Subgroups — #28350866

Order 13696.

1258 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293369	C₂	1	4	yes	maximal, complemented
2	#20293343	C₂	1	2	yes	maximal, complemented
3	#20293342	C₂	1	4	yes	maximal, complemented
4	D₄×D₄₂₈	C₂	1	4	yes	maximal, complemented
5	#20293262	C₂	1	2	yes	maximal, complemented
6	#20293193	C₂	1	8	yes	maximal, complemented
7	#20293191	C₂	1	4	yes	maximal, complemented
8	#20293177	C₂	1	2	yes	maximal, complemented
9	#532×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133818	C₂²	1	4	yes	complemented
11	C₂²×D₄₂₈	C₂²	1	4	yes	complemented
12	C₂×D₄×D₁₀₇	C₂²	1	8	yes	complemented
13	#16812109	C₂²	1	2	yes	complemented
14	#16812106	C₂²	1	2	yes	—
15	#16812105	C₂²	1	8	yes	complemented
16	#16812098	C₂²	1	8	yes	complemented
17	#16812097	C₂²	1	4	yes	complemented
18	#16812095	C₂²	1	16	yes	complemented
19	#16812084	C₂²	1	8	yes	complemented
20	#16812083	C₂²	1	4	yes	complemented
21	#16812083	C₂²	1	4	yes	complemented
22	#16812083	C₂²	1	8	yes	complemented
23	#16812080	C₂²	1	8	yes	complemented
24	#16812079	C₂²	1	16	yes	complemented
25	#16812077	C₂²	1	8	yes	complemented
26	#16812077	C₂²	1	8	yes	complemented
27	C₄×D₄₂₈	C₂²	1	4	yes	complemented
28	C₄×D₄₂₈	C₂²	1	4	yes	complemented
29	#16812068	C₂²	1	4	yes	complemented
30	#16812067	C₂²	1	8	yes	complemented
31	C₂×#41×C₁₀₇	C₂²	1	2	yes	complemented
32	C₄⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic, complemented
33	#127×C₁₀₇	C₂²	1	1	yes	characteristic
34	#124×C₁₀₇	C₂²	1	2	yes	complemented
35	#121×C₁₀₇	C₂²	1	4	yes	complemented
36	C₄×D₄×C₁₀₇	C₂²	1	4	yes	complemented
37	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic, complemented
38	C₂³×D₁₀₇	C₂³	1	8	yes	—
39	#13133818	—	4	8	no	—
40	C₂×Dic₂₁₄	C₂³	1	4	yes	—
41	C₂×D₄₂₈	—	4	8	no	—
42	C₂×D₄₂₈	C₂³	1	4	yes	—
43	C₂×D₄₂₈	C₂³	1	4	yes	—
44	C₂×D₄₂₈	C₂³	1	4	yes	—
45	C₂×D₄₂₈	C₂³	1	8	yes	—
46	C₂×C₂²⋊D₁₀₇	C₂³	1	8	yes	—
47	C₂×C₂²⋊D₁₀₇	C₂³	1	16	yes	—
48	D₄×D₁₀₇	—	4	16	no	—
49	C₂×C₄×D₁₀₇	C₂³	1	8	yes	—
50	C₄⋊Dic₁₀₇	C₂³	1	4	yes	—
51	C₄⋊Dic₁₀₇	C₂³	1	4	yes	—
52	C₂²⋊Dic₁₀₇	C₂³	1	8	yes	—
53	(C₂×D₁₀₇)⋊C₄	C₂³	1	16	yes	—
54	(C₂×D₁₀₇)⋊C₄	C₂³	1	8	yes	—
55	(C₂×D₁₀₇)⋊C₄	C₂³	1	8	yes	—
56	C₄×Dic₁₀₇	C₂³	1	8	yes	—
57	#41×C₁₀₇	—	4	4	no	—
58	C₂×Q₈×C₁₀₇	C₂³	1	2	yes	—
59	C₂×D₄×C₁₀₇	C₂³	1	4	yes	—
60	C₂×D₄×C₁₀₇	C₂³	1	2	yes	—
61	C₂×D₄×C₁₀₇	C₂³	1	4	yes	—
62	C₂²×C₄×C₁₀₇	C₂³	1	4	yes	—
63	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₇	C₂³	1	2	yes	—
65	C₄⋊C₄×C₁₀₇	C₂³	1	4	yes	—
66	C₂²⋊C₄×C₁₀₇	C₂³	1	8	yes	—
67	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
68	C₄²×C₁₀₇	C₂³	1	2	yes	—
69	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
70	C₂³×C₁₀₇	C₂⁴	1	4	yes	—
71	C₂³×C₁₀₇	C₂⁴	1	1	yes	characteristic
72	C₂²×D₁₀₇	—	4	16	no	—
73	C₂²×D₁₀₇	—	4	8	no	—
74	C₂²×D₁₀₇	C₂⁴	1	8	yes	—
75	Q₈×C₁₀₇	—	4	2	no	—
76	D₄×C₁₀₇	—	2	8	no	—
77	D₄×C₁₀₇	—	4	4	no	—
78	D₄×C₁₀₇	—	4	2	no	—
79	C₂×C₄×C₁₀₇	—	2	2	no	—
80	C₂×C₄×C₁₀₇	—	2	8	no	—
81	C₂×C₄×C₁₀₇	—	2	4	no	—
82	C₂×C₄×C₁₀₇	C₂⁴	1	2	yes	—
83	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
84	C₂×C₄×C₁₀₇	C₂⁴	1	4	yes	—
85	C₂×C₄×C₁₀₇	C₂⁴	1	2	yes	—
86	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
87	C₂²⋊D₁₀₇	—	4	16	no	—
88	C₂²⋊D₁₀₇	—	4	8	no	—
89	C₂×Dic₁₀₇	C₂⁴	1	8	yes	—
90	D₄₂₈	—	2	8	no	—
91	D₄₂₈	—	2	8	no	—
92	D₄₂₈	—	4	8	no	—
93	D₄₂₈	—	4	4	no	—
94	C₄×D₁₀₇	—	2	16	no	—
95	Dic₂₁₄	—	4	4	no	—
96	C₂²×C₁₀₇	—	2	2	no	—
97	C₂²×C₁₀₇	—	2	8	no	—
98	C₂²×C₁₀₇	—	2	1	no	—
99	C₂²×C₁₀₇	—	2	4	no	—
100	C₂²×C₁₀₇	C₂⁵	1	1	yes	characteristic
101	C₂×D₁₀₇	—	2	8	no	—
102	C₂×D₁₀₇	—	2	8	no	—
103	C₂×D₁₀₇	—	2	8	no	—
104	C₂×D₁₀₇	—	8	16	no	—
105	C₄×C₁₀₇	—	2	2	no	—
106	C₄×C₁₀₇	—	2	4	no	—
107	C₄×C₁₀₇	—	2	1	no	—
108	C₄×C₁₀₇	H(5,𝔽₂)	1	2	yes	—
109	C₄×C₁₀₇	H(5,𝔽₂)	1	4	yes	—
110	Dic₁₀₇	—	2	8	no	—
111	C₂×C₁₀₇	—	4	4	no	—
112	C₂×C₁₀₇	—	4	1	no	—
113	C₂×C₁₀₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
114	C₂×C₁₀₇	C₂×H(5,𝔽₂)	1	2	yes	—
115	D₁₀₇	—	4	8	no	—
116	#3549	—	107	1	no	maximal, Sylow
117	C₁₀₇	#3549	1	1	yes	characteristic, complemented, Sylow
118	H(3,𝔽₄)	—	107	4	no	—
119	#555	—	107	2	no	—
120	#555	—	107	4	no	—
121	D₄²	—	107	2	no	—
122	D₄²	—	107	4	no	—
123	#535	—	107	4	no	—
124	#535	—	107	8	no	—
125	#532	—	107	2	no	—
126	#532	D₁₀₇	1	1	yes	characteristic, complemented
127	C₂×#41	—	107	4	no	—
128	C₂×#41	C₂×D₁₀₇	1	2	yes	complemented
129	C₂²×D₄	—	107	8	no	—
130	C₂²×D₄	—	107	4	no	—
131	C₄⋊Q₈	—	107	2	no	—
132	C₄⋊Q₈	C₂×D₁₀₇	1	1	yes	characteristic, complemented
133	#127	—	107	8	no	—
134	#127	—	107	2	no	—
135	#127	C₂×D₁₀₇	1	1	yes	characteristic
136	#124	—	107	8	no	—
137	#124	—	107	16	no	—
138	#124	—	107	4	no	—
139	#124	C₂×D₁₀₇	1	2	yes	complemented
140	#121	—	107	8	no	—
141	#121	—	107	4	no	—
142	#121	—	107	4	no	—
143	#121	—	107	8	no	—
144	#121	—	107	8	no	—
145	#121	C₂×D₁₀₇	1	4	yes	complemented
146	#120	—	107	8	no	—
147	#120	—	107	8	no	—
148	#120	—	107	16	no	—
149	C₄×D₄	—	107	4	no	—
150	C₄×D₄	—	107	4	no	—
151	C₄×D₄	C₂×D₁₀₇	1	4	yes	complemented
152	#117	—	107	4	no	—
153	#117	—	107	8	no	—
154	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic, complemented
155	C₂⁴	—	107	8	no	—
156	#41	—	4	4	no	—
157	#41	—	428	8	no	—
158	C₂×Q₈	—	107	4	no	—
159	C₂×Q₈	C₂²×D₁₀₇	1	2	yes	—
160	C₂×D₄	—	107	16	no	—
161	C₂×D₄	—	107	8	no	—
162	C₂×D₄	—	107	4	no	—
163	C₂×D₄	—	107	8	no	—
164	C₂×D₄	—	107	4	no	—
165	C₂×D₄	—	107	4	no	—
166	C₂×D₄	—	428	8	no	—
167	C₂×D₄	—	428	16	no	—
168	C₂×D₄	C₂²×D₁₀₇	1	4	yes	—
169	C₂×D₄	C₂²×D₁₀₇	1	2	yes	—
170	C₂×D₄	C₂²×D₁₀₇	1	4	yes	—
171	C₂²×C₄	—	107	8	no	—
172	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₁₀₇	1	2	yes	—
174	C₂²×C₄	C₂²×D₁₀₇	1	4	yes	—
175	C₄⋊C₄	—	107	4	no	—
176	C₄⋊C₄	—	107	4	no	—
177	C₄⋊C₄	C₂²×D₁₀₇	1	4	yes	—
178	C₂²⋊C₄	—	107	8	no	—
179	C₂²⋊C₄	—	107	16	no	—
180	C₂²⋊C₄	—	107	8	no	—
181	C₂²⋊C₄	—	107	8	no	—
182	C₂²⋊C₄	C₂²×D₁₀₇	1	8	yes	—
183	C₄²	—	107	8	no	—
184	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
185	C₄²	C₂²×D₁₀₇	1	2	yes	—
186	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
187	C₂³	—	107	8	no	—
188	C₂³	—	428	8	no	—
189	C₂³	—	428	16	no	—
190	C₂³	C₂³×D₁₀₇	1	4	yes	—
191	C₂³	C₂³×D₁₀₇	1	1	yes	characteristic
192	Q₈	—	4	2	no	—
193	Q₈	—	428	4	no	—
194	D₄	—	2	8	no	—
195	D₄	—	4	2	no	—
196	D₄	—	4	4	no	—
197	D₄	—	428	8	no	—
198	D₄	—	428	4	no	—
199	D₄	—	428	16	no	—
200	D₄	—	428	8	no	—
201	D₄	—	214	8	no	—
202	D₄	—	214	8	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	107	8	no	—
207	C₂×C₄	—	214	16	no	—
208	C₂×C₄	C₂³×D₁₀₇	1	4	yes	—
209	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
210	C₂×C₄	C₂³×D₁₀₇	1	2	yes	—
211	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
212	C₂×C₄	C₂³×D₁₀₇	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	856	16	no	—
218	C₂²	—	214	8	no	—
219	C₂²	—	214	8	no	—
220	C₂²	—	214	8	no	—
221	C₂²	C₂⁴×D₁₀₇	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	4	no	—
224	C₄	—	2	2	no	—
225	C₄	—	214	8	no	—
226	C₄	#16812154	1	2	yes	—
227	C₄	#16812155	1	4	yes	—
228	C₂	—	4	4	no	—
229	C₂	—	4	1	no	—
230	C₂	—	428	8	no	—
231	C₂	C₂×#16812154	1	1	yes	characteristic
232	C₂	C₂×#16812155	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₇
commutator, socle, cosocle	C₂²×C₁₀₇	C₂⁵
Fitting	#532×C₁₀₇	C₂
hypercenter	#532	D₁₀₇
hypocenter	C₁₀₇	#3549
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ #532

Complements

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

N	H ≅ G/N	type
1. #20293369	C₂	semidirect
2. #20293343	C₂	semidirect
3. #20293342	C₂	semidirect
4. D₄×D₄₂₈	C₂	semidirect
5. #20293262	C₂	semidirect
6. #20293193	C₂	semidirect
7. #20293191	C₂	semidirect
8. #20293177	C₂	semidirect
9. #532×C₁₀₇	C₂	semidirect
10. C₂×#13133818	C₂²	semidirect
11. C₂²×D₄₂₈	C₂²	semidirect
12. C₂×D₄×D₁₀₇	C₂²	semidirect
13. #16812109	C₂²	semidirect
15. #16812105	C₂²	semidirect
16. #16812098	C₂²	semidirect
17. #16812097	C₂²	semidirect
18. #16812095	C₂²	semidirect
19. #16812084	C₂²	semidirect
20. #16812083	C₂²	semidirect
21. #16812083	C₂²	semidirect
22. #16812083	C₂²	semidirect
23. #16812080	C₂²	semidirect
24. #16812079	C₂²	semidirect
25. #16812077	C₂²	semidirect
26. #16812077	C₂²	semidirect
27. C₄×D₄₂₈	C₂²	semidirect
28. C₄×D₄₂₈	C₂²	semidirect
29. #16812068	C₂²	semidirect
30. #16812067	C₂²	semidirect
31. C₂×#41×C₁₀₇	C₂²	semidirect
32. C₄⋊Q₈×C₁₀₇	C₂²	semidirect
34. #124×C₁₀₇	C₂²	semidirect
35. #121×C₁₀₇	C₂²	semidirect
36. C₄×D₄×C₁₀₇	C₂²	semidirect
37. C₂×C₄²×C₁₀₇	C₂²	semidirect
117. C₁₀₇	#3549	semidirect
126. #532	D₁₀₇	semidirect
128. C₂×#41	C₂×D₁₀₇	semidirect
132. C₄⋊Q₈	C₂×D₁₀₇	semidirect
139. #124	C₂×D₁₀₇	semidirect
145. #121	C₂×D₁₀₇	semidirect
151. C₄×D₄	C₂×D₁₀₇	semidirect
154. C₂×C₄²	C₂×D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#20293369	2
#20293343	2
#20293342	2
D₄×D₄₂₈	2
#20293262	2
#20293193	2
#20293191	2
#20293177	2
#532×C₁₀₇	2
#3549	107