Subgroups — #34030337

Order 18688.

602 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	#22626310	C₂	1	6	yes	maximal, complemented
3	#22626291	C₂	1	3	yes	maximal, complemented
4	C₄×#14556664	C₂	1	3	yes	maximal, complemented
5	C₈×#11924029	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556664	C₂²	1	6	yes	—
7	C₄⋊#11924031	C₂²	1	6	yes	—
8	#17989482	C₂²	1	6	yes	—
9	#17989473	C₂²	1	1	yes	characteristic
10	#17989473	C₂²	1	3	yes	—
11	C₄×#11924031	C₂²	1	6	yes	—
12	C₄×#11924029	C₄	1	1	yes	characteristic
13	Q₈×#10596831	C₂²	1	1	yes	characteristic
14	C₈×D₂₉₂	C₄	1	3	yes	—
15	#17988652	C₂²	1	3	yes	—
16	#17988639	C₄	1	3	yes	—
17	C₄×C₈×D₇₃	C₂²	1	3	yes	—
18	C₈×Q₈×C₇₃	C₄	1	1	yes	characteristic
19	#14556664	—	2	12	no	—
20	C₂×#11924031	C₂³	1	6	yes	—
21	C₂×#11924031	C₂³	1	2	yes	—
22	C₂×#11924029	C₈	1	1	yes	characteristic
23	C₂×C₈×D₇₃	C₂×C₄	1	3	yes	—
24	C₂×C₈×D₇₃	C₂³	1	3	yes	—
25	Q₈×Dic₇₃	C₈	1	1	yes	characteristic
26	C₄×D₂₉₂	C₂×C₄	1	3	yes	—
27	#14556553	C₈	1	3	yes	—
28	#14556549	C₈	1	3	yes	—
29	C₄²×D₇₃	C₂×C₄	1	3	yes	—
30	C₄⋊#10596831	C₂×C₄	1	3	yes	—
31	#14556503	C₂×C₄	1	6	yes	—
32	C₄×#10596831	C₂³	1	3	yes	—
33	C₈×Dic₇₃	C₂×C₄	1	3	yes	—
34	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₇₃	C₂×C₄	1	3	yes	—
37	C₄×C₈×C₇₃	C₂×C₄	1	3	yes	—
38	#11924031	—	2	6	no	—
39	#11924031	#41	1	4	yes	complemented
40	#11924029	C₁₆	1	8	yes	complemented
41	C₂×D₂₉₂	C₂×C₈	1	3	yes	—
42	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
43	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
44	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
45	C₂×#10596831	C₂²×C₄	1	3	yes	—
46	C₂×#10596831	C₂⁴	1	1	yes	characteristic
47	C₈×D₇₃	—	2	6	no	—
48	C₈×D₇₃	—	2	6	no	—
49	C₄⋊Dic₇₃	C₂×C₈	1	3	yes	—
50	#11924005	C₂×C₈	1	6	yes	—
51	C₄×Dic₇₃	C₂×C₈	1	3	yes	—
52	C₄×Dic₇₃	C₂×C₈	1	3	yes	—
53	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
56	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₇₃	C₂×C₈	1	3	yes	—
58	C₄²×C₇₃	C₂²×C₄	1	3	yes	—
59	C₂²×D₇₃	C₂²×C₈	1	3	yes	—
60	Q₈×C₇₃	C₂×C₁₆	1	4	yes	complemented
61	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
62	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
63	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
64	C₂×Dic₇₃	C₂²×C₈	1	3	yes	—
65	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
67	D₂₉₂	C₂×C₁₆	1	12	yes	complemented
68	C₄×D₇₃	—	2	3	no	—
69	C₄×D₇₃	—	2	3	no	—
70	C₄×D₇₃	—	2	6	no	—
71	C₄×D₇₃	C₂×C₁₆	1	12	yes	complemented
72	C₈×C₇₃	—	2	3	no	—
73	C₈×C₇₃	#131	1	2	yes	—
74	#10596831	—	2	3	no	—
75	#10596831	C₂×#41	1	2	yes	—
76	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
77	C₂×D₇₃	—	2	3	no	—
78	C₂×D₇₃	—	2	3	no	—
79	C₂×D₇₃	C₂²×C₁₆	1	6	yes	—
80	C₄×C₇₃	—	2	3	no	—
81	C₄×C₇₃	C₂²×C₁₆	1	6	yes	—
82	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
83	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
84	Dic₇₃	—	2	3	no	—
85	Dic₇₃	C₂²×C₁₆	1	2	yes	—
86	Dic₇₃	#504	1	2	yes	—
87	C₁₆×#41	—	73	1	no	maximal, Sylow
88	C₂×C₇₃	C₈×#41	1	1	yes	characteristic
89	C₂×C₇₃	C₂³×C₁₆	1	1	yes	characteristic
90	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
91	D₇₃	—	2	6	no	—
92	C₈×#41	—	73	1	no	—
93	Q₈×C₁₆	—	73	2	no	—
94	D₄×C₁₆	—	73	6	no	—
95	#2162	—	73	3	no	—
96	C₂×C₄×C₁₆	—	73	3	no	—
97	C₇₃	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
98	C₄×#41	—	73	1	no	—
99	C₂²×C₁₆	—	73	6	no	—
100	C₈×Q₈	—	73	1	no	—
101	C₈×Q₈	#64408	1	1	yes	characteristic
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	C₄×C₁₆	—	73	1	no	—
109	C₄×C₁₆	—	73	6	no	—
110	C₄×C₁₆	—	73	3	no	—
111	C₂×#41	—	73	1	no	—
112	C₂²×C₈	—	73	3	no	—
113	C₂²×C₈	—	73	3	no	—
114	C₄×Q₈	—	73	1	no	—
115	C₄×Q₈	C₂×#64408	1	1	yes	characteristic
116	C₄×D₄	—	73	3	no	—
117	C₄×D₄	—	73	3	no	—
118	#117	—	73	3	no	—
119	C₂×C₄²	—	73	3	no	—
120	C₂×C₁₆	—	73	2	no	—
121	C₂×C₁₆	—	73	6	no	—
122	C₂×C₁₆	—	146	12	no	—
123	C₄⋊C₈	—	73	3	no	—
124	C₄⋊C₈	C₂×#64408	1	3	yes	—
125	C₂²⋊C₈	—	73	6	no	—
126	C₄×C₈	—	73	3	no	—
127	C₄×C₈	—	73	1	no	—
128	C₄×C₈	—	73	3	no	—
129	C₄×C₈	C₂×#64408	1	3	yes	—
130	#41	—	73	8	no	—
131	C₂×Q₈	#11924036	1	1	yes	characteristic
132	C₂×D₄	—	73	3	no	—
133	C₂²×C₄	—	73	3	no	—
134	C₂²×C₄	—	73	3	no	—
135	C₂²×C₄	—	73	3	no	—
136	C₂×C₈	—	73	1	no	—
137	C₂×C₈	—	73	3	no	—
138	C₂×C₈	—	146	6	no	—
139	C₂×C₈	—	146	6	no	—
140	C₂×C₈	C₂²×#64408	1	3	yes	—
141	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
142	C₄⋊C₄	—	73	3	no	—
143	C₄⋊C₄	#11924036	1	3	yes	—
144	C₂²⋊C₄	—	73	6	no	—
145	C₄²	—	73	3	no	—
146	C₄²	—	73	3	no	—
147	C₄²	—	73	1	no	—
148	C₄²	C₂²×#64408	1	3	yes	—
149	C₁₆	—	73	4	no	—
150	C₁₆	—	146	6	no	—
151	C₂³	—	73	3	no	—
152	Q₈	#14556664	1	4	yes	complemented
153	D₄	—	73	12	no	—
154	C₂×C₄	—	73	1	no	—
155	C₂×C₄	—	73	1	no	—
156	C₂×C₄	—	73	12	no	—
157	C₂×C₄	—	73	3	no	—
158	C₂×C₄	—	146	6	no	—
159	C₂×C₄	—	146	3	no	—
160	C₂×C₄	—	146	3	no	—
161	C₂×C₄	C₂×#11924036	1	3	yes	—
162	C₂×C₄	C₂×#11924036	1	3	yes	—
163	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
164	C₈	—	2	3	no	—
165	C₈	—	73	2	no	—
166	C₈	—	146	3	no	—
167	C₈	#14556679	1	2	yes	—
168	C₂²	—	73	6	no	—
169	C₂²	—	146	3	no	—
170	C₂²	—	146	3	no	—
171	C₂²	C₂²×#11924036	1	1	yes	characteristic
172	C₄	—	2	3	no	—
173	C₄	—	73	2	no	—
174	C₄	—	73	2	no	—
175	C₄	—	146	3	no	—
176	C₄	C₂×#14556664	1	6	yes	—
177	C₄	#41×#64408	1	1	yes	characteristic
178	C₄	C₂×#14556679	1	1	yes	characteristic
179	C₂	—	146	6	no	—
180	C₂	#22626737	1	1	yes	characteristic
181	C₂	C₂²×#14556664	1	1	yes	characteristic
182	C₂	#22626896	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×#64408
commutator	C₂×C₇₃	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₈×Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂³×C₈
prime radical	C₂×#11924029	C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₈×Q₈	#64408
hypocenter	C₇₃	C₁₆×#41
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#11924031	C₂	semidirect
2. #22626310	C₂	semidirect
3. #22626291	C₂	semidirect
4. C₄×#14556664	C₂	semidirect
39. #11924031	#41	semidirect
40. #11924029	C₁₆	semidirect
60. Q₈×C₇₃	C₂×C₁₆	semidirect
67. D₂₉₂	C₂×C₁₆	semidirect
71. C₄×D₇₃	C₂×C₁₆	semidirect
97. C₇₃	C₁₆×#41	semidirect
152. Q₈	#14556664	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
#22626310	2
#22626291	2
C₄×#14556664	2
C₈×#11924029	2
C₁₆×#41	73