Subgroups — #72323216

Order 16384.

Only the 353 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 199 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72323215	C₂	1	2	yes	maximal, complemented
2	#72323214	C₂	1	2	yes	maximal, complemented
3	#72323209	C₂	1	2	yes	maximal, complemented
4	#72322576	C₂	1	1	yes	maximal, characteristic, complemented
5	#72322450	C₂²	1	1	yes	characteristic
6	#72319803	C₂²	1	1	yes	characteristic
7	#72317280	C₂	1	1	yes	maximal, characteristic, complemented
8	#72317153	C₂²	1	2	yes	—
9	#72317060	C₂²	1	1	yes	characteristic, complemented
10	#72316428	C₂	1	2	yes	maximal, complemented
11	#72314070	C₂²	1	2	yes	complemented
12	#72313274	C₂²	1	2	yes	—
13	#72311473	C₂	1	2	yes	maximal, complemented
14	#72310618	C₂²	1	2	yes	—
15	#72310595	D₄	1	4	yes	complemented
16	#72310457	C₂²	1	2	yes	—
17	#72310419	D₄	1	2	yes	complemented
18	#72309403	C₂²	1	2	yes	complemented
19	#72305399	C₂	1	1	yes	maximal, characteristic, complemented
20	#72305323	C₂²	1	1	yes	characteristic
21	#72305244	D₄	1	1	yes	characteristic, complemented
22	#72304657	C₂²	1	1	yes	characteristic, complemented
23	#72303878	D₄	1	1	yes	characteristic, complemented
24	#72302662	D₄	1	1	yes	characteristic, complemented
25	#72302114	C₂²	1	1	yes	characteristic, complemented
26	#72266905	C₂²	1	2	yes	—
27	#72266887	D₄	1	2	yes	complemented
28	#72266881	D₄	1	2	yes	complemented
29	#72266873	D₄	1	2	yes	complemented
30	#72265757	D₄	1	2	yes	complemented
31	#72251277	D₄	1	2	yes	complemented
32	#72238380	C₂	1	2	yes	maximal, complemented
33	#72238363	C₂²	1	2	yes	complemented
34	#72238360	C₂²	1	2	yes	complemented
35	#72238358	C₂²	1	1	yes	characteristic, complemented
36	#72238353	C₂²	1	2	yes	—
37	#72238348	C₂²	1	2	yes	—
38	#72237969	D₄	1	4	yes	complemented
39	#72237631	C₂²	1	2	yes	—
40	#72237630	C₂³	1	2	yes	—
41	#72237628	C₂³	1	2	yes	—
42	#72237625	C₂×D₄	1	4	yes	—
43	#72237618	C₂×D₄	1	2	yes	—
44	#72237606	D₄	1	2	yes	complemented
45	#72237591	D₄	1	2	yes	complemented
46	#72237588	D₄	1	2	yes	complemented
47	#72237581	D₄	1	2	yes	complemented
48	#72237032	D₄	1	2	yes	complemented
49	#72234751	C₂²	1	2	yes	complemented
50	#72234688	C₂³	1	2	yes	—
51	#72234683	C₂³	1	1	yes	characteristic
52	#72234670	C₂×D₄	1	1	yes	characteristic
53	#72234668	C₂×D₄	1	2	yes	—
54	#72234665	C₂³	1	2	yes	—
55	#72234663	C₂×D₄	1	2	yes	—
56	#72234628	D₄	1	1	yes	characteristic, complemented
57	#72234625	C₂×D₄	1	2	yes	—
58	#72232884	C₂×D₄	1	2	yes	—
59	#72232592	C₂×D₄	1	2	yes	—
60	C₂×#8932885	C₂×D₄	1	1	yes	characteristic
61	#72226365	C₂²	1	2	yes	complemented
62	#72225781	C₂³	1	1	yes	characteristic
63	#72225777	C₂×D₄	1	1	yes	characteristic
64	#72225776	C₂×D₄	1	1	yes	characteristic
65	#72225666	C₂³	1	1	yes	characteristic
66	#72225658	C₂×D₄	1	1	yes	characteristic
67	#72225655	C₂×D₄	1	1	yes	characteristic
68	#72225629	C₂³	1	2	yes	—
69	#72219397	C₂³	1	2	yes	—
70	#72219340	C₂×D₄	1	4	yes	—
71	#72219311	C₂×D₄	1	2	yes	—
72	#72219311	C₂×D₄	1	2	yes	—
73	#72219311	C₂×D₄	1	2	yes	—
74	#72219310	C₂×D₄	1	2	yes	—
75	#72219301	C₂×D₄	1	2	yes	—
76	#72219301	C₂×D₄	1	2	yes	—
77	#72219301	C₂×D₄	1	2	yes	—
78	#72219298	D₄	1	2	yes	complemented
79	#72219227	C₂×D₄	1	2	yes	—
80	C₂×#7670203	C₂⁴	1	1	yes	characteristic
81	#72218518	C₂×D₄	1	2	yes	—
82	#72218518	C₂×D₄	1	2	yes	—
83	#72218375	C₂×D₄	1	2	yes	—
84	#72217789	C₂×D₄	1	2	yes	—
85	#72217511	C₂×D₄	1	2	yes	—
86	#72216064	C₂×D₄	1	2	yes	—
87	C₂×#62817	#127	1	2	yes	complemented
88	#10586066	#120	1	1	yes	characteristic
89	C₂²×#3484	C₂²×D₄	1	2	yes	—
90	C₂⁴×#120	C₂²×D₄	1	2	yes	—
91	C₂²×#2846	H(5,𝔽₂)	1	1	yes	characteristic
92	C₂×#35390	H(5,𝔽₂)	1	1	yes	characteristic
93	C₂×#35348	H(5,𝔽₂)	1	2	yes	—
94	C₂×#35346	C₂²×D₄	1	1	yes	characteristic
95	C₂×#34981	C₂²×D₄	1	2	yes	—
96	C₂×#34980	C₂²×D₄	1	2	yes	—
97	#7626202	#127	1	2	yes	complemented
98	C₂×#34198	H(5,𝔽₂)	1	2	yes	—
99	C₂×#21746	#127	1	2	yes	complemented
100	C₂×#21746	#127	1	2	yes	complemented
101	#6370693	#120	1	1	yes	characteristic
102	#6370693	#120	1	2	yes	—
103	#392109	#120	1	2	yes	—
104	#392109	#120	1	2	yes	—
105	#392109	#120	1	2	yes	—
106	#392109	#120	1	2	yes	—
107	#190147	#120	1	2	yes	—
108	#190116	#120	1	2	yes	—
109	C₂⁸	D₄²	1	1	yes	characteristic
110	C₂×H(7,𝔽₂)	B(4,𝔽₂)	1	1	yes	characteristic
111	C₂³×H(5,𝔽₂)	C₂×#120	1	1	yes	characteristic
112	C₂⁵×D₄	C₂×#120	1	2	yes	—
113	C₂²×#534	D₄²	1	2	yes	—
114	C₂²×#534	D₄²	1	2	yes	—
115	C₂²×#534	D₄²	1	1	yes	characteristic
116	C₂³×#120	C₂×#127	1	2	yes	—
117	C₂³×#120	#534	1	1	yes	characteristic
118	C₂³×#120	#535	1	2	yes	—
119	C₂³×#120	D₄²	1	2	yes	—
120	C₂³×#120	D₄²	1	2	yes	—
121	C₂³×#120	D₄²	1	2	yes	—
122	#62652	B(4,𝔽₂)	1	1	yes	characteristic
123	C₂²×#518	#534	1	2	yes	—
124	C₂⁴×C₂²⋊C₄	D₄²	1	2	yes	—
125	C₂×#2883	B(4,𝔽₂)	1	4	yes	—
126	C₂×#2315	#535	1	2	yes	—
127	C₂×#2308	C₂×#120	1	1	yes	characteristic
128	C₂×#2308	D₄²	1	2	yes	—
129	C₂×#2308	D₄²	1	2	yes	—
130	C₂²×#379	#535	1	2	yes	—
131	C₂²×#379	#560	1	1	yes	characteristic
132	C₂×#2273	#560	1	1	yes	characteristic
133	#20585	B(4,𝔽₂)	1	2	yes	—
134	C₂×#1781	B(4,𝔽₂)	1	4	yes	complemented
135	#14679	B(4,𝔽₂)	1	4	yes	complemented
136	#8747	B(4,𝔽₂)	1	4	yes	complemented
137	C₂⁷	#2433	1	2	yes	—
138	C₂⁷	#2669	1	1	yes	characteristic
139	C₂⁷	C₂×B(4,𝔽₂)	1	2	yes	—
140	C₂²×H(5,𝔽₂)	C₂×B(4,𝔽₂)	1	1	yes	characteristic
141	C₂²×H(5,𝔽₂)	C₂×B(4,𝔽₂)	1	1	yes	characteristic
142	C₂²×H(5,𝔽₂)	C₂×B(4,𝔽₂)	1	2	yes	—
143	C₂⁴×D₄	#2403	1	1	yes	characteristic
144	C₂⁴×D₄	#2403	1	2	yes	—
145	C₂⁴×D₄	#2433	1	2	yes	—
146	C₂⁴×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
147	C₂⁴×D₄	#3027	1	2	yes	—
148	C₂⁵×C₄	#3027	1	2	yes	—
149	C₂³×C₂²⋊C₄	#2433	1	2	yes	—
150	C₂³×C₂²⋊C₄	#2669	1	2	yes	—
151	C₂³×C₂²⋊C₄	#2669	1	2	yes	—
152	C₂³×C₂²⋊C₄	#2669	1	1	yes	characteristic
153	#2308	C₂×B(4,𝔽₂)	1	2	yes	—
154	#2308	#3027	1	2	yes	—
155	C₂⁶	#15258	1	1	yes	characteristic
156	C₂⁶	#23895	1	1	yes	characteristic
157	C₂⁶	#23900	1	2	yes	—
158	C₂⁶	#25771	1	2	yes	—
159	C₂⁶	#25771	1	2	yes	—
160	C₂⁶	#25771	1	2	yes	—
161	C₂³×D₄	#23895	1	2	yes	—
162	C₂³×D₄	#23895	1	2	yes	—
163	C₂³×D₄	#23900	1	2	yes	—
164	C₂³×D₄	#23900	1	1	yes	characteristic
165	C₂³×D₄	#23900	1	2	yes	—
166	C₂³×D₄	#23900	1	1	yes	characteristic
167	C₂³×D₄	#23900	1	1	yes	characteristic
168	C₂³×D₄	#23904	1	1	yes	characteristic
169	C₂⁴×C₄	#23895	1	1	yes	characteristic
170	C₂⁴×C₄	#23904	1	2	yes	—
171	C₂⁴×C₄	#25771	1	2	yes	—
172	C₂²×C₂²⋊C₄	#25771	1	4	yes	—
173	C₂⁵	#245225	1	2	yes	—
174	C₂⁵	#245225	1	1	yes	characteristic
175	C₂⁵	#245233	1	2	yes	—
176	C₂⁵	#245249	1	1	yes	characteristic
177	C₂⁵	#417744	1	1	yes	characteristic
178	C₂⁵	#417821	1	1	yes	characteristic
179	C₂⁵	#417821	1	2	yes	—
180	C₂⁵	#417824	1	1	yes	characteristic
181	C₂⁵	#417824	1	2	yes	—
182	C₂³×C₄	#245233	1	1	yes	characteristic
183	C₂³×C₄	#245249	1	2	yes	—
184	C₂³×C₄	#245252	1	1	yes	characteristic
185	C₂³×C₄	#245252	1	2	yes	—
186	C₂⁴	B(5,𝔽₂)	1	4	yes	—
187	C₂⁴	#72210578	1	2	yes	—
188	C₂⁴	#72210578	1	1	yes	characteristic
189	C₂⁴	#72234637	1	1	yes	characteristic
190	C₂⁴	#72237501	1	2	yes	—
191	C₂⁴	#72237501	1	1	yes	characteristic
192	C₂³	#72256024	1	2	yes	—
193	C₂³	#72264779	1	2	yes	—
194	C₂³	#72265084	1	1	yes	characteristic
195	C₂²	#72265820	1	2	yes	—
196	C₂²	#72265931	1	1	yes	characteristic
197	C₂	#72266287	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266287
commutator, Frattini, cosocle	C₂×#7670203	C₂⁴
LCS	G ▷ C₂×#7670203 ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#7670203 ◁ G
LECS	G ▷ C₂×#7670203 ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72323215	C₂	semidirect
2. #72323214	C₂	semidirect
3. #72323209	C₂	semidirect
4. #72322576	C₂	semidirect
7. #72317280	C₂	semidirect
9. #72317060	C₂²	semidirect
10. #72316428	C₂	semidirect
11. #72314070	C₂²	semidirect
13. #72311473	C₂	semidirect
15. #72310595	D₄	semidirect
17. #72310419	D₄	semidirect
18. #72309403	C₂²	semidirect
19. #72305399	C₂	semidirect
21. #72305244	D₄	semidirect
22. #72304657	C₂²	semidirect
23. #72303878	D₄	semidirect
24. #72302662	D₄	semidirect
25. #72302114	C₂²	semidirect
27. #72266887	D₄	semidirect
28. #72266881	D₄	semidirect
29. #72266873	D₄	semidirect
30. #72265757	D₄	semidirect
31. #72251277	D₄	semidirect
32. #72238380	C₂	semidirect
33. #72238363	C₂²	semidirect
34. #72238360	C₂²	semidirect
35. #72238358	C₂²	semidirect
38. #72237969	D₄	semidirect
44. #72237606	D₄	semidirect
45. #72237591	D₄	semidirect
46. #72237588	D₄	semidirect
47. #72237581	D₄	semidirect
48. #72237032	D₄	semidirect
49. #72234751	C₂²	semidirect
56. #72234628	D₄	semidirect
61. #72226365	C₂²	semidirect
78. #72219298	D₄	semidirect
87. C₂×#62817	#127	semidirect
97. #7626202	#127	semidirect
99. C₂×#21746	#127	semidirect
100. C₂×#21746	#127	semidirect
134. C₂×#1781	B(4,𝔽₂)	semidirect
135. #14679	B(4,𝔽₂)	semidirect
136. #8747	B(4,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
#72323215	2
#72323214	2
#72323209	2
#72322576	2
#72317280	2
#72316428	2
#72311473	2
#72305399	2
#72238380	2