Subgroups — #34030374

Order 18688.

548 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11924031	C₂	1	2	yes	maximal, complemented
2	#22626315	C₂	1	6	yes	maximal, complemented
3	#22626292	C₂	1	3	yes	maximal, complemented
4	C₄×#14556665	C₂	1	3	yes	maximal, complemented
5	#41×#10596831	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556665	C₂²	1	6	yes	—
7	C₄⋊#11924031	C₂²	1	6	yes	—
8	C₂²⋊#11924031	C₂²	1	6	yes	—
9	#17989475	C₂²	1	1	yes	characteristic
10	#17989475	C₂²	1	3	yes	—
11	C₄×#11924031	C₂²	1	6	yes	—
12	Q₈×#10596831	C₄	1	1	yes	characteristic
13	Q₈×#10596831	C₂²	1	1	yes	characteristic
14	D₄×#10596831	C₄	1	3	yes	—
15	D₄×#10596831	C₂²	1	3	yes	—
16	#17988640	C₄	1	3	yes	—
17	C₂×C₄×#10596831	C₂²	1	3	yes	—
18	C₄×#41×C₇₃	C₄	1	1	yes	characteristic
19	#14556665	—	2	12	no	—
20	C₂×#11924031	C₂³	1	6	yes	—
21	C₂×#11924031	C₂³	1	2	yes	—
22	C₂²×#10596831	C₂×C₄	1	3	yes	—
23	C₂²×#10596831	C₂³	1	3	yes	—
24	C₄⋊#10596831	C₂×C₄	1	3	yes	—
25	C₄⋊#10596831	C₂×C₄	1	3	yes	—
26	C₂²⋊#10596831	C₂×C₄	1	6	yes	—
27	C₄×#10596831	C₂×C₄	1	3	yes	—
28	C₄×#10596831	C₂×C₄	1	3	yes	—
29	C₄×#10596831	C₂³	1	3	yes	—
30	C₄×#10596831	C₂³	1	1	yes	characteristic
31	C₂×#41×C₇₃	C₈	1	1	yes	characteristic
32	C₄×Q₈×C₇₃	C₈	1	1	yes	characteristic
33	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₄×D₄×C₇₃	C₈	1	3	yes	—
35	C₄×D₄×C₇₃	C₂×C₄	1	3	yes	—
36	#117×C₇₃	C₈	1	3	yes	—
37	C₂×C₄²×C₇₃	C₂×C₄	1	3	yes	—
38	#11924031	—	2	6	no	—
39	#11924031	#41	1	4	yes	complemented
40	C₂×#10596831	—	2	6	no	—
41	C₂×#10596831	—	2	6	no	—
42	C₂×#10596831	C₂²×C₄	1	3	yes	—
43	C₂×#10596831	C₂²×C₄	1	3	yes	—
44	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
45	C₂×#10596831	C₂⁴	1	1	yes	characteristic
46	#41×C₇₃	—	2	4	no	—
47	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
48	C₂×D₄×C₇₃	C₂×C₈	1	3	yes	—
49	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
50	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
51	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
52	C₄⋊C₄×C₇₃	C₂×C₈	1	3	yes	—
53	C₄⋊C₄×C₇₃	C₂×C₈	1	3	yes	—
54	C₂²⋊C₄×C₇₃	C₂×C₈	1	6	yes	—
55	C₄²×C₇₃	C₂×C₈	1	3	yes	—
56	C₄²×C₇₃	C₂×C₈	1	3	yes	—
57	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₇₃	C₂²×C₄	1	3	yes	—
59	C₂³×C₇₃	C₂²×C₈	1	3	yes	—
60	Q₈×C₇₃	M₃₂	1	4	yes	complemented
61	D₄×C₇₃	—	2	6	no	—
62	C₂×C₄×C₇₃	—	2	3	no	—
63	C₂×C₄×C₇₃	—	2	3	no	—
64	C₂×C₄×C₇₃	—	2	6	no	—
65	C₂×C₄×C₇₃	—	4	3	no	—
66	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
71	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
72	#10596831	—	2	3	no	—
73	#10596831	—	2	3	no	—
74	#10596831	#131	1	2	yes	—
75	#10596831	C₂×#41	1	2	yes	—
76	C₂²×C₇₃	—	2	3	no	—
77	C₂²×C₇₃	—	2	3	no	—
78	C₂²×C₇₃	—	2	3	no	—
79	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
80	C₄×C₇₃	—	2	1	no	—
81	C₄×C₇₃	—	2	1	no	—
82	C₄×C₇₃	—	2	3	no	—
83	C₄×C₇₃	—	2	3	no	—
84	C₄×C₇₃	C₂×M₃₂	1	6	yes	—
85	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
86	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
87	#33485	—	73	1	no	maximal, Sylow
88	C₂×C₇₃	—	4	3	no	—
89	C₂×C₇₃	C₈×#41	1	1	yes	characteristic
90	C₂×C₇₃	C₂²×M₃₂	1	1	yes	characteristic
91	C₂×C₇₃	#3407	1	1	yes	characteristic
92	C₈×#41	—	73	1	no	—
93	Q₈×C₁₆	—	73	2	no	—
94	#2169	—	73	6	no	—
95	#2163	—	73	3	no	—
96	C₄×M₃₂	—	73	3	no	—
97	C₇₃	#33485	1	1	yes	characteristic, complemented, Sylow
98	C₄×#41	#64408	1	1	yes	characteristic
99	C₂×M₃₂	—	73	6	no	—
100	C₈×Q₈	—	73	1	no	—
101	C₈×Q₈	—	73	1	no	—
102	C₈×D₄	—	73	3	no	—
103	C₈×D₄	—	73	3	no	—
104	#431	—	73	3	no	—
105	C₂×C₄×C₈	—	73	3	no	—
106	C₄⋊C₁₆	—	73	6	no	—
107	C₂²⋊C₁₆	—	73	6	no	—
108	#346	—	73	1	no	—
109	#346	—	73	3	no	—
110	C₄×C₁₆	—	73	6	no	—
111	C₂×#41	#10596833	1	1	yes	characteristic
112	C₂²×C₈	—	73	3	no	—
113	C₂²×C₈	—	73	3	no	—
114	C₄×Q₈	#10596833	1	1	yes	characteristic
115	C₄×Q₈	C₂×#64408	1	1	yes	characteristic
116	C₄×D₄	#10596833	1	3	yes	—
117	C₄×D₄	C₂×#64408	1	3	yes	—
118	#117	#10596833	1	3	yes	—
119	C₂×C₄²	C₂×#64408	1	3	yes	—
120	M₃₂	—	146	12	no	—
121	C₂×C₁₆	—	73	2	no	—
122	C₂×C₁₆	—	73	6	no	—
123	C₄⋊C₈	—	73	3	no	—
124	C₄⋊C₈	—	73	3	no	—
125	C₂²⋊C₈	—	73	6	no	—
126	C₄×C₈	—	73	3	no	—
127	C₄×C₈	—	73	3	no	—
128	C₄×C₈	—	73	3	no	—
129	C₄×C₈	—	73	1	no	—
130	#41	—	2	4	no	—
131	C₂×Q₈	C₂×#10596833	1	1	yes	characteristic
132	C₂×D₄	C₂×#10596833	1	3	yes	—
133	C₂²×C₄	C₂×#10596833	1	3	yes	—
134	C₂²×C₄	C₂×#10596833	1	3	yes	—
135	C₂²×C₄	C₂²×#64408	1	3	yes	—
136	C₂×C₈	—	73	1	no	—
137	C₂×C₈	—	73	1	no	—
138	C₂×C₈	—	73	3	no	—
139	C₂×C₈	—	73	3	no	—
140	C₂×C₈	—	146	6	no	—
141	C₂×C₈	—	146	6	no	—
142	C₄⋊C₄	C₂×#10596833	1	3	yes	—
143	C₄⋊C₄	C₂×#10596833	1	3	yes	—
144	C₂²⋊C₄	C₂×#10596833	1	6	yes	—
145	C₄²	C₂×#10596833	1	3	yes	—
146	C₄²	C₂×#10596833	1	3	yes	—
147	C₄²	C₂²×#64408	1	1	yes	characteristic
148	C₄²	C₂²×#64408	1	3	yes	—
149	C₁₆	—	73	4	no	—
150	C₁₆	—	146	6	no	—
151	C₂³	C₂²×#10596833	1	3	yes	—
152	Q₈	#14556665	1	4	yes	complemented
153	D₄	—	2	6	no	—
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	3	no	—
156	C₂×C₄	—	2	3	no	—
157	C₂×C₄	—	4	3	no	—
158	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
159	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
160	C₂×C₄	C₂²×#10596833	1	3	yes	—
161	C₂×C₄	C₂²×#10596833	1	3	yes	—
162	C₂×C₄	C₂²×#10596833	1	3	yes	—
163	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
164	C₈	—	73	2	no	—
165	C₈	—	73	2	no	—
166	C₈	—	146	3	no	—
167	C₈	—	146	3	no	—
168	C₂²	—	2	3	no	—
169	C₂²	—	2	3	no	—
170	C₂²	—	2	3	no	—
171	C₂²	C₂³×#10596833	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	3	no	—
175	C₄	—	2	3	no	—
176	C₄	C₂×#14556665	1	6	yes	—
177	C₄	#41×#64408	1	1	yes	characteristic
178	C₄	D₇₃⋊#131	1	1	yes	characteristic
179	C₂	—	4	3	no	—
180	C₂	#41×#10596833	1	1	yes	characteristic
181	C₂	C₂²×#14556665	1	1	yes	characteristic
182	C₂	#22626898	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₄×#41×C₇₃	C₄
prime radical	C₂×#41×C₇₃	C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₄×#41	#64408
hypocenter	C₇₃	#33485
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×#11924031	C₂	semidirect
2. #22626315	C₂	semidirect
3. #22626292	C₂	semidirect
4. C₄×#14556665	C₂	semidirect
39. #11924031	#41	semidirect
60. Q₈×C₇₃	M₃₂	semidirect
97. C₇₃	#33485	semidirect
152. Q₈	#14556665	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11924031	2
#22626315	2
#22626292	2
C₄×#14556665	2
#41×#10596831	2
#33485	73