Subgroups — #28349794

Order 13696.

1556 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133818	C₂	1	2	yes	maximal, complemented
2	#20293208	C₂	1	4	yes	maximal, complemented
3	#20293203	C₂	1	4	yes	maximal, complemented
4	#20293196	C₂	1	4	yes	maximal, complemented
5	#20293189	C₂	1	4	yes	maximal, complemented
6	#20293177	C₂	1	4	yes	maximal, complemented
7	#20293139	C₂	1	8	yes	maximal, complemented
8	C₂×#117×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133818	—	2	16	no	—
10	C₂×#13133818	C₂²	1	4	yes	—
11	C₂×#13133818	C₂²	1	4	yes	—
12	C₂×#13133818	C₂²	1	4	yes	complemented
13	C₂×#13133818	C₂²	1	4	yes	complemented
14	C₂²×Dic₂₁₄	C₂²	1	2	yes	complemented
15	C₂²×D₄₂₈	C₂²	1	2	yes	complemented
16	C₂²×C₂²⋊D₁₀₇	C₂²	1	4	yes	—
17	C₂²×C₄×D₁₀₇	C₂²	1	4	yes	complemented
18	#16812109	C₂²	1	4	yes	complemented
19	#16812106	C₂²	1	4	yes	complemented
20	#16812097	C₂²	1	8	yes	complemented
21	#16812093	C₂²	1	16	yes	complemented
22	#16812089	C₂²	1	16	yes	complemented
23	#16812084	C₂²	1	16	yes	complemented
24	#16812083	C₂²	1	16	yes	complemented
25	#16812079	C₂²	1	16	yes	complemented
26	C₄×D₄₂₈	C₂²	1	16	yes	complemented
27	C₂³×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
28	#117×C₁₀₇	C₂²	1	4	yes	complemented
29	#117×C₁₀₇	C₂²	1	4	yes	complemented
30	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	complemented
31	C₂×C₂²⋊C₄×C₁₀₇	C₂²	1	2	yes	—
32	C₂×C₄²×C₁₀₇	C₂²	1	2	yes	complemented
33	C₂³×D₁₀₇	C₂³	1	4	yes	—
34	#13133818	—	2	16	no	—
35	#13133818	—	2	16	no	—
36	#13133818	—	2	16	no	—
37	#13133818	—	2	16	no	—
38	C₂×Dic₂₁₄	—	2	8	no	—
39	C₂×Dic₂₁₄	C₂³	1	4	yes	—
40	C₂×Dic₂₁₄	C₂³	1	4	yes	—
41	C₂×D₄₂₈	—	2	8	no	—
42	C₂×D₄₂₈	C₂³	1	4	yes	—
43	C₂×D₄₂₈	C₂³	1	4	yes	—
44	C₂×D₄₂₈	C₂³	1	16	yes	—
45	C₂×C₂²⋊D₁₀₇	—	2	16	no	—
46	C₂×C₂²⋊D₁₀₇	C₂³	1	8	yes	—
47	C₂×C₂²⋊D₁₀₇	C₂³	1	8	yes	—
48	C₂²×Dic₁₀₇	C₂³	1	4	yes	—
49	C₂×C₄×D₁₀₇	—	2	16	no	—
50	C₂×C₄×D₁₀₇	C₂³	1	8	yes	—
51	C₂×C₄×D₁₀₇	C₂³	1	8	yes	—
52	C₄⋊Dic₁₀₇	C₂³	1	16	yes	—
53	(C₂×D₁₀₇)⋊C₄	C₂³	1	32	yes	—
54	C₂⁴×C₁₀₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₇	D₄	1	8	yes	complemented
56	C₂²×C₄×C₁₀₇	C₂³	1	4	yes	—
57	C₂²×C₄×C₁₀₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₀₇	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₁₀₇	C₂³	1	8	yes	—
62	C₂²⋊C₄×C₁₀₇	C₂³	1	8	yes	—
63	C₄²×C₁₀₇	C₂³	1	4	yes	—
64	C₄²×C₁₀₇	C₂³	1	4	yes	—
65	C₂³×C₁₀₇	—	4	2	no	—
66	C₂³×C₁₀₇	C₂×D₄	1	4	yes	—
67	C₂³×C₁₀₇	C₂⁴	1	2	yes	—
68	C₂³×C₁₀₇	C₂⁴	1	1	yes	characteristic
69	C₂²×D₁₀₇	—	2	8	no	—
70	C₂²×D₁₀₇	—	4	8	no	—
71	C₂²×D₁₀₇	C₂⁴	1	8	yes	—
72	C₂×C₄×C₁₀₇	—	4	4	no	—
73	C₂×C₄×C₁₀₇	C₂×D₄	1	8	yes	—
74	C₂×C₄×C₁₀₇	C₂×D₄	1	8	yes	—
75	C₂×C₄×C₁₀₇	C₂×D₄	1	4	yes	—
76	C₂×C₄×C₁₀₇	C₂×D₄	1	4	yes	—
77	C₂×C₄×C₁₀₇	C₂⁴	1	8	yes	—
78	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
79	C₂×C₄×C₁₀₇	C₂⁴	1	2	yes	—
80	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
81	C₂²⋊D₁₀₇	—	2	16	no	—
82	C₂²⋊D₁₀₇	—	2	16	no	—
83	C₂×Dic₁₀₇	—	2	8	no	—
84	C₂×Dic₁₀₇	C₂⁴	1	8	yes	—
85	D₄₂₈	—	2	8	no	—
86	D₄₂₈	—	2	8	no	—
87	D₄₂₈	—	4	16	no	—
88	C₄×D₁₀₇	—	2	16	no	—
89	C₄×D₁₀₇	—	2	16	no	—
90	Dic₂₁₄	—	2	8	no	—
91	Dic₂₁₄	—	2	8	no	—
92	C₂²×C₁₀₇	—	2	2	no	—
93	C₂²×C₁₀₇	—	2	4	no	—
94	C₂²×C₁₀₇	—	4	4	no	—
95	C₂²×C₁₀₇	C₂²×D₄	1	4	yes	—
96	C₂²×C₁₀₇	C₂²×D₄	1	2	yes	—
97	C₂²×C₁₀₇	C₂⁵	1	1	yes	characteristic
98	C₂×D₁₀₇	—	2	8	no	—
99	C₂×D₁₀₇	—	2	16	no	—
100	C₂×D₁₀₇	—	4	16	no	—
101	C₄×C₁₀₇	—	2	8	no	—
102	C₄×C₁₀₇	C₂²×D₄	1	2	yes	—
103	C₄×C₁₀₇	C₂²×D₄	1	4	yes	—
104	C₄×C₁₀₇	C₂²×D₄	1	2	yes	—
105	Dic₁₀₇	—	2	8	no	—
106	C₂×C₁₀₇	—	2	4	no	—
107	C₂×C₁₀₇	—	2	2	no	—
108	C₂×C₁₀₇	C₂³×D₄	1	1	yes	characteristic
109	C₂×C₁₀₇	#585	1	2	yes	—
110	D₁₀₇	—	4	8	no	—
111	#3449	—	107	1	no	maximal, Sylow
112	C₁₀₇	#3449	1	1	yes	characteristic, complemented, Sylow
113	C₂²×#41	—	107	2	no	—
114	#537	—	107	4	no	—
115	#536	—	107	4	no	—
116	#535	—	107	4	no	—
117	#534	—	107	4	no	—
118	#532	—	107	4	no	—
119	#525	—	107	8	no	—
120	C₂×#117	D₁₀₇	1	1	yes	characteristic, complemented
121	C₂×#41	—	107	4	no	—
122	C₂×#41	—	107	4	no	—
123	C₂×#41	—	107	4	no	—
124	C₂×#41	—	107	4	no	—
125	C₂×#41	—	214	16	no	—
126	C₂²×Q₈	—	107	2	no	—
127	C₂²×D₄	—	107	4	no	—
128	C₂²×D₄	—	107	2	no	—
129	C₂³×C₄	—	107	4	no	—
130	C₂³×C₄	C₂×D₁₀₇	1	1	yes	characteristic
131	C₄⋊Q₈	—	107	4	no	—
132	#127	—	107	4	no	—
133	#124	—	107	8	no	—
134	#123	—	107	16	no	—
135	C₂²⋊Q₈	—	107	16	no	—
136	#121	—	107	16	no	—
137	#121	—	107	16	no	—
138	#120	—	107	16	no	—
139	C₄×D₄	—	107	16	no	—
140	#117	C₂×D₁₀₇	1	4	yes	complemented
141	#117	C₂×D₁₀₇	1	4	yes	complemented
142	C₂×C₄⋊C₄	C₂×D₁₀₇	1	2	yes	complemented
143	C₂×C₂²⋊C₄	C₂×D₁₀₇	1	2	yes	—
144	C₂×C₄²	C₂×D₁₀₇	1	2	yes	complemented
145	C₂⁴	—	107	4	no	—
146	C₂⁴	C₂²×D₁₀₇	1	1	yes	characteristic
147	#41	—	214	16	no	—
148	#41	—	214	16	no	—
149	#41	—	214	16	no	—
150	#41	—	214	16	no	—
151	C₂×Q₈	—	107	4	no	—
152	C₂×Q₈	—	107	4	no	—
153	C₂×Q₈	—	214	8	no	—
154	C₂×D₄	—	107	8	no	—
155	C₂×D₄	—	107	8	no	—
156	C₂×D₄	—	107	16	no	—
157	C₂×D₄	—	107	4	no	—
158	C₂×D₄	—	107	4	no	—
159	C₂×D₄	—	214	16	no	—
160	C₂×D₄	—	214	8	no	—
161	C₂²×C₄	—	107	4	no	—
162	C₂²×C₄	—	107	8	no	—
163	C₂²×C₄	—	107	8	no	—
164	C₂²×C₄	—	214	16	no	—
165	C₂²×C₄	D₄₂₈	1	8	yes	complemented
166	C₂²×C₄	C₂²×D₁₀₇	1	4	yes	—
167	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₁₀₇	1	2	yes	—
169	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₀₇	1	2	yes	—
171	C₄⋊C₄	—	107	16	no	—
172	C₄⋊C₄	C₂²×D₁₀₇	1	8	yes	—
173	C₂²⋊C₄	—	107	32	no	—
174	C₂²⋊C₄	C₂²×D₁₀₇	1	8	yes	—
175	C₄²	C₂²×D₁₀₇	1	4	yes	—
176	C₄²	C₂²×D₁₀₇	1	4	yes	—
177	C₂³	—	4	2	no	—
178	C₂³	—	107	8	no	—
179	C₂³	—	428	8	no	—
180	C₂³	—	214	8	no	—
181	C₂³	C₂×D₄₂₈	1	4	yes	—
182	C₂³	C₂³×D₁₀₇	1	2	yes	—
183	C₂³	C₂³×D₁₀₇	1	1	yes	characteristic
184	Q₈	—	214	8	no	—
185	Q₈	—	214	8	no	—
186	D₄	—	428	16	no	—
187	D₄	—	214	16	no	—
188	D₄	—	214	16	no	—
189	D₄	—	214	8	no	—
190	D₄	—	214	8	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	107	8	no	—
193	C₂×C₄	—	214	8	no	—
194	C₂×C₄	—	214	16	no	—
195	C₂×C₄	—	214	16	no	—
196	C₂×C₄	C₂×D₄₂₈	1	8	yes	—
197	C₂×C₄	C₂×D₄₂₈	1	8	yes	—
198	C₂×C₄	C₂×D₄₂₈	1	4	yes	—
199	C₂×C₄	C₂×D₄₂₈	1	4	yes	—
200	C₂×C₄	C₂³×D₁₀₇	1	8	yes	—
201	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₁₀₇	1	2	yes	—
203	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
204	C₂²	—	2	2	no	—
205	C₂²	—	2	4	no	—
206	C₂²	—	4	4	no	—
207	C₂²	—	428	16	no	—
208	C₂²	—	214	8	no	—
209	C₂²	—	214	16	no	—
210	C₂²	C₂²×D₄₂₈	1	4	yes	—
211	C₂²	C₂²×D₄₂₈	1	2	yes	—
212	C₂²	C₂⁴×D₁₀₇	1	1	yes	characteristic
213	C₄	—	2	8	no	—
214	C₄	—	214	8	no	—
215	C₄	C₂²×D₄₂₈	1	2	yes	—
216	C₄	C₂²×D₄₂₈	1	4	yes	—
217	C₄	C₂²×D₄₂₈	1	2	yes	—
218	C₂	—	2	4	no	—
219	C₂	—	2	2	no	—
220	C₂	—	428	8	no	—
221	C₂	C₂³×D₄₂₈	1	1	yes	characteristic
222	C₂	#20293463	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₀₇
commutator, socle, cosocle	C₂²×C₁₀₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₇
Fitting	C₂×#117×C₁₀₇	C₂
hypercenter	C₂×#117	D₁₀₇
hypocenter	C₁₀₇	#3449
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133818	C₂	semidirect
2. #20293208	C₂	semidirect
3. #20293203	C₂	semidirect
4. #20293196	C₂	semidirect
5. #20293189	C₂	semidirect
6. #20293177	C₂	semidirect
7. #20293139	C₂	semidirect
8. C₂×#117×C₁₀₇	C₂	semidirect
12. C₂×#13133818	C₂²	semidirect
13. C₂×#13133818	C₂²	semidirect
14. C₂²×Dic₂₁₄	C₂²	semidirect
15. C₂²×D₄₂₈	C₂²	semidirect
17. C₂²×C₄×D₁₀₇	C₂²	semidirect
18. #16812109	C₂²	semidirect
19. #16812106	C₂²	semidirect
20. #16812097	C₂²	semidirect
21. #16812093	C₂²	semidirect
22. #16812089	C₂²	semidirect
23. #16812084	C₂²	semidirect
24. #16812083	C₂²	semidirect
25. #16812079	C₂²	semidirect
26. C₄×D₄₂₈	C₂²	semidirect
28. #117×C₁₀₇	C₂²	semidirect
29. #117×C₁₀₇	C₂²	semidirect
30. C₂×C₄⋊C₄×C₁₀₇	C₂²	semidirect
32. C₂×C₄²×C₁₀₇	C₂²	semidirect
55. C₂²×C₄×C₁₀₇	D₄	semidirect
112. C₁₀₇	#3449	semidirect
120. C₂×#117	D₁₀₇	semidirect
140. #117	C₂×D₁₀₇	semidirect
141. #117	C₂×D₁₀₇	semidirect
142. C₂×C₄⋊C₄	C₂×D₁₀₇	semidirect
144. C₂×C₄²	C₂×D₁₀₇	semidirect
165. C₂²×C₄	D₄₂₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133818	2
#20293208	2
#20293203	2
#20293196	2
#20293189	2
#20293177	2
#20293139	2
C₂×#117×C₁₀₇	2
#3449	107