Subgroups — #28345331

Order 13696.

484 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812128	C₂	1	1	yes	maximal, characteristic
2	#20293106	C₂	1	2	yes	maximal, complemented
3	#20292727	C₂	1	2	yes	maximal, complemented
4	#20292724	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₄₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#20292721	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×Dic₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#107×D₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#20292625	C₂	1	2	yes	maximal, complemented
10	#20292581	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133817	C₂²	1	2	yes	—
13	#16812128	C₄	1	8	yes	complemented
14	C₂×Dic₄₂₈	C₂²	1	1	yes	characteristic
15	C₂×#13133806	C₂²	1	2	yes	—
16	C₂×#13133801	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₀₇	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₁₄	C₂²	1	2	yes	—
19	#16812074	C₂²	1	2	yes	—
20	D₄×Dic₁₀₇	C₂²	1	2	yes	—
21	C₄×C₂²⋊D₁₀₇	—	2	2	no	—
22	(C₂²⋊D₁₀₇)⋊C₄	—	2	2	no	—
23	#16812066	C₂²	1	1	yes	characteristic
24	#16812065	—	2	2	no	—
25	C₂×Dic₁₀₇⋊C₄	—	2	2	no	—
26	C₄⋊C₄×D₁₀₇	C₂²	1	2	yes	—
27	#16812043	C₂²	1	1	yes	characteristic
28	#16812042	C₂²	1	2	yes	—
29	#16812034	C₂²	1	2	yes	—
30	#16812032	C₂²	1	2	yes	—
31	#16812030	C₂²	1	2	yes	—
32	#16812020	C₂²	1	1	yes	characteristic
33	#16812019	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₀₇	C₂²	1	1	yes	characteristic
35	C₂×D₈×C₁₀₇	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₁₀₇	C₂²	1	2	yes	—
37	#107×C₁₀₇	C₂²	1	1	yes	characteristic
38	#102×C₁₀₇	C₂²	1	2	yes	—
39	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
40	#13133817	—	2	4	no	—
41	#13133817	C₂×C₄	1	8	yes	complemented
42	C₂×Dic₂₁₄	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₁₀₇	—	2	2	no	—
44	C₂²×Dic₁₀₇	—	2	2	no	—
45	C₂²×Dic₁₀₇	—	2	2	no	—
46	C₂×C₄×D₁₀₇	—	2	2	no	—
47	C₂×C₄×D₁₀₇	C₂³	1	1	yes	characteristic
48	Dic₄₂₈	C₂×C₄	1	4	yes	complemented
49	#13133806	C₂×C₄	1	8	yes	complemented
50	#13133801	—	2	2	no	—
51	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
52	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
53	C₈×D₁₀₇	C₂×C₄	1	4	yes	complemented
54	C₄⋊Dic₁₀₇	C₂³	1	2	yes	—
55	Dic₁₀₇⋊C₄	—	2	2	no	—
56	Dic₁₀₇⋊C₄	—	2	2	no	—
57	Dic₁₀₇⋊C₄	—	2	2	no	—
58	Dic₁₀₇⋊C₄	D₄	1	1	yes	characteristic
59	Dic₁₀₇⋊C₄	D₄	1	1	yes	characteristic
60	C₂²⋊Dic₁₀₇	—	2	2	no	—
61	(C₂×D₁₀₇)⋊C₄	D₄	1	1	yes	characteristic
62	(C₂×D₁₀₇)⋊C₄	D₄	1	1	yes	characteristic
63	C₄×Dic₁₀₇	—	2	2	no	—
64	C₄×Dic₁₀₇	—	2	2	no	—
65	C₄×Dic₁₀₇	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₁₀₇	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₀₇	—	2	2	no	—
68	D₈×C₁₀₇	C₂×C₄	1	4	yes	complemented
69	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₀₇	C₂³	1	2	yes	—
72	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
73	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
74	C₂³×C₁₀₇	—	2	2	no	—
75	C₂²×D₁₀₇	C₂×D₄	1	1	yes	characteristic
76	D₄×C₁₀₇	—	2	2	no	—
77	D₄×C₁₀₇	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₁₀₇	—	2	2	no	—
79	C₂×C₄×C₁₀₇	—	8	2	no	—
80	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₁₀₇	—	2	8	no	—
84	C₂×Dic₁₀₇	—	2	2	no	—
85	C₂×Dic₁₀₇	—	2	8	no	—
86	C₂×Dic₁₀₇	—	2	2	no	—
87	C₂×Dic₁₀₇	—	8	2	no	—
88	C₂×Dic₁₀₇	C₂×D₄	1	1	yes	characteristic
89	C₂×Dic₁₀₇	C₂×D₄	1	1	yes	characteristic
90	C₂×Dic₁₀₇	C₂×D₄	1	1	yes	characteristic
91	C₄×D₁₀₇	—	8	2	no	—
92	C₄×D₁₀₇	C₂²×C₄	1	2	yes	—
93	C₄×D₁₀₇	#41	1	2	yes	—
94	Dic₂₁₄	—	2	2	no	—
95	Dic₂₁₄	C₂²×C₄	1	4	yes	—
96	C₈×C₁₀₇	—	2	1	no	—
97	C₈×C₁₀₇	C₂²×C₄	1	2	yes	—
98	C₁₀₇⋊C₈	—	2	1	no	—
99	C₁₀₇⋊C₈	C₂²×C₄	1	2	yes	—
100	C₂²×C₁₀₇	—	2	4	no	—
101	C₂²×C₁₀₇	—	4	2	no	—
102	C₂²×C₁₀₇	—	4	2	no	—
103	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×D₁₀₇	—	2	1	no	—
105	C₂×D₁₀₇	—	2	1	no	—
106	C₂×D₁₀₇	C₄×D₄	1	2	yes	—
107	C₄×C₁₀₇	—	2	1	no	—
108	C₄×C₁₀₇	—	2	1	no	—
109	C₄×C₁₀₇	—	4	2	no	—
110	C₄×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
111	C₄×C₁₀₇	C₂×#41	1	1	yes	characteristic
112	Dic₁₀₇	—	2	4	no	—
113	Dic₁₀₇	—	2	1	no	—
114	Dic₁₀₇	—	2	1	no	—
115	Dic₁₀₇	—	4	2	no	—
116	Dic₁₀₇	C₄×D₄	1	2	yes	—
117	C₂×C₁₀₇	—	4	4	no	—
118	C₂×C₁₀₇	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂×C₁₀₇	#576	1	1	yes	characteristic
120	C₂×C₁₀₇	#578	1	1	yes	characteristic
121	D₁₀₇	—	2	2	no	—
122	#2951	—	107	1	no	maximal, Sylow
123	C₁₀₇	#2951	1	1	yes	characteristic, complemented, Sylow
124	C₂×#135	—	107	1	no	—
125	#520	—	107	2	no	—
126	#440	—	107	2	no	—
127	#440	—	107	2	no	—
128	C₄×Q₁₆	—	107	1	no	—
129	C₄×Q₁₆	—	107	1	no	—
130	C₄×D₈	—	107	1	no	—
131	C₄×D₈	D₁₀₇	1	1	yes	characteristic, complemented
132	C₂×#107	—	107	1	no	—
133	#417	—	107	2	no	—
134	#405	—	107	1	no	—
135	C₂×#41	—	107	2	no	—
136	#135	—	107	8	no	—
137	C₂×Q₁₆	—	107	1	no	—
138	C₂×SD₁₆	—	107	2	no	—
139	C₂×D₈	C₂×D₁₀₇	1	1	yes	characteristic
140	C₂×M₁₆	—	107	1	no	—
141	C₂²×C₈	—	107	1	no	—
142	C₄×Q₈	—	107	2	no	—
143	C₄×Q₈	—	107	2	no	—
144	C₄×D₄	—	107	2	no	—
145	C₄×D₄	—	214	2	no	—
146	C₄×D₄	—	214	2	no	—
147	C₄×D₄	C₂×D₁₀₇	1	2	yes	—
148	#117	—	107	1	no	—
149	#117	—	214	2	no	—
150	C₂×C₄⋊C₄	—	107	2	no	—
151	C₂×C₄⋊C₄	—	214	2	no	—
152	#107	—	107	1	no	—
153	#107	—	107	2	no	—
154	#107	C₂×D₁₀₇	1	1	yes	characteristic
155	#103	—	107	2	no	—
156	#103	—	107	2	no	—
157	#102	—	107	2	no	—
158	#102	C₂×D₁₀₇	1	2	yes	—
159	#97	—	107	1	no	—
160	#97	—	107	1	no	—
161	C₄×C₈	—	107	1	no	—
162	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
163	#41	—	107	8	no	—
164	#41	—	214	4	no	—
165	C₂×Q₈	—	107	2	no	—
166	C₂×D₄	—	214	2	no	—
167	C₂×D₄	C₂²×D₁₀₇	1	2	yes	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	107	1	no	—
170	C₂²×C₄	—	214	2	no	—
171	C₂²×C₄	—	214	2	no	—
172	C₂²×C₄	—	214	2	no	—
173	Q₁₆	—	107	4	no	—
174	SD₁₆	—	107	8	no	—
175	D₈	C₄×D₁₀₇	1	4	yes	complemented
176	M₁₆	—	214	2	no	—
177	C₂×C₈	—	107	1	no	—
178	C₂×C₈	—	107	4	no	—
179	C₂×C₈	—	107	1	no	—
180	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
181	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
182	C₄⋊C₄	—	107	1	no	—
183	C₄⋊C₄	—	107	1	no	—
184	C₄⋊C₄	—	107	2	no	—
185	C₄⋊C₄	—	214	2	no	—
186	C₄⋊C₄	—	214	2	no	—
187	C₄⋊C₄	—	214	2	no	—
188	C₄⋊C₄	C₂²×D₁₀₇	1	2	yes	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	107	1	no	—
191	C₂²⋊C₄	—	107	1	no	—
192	C₂²⋊C₄	—	214	2	no	—
193	C₄²	—	107	1	no	—
194	C₄²	—	214	2	no	—
195	C₄²	—	214	2	no	—
196	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
197	C₂³	—	2	2	no	—
198	C₂³	—	107	1	no	—
199	Q₈	—	107	4	no	—
200	Q₈	—	214	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	214	8	no	—
203	D₄	C₂×C₄×D₁₀₇	1	4	yes	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	—	856	2	no	—
207	C₂×C₄	—	856	2	no	—
208	C₂×C₄	—	107	1	no	—
209	C₂×C₄	—	107	2	no	—
210	C₂×C₄	—	107	2	no	—
211	C₂×C₄	—	107	1	no	—
212	C₂×C₄	—	107	1	no	—
213	C₂×C₄	—	214	8	no	—
214	C₂×C₄	—	214	2	no	—
215	C₂×C₄	—	214	2	no	—
216	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
217	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
218	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	107	2	no	—
221	C₈	—	214	1	no	—
222	C₈	C₂×C₄×D₁₀₇	1	2	yes	—
223	C₂²	—	2	4	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	107	2	no	—
227	C₂²	—	214	1	no	—
228	C₂²	—	214	1	no	—
229	C₂²	C₂×D₄×D₁₀₇	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	2	no	—
233	C₄	—	107	2	no	—
234	C₄	—	428	2	no	—
235	C₄	—	214	1	no	—
236	C₄	—	214	1	no	—
237	C₄	—	214	4	no	—
238	C₄	C₂²×C₄×D₁₀₇	1	1	yes	characteristic
239	C₄	#41×D₁₀₇	1	1	yes	characteristic
240	C₂	—	4	4	no	—
241	C₂	—	214	2	no	—
242	C₂	C₄×D₄×D₁₀₇	1	1	yes	characteristic
243	C₂	#20293428	1	1	yes	characteristic
244	C₂	#20293441	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₇
commutator	C₄×C₁₀₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting	C₄×D₈×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂²×D₄
prime radical	C₂×#16812128	C₂
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	C₄×D₈	D₁₀₇
hypocenter	C₁₀₇	#2951
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #20293106	C₂	semidirect
3. #20292727	C₂	semidirect
4. #20292724	C₂	semidirect
5. C₄×Dic₄₂₈	C₂	semidirect
6. #20292721	C₂	semidirect
7. D₈×Dic₁₀₇	C₂	semidirect
8. #107×D₁₀₇	C₂	semidirect
9. #20292625	C₂	semidirect
10. #20292581	C₂	semidirect
11. C₄×D₈×C₁₀₇	C₂	semidirect
13. #16812128	C₄	semidirect
41. #13133817	C₂×C₄	semidirect
48. Dic₄₂₈	C₂×C₄	semidirect
49. #13133806	C₂×C₄	semidirect
53. C₈×D₁₀₇	C₂×C₄	semidirect
68. D₈×C₁₀₇	C₂×C₄	semidirect
123. C₁₀₇	#2951	semidirect
131. C₄×D₈	D₁₀₇	semidirect
175. D₈	C₄×D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812128	2
#20293106	2
#20292727	2
#20292724	2
C₄×Dic₄₂₈	2
#20292721	2
D₈×Dic₁₀₇	2
#107×D₁₀₇	2
#20292625	2
#20292581	2
C₄×D₈×C₁₀₇	2
#2951	107