Subgroups — #28346251

Order 13696.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293184	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#13133818	C₂	1	1	yes	maximal, characteristic
3	#20292935	C₂	1	2	yes	maximal, complemented
4	#20292928	C₂	1	2	yes	maximal, complemented
5	#20292923	C₂	1	1	yes	maximal, characteristic, complemented
6	#20292918	C₂	1	1	yes	maximal, characteristic, complemented
7	#20292913	C₂	1	1	yes	maximal, characteristic, complemented
8	#20292908	C₂	1	1	yes	maximal, characteristic, complemented
9	#20292665	C₂	1	2	yes	maximal, complemented
10	#20292624	C₂	1	2	yes	maximal
11	C₂×C₄⋊C₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133818	C₂²	1	1	yes	characteristic
13	#16812109	C₂²	1	1	yes	characteristic
14	#16812109	C₂²	1	1	yes	characteristic
15	#16812101	C₂²	1	2	yes	—
16	#16812088	—	2	1	no	—
17	#16812088	—	2	1	no	—
18	C₄×Dic₂₁₄	—	2	1	no	—
19	C₄×Dic₂₁₄	—	2	1	no	—
20	C₄×Dic₂₁₄	C₂²	1	1	yes	characteristic
21	C₄×Dic₂₁₄	C₂²	1	1	yes	characteristic
22	C₄×D₄₂₈	C₂²	1	1	yes	characteristic
23	C₄×D₄₂₈	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊D₁₀₇	—	2	1	no	—
25	C₄×C₂²⋊D₁₀₇	—	2	1	no	—
26	#16812068	C₂²	1	2	yes	—
27	#16812068	C₂²	1	1	yes	characteristic
28	#16812066	—	2	1	no	—
29	C₄²×D₁₀₇	—	2	1	no	—
30	#16812043	C₂²	1	2	yes	—
31	#16812043	C₂²	1	2	yes	—
32	#16812041	C₂²	1	2	yes	—
33	#16812041	C₂²	1	2	yes	—
34	#16812034	C₂²	1	2	yes	—
35	#16812034	C₂²	1	2	yes	—
36	#16812031	C₂²	1	2	yes	—
37	#16812031	C₂²	1	2	yes	—
38	C₂²×C₈×C₁₀₇	C₂²	1	2	yes	—
39	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
40	C₄⋊C₈×C₁₀₇	C₂²	1	1	yes	characteristic
41	C₄⋊C₈×C₁₀₇	C₂²	1	1	yes	characteristic
42	C₄⋊C₈×C₁₀₇	C₂²	1	2	yes	—
43	#13133818	—	2	2	no	—
44	#13133818	Q₈	1	4	yes	complemented
45	C₂×Dic₂₁₄	—	2	1	no	—
46	C₂×Dic₂₁₄	—	2	1	no	—
47	C₂×Dic₂₁₄	C₂³	1	1	yes	characteristic
48	C₂×D₄₂₈	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊D₁₀₇	—	2	1	no	—
50	C₂×C₄×D₁₀₇	—	2	1	no	—
51	C₂×C₄×D₁₀₇	—	2	1	no	—
52	C₂×C₄×D₁₀₇	—	2	1	no	—
53	C₄⋊Dic₁₀₇	—	2	1	no	—
54	C₄⋊Dic₁₀₇	—	2	1	no	—
55	C₄⋊Dic₁₀₇	C₂³	1	2	yes	—
56	C₄⋊Dic₁₀₇	C₂³	1	2	yes	—
57	C₄⋊Dic₁₀₇	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₀₇	C₂³	1	1	yes	characteristic
59	Dic₁₀₇⋊C₄	—	2	1	no	—
60	Dic₁₀₇⋊C₄	—	2	1	no	—
61	Dic₁₀₇⋊C₄	—	2	2	no	—
62	Dic₁₀₇⋊C₄	—	2	2	no	—
63	C₂²⋊Dic₁₀₇	—	2	1	no	—
64	C₂²⋊Dic₁₀₇	—	2	2	no	—
65	(C₂×D₁₀₇)⋊C₄	—	2	1	no	—
66	(C₂×D₁₀₇)⋊C₄	—	2	1	no	—
67	C₄×Dic₁₀₇	—	2	1	no	—
68	C₄×Dic₁₀₇	—	2	1	no	—
69	C₄×Dic₁₀₇	—	2	1	no	—
70	C₄×Dic₁₀₇	—	2	2	no	—
71	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₀₇	—	2	4	no	—
75	C₂×C₈×C₁₀₇	C₂³	1	2	yes	—
76	C₂×C₈×C₁₀₇	C₂³	1	2	yes	—
77	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
78	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
79	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
80	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
81	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₁₀₇	—	2	1	no	—
83	C₂×C₄×C₁₀₇	—	2	2	no	—
84	C₂×C₄×C₁₀₇	—	4	1	no	—
85	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
91	C₂×C₄×C₁₀₇	#41	1	2	yes	—
92	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
93	C₂²⋊D₁₀₇	—	2	4	no	—
94	C₂×Dic₁₀₇	—	2	1	no	—
95	C₂×Dic₁₀₇	—	2	1	no	—
96	C₂×Dic₁₀₇	—	2	1	no	—
97	C₂×Dic₁₀₇	—	2	2	no	—
98	C₂×Dic₁₀₇	—	2	2	no	—
99	D₄₂₈	—	2	1	no	—
100	D₄₂₈	C₂×Q₈	1	2	yes	—
101	C₄×D₁₀₇	—	2	4	no	—
102	C₄×D₁₀₇	—	4	2	no	—
103	C₄×D₁₀₇	—	4	2	no	—
104	Dic₂₁₄	—	2	1	no	—
105	Dic₂₁₄	—	4	2	no	—
106	Dic₂₁₄	—	4	2	no	—
107	Dic₂₁₄	C₂×Q₈	1	2	yes	—
108	C₈×C₁₀₇	—	2	2	no	—
109	C₈×C₁₀₇	—	2	2	no	—
110	C₂²×C₁₀₇	—	2	1	no	—
111	C₂²×C₁₀₇	—	2	1	no	—
112	C₂²×C₁₀₇	C₂²⋊Q₈	1	2	yes	—
113	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×D₁₀₇	—	2	2	no	—
115	C₂×D₁₀₇	—	4	1	no	—
116	C₂×D₁₀₇	—	4	1	no	—
117	C₄×C₁₀₇	—	2	1	no	—
118	C₄×C₁₀₇	—	2	1	no	—
119	C₄×C₁₀₇	—	2	1	no	—
120	C₄×C₁₀₇	C₂²⋊Q₈	1	2	yes	—
121	C₄×C₁₀₇	#135	1	2	yes	—
122	C₄×C₁₀₇	C₂²×Q₈	1	1	yes	characteristic
123	C₄×C₁₀₇	C₂×#41	1	1	yes	characteristic
124	Dic₁₀₇	—	2	2	no	—
125	Dic₁₀₇	—	4	2	no	—
126	Dic₁₀₇	—	4	1	no	—
127	Dic₁₀₇	—	4	2	no	—
128	Dic₁₀₇	—	4	1	no	—
129	C₂×C₁₀₇	—	2	2	no	—
130	C₂×C₁₀₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₁₀₇	C₂×#135	1	1	yes	characteristic
132	C₂×C₁₀₇	#575	1	1	yes	characteristic
133	D₁₀₇	—	4	2	no	—
134	#3076	—	107	1	no	maximal, Sylow
135	C₁₀₇	#3076	1	1	yes	characteristic, complemented, Sylow
136	#533	—	107	1	no	—
137	C₄×#41	—	107	1	no	—
138	#479	—	107	2	no	—
139	#478	—	107	2	no	—
140	#477	—	107	1	no	—
141	#476	—	107	1	no	—
142	#475	—	107	1	no	—
143	#474	—	107	1	no	—
144	#427	—	107	2	no	—
145	C₂×C₄⋊C₈	D₁₀₇	1	1	yes	characteristic, complemented
146	#416	—	107	2	no	—
147	C₂×#41	—	107	1	no	—
148	C₂²×C₈	C₂×D₁₀₇	1	2	yes	—
149	C₄⋊Q₈	—	107	1	no	—
150	C₄⋊Q₈	—	107	1	no	—
151	#125	—	107	2	no	—
152	C₂²⋊Q₈	—	214	1	no	—
153	C₂²⋊Q₈	—	214	1	no	—
154	C₄×Q₈	—	107	1	no	—
155	C₄×Q₈	—	107	1	no	—
156	C₄×Q₈	—	214	1	no	—
157	C₄×Q₈	—	214	1	no	—
158	C₄×D₄	—	107	1	no	—
159	C₄×D₄	—	107	1	no	—
160	C₄×D₄	—	214	1	no	—
161	C₄×D₄	—	214	1	no	—
162	#117	—	107	1	no	—
163	#117	—	107	2	no	—
164	#117	—	214	1	no	—
165	C₂×C₄²	—	214	1	no	—
166	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
167	#107	—	107	2	no	—
168	#107	—	107	2	no	—
169	#106	—	107	2	no	—
170	#106	—	107	2	no	—
171	C₄⋊C₈	C₂×D₁₀₇	1	1	yes	characteristic
172	C₄⋊C₈	C₂×D₁₀₇	1	1	yes	characteristic
173	C₄⋊C₈	C₂×D₁₀₇	1	2	yes	—
174	#103	—	107	2	no	—
175	#103	—	107	2	no	—
176	#102	—	107	2	no	—
177	#102	—	107	2	no	—
178	#41	—	107	4	no	—
179	#41	—	214	2	no	—
180	C₂×Q₈	—	107	1	no	—
181	C₂×Q₈	—	214	1	no	—
182	C₂×Q₈	—	214	1	no	—
183	C₂×D₄	—	107	1	no	—
184	C₂×D₄	—	214	1	no	—
185	C₂²×C₄	—	214	1	no	—
186	C₂²×C₄	—	214	1	no	—
187	C₂²×C₄	—	214	1	no	—
188	C₂²×C₄	D₄₂₈	1	1	yes	characteristic
189	C₂²×C₄	D₄₂₈	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	C₂²×D₁₀₇	1	2	yes	—
193	C₂×C₈	C₂²×D₁₀₇	1	2	yes	—
194	C₄⋊C₄	—	107	2	no	—
195	C₄⋊C₄	—	107	2	no	—
196	C₄⋊C₄	—	107	1	no	—
197	C₄⋊C₄	—	107	1	no	—
198	C₄⋊C₄	—	214	1	no	—
199	C₄⋊C₄	—	214	1	no	—
200	C₄⋊C₄	—	214	2	no	—
201	C₄⋊C₄	—	214	2	no	—
202	C₄⋊C₄	—	214	1	no	—
203	C₄⋊C₄	—	214	1	no	—
204	C₂²⋊C₄	—	214	1	no	—
205	C₂²⋊C₄	—	214	2	no	—
206	C₂²⋊C₄	—	214	1	no	—
207	C₂²⋊C₄	—	214	1	no	—
208	C₄²	—	214	1	no	—
209	C₄²	—	214	1	no	—
210	C₄²	—	214	1	no	—
211	C₄²	—	214	2	no	—
212	C₄²	D₄₂₈	1	1	yes	characteristic
213	C₄²	D₄₂₈	1	1	yes	characteristic
214	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
215	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
216	C₂³	—	214	1	no	—
217	C₂³	C₂×D₄₂₈	1	1	yes	characteristic
218	Q₈	—	107	2	no	—
219	Q₈	—	428	2	no	—
220	Q₈	—	428	2	no	—
221	Q₈	—	214	1	no	—
222	D₄	—	107	2	no	—
223	D₄	—	214	4	no	—
224	D₄	—	214	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	428	2	no	—
228	C₂×C₄	—	428	2	no	—
229	C₂×C₄	—	214	1	no	—
230	C₂×C₄	—	214	1	no	—
231	C₂×C₄	—	214	2	no	—
232	C₂×C₄	—	214	2	no	—
233	C₂×C₄	—	214	1	no	—
234	C₂×C₄	—	214	4	no	—
235	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
238	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
239	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
240	C₂×C₄	Q₈×D₁₀₇	1	2	yes	—
241	C₂×C₄	#13133817	1	2	yes	—
242	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	428	1	no	—
248	C₂²	—	428	1	no	—
249	C₂²	—	214	2	no	—
250	C₂²	#16812089	1	2	yes	—
251	C₂²	C₂²×D₄₂₈	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	428	1	no	—
256	C₄	—	428	2	no	—
257	C₄	—	428	1	no	—
258	C₄	—	428	2	no	—
259	C₄	—	214	2	no	—
260	C₄	#16812089	1	2	yes	—
261	C₄	#16812131	1	2	yes	—
262	C₄	C₂×Q₈×D₁₀₇	1	1	yes	characteristic
263	C₄	C₂×#13133817	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	428	2	no	—
266	C₂	C₂×#16812089	1	1	yes	characteristic
267	C₂	C₂×#16812131	1	1	yes	characteristic
268	C₂	#20293427	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₈
commutator, cosocle	C₂×C₄×C₁₀₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting	C₂×C₄⋊C₈×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂²×D₄
prime radical	C₂×#13133818	C₂²
hypercenter	C₂×C₄⋊C₈	D₁₀₇
hypocenter	C₁₀₇	#3076
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. #20293184	C₂	semidirect
3. #20292935	C₂	semidirect
4. #20292928	C₂	semidirect
5. #20292923	C₂	semidirect
6. #20292918	C₂	semidirect
7. #20292913	C₂	semidirect
8. #20292908	C₂	semidirect
9. #20292665	C₂	semidirect
11. C₂×C₄⋊C₈×C₁₀₇	C₂	semidirect
44. #13133818	Q₈	semidirect
135. C₁₀₇	#3076	semidirect
145. C₂×C₄⋊C₈	D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#20293184	2
C₄×#13133818	2
#20292935	2
#20292928	2
#20292923	2
#20292918	2
#20292913	2
#20292908	2
#20292665	2
#20292624	2
C₂×C₄⋊C₈×C₁₀₇	2
#3076	107