Subgroups — #117×#89137 #28379465

Order 13952.

752 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812778	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#89137	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×#89137	C₂	1	4	yes	maximal, complemented
4	C₂²⋊C₄×#89137	C₂	1	4	yes	maximal
5	#20295962	C₂	1	2	yes	maximal, complemented
6	C₄²×#89137	C₂	1	2	yes	maximal, complemented
7	#117×D₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂³×#89137	C₂²	1	2	yes	—
9	#16812778	C₄	1	8	yes	complemented
10	C₂×C₄⋊#89137	C₂²	1	2	yes	—
11	C₂×C₂²⋊#89137	C₂²	1	2	yes	—
12	C₂×C₄×#89137	C₄	1	8	yes	complemented
13	C₂×C₄×#89137	C₂²	1	8	yes	—
14	C₂×C₄×#89137	C₂²	1	1	yes	characteristic
15	C₂×C₄×#89137	C₂²	1	1	yes	characteristic
16	C₂×C₄×#89137	C₂²	1	2	yes	—
17	C₂×C₄×#89137	C₂²	1	2	yes	—
18	(C₂×#89137)⋊C₄	C₂²	1	8	yes	—
19	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
20	#16812658	C₄	1	1	yes	characteristic, complemented
21	#16812657	C₄	1	2	yes	complemented
22	#16812656	C₄	1	2	yes	complemented
23	#16812655	C₄	1	2	yes	complemented
24	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
25	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	—
26	C₄²×D₁₀₉	C₂²	1	2	yes	—
27	#117×C₁₀₉	C₄	1	1	yes	characteristic, complemented
28	C₂²×#89137	—	2	8	no	—
29	C₂²×#89137	C₂×C₄	1	8	yes	—
30	C₂²×#89137	C₂³	1	2	yes	—
31	C₂²×#89137	C₂³	1	4	yes	—
32	C₂²×#89137	C₂³	1	2	yes	—
33	C₄⋊#89137	C₂×C₄	1	8	yes	complemented
34	C₂²⋊#89137	C₂×C₄	1	8	yes	—
35	C₄×#89137	—	2	16	no	—
36	C₄×#89137	C₂×C₄	1	4	yes	—
37	C₄×#89137	C₂×C₄	1	8	yes	—
38	C₄×#89137	C₂×C₄	1	4	yes	—
39	C₄×#89137	C₂×C₄	1	8	yes	complemented
40	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₀₉	C₂×C₄	1	4	yes	—
45	C₂×C₄×D₁₀₉	C₂×C₄	1	4	yes	—
46	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
47	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
49	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	complemented
50	#13182008	C₂×C₄	1	4	yes	complemented
51	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
52	#13182006	C₂×C₄	1	4	yes	—
53	C₄×Dic₁₀₉	C₂×C₄	1	4	yes	complemented
54	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	complemented
55	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	complemented
57	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
58	C₄²×C₁₀₉	C₂×C₄	1	2	yes	complemented
59	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×D₁₀₉	—	2	4	no	—
61	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
63	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
64	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
65	C₂×#89137	—	2	8	no	—
66	C₂×#89137	—	2	4	no	—
67	C₂×#89137	—	2	16	no	—
68	C₂×#89137	C₂²×C₄	1	4	yes	—
69	C₂×#89137	C₂²×C₄	1	8	yes	—
70	C₂×#89137	C₂²×C₄	1	4	yes	—
71	C₂×#89137	#41	1	8	yes	—
72	C₂×C₄×C₁₀₉	C₄²	1	4	yes	complemented
73	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
74	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₁₀₉	C₄²	1	4	yes	complemented
77	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
78	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
80	C₄×D₁₀₉	—	2	8	no	—
81	C₄×D₁₀₉	C₄²	1	4	yes	complemented
82	C₄×D₁₀₉	C₄²	1	4	yes	complemented
83	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
84	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
85	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
86	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
87	C₂²×C₁₀₉	—	2	2	no	—
88	C₂²×C₁₀₉	C₂×C₄²	1	2	yes	—
89	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
90	C₂×D₁₀₉	—	2	2	no	—
91	C₂×D₁₀₉	—	2	4	no	—
92	C₂×D₁₀₉	—	2	4	no	—
93	C₂×D₁₀₉	C₂×C₄²	1	2	yes	—
94	C₂×D₁₀₉	C₂³×C₄	1	1	yes	characteristic
95	C₂×D₁₀₉	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₁₀₉	C₂×#41	1	4	yes	—
97	#89137	—	2	4	no	—
98	#89137	—	2	8	no	—
99	#89137	#117	1	8	yes	direct product factor
100	C₄×C₁₀₉	—	2	4	no	—
101	C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
102	C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
103	Dic₁₀₉	—	2	4	no	—
104	Dic₁₀₉	C₂×C₄²	1	2	yes	—
105	Dic₁₀₉	C₂×C₄²	1	2	yes	—
106	C₂×C₁₀₉	—	2	2	no	—
107	C₂×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
108	C₂×C₁₀₉	C₄×#41	1	2	yes	—
109	D₁₀₉	—	2	2	no	—
110	D₁₀₉	C₂×#117	1	2	yes	—
111	D₁₀₉	C₄×#41	1	2	yes	—
112	C₄×#117	—	109	1	no	maximal, Sylow
113	C₁₀₉	C₄×#117	1	1	yes	characteristic, complemented, Sylow
114	C₂×#117	—	109	1	no	—
115	C₂×#117	—	109	1	no	—
116	C₂²×C₄²	—	109	1	no	—
117	C₄×C₄⋊C₄	—	109	4	no	—
118	C₄×C₂²⋊C₄	—	109	4	no	—
119	#376	—	109	2	no	—
120	C₄³	—	109	2	no	—
121	C₂³×C₄	—	109	1	no	—
122	C₂³×C₄	—	109	2	no	—
123	#117	—	109	8	no	—
124	#117	—	109	1	no	—
125	#117	—	109	2	no	—
126	#117	—	109	2	no	—
127	#117	—	109	2	no	—
128	#117	#89137	1	1	yes	characteristic, direct product factor
129	C₂×C₄⋊C₄	—	109	2	no	—
130	C₂×C₄⋊C₄	—	109	2	no	—
131	C₂×C₂²⋊C₄	—	109	2	no	—
132	C₂×C₂²⋊C₄	—	109	2	no	—
133	C₂×C₄²	—	109	2	no	—
134	C₂×C₄²	—	109	1	no	—
135	C₂×C₄²	—	109	1	no	—
136	C₂×C₄²	—	109	2	no	—
137	C₂×C₄²	—	109	8	no	—
138	C₂×C₄²	—	109	2	no	—
139	C₂×C₄²	—	109	8	no	—
140	#95	—	109	8	no	—
141	C₂⁴	—	109	1	no	—
142	C₂²×C₄	—	109	2	no	—
143	C₂²×C₄	—	109	1	no	—
144	C₂²×C₄	—	109	4	no	—
145	C₂²×C₄	—	109	4	no	—
146	C₂²×C₄	—	109	1	no	—
147	C₂²×C₄	—	109	1	no	—
148	C₂²×C₄	—	109	1	no	—
149	C₂²×C₄	—	109	1	no	—
150	C₂²×C₄	—	109	8	no	—
151	C₂²×C₄	—	109	2	no	—
152	C₂²×C₄	—	109	4	no	—
153	C₂²×C₄	—	109	4	no	—
154	C₂²×C₄	—	218	8	no	—
155	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
156	C₄⋊C₄	—	109	8	no	—
157	C₄⋊C₄	—	109	4	no	—
158	C₄⋊C₄	—	109	2	no	—
159	C₄⋊C₄	C₂×#89137	1	2	yes	complemented
160	C₂²⋊C₄	—	109	8	no	—
161	C₂²⋊C₄	—	109	2	no	—
162	C₂²⋊C₄	—	109	4	no	—
163	C₂²⋊C₄	C₂×#89137	1	2	yes	—
164	C₄²	—	109	4	no	—
165	C₄²	—	109	8	no	—
166	C₄²	—	109	4	no	—
167	C₄²	—	109	8	no	—
168	C₄²	—	109	4	no	—
169	C₄²	—	109	2	no	—
170	C₄²	—	218	16	no	—
171	C₄²	C₂×#89137	1	2	yes	complemented
172	C₂³	—	109	1	no	—
173	C₂³	—	109	2	no	—
174	C₂³	—	109	2	no	—
175	C₂³	—	109	1	no	—
176	C₂³	—	218	4	no	—
177	C₂³	C₂²×#89137	1	1	yes	characteristic
178	C₂×C₄	—	109	4	no	—
179	C₂×C₄	—	109	8	no	—
180	C₂×C₄	—	109	4	no	—
181	C₂×C₄	—	109	1	no	—
182	C₂×C₄	—	109	1	no	—
183	C₂×C₄	—	109	2	no	—
184	C₂×C₄	—	109	2	no	—
185	C₂×C₄	—	109	2	no	—
186	C₂×C₄	—	109	2	no	—
187	C₂×C₄	—	109	4	no	—
188	C₂×C₄	—	109	4	no	—
189	C₂×C₄	—	109	4	no	—
190	C₂×C₄	—	109	8	no	—
191	C₂×C₄	—	109	4	no	—
192	C₂×C₄	—	218	8	no	—
193	C₂×C₄	—	218	16	no	—
194	C₂×C₄	—	218	8	no	—
195	C₂×C₄	—	218	4	no	—
196	C₂×C₄	C₄×#89137	1	4	yes	complemented
197	C₂×C₄	C₂²×#89137	1	4	yes	—
198	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
199	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
200	C₂²	—	2	2	no	—
201	C₂²	—	109	1	no	—
202	C₂²	—	109	1	no	—
203	C₂²	—	109	4	no	—
204	C₂²	—	109	2	no	—
205	C₂²	—	218	4	no	—
206	C₂²	—	218	4	no	—
207	C₂²	—	218	2	no	—
208	C₂²	C₂×C₄×#89137	1	2	yes	—
209	C₂²	C₂³×#89137	1	1	yes	characteristic
210	C₄	—	2	4	no	—
211	C₄	—	109	8	no	—
212	C₄	—	109	2	no	—
213	C₄	—	109	2	no	—
214	C₄	—	218	4	no	—
215	C₄	—	218	8	no	—
216	C₄	—	218	4	no	—
217	C₄	C₂×C₄×#89137	1	2	yes	—
218	C₄	C₂×C₄×#89137	1	2	yes	—
219	C₂	—	2	2	no	—
220	C₂	—	109	2	no	—
221	C₂	—	109	2	no	—
222	C₂	—	218	2	no	—
223	C₂	C₂²×C₄×#89137	1	1	yes	characteristic
224	C₂	#41×#89137	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×#89137
commutator	C₂×C₁₀₉	C₂²×C₄²
Frattini	C₂²	C₂³×#89137
Fitting	#117×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂³×C₄
prime radical	C₂³×D₁₀₉	C₂³
cosocle	C₂²×D₁₀₉	C₂⁴
hypercenter	#117	#89137
hypocenter	C₁₀₉	C₄×#117
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. C₄⋊C₄×#89137	C₂	semidirect
5. #20295962	C₂	semidirect
6. C₄²×#89137	C₂	semidirect
9. #16812778	C₄	semidirect
12. C₂×C₄×#89137	C₄	semidirect
20. #16812658	C₄	semidirect
21. #16812657	C₄	semidirect
22. #16812656	C₄	semidirect
23. #16812655	C₄	semidirect
27. #117×C₁₀₉	C₄	semidirect
33. C₄⋊#89137	C₂×C₄	semidirect
39. C₄×#89137	C₂×C₄	semidirect
49. C₄⋊Dic₁₀₉	C₂×C₄	semidirect
50. #13182008	C₂×C₄	semidirect
53. C₄×Dic₁₀₉	C₂×C₄	semidirect
54. C₄×Dic₁₀₉	C₂×C₄	semidirect
56. C₄⋊C₄×C₁₀₉	C₂×C₄	semidirect
58. C₄²×C₁₀₉	C₂×C₄	semidirect
72. C₂×C₄×C₁₀₉	C₄²	semidirect
76. C₂×Dic₁₀₉	C₄²	semidirect
81. C₄×D₁₀₉	C₄²	semidirect
82. C₄×D₁₀₉	C₄²	semidirect
99. #89137	#117	direct (N × H)
113. C₁₀₉	C₄×#117	semidirect
128. #117	#89137	direct (N × H)
159. C₄⋊C₄	C₂×#89137	semidirect
171. C₄²	C₂×#89137	semidirect
196. C₂×C₄	C₄×#89137	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812778	2
C₂²×C₄×#89137	2
C₄⋊C₄×#89137	2
C₂²⋊C₄×#89137	2
#20295962	2
C₄²×#89137	2
#117×D₁₀₉	2
C₄×#117	109