Subgroups — #28379982

Order 13952.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20296077	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#11724786	C₂	1	1	yes	maximal, characteristic, complemented
3	#20296061	C₂	1	1	yes	maximal, characteristic, complemented
4	#20296058	C₂	1	2	yes	maximal, complemented
5	#20296053	C₂	1	2	yes	maximal, complemented
6	D₄×#11724786	C₂	1	1	yes	maximal, characteristic, complemented
7	#20296050	C₂	1	2	yes	maximal, complemented
8	#20296045	C₂	1	1	yes	maximal, characteristic, complemented
9	#20296032	C₂	1	2	yes	maximal, complemented
10	D₁₀₉⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295555	C₂	1	1	yes	maximal, characteristic
12	C₂×#13182038	C₂²	1	2	yes	—
13	C₂×D₁₀₉⋊C₈	C₂²	1	2	yes	—
14	C₂²×#11724786	C₂²	1	2	yes	—
15	#16812766	C₂²	1	2	yes	—
16	#16812766	C₂²	1	2	yes	—
17	C₄⋊#11724786	C₂²	1	1	yes	characteristic
18	C₄⋊#11724786	C₂²	1	1	yes	characteristic
19	C₄⋊#11724786	C₂²	1	1	yes	characteristic
20	C₄⋊#11724786	C₂²	1	2	yes	—
21	C₄⋊#11724786	C₂²	1	1	yes	characteristic
22	#16812756	C₂²	1	1	yes	characteristic
23	#16812756	C₂²	1	1	yes	characteristic
24	C₂²⋊#11724786	C₂²	1	2	yes	—
25	C₂²⋊#11724786	C₂²	1	2	yes	—
26	#16812754	C₂²	1	2	yes	—
27	C₄×#11724786	C₂²	1	1	yes	characteristic
28	C₄×#11724786	C₂²	1	1	yes	characteristic
29	C₄×#11724786	C₂²	1	2	yes	—
30	#16812692	C₄	1	2	yes	—
31	#16812691	C₄	1	1	yes	characteristic
32	#16812681	C₄	1	2	yes	—
33	#16812675	C₄	1	1	yes	characteristic
34	#16812670	C₄	1	1	yes	characteristic
35	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
36	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
37	#16812659	C₂²	1	2	yes	—
38	#16812655	C₂²	1	2	yes	—
39	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
40	#124×C₁₀₉	C₄	1	1	yes	characteristic
41	#13182038	—	4	2	no	—
42	D₁₀₉⋊C₈	—	4	2	no	—
43	C₂×#11724786	—	4	2	no	—
44	C₂×#11724786	C₂³	1	1	yes	characteristic
45	C₂×#11724786	C₂³	1	1	yes	characteristic
46	C₂×#11724786	C₂³	1	2	yes	—
47	C₂×#11724786	C₂³	1	1	yes	characteristic
48	C₂×#11724786	C₂³	1	1	yes	characteristic
49	C₂×#11724786	C₂³	1	2	yes	—
50	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
51	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
52	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
55	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
56	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
58	#13182008	C₂×C₄	1	2	yes	—
59	#13182008	C₂×C₄	1	2	yes	—
60	#13182008	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
62	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
63	#13182006	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
66	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
67	C₄×Dic₁₀₉	C₂³	1	2	yes	—
68	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
69	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
70	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
71	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
72	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
73	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
74	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₁₀₉	—	2	2	no	—
76	D₄×C₁₀₉	—	2	2	no	—
77	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
81	C₂²⋊D₁₀₉	—	2	4	no	—
82	C₂×Dic₁₀₉	—	2	2	no	—
83	C₂×Dic₁₀₉	—	2	2	no	—
84	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
87	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
88	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
89	C₄×D₁₀₉	—	2	2	no	—
90	C₄×D₁₀₉	—	2	2	no	—
91	C₄×D₁₀₉	—	2	1	no	—
92	C₄×D₁₀₉	—	2	1	no	—
93	#11724786	—	2	1	no	—
94	#11724786	—	2	2	no	—
95	#11724786	—	2	2	no	—
96	#11724786	—	2	1	no	—
97	#11724786	—	2	1	no	—
98	#11724786	—	2	1	no	—
99	C₂²×C₁₀₉	—	2	2	no	—
100	C₂²×C₁₀₉	—	2	2	no	—
101	C₂²×C₁₀₉	—	2	2	no	—
102	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₁₀₉	—	2	1	no	—
104	C₂×D₁₀₉	—	2	1	no	—
105	C₂×D₁₀₉	#131	1	2	yes	—
106	C₄×C₁₀₉	—	2	1	no	—
107	C₄×C₁₀₉	—	2	1	no	—
108	C₄×C₁₀₉	—	2	2	no	—
109	C₄×C₁₀₉	#131	1	2	yes	—
110	Dic₁₀₉	—	2	2	no	—
111	Dic₁₀₉	—	2	1	no	—
112	Dic₁₀₉	—	2	1	no	—
113	Dic₁₀₉	—	2	2	no	—
114	Dic₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
115	Dic₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×C₁₀₉	—	4	2	no	—
117	C₂×C₁₀₉	#520	1	1	yes	characteristic
118	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
119	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
120	D₁₀₉	—	2	2	no	—
121	#2994	—	109	1	no	maximal, Sylow
122	C₁₀₉	#2994	1	1	yes	characteristic, complemented, Sylow
123	#529	—	109	1	no	—
124	#446	—	109	1	no	—
125	C₈×Q₈	—	109	1	no	—
126	#436	—	109	1	no	—
127	#435	—	109	2	no	—
128	C₈×D₄	—	109	1	no	—
129	C₈×D₄	—	109	2	no	—
130	#433	—	109	2	no	—
131	#431	—	109	1	no	—
132	#424	—	109	2	no	—
133	C₂×C₄⋊C₈	—	109	1	no	—
134	C₂×M₁₆	—	109	2	no	—
135	C₂²×C₈	—	109	2	no	—
136	C₂²×C₈	—	109	2	no	—
137	#126	—	109	2	no	—
138	#125	—	109	1	no	—
139	#124	#89137	1	1	yes	characteristic
140	#123	—	109	2	no	—
141	C₂²⋊Q₈	—	109	1	no	—
142	#121	—	109	1	no	—
143	C₄×Q₈	—	109	1	no	—
144	C₄×D₄	—	109	1	no	—
145	C₄×D₄	—	109	2	no	—
146	#117	—	109	2	no	—
147	C₂×C₄²	—	109	1	no	—
148	C₄⋊C₈	—	109	2	no	—
149	C₄⋊C₈	—	109	1	no	—
150	C₄⋊C₈	—	109	1	no	—
151	C₄⋊C₈	—	109	1	no	—
152	C₄⋊C₈	—	109	2	no	—
153	C₄⋊C₈	—	109	2	no	—
154	C₄⋊C₈	—	109	1	no	—
155	C₂²⋊C₈	—	109	1	no	—
156	C₂²⋊C₈	—	109	1	no	—
157	C₂²⋊C₈	—	109	2	no	—
158	C₂²⋊C₈	—	109	2	no	—
159	#97	—	109	2	no	—
160	C₄×C₈	—	109	1	no	—
161	C₄×C₈	—	109	1	no	—
162	C₄×C₈	—	109	2	no	—
163	C₂×Q₈	C₂×#89137	1	1	yes	characteristic
164	C₂×D₄	—	109	2	no	—
165	C₂×D₄	C₂×#89137	1	1	yes	characteristic
166	C₂²×C₄	—	109	2	no	—
167	C₂²×C₄	—	109	1	no	—
168	C₂²×C₄	—	109	1	no	—
169	C₂²×C₄	—	109	1	no	—
170	M₁₆	—	436	2	no	—
171	C₂×C₈	—	109	1	no	—
172	C₂×C₈	—	109	1	no	—
173	C₂×C₈	—	109	2	no	—
174	C₂×C₈	—	109	1	no	—
175	C₂×C₈	—	109	1	no	—
176	C₂×C₈	—	109	2	no	—
177	C₂×C₈	—	436	2	no	—
178	C₂×C₈	—	436	2	no	—
179	C₄⋊C₄	—	109	2	no	—
180	C₄⋊C₄	—	109	2	no	—
181	C₄⋊C₄	—	109	2	no	—
182	C₄⋊C₄	—	109	1	no	—
183	C₄⋊C₄	—	109	1	no	—
184	C₄⋊C₄	—	109	2	no	—
185	C₂²⋊C₄	—	109	2	no	—
186	C₂²⋊C₄	—	109	2	no	—
187	C₂²⋊C₄	—	109	2	no	—
188	C₂²⋊C₄	C₂×#89137	1	2	yes	—
189	C₂²⋊C₄	C₂×#89137	1	2	yes	—
190	C₄²	—	109	1	no	—
191	C₄²	—	109	1	no	—
192	C₄²	—	109	2	no	—
193	C₄²	—	109	1	no	—
194	C₄²	C₂×#89137	1	1	yes	characteristic
195	C₂³	—	109	1	no	—
196	C₂³	C₂²×#89137	1	2	yes	—
197	Q₈	—	2	2	no	—
198	D₄	—	2	2	no	—
199	D₄	—	218	4	no	—
200	C₂×C₄	—	109	1	no	—
201	C₂×C₄	—	109	1	no	—
202	C₂×C₄	—	109	1	no	—
203	C₂×C₄	—	109	2	no	—
204	C₂×C₄	—	109	2	no	—
205	C₂×C₄	—	218	1	no	—
206	C₂×C₄	—	218	1	no	—
207	C₂×C₄	—	218	2	no	—
208	C₂×C₄	—	218	2	no	—
209	C₂×C₄	—	218	2	no	—
210	C₂×C₄	—	218	2	no	—
211	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
212	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
213	C₂×C₄	C₂²×#89137	1	2	yes	—
214	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
215	C₈	—	218	1	no	—
216	C₈	—	218	2	no	—
217	C₈	—	218	2	no	—
218	C₈	—	218	1	no	—
219	C₈	—	218	1	no	—
220	C₈	—	218	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	109	2	no	—
225	C₂²	—	218	1	no	—
226	C₂²	—	218	1	no	—
227	C₂²	C₂³×#89137	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	109	1	no	—
232	C₄	—	109	1	no	—
233	C₄	—	218	2	no	—
234	C₄	—	218	1	no	—
235	C₄	—	218	1	no	—
236	C₄	—	218	2	no	—
237	C₄	#16812786	1	2	yes	—
238	C₂	—	4	2	no	—
239	C₂	—	218	2	no	—
240	C₂	#20296095	1	1	yes	characteristic
241	C₂	C₂×#16812786	1	1	yes	characteristic
242	C₂	D₁₀₉⋊#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#89137
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#124×C₁₀₉	C₄
prime radical	#16812670	C₄
cosocle	C₂×Dic₁₀₉	C₂⁴
hypercenter	#124	#89137
hypocenter	C₁₀₉	#2994
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
1. #20296077	C₂	semidirect
2. Q₈×#11724786	C₂	semidirect
3. #20296061	C₂	semidirect
4. #20296058	C₂	semidirect
5. #20296053	C₂	semidirect
6. D₄×#11724786	C₂	semidirect
7. #20296050	C₂	semidirect
8. #20296045	C₂	semidirect
9. #20296032	C₂	semidirect
10. D₁₀₉⋊(C₄⋊C₈)	C₂	semidirect
122. C₁₀₉	#2994	semidirect

Maximal subgroups

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

subgroup	index
#20296077	2
Q₈×#11724786	2
#20296061	2
#20296058	2
#20296053	2
D₄×#11724786	2
#20296050	2
#20296045	2
#20296032	2
D₁₀₉⋊(C₄⋊C₈)	2
#20295555	2
#2994	109