Subgroups — #28372067

Order 13952.

464 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295104	C₂	1	2	yes	maximal, complemented
2	#20295098	C₂	1	4	yes	maximal, complemented
3	D₄×#11724784	C₂	1	2	yes	maximal, complemented
4	#20295083	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295079	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294990	C₂	1	2	yes	maximal, complemented
7	C₂×C₂²⋊#11724784	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#13182013	C₂²	1	4	yes	complemented
10	C₂²×#11724784	C₂²	1	4	yes	complemented
11	C₄⋊#11724784	C₂²	1	2	yes	complemented
12	C₄⋊#11724784	C₂²	1	2	yes	complemented
13	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
14	C₂²⋊#11724784	C₂²	1	2	yes	—
15	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
16	C₂²⋊#11724784	C₂²	1	4	yes	complemented
17	C₂²⋊#11724784	C₂²	1	4	yes	complemented
18	#16812610	C₂²	1	2	yes	complemented
19	C₄×#11724784	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₁₀₉	C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
22	C₄×D₄×C₁₀₉	C₄	1	2	yes	—
23	C₄×D₄×C₁₀₉	C₄	1	1	yes	characteristic
24	C₄×D₄×C₁₀₉	C₄	1	1	yes	characteristic
25	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
27	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₁₀₉	C₄	1	2	yes	—
30	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
31	#13182013	—	4	4	no	—
32	C₂×#11724784	—	4	4	no	—
33	C₂×#11724784	C₂³	1	2	yes	—
34	C₂×#11724784	C₂³	1	2	yes	—
35	C₂×#11724784	C₂³	1	4	yes	—
36	C₂⁴×C₁₀₉	C₂×C₄	1	2	yes	—
37	C₂×D₄×C₁₀₉	—	2	4	no	—
38	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
40	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₀₉	—	4	4	no	—
42	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
45	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
47	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
50	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
51	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
53	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
54	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
55	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
56	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
57	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
60	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
61	C₂³×C₁₀₉	—	2	4	no	—
62	C₂³×C₁₀₉	—	2	4	no	—
63	C₂³×C₁₀₉	—	4	2	no	—
64	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
65	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
66	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
67	D₄×C₁₀₉	—	2	4	no	—
68	D₄×C₁₀₉	—	2	2	no	—
69	D₄×C₁₀₉	M₁₆	1	4	yes	complemented
70	C₂×C₄×C₁₀₉	—	2	2	no	—
71	C₂×C₄×C₁₀₉	—	2	2	no	—
72	C₂×C₄×C₁₀₉	—	2	2	no	—
73	C₂×C₄×C₁₀₉	—	2	4	no	—
74	C₂×C₄×C₁₀₉	—	2	4	no	—
75	C₂×C₄×C₁₀₉	—	4	2	no	—
76	C₂×C₄×C₁₀₉	—	4	2	no	—
77	C₂×C₄×C₁₀₉	—	4	2	no	—
78	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
86	#11724784	—	2	2	no	—
87	#11724784	—	2	4	no	—
88	#11724784	—	2	2	no	—
89	C₂²×C₁₀₉	—	2	1	no	—
90	C₂²×C₁₀₉	—	2	2	no	—
91	C₂²×C₁₀₉	—	2	1	no	—
92	C₂²×C₁₀₉	—	2	2	no	—
93	C₂²×C₁₀₉	—	2	2	no	—
94	C₂²×C₁₀₉	—	2	2	no	—
95	C₂²×C₁₀₉	—	2	2	no	—
96	C₂²×C₁₀₉	—	4	8	no	—
97	C₂²×C₁₀₉	C₂×M₁₆	1	4	yes	—
98	C₂²×C₁₀₉	#131	1	2	yes	—
99	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
100	C₄×C₁₀₉	—	2	1	no	—
101	C₄×C₁₀₉	—	2	2	no	—
102	C₄×C₁₀₉	—	2	2	no	—
103	C₄×C₁₀₉	—	2	1	no	—
104	C₄×C₁₀₉	—	2	1	no	—
105	C₄×C₁₀₉	—	2	1	no	—
106	C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
107	C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
108	C₂×C₁₀₉	—	2	2	no	—
109	C₂×C₁₀₉	—	2	4	no	—
110	C₂×C₁₀₉	—	4	2	no	—
111	C₂×C₁₀₉	#518	1	1	yes	characteristic
112	C₂×C₁₀₉	C₂²×M₁₆	1	1	yes	characteristic
113	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
114	#2974	—	109	1	no	maximal, Sylow
115	C₁₀₉	#2974	1	1	yes	characteristic, complemented, Sylow
116	C₂×C₄×D₄	D₁₀₉	1	1	yes	characteristic
117	#436	—	109	2	no	—
118	#435	—	109	4	no	—
119	C₈×D₄	—	109	2	no	—
120	#432	—	109	1	no	—
121	#431	—	109	1	no	—
122	#407	—	109	2	no	—
123	C₂×C₂²⋊C₈	—	109	2	no	—
124	C₂²×D₄	Dic₁₀₉	1	1	yes	characteristic
125	C₂³×C₄	C₂×D₁₀₉	1	2	yes	—
126	C₂×M₁₆	—	109	4	no	—
127	C₂²×C₈	—	109	4	no	—
128	C₄×D₄	Dic₁₀₉	1	2	yes	—
129	C₄×D₄	Dic₁₀₉	1	1	yes	characteristic
130	C₄×D₄	Dic₁₀₉	1	1	yes	characteristic
131	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
132	C₄×D₄	C₂×D₁₀₉	1	2	yes	—
133	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
134	C₂×C₄⋊C₄	Dic₁₀₉	1	1	yes	characteristic
135	C₂×C₂²⋊C₄	Dic₁₀₉	1	2	yes	—
136	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
137	C₄⋊C₈	—	109	2	no	—
138	C₄⋊C₈	—	109	2	no	—
139	C₂²⋊C₈	—	109	1	no	—
140	C₂²⋊C₈	—	109	2	no	—
141	C₂²⋊C₈	—	109	1	no	—
142	C₂²⋊C₈	—	109	4	no	—
143	C₂²⋊C₈	—	109	4	no	—
144	#97	—	109	2	no	—
145	C₄×C₈	—	109	2	no	—
146	C₂⁴	C₂×Dic₁₀₉	1	2	yes	—
147	C₂×D₄	—	2	4	no	—
148	C₂×D₄	C₂×Dic₁₀₉	1	2	yes	—
149	C₂×D₄	C₂×Dic₁₀₉	1	1	yes	characteristic
150	C₂×D₄	C₂×Dic₁₀₉	1	1	yes	characteristic
151	C₂²×C₄	—	4	4	no	—
152	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
153	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
154	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
155	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
156	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
157	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
159	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
160	M₁₆	—	436	4	no	—
161	C₂×C₈	—	109	2	no	—
162	C₂×C₈	—	109	2	no	—
163	C₂×C₈	—	109	4	no	—
164	C₂×C₈	—	436	4	no	—
165	C₄⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
166	C₄⋊C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
167	C₄⋊C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
168	C₂²⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
169	C₂²⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
170	C₂²⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
171	C₂²⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
172	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
173	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
174	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
175	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
176	C₂³	—	2	4	no	—
177	C₂³	—	2	4	no	—
178	C₂³	—	4	2	no	—
179	C₂³	C₂²×Dic₁₀₉	1	2	yes	—
180	C₂³	C₂²×Dic₁₀₉	1	2	yes	—
181	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
182	D₄	—	2	4	no	—
183	D₄	—	2	2	no	—
184	D₄	#13182013	1	4	yes	complemented
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
194	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
195	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
196	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
197	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
198	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
199	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
200	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
201	C₈	—	218	2	no	—
202	C₈	—	218	4	no	—
203	C₈	—	218	2	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	4	8	no	—
212	C₂²	C₂×#13182013	1	4	yes	—
213	C₂²	#16812706	1	2	yes	—
214	C₂²	C₂³×Dic₁₀₉	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	C₂×#13182013	1	2	yes	—
222	C₄	#16812744	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	2	4	no	—
225	C₂	—	4	2	no	—
226	C₂	#20295501	1	1	yes	characteristic
227	C₂	C₂²×#13182013	1	1	yes	characteristic
228	C₂	C₂×#16812706	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	C₂×C₄×D₄×C₁₀₉	C₂
prime radical	C₂²×D₄×C₁₀₉	C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂×C₄×D₄	D₁₀₉
hypocenter	C₁₀₉	#2974
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #20295104	C₂	semidirect
2. #20295098	C₂	semidirect
3. D₄×#11724784	C₂	semidirect
4. #20295083	C₂	semidirect
5. #20295079	C₂	semidirect
6. #20294990	C₂	semidirect
7. C₂×C₂²⋊#11724784	C₂	semidirect
9. C₂×#13182013	C₂²	semidirect
10. C₂²×#11724784	C₂²	semidirect
11. C₄⋊#11724784	C₂²	semidirect
12. C₄⋊#11724784	C₂²	semidirect
16. C₂²⋊#11724784	C₂²	semidirect
17. C₂²⋊#11724784	C₂²	semidirect
18. #16812610	C₂²	semidirect
19. C₄×#11724784	C₂²	semidirect
69. D₄×C₁₀₉	M₁₆	semidirect
115. C₁₀₉	#2974	semidirect
184. D₄	#13182013	semidirect

Maximal subgroups

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

subgroup	index
#20295104	2
#20295098	2
D₄×#11724784	2
#20295083	2
#20295079	2
#20294990	2
C₂×C₂²⋊#11724784	2
C₂×C₄×D₄×C₁₀₉	2
#2974	109