223	C₂×C₄	—	109	1	no	—
224	C₂×C₄	—	109	1	no	—
225	C₂×C₄	—	109	2	no	—
226	C₂×C₄	—	436	8	no	—
227	C₂×C₄	—	218	2	no	—
228	C₂×C₄	—	218	2	no	—
229	C₂×C₄	—	218	2	no	—
230	C₂×C₄	—	218	4	no	—
231	C₂×C₄	—	218	4	no	—
232	C₂×C₄	—	218	2	no	—
233	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
234	C₈	—	109	1	no	—
235	C₈	—	109	1	no	—
236	C₈	—	218	2	no	—
237	C₈	C₂²×#89137	1	1	yes	characteristic
238	C₈	C₂²×#89137	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	109	1	no	—
241	C₂²	—	109	1	no	—
242	C₂²	—	436	4	no	—
243	C₂²	—	218	2	no	—
244	C₂²	—	218	2	no	—
245	C₂²	D₄×#89137	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	109	4	no	—
248	C₄	—	109	1	no	—
249	C₄	—	109	1	no	—
250	C₄	—	436	4	no	—
251	C₄	—	218	2	no	—
252	C₄	—	218	2	no	—
253	C₄	D₄×#89137	1	1	yes	characteristic
254	C₄	C₂³×#89137	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	—	109	2	no	—
258	C₂	—	436	2	no	—
259	C₂	—	218	1	no	—
260	C₂	C₂×D₄×#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#89137
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₄	C₂³×#89137
Fitting	#135×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×C₄×D₄
prime radical	#135×D₁₀₉	C₂
cosocle	C₄×D₁₀₉	C₂⁴
hypercenter	#135	#89137
hypocenter	C₁₀₉	C₄×#135
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #41×#89137	C₂	semidirect
2. Q₁₆×#89137	C₂	semidirect
3. SD₁₆×#89137	C₂	semidirect
4. D₈×#89137	C₂	semidirect
5. #20296036	C₂	semidirect
6. #20296016	C₂	semidirect
7. C₂×C₈×#89137	C₂	semidirect
24. #16812722	C₄	semidirect
25. #16812721	C₄	semidirect
26. #16812720	C₄	semidirect
27. #16812719	C₄	semidirect
28. #16812718	C₄	semidirect
33. #135×C₁₀₉	C₄	semidirect
48. #13182030	C₂×C₄	semidirect
50. #13182029	C₂×C₄	semidirect
52. #13182028	C₂×C₄	semidirect
58. Dic₄₃₆	C₂×C₄	semidirect
59. #13182019	C₂×C₄	semidirect
60. #13182018	C₂×C₄	semidirect
61. #13182017	C₂×C₄	semidirect
62. #13182016	C₂×C₄	semidirect
63. D₈₇₂	C₂×C₄	semidirect
64. #13182014	C₂×C₄	semidirect
65. C₂×#11724784	C₂×C₄	semidirect
66. C₈×D₁₀₉	C₂×C₄	semidirect
67. C₈×D₁₀₉	C₂×C₄	semidirect
70. #41×C₁₀₉	C₂×C₄	semidirect
71. Q₁₆×C₁₀₉	C₂×C₄	semidirect
72. SD₁₆×C₁₀₉	C₂×C₄	semidirect
73. D₈×C₁₀₉	C₂×C₄	semidirect
74. C₂×C₈×C₁₀₉	C₂×C₄	semidirect
117. #89137	#135	direct (N × H)
131. C₁₀₉	C₄×#135	semidirect
146. #135	#89137	direct (N × H)
170. #41	C₂×#89137	semidirect
180. Q₁₆	C₂×#89137	semidirect
184. SD₁₆	C₂×#89137	semidirect
187. D₈	C₂×#89137	semidirect
196. C₂×C₈	C₂×#89137	semidirect

Maximal subgroups

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

subgroup	index
#41×#89137	2
Q₁₆×#89137	2
SD₁₆×#89137	2
D₈×#89137	2
#20296036	2
#20296016	2
C₂×C₈×#89137	2
#135×D₁₀₉	2
C₄×#135	109