Finite Group Databasebeta

Subgroups — #30776335

Order 17152.

194 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863229	C₂	1	1	yes	maximal, characteristic
2	#22561990	C₂	1	1	yes	maximal, characteristic
3	#366×D₆₇	C₂	1	1	yes	maximal, characteristic
4	#22558411	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558410	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558408	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
9	#16863433	—	2	1	no	—
10	#107×D₆₇	—	2	1	no	—
11	#16863229	C₄	1	4	yes	complemented
12	C₂×C₆₇⋊C₃₂	C₂²	1	1	yes	characteristic
13	#16863223	—	2	1	no	—
14	#16863222	C₂²	1	1	yes	characteristic
15	#16863222	C₂²	1	1	yes	characteristic
16	#16863221	—	2	1	no	—
17	C₂×C₃₂×C₆₇	C₂²	1	1	yes	characteristic
18	#366×C₆₇	C₂²	1	1	yes	characteristic
19	#366×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
21	#14524001	—	4	1	no	—
22	C₄⋊C₄×D₆₇	—	4	1	no	—
23	C₂×C₆₇⋊C₁₆	D₄	1	1	yes	characteristic
24	C₁₆×D₆₇	C₂×C₄	1	2	yes	—
25	C₁₆×D₆₇	Q₈	1	2	yes	—
26	#14523977	—	2	1	no	—
27	#14523977	—	2	1	no	—
28	#14523976	—	4	1	no	—
29	#14523974	—	4	1	no	—
30	C₆₇⋊C₃₂	C₂×C₄	1	2	yes	complemented
31	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
32	#107×C₆₇	—	2	1	no	—
33	#107×C₆₇	—	2	1	no	—
34	C₃₂×C₆₇	C₂×C₄	1	2	yes	complemented
35	C₂×C₄×D₆₇	—	8	1	no	—
36	C₂×C₄×D₆₇	D₈	1	1	yes	characteristic
37	C₂×C₆₇⋊C₈	D₈	1	1	yes	characteristic
38	C₈×D₆₇	C₄⋊C₄	1	2	yes	—
39	C₈×D₆₇	Q₁₆	1	2	yes	—
40	C₄⋊Dic₆₇	—	4	1	no	—
41	C₄⋊Dic₆₇	—	4	1	no	—
42	Dic₆₇⋊C₄	—	8	1	no	—
43	(C₂×D₆₇)⋊C₄	—	8	1	no	—
44	C₄×Dic₆₇	—	8	1	no	—
45	C₆₇⋊C₁₆	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₄⋊C₄×C₆₇	—	4	1	no	—
48	C₄⋊C₄×C₆₇	—	4	1	no	—
49	C₁₆×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₁₆×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₂²×D₆₇	D₁₆	1	1	yes	characteristic
52	C₂×C₄×C₆₇	—	8	1	no	—
53	C₂×C₄×C₆₇	—	8	1	no	—
54	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₆₇	—	8	1	no	—
56	C₂×Dic₆₇	—	8	1	no	—
57	C₂×Dic₆₇	D₁₆	1	1	yes	characteristic
58	C₄×D₆₇	—	16	2	no	—
59	C₄×D₆₇	#107	1	2	yes	—
60	C₄×D₆₇	Q₃₂	1	2	yes	—
61	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₆₇⋊C₈	#107	1	2	yes	—
64	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
65	C₂×D₆₇	—	2	1	no	—
66	C₂×D₆₇	—	2	1	no	—
67	C₂×D₆₇	#366	1	2	yes	—
68	C₄×C₆₇	—	16	1	no	—
69	C₄×C₆₇	—	16	1	no	—
70	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
71	C₄×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
72	Dic₆₇	—	16	1	no	—
73	Dic₆₇	—	16	1	no	—
74	Dic₆₇	#366	1	2	yes	—
75	#13634	—	67	1	no	maximal, Sylow
76	C₂×C₆₇	C₂×#366	1	1	yes	characteristic
77	C₂×C₆₇	#2263	1	1	yes	characteristic
78	C₂×C₆₇	#2264	1	1	yes	characteristic
79	D₆₇	—	2	2	no	—
80	C₂×M₆₄	—	67	1	no	—
81	#2158	—	67	1	no	—
82	C₂×#366	—	67	1	no	—
83	#1424	—	67	1	no	—
84	#1424	—	67	1	no	—
85	#1423	—	67	1	no	—
86	#1423	D₆₇	1	1	yes	characteristic, complemented
87	C₆₇	#13634	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	—	67	1	no	—
89	#427	—	134	1	no	—
90	C₂×#107	—	134	1	no	—
91	M₆₄	—	67	4	no	—
92	C₂×C₃₂	—	67	1	no	—
93	C₂×C₃₂	C₂×D₆₇	1	1	yes	characteristic
94	#367	—	134	1	no	—
95	#366	—	67	1	no	—
96	#366	—	67	1	no	—
97	#366	—	134	1	no	—
98	#366	C₂×D₆₇	1	1	yes	characteristic
99	#366	C₂×D₆₇	1	1	yes	characteristic
100	C₂²×C₈	—	67	1	no	—
101	#117	—	268	1	no	—
102	C₂×C₄⋊C₄	—	268	1	no	—
103	C₂×C₁₆	—	67	1	no	—
104	C₂×C₁₆	—	67	2	no	—
105	C₂×C₁₆	—	67	2	no	—
106	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
107	#107	—	2	1	no	—
108	#107	—	2	1	no	—
109	#107	—	268	1	no	—
110	#107	—	134	1	no	—
111	#107	—	134	1	no	—
112	#106	—	268	1	no	—
113	C₃₂	—	67	2	no	—
114	C₃₂	C₄×D₆₇	1	2	yes	complemented
115	C₂²×C₄	—	536	1	no	—
116	C₂²×C₄	—	67	1	no	—
117	C₂×C₈	—	67	1	no	—
118	C₂×C₈	—	67	2	no	—
119	C₂×C₈	—	67	2	no	—
120	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	4	1	no	—
123	C₄⋊C₄	—	268	1	no	—
124	C₄⋊C₄	—	268	1	no	—
125	C₄⋊C₄	—	536	1	no	—
126	C₂²⋊C₄	—	536	1	no	—
127	C₄²	—	536	1	no	—
128	C₁₆	—	67	2	no	—
129	C₁₆	C₂×C₄×D₆₇	1	1	yes	characteristic
130	C₁₆	Q₈×D₆₇	1	1	yes	characteristic
131	C₂³	—	67	1	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	8	1	no	—
134	C₂×C₄	—	536	1	no	—
135	C₂×C₄	—	536	1	no	—
136	C₂×C₄	—	1072	2	no	—
137	C₂×C₄	—	67	1	no	—
138	C₂×C₄	—	67	2	no	—
139	C₂×C₄	—	67	2	no	—
140	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
141	C₈	—	67	2	no	—
142	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
143	C₈	Q₁₆×D₆₇	1	1	yes	characteristic
144	C₂²	—	67	2	no	—
145	C₂²	—	134	1	no	—
146	C₂²	—	134	1	no	—
147	C₂²	D₁₆×D₆₇	1	1	yes	characteristic
148	C₄	—	16	1	no	—
149	C₄	—	16	1	no	—
150	C₄	—	1072	1	no	—
151	C₄	—	1072	1	no	—
152	C₄	—	67	2	no	—
153	C₄	#107×D₆₇	1	1	yes	characteristic
154	C₄	Q₃₂×D₆₇	1	1	yes	characteristic
155	C₂	—	134	2	no	—
156	C₂	#366×D₆₇	1	1	yes	characteristic
157	C₂	#22562520	1	1	yes	characteristic
158	C₂	#22562528	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₆₇
commutator	C₁₆×C₆₇	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₆₇
Fitting	C₆₇×#1423	C₂
socle	C₂²×C₆₇	C₂×D₁₆
prime radical	C₂²×D₆₇	D₁₆
cosocle	C₂×C₁₆×C₆₇	C₂³
hypercenter	#1423	D₆₇
hypocenter	C₆₇	#13634
LCS	G ▷ C₁₆×C₆₇ ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1423

Complements

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

N	H ≅ G/N	type
4. #22558411	C₂	semidirect
5. #22558410	C₂	semidirect
6. #22558408	C₂	semidirect
7. C₆₇×#1423	C₂	semidirect
11. #16863229	C₄	semidirect
30. C₆₇⋊C₃₂	C₂×C₄	semidirect
34. C₃₂×C₆₇	C₂×C₄	semidirect
86. #1423	D₆₇	semidirect
87. C₆₇	#13634	semidirect
114. C₃₂	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863229	2
#22561990	2
#366×D₆₇	2
#22558411	2
#22558410	2
#22558408	2
C₆₇×#1423	2
#13634	67