Subgroups — #28345489

Order 13696.

456 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄₂₈	C₂	1	1	yes	maximal, characteristic
2	#20292700	C₂	1	4	yes	maximal, complemented
3	#20292697	C₂	1	4	yes	maximal, complemented
4	#20292684	C₂	1	1	yes	maximal, characteristic, complemented
5	#20292589	C₂	1	2	yes	maximal, complemented
6	C₂²⋊C₈×D₁₀₇	C₂	1	2	yes	maximal, complemented
7	#432×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₂₈	C₄	1	1	yes	characteristic
9	C₂²×C₄×D₁₀₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₁₀₇	C₂²	1	1	yes	characteristic
11	C₂×#13133801	C₂²	1	4	yes	complemented
12	C₂×C₈×D₁₀₇	C₂²	1	4	yes	complemented
13	C₄×D₄₂₈	C₄	1	2	yes	—
14	C₄×D₄₂₈	C₄	1	2	yes	—
15	C₄×D₄₂₈	C₂²	1	2	yes	—
16	C₄×D₄₂₈	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₀₇	C₄	1	1	yes	characteristic
18	C₂×(C₂×D₁₀₇)⋊C₄	C₄	1	1	yes	characteristic
19	C₂×(C₂×D₁₀₇)⋊C₄	C₄	1	1	yes	characteristic
20	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	2	yes	complemented
21	C₂²⋊(C₁₀₇⋊C₈)	C₂²	1	2	yes	—
22	(C₂×D₁₀₇)⋊C₈	C₂²	1	4	yes	complemented
23	(C₂×D₁₀₇)⋊C₈	C₂²	1	4	yes	complemented
24	#16812020	C₂²	1	2	yes	complemented
25	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₁₀₇	C₂²	1	2	yes	complemented
27	C₂²⋊C₈×C₁₀₇	C₂²	1	2	yes	—
28	#97×C₁₀₇	C₂²	1	2	yes	complemented
29	C₂³×D₁₀₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×D₁₀₇	C₂×C₄	1	1	yes	characteristic
31	C₂×D₄₂₈	—	2	4	no	—
32	C₂×D₄₂₈	C₂×C₄	1	2	yes	—
33	C₂×D₄₂₈	C₂×C₄	1	2	yes	—
34	C₂²×Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₀₇	—	4	2	no	—
37	C₂×C₄×D₁₀₇	—	4	2	no	—
38	C₂×C₄×D₁₀₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₁₀₇	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₁₀₇	C₂³	1	2	yes	—
41	C₂×C₄×D₁₀₇	C₂³	1	2	yes	—
42	#13133801	—	4	4	no	—
43	C₂×C₁₀₇⋊C₈	C₂³	1	4	yes	—
44	C₈×D₁₀₇	—	4	4	no	—
45	C₄⋊Dic₁₀₇	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₁₀₇	C₂×C₄	1	2	yes	—
47	(C₂×D₁₀₇)⋊C₄	C₂×C₄	1	2	yes	—
48	(C₂×D₁₀₇)⋊C₄	C₂×C₄	1	2	yes	—
49	(C₂×D₁₀₇)⋊C₄	C₂×C₄	1	2	yes	—
50	(C₂×D₁₀₇)⋊C₄	C₂×C₄	1	2	yes	—
51	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₁₀₇	C₂³	1	4	yes	—
55	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
58	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
59	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×D₁₀₇	—	2	2	no	—
61	C₂²×D₁₀₇	—	4	1	no	—
62	C₂²×D₁₀₇	—	4	1	no	—
63	C₂²×D₁₀₇	—	4	1	no	—
64	C₂²×D₁₀₇	—	4	1	no	—
65	C₂²×D₁₀₇	M₁₆	1	4	yes	complemented
66	C₂²×D₁₀₇	C₂²×C₄	1	2	yes	—
67	C₂²×D₁₀₇	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₁₀₇	—	2	2	no	—
69	C₂×C₄×C₁₀₇	—	2	2	no	—
70	C₂×C₄×C₁₀₇	—	4	1	no	—
71	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
77	C₂×Dic₁₀₇	—	4	1	no	—
78	C₂×Dic₁₀₇	—	4	1	no	—
79	C₂×Dic₁₀₇	C₂²×C₄	1	2	yes	—
80	C₂×Dic₁₀₇	C₂²×C₄	1	2	yes	—
81	D₄₂₈	—	2	4	no	—
82	D₄₂₈	—	2	4	no	—
83	C₄×D₁₀₇	—	2	4	no	—
84	C₄×D₁₀₇	—	2	4	no	—
85	C₄×D₁₀₇	—	4	2	no	—
86	C₄×D₁₀₇	—	4	2	no	—
87	C₈×C₁₀₇	—	2	4	no	—
88	C₁₀₇⋊C₈	—	2	4	no	—
89	C₂²×C₁₀₇	—	2	1	no	—
90	C₂²×C₁₀₇	—	2	1	no	—
91	C₂²×C₁₀₇	C₂×M₁₆	1	2	yes	—
92	C₂²×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×D₁₀₇	—	2	2	no	—
94	C₂×D₁₀₇	—	2	2	no	—
95	C₂×D₁₀₇	—	2	2	no	—
96	C₂×D₁₀₇	—	2	2	no	—
97	C₂×D₁₀₇	—	2	2	no	—
98	C₂×D₁₀₇	—	4	4	no	—
99	C₂×D₁₀₇	—	4	4	no	—
100	C₂×D₁₀₇	C₂×M₁₆	1	4	yes	—
101	C₄×C₁₀₇	—	2	1	no	—
102	C₄×C₁₀₇	—	2	1	no	—
103	C₄×C₁₀₇	—	2	1	no	—
104	C₄×C₁₀₇	—	2	1	no	—
105	C₄×C₁₀₇	—	2	1	no	—
106	C₄×C₁₀₇	H(5,𝔽₂)	1	2	yes	—
107	Dic₁₀₇	—	2	2	no	—
108	Dic₁₀₇	—	2	2	no	—
109	C₂×C₁₀₇	—	2	2	no	—
110	C₂×C₁₀₇	#518	1	1	yes	characteristic
111	C₂×C₁₀₇	C₂²×M₁₆	1	1	yes	characteristic
112	C₂×C₁₀₇	#568	1	1	yes	characteristic
113	D₁₀₇	—	2	4	no	—
114	D₁₀₇	—	4	2	no	—
115	#2973	—	107	1	no	maximal, Sylow
116	C₁₀₇	#2973	1	1	yes	characteristic, complemented, Sylow
117	C₂×C₄×D₄	—	107	1	no	—
118	#435	—	107	4	no	—
119	#435	—	107	4	no	—
120	#432	—	107	1	no	—
121	#432	D₁₀₇	1	1	yes	characteristic, complemented
122	#407	—	107	2	no	—
123	C₂×C₂²⋊C₈	—	107	2	no	—
124	C₂²×D₄	—	107	1	no	—
125	C₂³×C₄	—	107	1	no	—
126	C₂³×C₄	—	107	1	no	—
127	C₂×M₁₆	—	107	4	no	—
128	C₂²×C₈	—	107	4	no	—
129	C₄×D₄	—	107	2	no	—
130	C₄×D₄	—	107	2	no	—
131	C₄×D₄	—	107	2	no	—
132	C₄×D₄	—	107	2	no	—
133	C₂×C₄⋊C₄	—	107	1	no	—
134	C₂×C₂²⋊C₄	—	107	1	no	—
135	C₂×C₂²⋊C₄	—	107	1	no	—
136	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
137	C₄⋊C₈	—	107	2	no	—
138	C₄⋊C₈	C₂×D₁₀₇	1	2	yes	complemented
139	C₂²⋊C₈	—	107	2	no	—
140	C₂²⋊C₈	—	107	4	no	—
141	C₂²⋊C₈	—	107	4	no	—
142	C₂²⋊C₈	C₂×D₁₀₇	1	2	yes	—
143	#97	—	107	2	no	—
144	#97	C₂×D₁₀₇	1	2	yes	complemented
145	C₂⁴	—	107	1	no	—
146	C₂⁴	—	107	1	no	—
147	C₂×D₄	—	107	2	no	—
148	C₂×D₄	—	107	2	no	—
149	C₂×D₄	—	214	4	no	—
150	C₂²×C₄	—	107	1	no	—
151	C₂²×C₄	—	107	1	no	—
152	C₂²×C₄	—	107	2	no	—
153	C₂²×C₄	—	107	2	no	—
154	C₂²×C₄	—	107	2	no	—
155	C₂²×C₄	—	107	2	no	—
156	C₂²×C₄	—	428	2	no	—
157	C₂²×C₄	—	428	2	no	—
158	C₂²×C₄	C₄×D₁₀₇	1	1	yes	characteristic
159	C₂²×C₄	C₄×D₁₀₇	1	1	yes	characteristic
160	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
161	M₁₆	—	428	4	no	—
162	C₂×C₈	—	107	4	no	—
163	C₂×C₈	—	428	4	no	—
164	C₂×C₈	C₂²×D₁₀₇	1	4	yes	—
165	C₄⋊C₄	—	107	2	no	—
166	C₄⋊C₄	—	107	2	no	—
167	C₂²⋊C₄	—	107	2	no	—
168	C₂²⋊C₄	—	107	2	no	—
169	C₂²⋊C₄	—	107	2	no	—
170	C₂²⋊C₄	—	107	2	no	—
171	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
172	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
173	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
174	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
175	C₂³	—	107	4	no	—
176	C₂³	—	107	2	no	—
177	C₂³	—	107	2	no	—
178	C₂³	—	428	1	no	—
179	C₂³	—	428	1	no	—
180	C₂³	—	428	1	no	—
181	C₂³	—	428	1	no	—
182	C₂³	—	214	2	no	—
183	C₂³	C₂×C₄×D₁₀₇	1	1	yes	characteristic
184	D₄	—	214	4	no	—
185	D₄	—	214	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	—	107	2	no	—
190	C₂×C₄	—	107	2	no	—
191	C₂×C₄	—	428	2	no	—
192	C₂×C₄	—	428	2	no	—
193	C₂×C₄	—	428	1	no	—
194	C₂×C₄	—	428	1	no	—
195	C₂×C₄	—	214	4	no	—
196	C₂×C₄	—	214	4	no	—
197	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
203	C₈	—	2	4	no	—
204	C₈	—	214	4	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	107	4	no	—
208	C₂²	—	428	4	no	—
209	C₂²	—	428	4	no	—
210	C₂²	—	214	2	no	—
211	C₂²	—	214	2	no	—
212	C₂²	—	214	2	no	—
213	C₂²	—	214	2	no	—
214	C₂²	—	214	2	no	—
215	C₂²	M₁₆×D₁₀₇	1	2	yes	—
216	C₂²	C₂²×C₄×D₁₀₇	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	214	2	no	—
223	C₄	—	214	2	no	—
224	C₄	#16812155	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	—	428	2	no	—
227	C₂	—	214	4	no	—
228	C₂	#20293101	1	1	yes	characteristic
229	C₂	C₂×M₁₆×D₁₀₇	1	1	yes	characteristic
230	C₂	#20293393	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂³×C₄
Fitting	#432×C₁₀₇	C₂
prime radical	C₂²×D₄₂₈	C₄
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	#432	D₁₀₇
hypocenter	C₁₀₇	#2973
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #20292700	C₂	semidirect
3. #20292697	C₂	semidirect
4. #20292684	C₂	semidirect
5. #20292589	C₂	semidirect
6. C₂²⋊C₈×D₁₀₇	C₂	semidirect
7. #432×C₁₀₇	C₂	semidirect
11. C₂×#13133801	C₂²	semidirect
12. C₂×C₈×D₁₀₇	C₂²	semidirect
20. C₄⋊(C₁₀₇⋊C₈)	C₂²	semidirect
22. (C₂×D₁₀₇)⋊C₈	C₂²	semidirect
23. (C₂×D₁₀₇)⋊C₈	C₂²	semidirect
24. #16812020	C₂²	semidirect
26. C₄⋊C₈×C₁₀₇	C₂²	semidirect
28. #97×C₁₀₇	C₂²	semidirect
65. C₂²×D₁₀₇	M₁₆	semidirect
116. C₁₀₇	#2973	semidirect
121. #432	D₁₀₇	semidirect
138. C₄⋊C₈	C₂×D₁₀₇	semidirect
144. #97	C₂×D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄₂₈	2
#20292700	2
#20292697	2
#20292684	2
#20292589	2
C₂²⋊C₈×D₁₀₇	2
#432×C₁₀₇	2
#2973	107