Subgroups — #72324182

Order 8192.

At least 43499 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

The subgroup lattice has 217 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72323694	C₂	1	1	yes	maximal, characteristic, complemented
2	#72323229	C₂	1	1	yes	maximal, characteristic, complemented
3	#72323220	C₂	1	1	yes	maximal, characteristic, complemented
4	#72317154	C₂	1	1	yes	maximal, characteristic, complemented
5	#72314319	C₂	1	1	yes	maximal, characteristic, complemented
6	#72309405	C₂	1	1	yes	maximal, characteristic, complemented
7	#72309397	C₂²	1	1	yes	characteristic
8	C₂×#97555	D₄	1	2	yes	complemented
9	#72266905	C₂	1	1	yes	maximal, characteristic
10	#72266887	C₄	1	1	yes	characteristic, complemented
11	#72266881	C₄	1	1	yes	characteristic, complemented
12	#72266873	C₄	1	1	yes	characteristic, complemented
13	#72265785	C₂²	1	1	yes	characteristic
14	#72259029	C₂²	1	1	yes	characteristic
15	#72259008	D₄	1	1	yes	characteristic, complemented
16	#72251277	C₄	1	1	yes	characteristic, complemented
17	#72235847	D₄	1	1	yes	characteristic, complemented
18	#72234688	C₂²	1	1	yes	characteristic
19	#72234665	C₂²	1	1	yes	characteristic
20	#72234663	C₂×C₄	1	1	yes	characteristic
21	#72234659	D₄	1	2	yes	—
22	#72234658	D₄	1	2	yes	—
23	#72232884	C₂×C₄	1	1	yes	characteristic
24	#72225883	C₂²	1	1	yes	characteristic
25	#72225666	C₂²	1	1	yes	characteristic
26	#72225658	D₄	1	1	yes	characteristic
27	#72225655	D₄	1	1	yes	characteristic
28	#72219311	D₄	1	1	yes	characteristic
29	#72219311	D₄	1	1	yes	characteristic
30	#72219301	C₂×C₄	1	1	yes	characteristic
31	#72219301	C₂×C₄	1	1	yes	characteristic
32	#72219301	C₂×C₄	1	1	yes	characteristic
33	#72219301	D₄	1	1	yes	characteristic
34	C₂×#7670203	C₂³	1	1	yes	characteristic
35	#72218518	D₄	1	1	yes	characteristic
36	#72218518	D₄	1	1	yes	characteristic
37	#72218518	D₄	1	1	yes	characteristic, complemented
38	#72218291	D₄	1	1	yes	characteristic, complemented
39	C₂×#134033	D₄	1	2	yes	complemented
40	#72216064	C₂×C₄	1	1	yes	characteristic
41	#72216064	D₄	1	1	yes	characteristic
42	#10586069	C₂²⋊C₄	1	2	yes	—
43	#10586066	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×#3484	C₂²×C₄	1	1	yes	characteristic
45	C₂²×#3484	C₂×D₄	1	1	yes	characteristic
46	C₂⁴×#120	C₂×D₄	1	1	yes	characteristic
47	C₂⁴×#120	C₂×D₄	1	1	yes	characteristic
48	C₂²×#2846	C₂×D₄	1	1	yes	characteristic
49	C₂×#35390	C₂×D₄	1	1	yes	characteristic
50	C₂×#35348	C₂×D₄	1	1	yes	characteristic
51	C₂×#35348	#41	1	1	yes	characteristic
52	C₂×#35346	C₂×D₄	1	1	yes	characteristic
53	C₂×#34981	#41	1	1	yes	characteristic
54	C₂×#34981	#41	1	1	yes	characteristic
55	C₂×#34980	C₂×D₄	1	1	yes	characteristic
56	C₂×#34980	C₂×D₄	1	1	yes	characteristic
57	C₂×#34198	C₂×D₄	1	1	yes	characteristic
58	C₂×#34198	#41	1	1	yes	characteristic
59	#6371007	C₂²⋊C₄	1	2	yes	—
60	#6370693	C₂²⋊C₄	1	1	yes	characteristic
61	#6370693	C₂²⋊C₄	1	1	yes	characteristic
62	#6370693	C₂²⋊C₄	1	1	yes	characteristic
63	#392109	C₂²⋊C₄	1	1	yes	characteristic
64	#392109	C₂²⋊C₄	1	1	yes	characteristic
65	#392109	C₂²⋊C₄	1	1	yes	characteristic
66	#392109	C₂²⋊C₄	1	1	yes	characteristic
67	C₂⁸	#127	1	1	yes	characteristic
68	C₂×H(7,𝔽₂)	#99	1	1	yes	characteristic
69	C₂³×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂⁵×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂⁵×D₄	#120	1	1	yes	characteristic
72	#62843	#99	1	2	yes	complemented
73	#62842	#99	1	2	yes	complemented
74	C₂²×#534	C₄×D₄	1	1	yes	characteristic
75	C₂²×#534	C₄×D₄	1	1	yes	characteristic
76	C₂²×#534	#120	1	1	yes	characteristic
77	C₂²×#534	#121	1	1	yes	characteristic
78	C₂²×#534	#127	1	1	yes	characteristic
79	C₂³×#120	C₄×D₄	1	1	yes	characteristic
80	C₂³×#120	C₄×D₄	1	1	yes	characteristic
81	C₂³×#120	#120	1	1	yes	characteristic
82	C₂³×#120	#120	1	1	yes	characteristic
83	C₂³×#120	#120	1	1	yes	characteristic
84	C₂³×#120	#120	1	1	yes	characteristic
85	C₂³×#120	#121	1	1	yes	characteristic
86	C₂³×#120	#121	1	1	yes	characteristic
87	C₂³×#120	#121	1	1	yes	characteristic
88	C₂³×#120	#121	1	1	yes	characteristic
89	C₂³×#120	#121	1	1	yes	characteristic
90	#62652	#99	1	1	yes	characteristic
91	C₂²×#518	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×#518	#123	1	1	yes	characteristic
93	C₂⁴×C₂²⋊C₄	#121	1	1	yes	characteristic
94	C₂⁴×C₂²⋊C₄	#124	1	1	yes	characteristic
95	C₂×#2883	#99	1	1	yes	characteristic
96	C₂×#2883	#99	1	1	yes	characteristic
97	C₂×#2883	#99	1	1	yes	characteristic
98	C₂×#2883	#99	1	1	yes	characteristic
99	C₂×#2315	#123	1	1	yes	characteristic
100	C₂×#2315	#123	1	1	yes	characteristic
101	C₂×#2308	#120	1	1	yes	characteristic
102	C₂×#2308	#121	1	1	yes	characteristic
103	C₂×#2308	#121	1	1	yes	characteristic
104	C₂×#2308	#123	1	1	yes	characteristic
105	C₂×#2308	#124	1	1	yes	characteristic
106	C₂²×#379	#120	1	1	yes	characteristic
107	C₂²×#379	#121	1	1	yes	characteristic
108	C₂²×#379	#121	1	1	yes	characteristic
109	C₂×#2273	#121	1	1	yes	characteristic
110	#20585	#99	1	1	yes	characteristic
111	#20585	#99	1	1	yes	characteristic
112	C₂×#1781	#99	1	2	yes	complemented
113	#20392	#99	1	4	yes	complemented
114	#14679	#99	1	2	yes	complemented
115	C₂⁷	#390	1	1	yes	characteristic
116	C₂⁷	#392	1	1	yes	characteristic
117	C₂⁷	#392	1	1	yes	characteristic
118	C₂⁷	C₂×#99	1	1	yes	characteristic
119	C₂⁷	B(4,𝔽₂)	1	1	yes	characteristic
120	C₂²×H(5,𝔽₂)	C₂×#99	1	1	yes	characteristic
121	C₂²×H(5,𝔽₂)	C₂×#99	1	1	yes	characteristic
122	C₂²×H(5,𝔽₂)	C₂×#99	1	1	yes	characteristic
123	C₂²×H(5,𝔽₂)	C₂×#99	1	1	yes	characteristic
124	C₂⁴×D₄	#379	1	1	yes	characteristic
125	C₂⁴×D₄	#386	1	1	yes	characteristic
126	C₂⁴×D₄	#386	1	1	yes	characteristic
127	C₂⁴×D₄	#390	1	1	yes	characteristic
128	C₂⁴×D₄	#392	1	1	yes	characteristic
129	C₂⁴×D₄	C₂×#99	1	1	yes	characteristic
130	C₂⁴×D₄	#410	1	1	yes	characteristic
131	C₂⁴×D₄	B(4,𝔽₂)	1	1	yes	characteristic
132	C₂⁴×D₄	B(4,𝔽₂)	1	1	yes	characteristic
133	C₂⁵×C₄	#410	1	1	yes	characteristic
134	C₂⁵×C₄	B(4,𝔽₂)	1	1	yes	characteristic
135	C₂²×#120	B(4,𝔽₂)	1	2	yes	—
136	C₂²×#120	B(4,𝔽₂)	1	2	yes	—
137	C₂×#518	C₂×#99	1	2	yes	—
138	C₂×#518	C₂×#99	1	2	yes	—
139	C₂³×C₂²⋊C₄	#386	1	1	yes	characteristic
140	C₂³×C₂²⋊C₄	#386	1	1	yes	characteristic
141	C₂³×C₂²⋊C₄	#392	1	1	yes	characteristic
142	C₂³×C₂²⋊C₄	#394	1	1	yes	characteristic
143	C₂³×C₂²⋊C₄	#394	1	1	yes	characteristic
144	C₂³×C₂²⋊C₄	#394	1	1	yes	characteristic
145	C₂³×C₂²⋊C₄	#394	1	1	yes	characteristic
146	C₂⁶	#1438	1	1	yes	characteristic
147	C₂⁶	#1781	1	1	yes	characteristic
148	C₂⁶	#1889	1	1	yes	characteristic
149	C₂⁶	#1913	1	1	yes	characteristic
150	C₂⁶	#1964	1	1	yes	characteristic
151	C₂⁶	#1964	1	1	yes	characteristic
152	C₂⁶	#1964	1	1	yes	characteristic
153	C₂⁶	#2021	1	1	yes	characteristic
154	C₂⁶	#2021	1	1	yes	characteristic
155	C₂⁶	#2021	1	1	yes	characteristic
156	C₂⁶	#2021	1	2	yes	—
157	C₂³×D₄	#1781	1	1	yes	characteristic
158	C₂³×D₄	#1781	1	1	yes	characteristic
159	C₂³×D₄	#1781	1	1	yes	characteristic
160	C₂³×D₄	#1781	1	1	yes	characteristic
161	C₂³×D₄	#1889	1	1	yes	characteristic
162	C₂³×D₄	#1889	1	1	yes	characteristic
163	C₂³×D₄	#1898	1	1	yes	characteristic
164	C₂³×D₄	#1898	1	1	yes	characteristic
165	C₂³×D₄	#1898	1	1	yes	characteristic
166	C₂³×D₄	#1898	1	1	yes	characteristic
167	C₂³×D₄	#1913	1	1	yes	characteristic
168	C₂³×D₄	#1913	1	1	yes	characteristic
169	C₂⁴×C₄	#1782	1	1	yes	characteristic
170	C₂⁴×C₄	#1889	1	1	yes	characteristic
171	C₂⁴×C₄	#1913	1	1	yes	characteristic
172	C₂⁴×C₄	#1964	1	1	yes	characteristic
173	C₂⁴×C₄	#2021	1	1	yes	characteristic
174	C₂²×C₂²⋊C₄	#1898	1	2	yes	—
175	C₂²×C₂²⋊C₄	#1898	1	2	yes	—
176	C₂²×C₂²⋊C₄	#2021	1	2	yes	—
177	C₂⁵	#8541	1	1	yes	characteristic
178	C₂⁵	#8546	1	1	yes	characteristic
179	C₂⁵	#9495	1	1	yes	characteristic
180	C₂⁵	#9495	1	1	yes	characteristic
181	C₂⁵	#9497	1	1	yes	characteristic
182	C₂⁵	#9497	1	1	yes	characteristic
183	C₂⁵	#10965	1	1	yes	characteristic
184	C₂⁵	#10965	1	1	yes	characteristic
185	C₂⁵	#10972	1	1	yes	characteristic
186	C₂⁵	#10972	1	1	yes	characteristic
187	C₂⁵	#10980	1	1	yes	characteristic
188	C₂⁵	#10991	1	1	yes	characteristic
189	C₂⁵	#10991	1	1	yes	characteristic
190	C₂³×C₄	#8542	1	1	yes	characteristic
191	C₂³×C₄	#8542	1	1	yes	characteristic
192	C₂³×C₄	#9495	1	1	yes	characteristic
193	C₂³×C₄	#9495	1	1	yes	characteristic
194	C₂³×C₄	#9497	1	1	yes	characteristic
195	C₂³×C₄	#9497	1	1	yes	characteristic
196	C₂⁴	#111516	1	1	yes	characteristic
197	C₂⁴	#115133	1	1	yes	characteristic
198	C₂⁴	#115133	1	1	yes	characteristic
199	C₂⁴	#115163	1	1	yes	characteristic
200	C₂⁴	#115163	1	1	yes	characteristic
201	C₂⁴	#115631	1	1	yes	characteristic
202	C₂⁴	#115690	1	1	yes	characteristic
203	C₂⁴	#133839	1	2	yes	—
204	C₂⁴	#139750	1	2	yes	—
205	C₂²×C₄	#139750	1	2	yes	—
206	C₂³	#72219158	1	1	yes	characteristic
207	C₂³	#72219158	1	1	yes	characteristic
208	C₂³	#72219443	1	1	yes	characteristic
209	C₂³	#72251552	1	1	yes	characteristic
210	C₂³	#72264770	1	1	yes	characteristic
211	C₂³	#72265913	1	2	yes	—
212	C₂²	#72219455	1	1	yes	characteristic
213	C₂²	#72265787	1	1	yes	characteristic
214	C₂²	#72265914	1	1	yes	characteristic
215	C₂	#72266322	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266322
commutator	C₂²×#3484	C₂²×C₄
Frattini, cosocle	C₂×#7670203	C₂³
prime radical	#72266905	C₂
LCS	G ▷ C₂²×#3484 ▷ 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. #72323694	C₂	semidirect
2. #72323229	C₂	semidirect
3. #72323220	C₂	semidirect
4. #72317154	C₂	semidirect
5. #72314319	C₂	semidirect
6. #72309405	C₂	semidirect
8. C₂×#97555	D₄	semidirect
10. #72266887	C₄	semidirect
11. #72266881	C₄	semidirect
12. #72266873	C₄	semidirect
15. #72259008	D₄	semidirect
16. #72251277	C₄	semidirect
17. #72235847	D₄	semidirect
37. #72218518	D₄	semidirect
38. #72218291	D₄	semidirect
39. C₂×#134033	D₄	semidirect
72. #62843	#99	semidirect
73. #62842	#99	semidirect
112. C₂×#1781	#99	semidirect
113. #20392	#99	semidirect
114. #14679	#99	semidirect

Maximal subgroups

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

subgroup	index
#72323694	2
#72323229	2
#72323220	2
#72317154	2
#72314319	2
#72309405	2
#72266905	2