Subgroups — #28345552

Order 13696.

322 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293181	C₂	1	1	yes	maximal, characteristic
2	#20292758	C₂	1	1	yes	maximal, characteristic, complemented
3	#20292757	C₂	1	2	yes	maximal, complemented
4	#20292756	C₂	1	2	yes	maximal, complemented
5	#20292754	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₂₁₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#20292687	C₂	1	2	yes	maximal
8	#20292686	C₂	1	2	yes	maximal
9	#20292683	C₂	1	1	yes	maximal, characteristic
10	#20292678	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#16812108	C₄	1	1	yes	characteristic
13	#16812107	C₄	1	1	yes	characteristic
14	#16812099	C₄	1	2	yes	—
15	#16812089	C₄	1	1	yes	characteristic
16	#16812087	C₄	1	2	yes	—
17	#16812085	C₄	1	1	yes	characteristic
18	C₄×Dic₂₁₄	C₂²	1	1	yes	characteristic
19	C₄×Dic₂₁₄	C₂²	1	2	yes	—
20	#16812066	C₂²	1	2	yes	—
21	C₄²×D₁₀₇	C₂²	1	1	yes	characteristic
22	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	2	yes	—
24	Dic₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
25	Dic₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
26	Dic₁₀₇⋊C₈	C₂²	1	2	yes	—
27	Dic₁₀₇⋊C₈	C₂²	1	2	yes	—
28	(C₂×D₁₀₇)⋊C₈	C₂²	1	1	yes	characteristic
29	(C₂×D₁₀₇)⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₁₀₇)⋊C₈	C₂²	1	2	yes	—
31	#16812020	C₂²	1	2	yes	—
32	#16812019	C₂²	1	2	yes	—
33	#16812019	C₂²	1	2	yes	—
34	C₄×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×Dic₁₀₇	C₂²	1	2	yes	—
36	C₄×Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₀₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₀₇	C₂²	1	2	yes	—
39	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₀₇	C₂²	1	2	yes	—
41	C₂×Dic₂₁₄	C₂×C₄	1	1	yes	characteristic
42	C₂×Dic₂₁₄	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₁₀₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₀₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₀₇	C₂³	1	1	yes	characteristic
46	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₂×C₁₀₇⋊C₈	C₂³	1	2	yes	—
48	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
49	C₄⋊Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₁₀₇	C₂×C₄	1	2	yes	—
51	Dic₁₀₇⋊C₄	C₂×C₄	1	2	yes	—
52	Dic₁₀₇⋊C₄	C₂×C₄	1	2	yes	—
53	Dic₁₀₇⋊C₄	C₂×C₄	1	2	yes	—
54	Dic₁₀₇⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₁₀₇⋊C₄	C₂×C₄	1	2	yes	—
56	(C₂×D₁₀₇)⋊C₄	C₂×C₄	1	2	yes	—
57	(C₂×D₁₀₇)⋊C₄	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₀₇	C₂³	1	1	yes	characteristic
61	C₄×Dic₁₀₇	C₂³	1	2	yes	—
62	C₂×Q₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₁₀₇	C₂³	1	2	yes	—
66	C₄⋊C₄×C₁₀₇	C₂×C₄	1	2	yes	—
67	C₄⋊C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
68	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
69	C₄²×C₁₀₇	C₂³	1	2	yes	—
70	C₂²×D₁₀₇	C₂²×C₄	1	1	yes	characteristic
71	Q₈×C₁₀₇	—	2	2	no	—
72	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
77	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₀₇	C₂²×C₄	1	2	yes	—
81	C₂×Dic₁₀₇	C₂²×C₄	1	2	yes	—
82	C₄×D₁₀₇	—	2	2	no	—
83	C₄×D₁₀₇	—	2	1	no	—
84	C₄×D₁₀₇	—	2	1	no	—
85	C₄×D₁₀₇	—	4	1	no	—
86	Dic₂₁₄	—	2	2	no	—
87	Dic₂₁₄	—	4	2	no	—
88	C₈×C₁₀₇	—	2	1	no	—
89	C₈×C₁₀₇	—	2	2	no	—
90	C₈×C₁₀₇	—	2	1	no	—
91	C₁₀₇⋊C₈	—	2	1	no	—
92	C₁₀₇⋊C₈	—	2	2	no	—
93	C₁₀₇⋊C₈	—	2	1	no	—
94	C₂²×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×D₁₀₇	—	2	1	no	—
96	C₂×D₁₀₇	—	2	1	no	—
97	C₂×D₁₀₇	—	2	1	no	—
98	C₄×C₁₀₇	—	2	2	no	—
99	C₄×C₁₀₇	—	2	2	no	—
100	C₄×C₁₀₇	—	2	1	no	—
101	C₄×C₁₀₇	#131	1	2	yes	—
102	C₄×C₁₀₇	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₁₀₇	2₋¹⁺⁴	1	1	yes	characteristic
104	Dic₁₀₇	—	2	1	no	—
105	Dic₁₀₇	—	2	2	no	—
106	Dic₁₀₇	—	2	1	no	—
107	Dic₁₀₇	—	2	1	no	—
108	Dic₁₀₇	—	2	2	no	—
109	C₂×C₁₀₇	#519	1	1	yes	characteristic
110	C₂×C₁₀₇	C₂×#131	1	1	yes	characteristic
111	C₂×C₁₀₇	#568	1	1	yes	characteristic
112	D₁₀₇	—	4	1	no	—
113	#2985	—	107	1	no	maximal, Sylow
114	C₁₀₇	#2985	1	1	yes	characteristic, complemented, Sylow
115	#533	—	107	1	no	—
116	#446	—	107	1	no	—
117	#446	—	107	1	no	—
118	#446	—	107	2	no	—
119	#446	—	107	2	no	—
120	C₈×Q₈	—	107	1	no	—
121	C₈×Q₈	D₁₀₇	1	1	yes	characteristic, complemented
122	#433	—	107	2	no	—
123	#433	—	107	2	no	—
124	#432	—	107	1	no	—
125	#431	—	107	1	no	—
126	C₄⋊Q₈	—	107	1	no	—
127	C₄⋊Q₈	—	107	1	no	—
128	#125	—	107	2	no	—
129	C₂²⋊Q₈	—	107	1	no	—
130	C₂²⋊Q₈	—	107	2	no	—
131	C₂²⋊Q₈	—	107	1	no	—
132	C₄×Q₈	—	107	1	no	—
133	C₄×Q₈	—	107	2	no	—
134	C₄×Q₈	C₂×D₁₀₇	1	1	yes	characteristic
135	#117	—	107	2	no	—
136	C₂×C₄²	—	107	1	no	—
137	C₄⋊C₈	—	107	1	no	—
138	C₄⋊C₈	—	107	1	no	—
139	C₄⋊C₈	—	107	2	no	—
140	C₄⋊C₈	—	107	2	no	—
141	C₄⋊C₈	—	107	1	no	—
142	C₄⋊C₈	—	107	2	no	—
143	C₄⋊C₈	C₂×D₁₀₇	1	1	yes	characteristic
144	C₄⋊C₈	C₂×D₁₀₇	1	2	yes	—
145	C₂²⋊C₈	—	107	1	no	—
146	C₂²⋊C₈	—	107	1	no	—
147	C₂²⋊C₈	—	107	2	no	—
148	#97	—	107	2	no	—
149	#97	—	107	2	no	—
150	#97	—	107	2	no	—
151	C₄×C₈	—	107	2	no	—
152	C₄×C₈	—	107	1	no	—
153	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
154	C₄×C₈	C₂×D₁₀₇	1	2	yes	—
155	C₂×Q₈	—	107	2	no	—
156	C₂×Q₈	—	107	1	no	—
157	C₂×Q₈	C₄×D₁₀₇	1	1	yes	characteristic
158	C₂²×C₄	—	107	1	no	—
159	C₂²×C₄	—	107	1	no	—
160	C₂²×C₄	—	107	1	no	—
161	C₂×C₈	—	107	1	no	—
162	C₂×C₈	—	107	2	no	—
163	C₂×C₈	—	107	1	no	—
164	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
165	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
166	C₂×C₈	C₂²×D₁₀₇	1	2	yes	—
167	C₄⋊C₄	—	107	2	no	—
168	C₄⋊C₄	—	107	2	no	—
169	C₄⋊C₄	—	107	2	no	—
170	C₄⋊C₄	—	107	2	no	—
171	C₄⋊C₄	—	107	2	no	—
172	C₄⋊C₄	—	107	2	no	—
173	C₄⋊C₄	—	107	1	no	—
174	C₄⋊C₄	C₄×D₁₀₇	1	2	yes	—
175	C₄⋊C₄	C₄×D₁₀₇	1	1	yes	characteristic
176	C₂²⋊C₄	—	107	2	no	—
177	C₂²⋊C₄	—	107	2	no	—
178	C₄²	—	107	1	no	—
179	C₄²	—	107	1	no	—
180	C₄²	—	107	2	no	—
181	C₄²	—	107	1	no	—
182	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
183	C₄²	C₂²×D₁₀₇	1	2	yes	—
184	C₂³	—	107	1	no	—
185	Q₈	—	2	2	no	—
186	Q₈	—	428	2	no	—
187	Q₈	—	214	2	no	—
188	C₂×C₄	—	107	1	no	—
189	C₂×C₄	—	107	1	no	—
190	C₂×C₄	—	107	1	no	—
191	C₂×C₄	—	107	2	no	—
192	C₂×C₄	—	107	2	no	—
193	C₂×C₄	—	428	1	no	—
194	C₂×C₄	—	214	2	no	—
195	C₂×C₄	—	214	1	no	—
196	C₂×C₄	—	214	1	no	—
197	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
200	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	2	no	—
204	C₈	—	2	1	no	—
205	C₈	—	214	1	no	—
206	C₈	—	214	2	no	—
207	C₈	—	214	1	no	—
208	C₂²	—	214	1	no	—
209	C₂²	—	214	1	no	—
210	C₂²	—	214	1	no	—
211	C₂²	C₂²×C₄×D₁₀₇	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	214	1	no	—
216	C₄	—	214	2	no	—
217	C₄	—	214	1	no	—
218	C₄	—	214	2	no	—
219	C₄	—	214	1	no	—
220	C₄	#16812115	1	2	yes	—
221	C₄	#16812156	1	1	yes	characteristic
222	C₄	#16812157	1	1	yes	characteristic
223	C₂	—	428	1	no	—
224	C₂	#20293103	1	1	yes	characteristic
225	C₂	C₂×#16812115	1	1	yes	characteristic
226	C₂	#20293392	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂³×C₄
Fitting	C₈×Q₈×C₁₀₇	C₂
prime radical	C₂²×D₁₀₇	C₂²×C₄
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	C₈×Q₈	D₁₀₇
hypocenter	C₁₀₇	#2985
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #20292758	C₂	semidirect
3. #20292757	C₂	semidirect
4. #20292756	C₂	semidirect
5. #20292754	C₂	semidirect
6. C₈×Dic₂₁₄	C₂	semidirect
11. C₈×Q₈×C₁₀₇	C₂	semidirect
114. C₁₀₇	#2985	semidirect
121. C₈×Q₈	D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#20293181	2
#20292758	2
#20292757	2
#20292756	2
#20292754	2
C₈×Dic₂₁₄	2
#20292687	2
#20292686	2
#20292683	2
#20292678	2
C₈×Q₈×C₁₀₇	2
#2985	107