Subgroups — #28379469

Order 13952.

696 conjugacy classes of subgroups.

The subgroup lattice has 224 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	C₄×C₄⋊#89137	C₂	1	4	yes	maximal, complemented
5	C₂²⋊C₄×#89137	C₂	1	4	yes	maximal
6	#117×D₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×#89137	C₂²	1	2	yes	—
8	#16812778	C₄	1	8	yes	complemented
9	C₂×C₄⋊#89137	C₄	1	8	yes	complemented
10	C₂×C₄⋊#89137	C₂²	1	2	yes	—
11	C₂×C₄⋊#89137	C₂²	1	2	yes	—
12	C₂×C₄⋊#89137	C₂²	1	2	yes	—
13	C₂×C₂²⋊#89137	C₂²	1	2	yes	—
14	C₂×C₄×#89137	C₂²	1	8	yes	—
15	C₂×C₄×#89137	C₂²	1	2	yes	—
16	(C₂×#89137)⋊C₄	C₂²	1	8	yes	—
17	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
18	#16812658	C₄	1	1	yes	characteristic, complemented
19	#16812657	C₄	1	2	yes	complemented
20	#16812656	C₄	1	2	yes	complemented
21	#16812655	C₄	1	2	yes	complemented
22	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
23	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	—
24	C₄²×D₁₀₉	C₂²	1	2	yes	—
25	#117×C₁₀₉	C₄	1	1	yes	characteristic, complemented
26	C₂²×#89137	—	4	4	no	—
27	C₂²×#89137	C₂×C₄	1	8	yes	—
28	C₂²×#89137	C₂³	1	4	yes	—
29	C₂²×#89137	C₂³	1	4	yes	—
30	C₄⋊#89137	C₂×C₄	1	8	yes	—
31	C₄⋊#89137	C₂×C₄	1	8	yes	—
32	C₄⋊#89137	C₂×C₄	1	8	yes	complemented
33	C₂²⋊#89137	C₂×C₄	1	8	yes	—
34	C₄×#89137	—	4	8	no	—
35	C₄×#89137	C₂×C₄	1	8	yes	complemented
36	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×D₁₀₉	C₂×C₄	1	4	yes	—
41	C₂×C₄×D₁₀₉	C₂×C₄	1	4	yes	—
42	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
43	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
44	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
45	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	complemented
46	#13182008	C₂×C₄	1	4	yes	complemented
47	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
48	#13182006	C₂×C₄	1	4	yes	—
49	C₄×Dic₁₀₉	C₂×C₄	1	4	yes	complemented
50	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	complemented
51	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	complemented
53	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
54	C₄²×C₁₀₉	C₂×C₄	1	2	yes	complemented
55	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×D₁₀₉	—	2	2	no	—
57	C₂²×D₁₀₉	—	2	2	no	—
58	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
60	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
61	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
62	C₂×#89137	—	2	4	no	—
63	C₂×#89137	—	2	8	no	—
64	C₂×#89137	—	2	4	no	—
65	C₂×#89137	—	4	8	no	—
66	C₂×#89137	C₂²×C₄	1	8	yes	—
67	C₂×#89137	C₂²×C₄	1	8	yes	—
68	C₂×C₄×C₁₀₉	C₄²	1	4	yes	complemented
69	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
70	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×Dic₁₀₉	C₄²	1	4	yes	complemented
73	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
74	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
76	C₄×D₁₀₉	—	2	4	no	—
77	C₄×D₁₀₉	—	2	4	no	—
78	C₄×D₁₀₉	C₄²	1	4	yes	complemented
79	C₄×D₁₀₉	C₄²	1	4	yes	complemented
80	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
81	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
82	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
83	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
84	C₂²×C₁₀₉	—	2	2	no	—
85	C₂²×C₁₀₉	C₂×C₄²	1	2	yes	—
86	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
87	C₂×D₁₀₉	—	2	2	no	—
88	C₂×D₁₀₉	—	2	2	no	—
89	C₂×D₁₀₉	—	2	2	no	—
90	C₂×D₁₀₉	—	2	4	no	—
91	C₂×D₁₀₉	C₂×C₄²	1	2	yes	—
92	C₂×D₁₀₉	C₂³×C₄	1	1	yes	characteristic
93	C₂×D₁₀₉	C₂³×C₄	1	1	yes	characteristic
94	C₂×D₁₀₉	H(5,𝔽₂)	1	2	yes	—
95	C₂×D₁₀₉	2₋¹⁺⁴	1	2	yes	—
96	#89137	—	2	8	no	—
97	#89137	—	2	8	no	—
98	C₄×C₁₀₉	—	2	4	no	—
99	C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
100	C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
101	Dic₁₀₉	—	2	4	no	—
102	Dic₁₀₉	C₂×C₄²	1	2	yes	—
103	Dic₁₀₉	C₂×C₄²	1	2	yes	—
104	C₂×C₁₀₉	—	2	2	no	—
105	C₂×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
106	C₂×C₁₀₉	#520	1	2	yes	—
107	D₁₀₉	—	2	2	no	—
108	D₁₀₉	#518	1	1	yes	characteristic
109	D₁₀₉	#519	1	1	yes	characteristic
110	D₁₀₉	#520	1	2	yes	—
111	#2274	—	109	1	no	maximal, Sylow
112	C₁₀₉	#2274	1	1	yes	characteristic, complemented, Sylow
113	C₂×#117	—	109	1	no	—
114	C₂×#117	—	109	1	no	—
115	C₂²×C₄⋊C₄	—	109	1	no	—
116	C₄×C₄⋊C₄	—	109	4	no	—
117	C₄×C₄⋊C₄	—	109	4	no	—
118	C₄×C₂²⋊C₄	—	109	4	no	—
119	C₂³×C₄	—	109	1	no	—
120	C₂³×C₄	—	109	2	no	—
121	#117	—	109	8	no	—
122	#117	—	109	1	no	—
123	#117	—	109	2	no	—
124	#117	—	109	2	no	—
125	#117	—	109	2	no	—
126	#117	#89137	1	1	yes	characteristic, complemented
127	C₂×C₄⋊C₄	—	109	2	no	—
128	C₂×C₄⋊C₄	—	109	2	no	—
129	C₂×C₄⋊C₄	—	109	8	no	—
130	C₂×C₄⋊C₄	—	109	2	no	—
131	C₂×C₄⋊C₄	—	109	2	no	—
132	C₂×C₂²⋊C₄	—	109	2	no	—
133	C₂×C₂²⋊C₄	—	109	2	no	—
134	C₂×C₄²	—	109	2	no	—
135	C₂×C₄²	—	109	2	no	—
136	C₂×C₄²	—	109	8	no	—
137	#95	—	109	8	no	—
138	C₂⁴	—	109	1	no	—
139	C₂²×C₄	—	109	1	no	—
140	C₂²×C₄	—	109	4	no	—
141	C₂²×C₄	—	109	4	no	—
142	C₂²×C₄	—	109	1	no	—
143	C₂²×C₄	—	109	1	no	—
144	C₂²×C₄	—	109	1	no	—
145	C₂²×C₄	—	109	1	no	—
146	C₂²×C₄	—	109	8	no	—
147	C₂²×C₄	—	109	4	no	—
148	C₂²×C₄	—	109	4	no	—
149	C₂²×C₄	—	109	4	no	—
150	C₂²×C₄	—	436	4	no	—
151	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
152	C₄⋊C₄	—	109	8	no	—
153	C₄⋊C₄	—	109	8	no	—
154	C₄⋊C₄	—	109	8	no	—
155	C₄⋊C₄	—	109	4	no	—
156	C₄⋊C₄	—	109	2	no	—
157	C₄⋊C₄	C₂×#89137	1	2	yes	complemented
158	C₂²⋊C₄	—	109	8	no	—
159	C₂²⋊C₄	—	109	2	no	—
160	C₂²⋊C₄	—	109	4	no	—
161	C₂²⋊C₄	C₂×#89137	1	2	yes	—
162	C₄²	—	109	8	no	—
163	C₄²	—	109	4	no	—
164	C₄²	—	109	2	no	—
165	C₄²	—	436	8	no	—
166	C₄²	C₂×#89137	1	2	yes	complemented
167	C₂³	—	109	1	no	—
168	C₂³	—	109	2	no	—
169	C₂³	—	109	2	no	—
170	C₂³	—	109	1	no	—
171	C₂³	—	218	2	no	—
172	C₂³	—	218	2	no	—
173	C₂³	C₂²×#89137	1	1	yes	characteristic
174	C₂×C₄	—	109	4	no	—
175	C₂×C₄	—	109	1	no	—
176	C₂×C₄	—	109	1	no	—
177	C₂×C₄	—	109	2	no	—
178	C₂×C₄	—	109	2	no	—
179	C₂×C₄	—	109	2	no	—
180	C₂×C₄	—	109	2	no	—
181	C₂×C₄	—	109	4	no	—
182	C₂×C₄	—	109	4	no	—
183	C₂×C₄	—	109	8	no	—
184	C₂×C₄	—	109	8	no	—
185	C₂×C₄	—	109	4	no	—
186	C₂×C₄	—	436	8	no	—
187	C₂×C₄	—	218	4	no	—
188	C₂×C₄	—	218	4	no	—
189	C₂×C₄	—	218	4	no	—
190	C₂×C₄	—	218	8	no	—
191	C₂×C₄	—	218	4	no	—
192	C₂×C₄	C₄×#89137	1	4	yes	complemented
193	C₂×C₄	C₂²×#89137	1	4	yes	—
194	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
195	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	—	109	1	no	—
198	C₂²	—	109	1	no	—
199	C₂²	—	109	2	no	—
200	C₂²	—	109	2	no	—
201	C₂²	—	109	2	no	—
202	C₂²	—	218	2	no	—
203	C₂²	—	218	4	no	—
204	C₂²	—	218	2	no	—
205	C₂²	—	218	2	no	—
206	C₂²	C₂×C₄×#89137	1	2	yes	—
207	C₂²	C₂³×#89137	1	1	yes	characteristic
208	C₄	—	2	4	no	—
209	C₄	—	109	2	no	—
210	C₄	—	109	2	no	—
211	C₄	—	218	8	no	—
212	C₄	—	218	8	no	—
213	C₄	—	218	4	no	—
214	C₄	C₂×C₄×#89137	1	2	yes	—
215	C₄	C₂×C₄×#89137	1	2	yes	—
216	C₂	—	2	2	no	—
217	C₂	—	109	1	no	—
218	C₂	—	109	2	no	—
219	C₂	—	109	1	no	—
220	C₂	—	218	2	no	—
221	C₂	C₂²×C₄×#89137	1	1	yes	characteristic
222	C₂	#20296095	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#89137
commutator	C₂×C₁₀₉	C₂²×C₄²
Fitting	#117×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂³×C₄
prime radical	C₂³×D₁₀₉	C₂³
cosocle	C₂²×D₁₀₉	C₂⁴
hypercenter	#117	#89137
hypocenter	C₁₀₉	#2274
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
3. C₄⋊C₄×#89137	C₂	semidirect
4. C₄×C₄⋊#89137	C₂	semidirect
8. #16812778	C₄	semidirect
9. C₂×C₄⋊#89137	C₄	semidirect
18. #16812658	C₄	semidirect
19. #16812657	C₄	semidirect
20. #16812656	C₄	semidirect
21. #16812655	C₄	semidirect
25. #117×C₁₀₉	C₄	semidirect
32. C₄⋊#89137	C₂×C₄	semidirect
35. C₄×#89137	C₂×C₄	semidirect
45. C₄⋊Dic₁₀₉	C₂×C₄	semidirect
46. #13182008	C₂×C₄	semidirect
49. C₄×Dic₁₀₉	C₂×C₄	semidirect
50. C₄×Dic₁₀₉	C₂×C₄	semidirect
52. C₄⋊C₄×C₁₀₉	C₂×C₄	semidirect
54. C₄²×C₁₀₉	C₂×C₄	semidirect
68. C₂×C₄×C₁₀₉	C₄²	semidirect
72. C₂×Dic₁₀₉	C₄²	semidirect
78. C₄×D₁₀₉	C₄²	semidirect
79. C₄×D₁₀₉	C₄²	semidirect
112. C₁₀₉	#2274	semidirect
126. #117	#89137	semidirect
157. C₄⋊C₄	C₂×#89137	semidirect
166. C₄²	C₂×#89137	semidirect
192. 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
C₄×C₄⋊#89137	2
C₂²⋊C₄×#89137	2
#117×D₁₀₉	2
#2274	109