Subgroups — #28379795

Order 13952.

484 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20296095	C₂	1	2	yes	maximal, complemented
2	#20296067	C₂	1	2	yes	maximal, complemented
3	#20296066	C₂	1	2	yes	maximal, complemented
4	SD₁₆×#89137	C₂	1	4	yes	maximal, complemented
5	C₂×#16812767	C₂	1	1	yes	maximal, characteristic, complemented
6	#20296017	C₂	1	2	yes	maximal, complemented
7	#20295989	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₁₀₉	C₂	1	1	yes	maximal, characteristic
9	D₁₀₉⋊M₁₆	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	1	yes	characteristic
15	C₂×C₄⋊#89137	—	2	2	no	—
16	C₂×C₄⋊#89137	C₂²	1	2	yes	—
17	#16812767	C₂²	1	2	yes	—
18	#16812767	C₂²	1	2	yes	—
19	#16812762	C₂²	1	4	yes	—
20	#16812761	C₂²	1	4	yes	—
21	#16812753	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	2	no	—
35	C₂²×#89137	—	2	4	no	—
36	#13182039	—	2	1	no	—
37	#13182038	—	2	1	no	—
38	D₁₀₉⋊C₈	C₂³	1	2	yes	—
39	C₄⋊#89137	—	2	4	no	—
40	C₄⋊#89137	—	2	2	no	—
41	C₄⋊#89137	D₄	1	2	yes	—
42	C₄⋊#89137	C₂³	1	4	yes	—
43	C₂²⋊#89137	—	2	4	no	—
44	C₂²⋊#89137	D₄	1	2	yes	—
45	C₄×#89137	—	2	4	no	—
46	C₄×#89137	C₂³	1	2	yes	—
47	#13182030	C₂×C₄	1	2	yes	complemented
48	#13182029	—	2	2	no	—
49	#13182029	C₂×C₄	1	2	yes	complemented
50	#13182028	—	2	2	no	—
51	#13182028	C₂×C₄	1	2	yes	complemented
52	Q₈×D₁₀₉	C₂³	1	2	yes	—
53	D₄×D₁₀₉	—	2	2	no	—
54	D₄×D₁₀₉	C₂³	1	2	yes	—
55	C₂×C₄×D₁₀₉	—	2	2	no	—
56	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
57	Dic₄₃₆	C₂×C₄	1	1	yes	characteristic, complemented
58	#13182019	C₂×C₄	1	2	yes	complemented
59	#13182018	C₂×C₄	1	2	yes	complemented
60	#13182017	C₂×C₄	1	2	yes	complemented
61	#13182016	C₂×C₄	1	2	yes	complemented
62	D₈₇₂	C₂×C₄	1	1	yes	characteristic, complemented
63	#13182014	C₂×C₄	1	2	yes	complemented
64	C₂×#11724784	C₂×C₄	1	1	yes	characteristic, complemented
65	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
66	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
67	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
68	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
69	#41×C₁₀₉	C₂×C₄	1	2	yes	complemented
70	Q₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
71	SD₁₆×C₁₀₉	C₂×C₄	1	2	yes	complemented
72	D₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
73	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
74	C₂²×D₁₀₉	—	2	2	no	—
75	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
76	C₂×#89137	—	2	4	no	—
77	C₂×#89137	—	8	4	no	—
78	C₂×#89137	—	8	2	no	—
79	C₂×#89137	C₂×D₄	1	4	yes	—
80	Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
81	D₄×C₁₀₉	—	2	2	no	—
82	D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₀₉	—	2	2	no	—
84	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²⋊D₁₀₉	—	2	2	no	—
86	C₂²⋊D₁₀₉	—	2	2	no	—
87	C₂²⋊D₁₀₉	—	2	2	no	—
88	C₂×Dic₁₀₉	—	2	2	no	—
89	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
90	D₄₃₆	—	2	2	no	—
91	D₄₃₆	C₂²×C₄	1	2	yes	—
92	C₄×D₁₀₉	—	2	2	no	—
93	C₄×D₁₀₉	—	2	2	no	—
94	C₄×D₁₀₉	—	2	2	no	—
95	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
96	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
97	C₄×D₁₀₉	#41	1	1	yes	characteristic
98	C₄×D₁₀₉	C₂⁴	1	1	yes	characteristic
99	Dic₂₁₈	—	2	2	no	—
100	Dic₂₁₈	C₂²×C₄	1	2	yes	—
101	#11724786	—	2	2	no	—
102	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
103	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
104	#11724784	C₂²×C₄	1	1	yes	characteristic
105	#11724784	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₁₀₉	—	2	2	no	—
107	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
108	C₂×D₁₀₉	—	2	2	no	—
109	C₂×D₁₀₉	—	2	2	no	—
110	C₂×D₁₀₉	—	4	2	no	—
111	C₂×D₁₀₉	—	4	2	no	—
112	C₂×D₁₀₉	C₄×D₄	1	1	yes	characteristic
113	C₂×D₁₀₉	C₂²×D₄	1	1	yes	characteristic
114	#89137	—	2	4	no	—
115	#89137	—	4	4	no	—
116	C₄×C₁₀₉	—	2	2	no	—
117	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
118	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
119	Dic₁₀₉	—	2	2	no	—
120	Dic₁₀₉	C₄×D₄	1	1	yes	characteristic
121	Dic₁₀₉	C₂×#41	1	1	yes	characteristic
122	C₂×C₁₀₉	—	2	1	no	—
123	C₂×C₁₀₉	—	4	2	no	—
124	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
125	D₁₀₉	—	2	1	no	—
126	D₁₀₉	—	4	2	no	—
127	D₁₀₉	#577	1	2	yes	—
128	#2952	—	109	1	no	maximal, Sylow
129	C₁₀₉	#2952	1	1	yes	characteristic, complemented, Sylow
130	C₂×#135	—	109	1	no	—
131	#520	—	109	2	no	—
132	#442	—	109	2	no	—
133	#441	—	109	2	no	—
134	C₄×SD₁₆	—	109	4	no	—
135	C₂×#106	—	109	1	no	—
136	#417	—	109	2	no	—
137	#405	—	109	1	no	—
138	C₂×#41	—	109	2	no	—
139	#135	—	109	2	no	—
140	#135	—	109	1	no	—
141	#135	—	109	2	no	—
142	#135	—	109	1	no	—
143	#135	—	109	1	no	—
144	#135	#89137	1	1	yes	characteristic, complemented
145	C₂×Q₁₆	—	109	1	no	—
146	C₂×SD₁₆	—	109	2	no	—
147	C₂×D₈	—	109	1	no	—
148	C₂×M₁₆	—	109	1	no	—
149	C₂²×C₈	—	109	1	no	—
150	C₄×Q₈	—	109	4	no	—
151	C₄×D₄	—	109	4	no	—
152	C₄×D₄	—	218	4	no	—
153	#117	—	109	1	no	—
154	#117	—	218	2	no	—
155	C₂×C₄⋊C₄	—	109	2	no	—
156	C₂×C₄⋊C₄	—	218	2	no	—
157	#106	—	109	2	no	—
158	#106	—	109	2	no	—
159	#103	—	109	4	no	—
160	#102	—	109	4	no	—
161	#97	—	109	2	no	—
162	C₄×C₈	—	109	2	no	—
163	#41	—	109	2	no	—
164	#41	—	109	2	no	—
165	#41	—	109	2	no	—
166	#41	—	218	2	no	—
167	#41	—	218	2	no	—
168	#41	C₂×#89137	1	2	yes	complemented
169	C₂×Q₈	—	109	2	no	—
170	C₂×D₄	—	109	2	no	—
171	C₂×D₄	—	218	2	no	—
172	C₂²×C₄	—	109	1	no	—
173	C₂²×C₄	—	218	2	no	—
174	C₂²×C₄	—	218	4	no	—
175	C₂²×C₄	—	218	2	no	—
176	Q₁₆	—	109	2	no	—
177	Q₁₆	—	109	1	no	—
178	Q₁₆	C₂×#89137	1	1	yes	characteristic, complemented
179	SD₁₆	—	109	2	no	—
180	SD₁₆	—	109	2	no	—
181	SD₁₆	—	109	2	no	—
182	SD₁₆	C₂×#89137	1	2	yes	complemented
183	D₈	—	109	2	no	—
184	D₈	—	109	1	no	—
185	D₈	C₂×#89137	1	1	yes	characteristic, complemented
186	M₁₆	—	218	1	no	—
187	M₁₆	—	218	1	no	—
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	C₂×#89137	1	1	yes	characteristic, complemented
195	C₄⋊C₄	—	109	2	no	—
196	C₄⋊C₄	—	109	4	no	—
197	C₄⋊C₄	—	218	2	no	—
198	C₄⋊C₄	—	218	4	no	—
199	C₂²⋊C₄	—	109	2	no	—
200	C₂²⋊C₄	—	218	4	no	—
201	C₄²	—	109	2	no	—
202	C₄²	—	218	4	no	—
203	C₂³	—	109	1	no	—
204	C₂³	—	218	2	no	—
205	Q₈	—	109	2	no	—
206	Q₈	—	218	2	no	—
207	Q₈	C₂²×#89137	1	2	yes	—
208	D₄	—	2	2	no	—
209	D₄	—	109	2	no	—
210	D₄	—	218	2	no	—
211	D₄	—	218	2	no	—
212	D₄	—	218	2	no	—
213	D₄	—	218	2	no	—
214	D₄	C₂²×#89137	1	2	yes	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	872	2	no	—
217	C₂×C₄	—	872	4	no	—
218	C₂×C₄	—	109	1	no	—
219	C₂×C₄	—	109	1	no	—
220	C₂×C₄	—	109	1	no	—
221	C₂×C₄	—	109	1	no	—
222	C₂×C₄	—	109	1	no	—
223	C₂×C₄	—	109	4	no	—
224	C₂×C₄	—	218	2	no	—
225	C₂×C₄	—	218	2	no	—
226	C₂×C₄	—	218	2	no	—
227	C₂×C₄	—	218	4	no	—
228	C₂×C₄	—	218	2	no	—
229	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
230	C₈	—	109	1	no	—
231	C₈	—	109	1	no	—
232	C₈	—	218	2	no	—
233	C₈	C₂²×#89137	1	1	yes	characteristic
234	C₈	C₂²×#89137	1	1	yes	characteristic
235	C₂²	—	2	2	no	—
236	C₂²	—	109	1	no	—
237	C₂²	—	109	1	no	—
238	C₂²	—	436	2	no	—
239	C₂²	—	436	2	no	—
240	C₂²	—	218	2	no	—
241	C₂²	—	218	2	no	—
242	C₂²	D₄×#89137	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	109	1	no	—
245	C₄	—	109	1	no	—
246	C₄	—	436	4	no	—
247	C₄	—	218	4	no	—
248	C₄	—	218	2	no	—
249	C₄	D₄×#89137	1	1	yes	characteristic
250	C₄	C₂³×#89137	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	4	2	no	—
253	C₂	—	109	2	no	—
254	C₂	—	436	2	no	—
255	C₂	—	218	1	no	—
256	C₂	C₂×D₄×#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×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	#2952
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #20296095	C₂	semidirect
2. #20296067	C₂	semidirect
3. #20296066	C₂	semidirect
4. SD₁₆×#89137	C₂	semidirect
5. C₂×#16812767	C₂	semidirect
6. #20296017	C₂	semidirect
7. #20295989	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
47. #13182030	C₂×C₄	semidirect
49. #13182029	C₂×C₄	semidirect
51. #13182028	C₂×C₄	semidirect
57. Dic₄₃₆	C₂×C₄	semidirect
58. #13182019	C₂×C₄	semidirect
59. #13182018	C₂×C₄	semidirect
60. #13182017	C₂×C₄	semidirect
61. #13182016	C₂×C₄	semidirect
62. D₈₇₂	C₂×C₄	semidirect
63. #13182014	C₂×C₄	semidirect
64. C₂×#11724784	C₂×C₄	semidirect
65. C₈×D₁₀₉	C₂×C₄	semidirect
66. C₈×D₁₀₉	C₂×C₄	semidirect
69. #41×C₁₀₉	C₂×C₄	semidirect
70. Q₁₆×C₁₀₉	C₂×C₄	semidirect
71. SD₁₆×C₁₀₉	C₂×C₄	semidirect
72. D₈×C₁₀₉	C₂×C₄	semidirect
73. C₂×C₈×C₁₀₉	C₂×C₄	semidirect
129. C₁₀₉	#2952	semidirect
144. #135	#89137	semidirect
168. #41	C₂×#89137	semidirect
178. Q₁₆	C₂×#89137	semidirect
182. SD₁₆	C₂×#89137	semidirect
185. D₈	C₂×#89137	semidirect
194. 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
#20296095	2
#20296067	2
#20296066	2
SD₁₆×#89137	2
C₂×#16812767	2
#20296017	2
#20295989	2
#135×D₁₀₉	2
#2952	109