Subgroups — C₂³×#14523968 #31719344

Order 17152.

5502 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14523968	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₃₄	—	2	7	no	—
6	C₂³×Dic₁₃₄	—	2	8	no	—
7	C₂³×Dic₁₃₄	C₄	1	16	yes	complemented
8	C₂³×Dic₁₃₄	C₂²	1	14	yes	—
9	C₂⁴×Dic₆₇	—	2	1	no	—
10	C₂⁴×Dic₆₇	—	2	1	no	—
11	C₂²×C₄⋊Dic₆₇	—	2	7	no	—
12	C₂²×C₄⋊Dic₆₇	C₂²	1	14	yes	—
13	C₂×#14523968	C₂²	1	112	yes	direct product factor
14	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	—	2	8	no	—
16	C₂³×C₈×C₆₇	C₂²	1	14	yes	—
17	C₂²×Dic₁₃₄	—	2	112	no	—
18	C₂²×Dic₁₃₄	—	2	56	no	—
19	C₂²×Dic₁₃₄	—	2	42	no	—
20	C₂²×Dic₁₃₄	C₂×C₄	1	112	yes	complemented
21	C₂²×Dic₁₃₄	C₂³	1	28	yes	—
22	C₂³×Dic₆₇	—	2	16	no	—
23	C₂³×Dic₆₇	—	2	14	no	—
24	C₂³×Dic₆₇	—	2	14	no	—
25	C₂³×Dic₆₇	—	4	8	no	—
26	C₂×C₄⋊Dic₆₇	—	2	42	no	—
27	C₂×C₄⋊Dic₆₇	C₂³	1	28	yes	—
28	#14523968	C₂³	1	64	yes	direct product factor
29	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
31	C₂³×C₄×C₆₇	D₄	1	7	yes	—
32	C₂³×C₄×C₆₇	D₄	1	8	yes	—
33	C₂³×C₄×C₆₇	C₂³	1	7	yes	—
34	C₂²×C₈×C₆₇	—	2	56	no	—
35	C₂²×C₈×C₆₇	C₂³	1	28	yes	—
36	C₂×Dic₁₃₄	—	2	336	no	—
37	C₂×Dic₁₃₄	—	2	56	no	—
38	C₂×Dic₁₃₄	—	2	28	no	—
39	C₂×Dic₁₃₄	C₂²×C₄	1	112	yes	complemented
40	C₂×Dic₁₃₄	C₂⁴	1	8	yes	—
41	C₂²×Dic₆₇	—	2	112	no	—
42	C₂²×Dic₆₇	—	2	28	no	—
43	C₂²×Dic₆₇	—	2	28	no	—
44	C₂²×Dic₆₇	—	4	56	no	—
45	C₄⋊Dic₆₇	—	2	28	no	—
46	C₄⋊Dic₆₇	C₂⁴	1	8	yes	—
47	C₂⁴×C₆₇	C₂²⋊C₄	1	8	yes	—
48	C₂⁴×C₆₇	SD₁₆	1	8	yes	—
49	C₂⁴×C₆₇	Q₁₆	1	8	yes	—
50	C₂⁴×C₆₇	C₂×D₄	1	7	yes	—
51	C₂²×C₄×C₆₇	C₂²⋊C₄	1	56	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	28	yes	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	28	yes	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	21	yes	—
55	C₂²×C₄×C₆₇	C₂⁴	1	7	yes	—
56	C₂×C₈×C₆₇	—	2	56	no	—
57	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
58	C₂³×C₆₇	#103	1	64	yes	complemented
59	C₂³×C₆₇	C₂×C₂²⋊C₄	1	28	yes	—
60	C₂³×C₆₇	C₂×SD₁₆	1	28	yes	—
61	C₂³×C₆₇	C₂×Q₁₆	1	28	yes	—
62	C₂³×C₆₇	C₂²×D₄	1	7	yes	—
63	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	84	yes	—
64	C₂×C₄×C₆₇	C₂³×C₄	1	14	yes	—
65	C₂×C₄×C₆₇	C₂²×D₄	1	14	yes	—
66	C₂×C₄×C₆₇	C₂²×D₄	1	7	yes	—
67	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
68	C₂×Dic₆₇	—	2	112	no	—
69	C₂×Dic₆₇	—	2	8	no	—
70	C₂×Dic₆₇	—	2	8	no	—
71	C₂×Dic₆₇	—	4	56	no	—
72	Dic₁₃₄	—	2	112	no	—
73	Dic₁₃₄	—	2	8	no	—
74	Dic₁₃₄	C₂³×C₄	1	16	yes	complemented
75	C₈×C₆₇	—	2	8	no	—
76	C₂²×C₆₇	C₂×#103	1	112	yes	complemented
77	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	14	yes	—
78	C₂²×C₆₇	C₂²×SD₁₆	1	14	yes	—
79	C₂²×C₆₇	C₂²×Q₁₆	1	14	yes	—
80	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
81	C₄×C₆₇	C₂²×C₂²⋊C₄	1	14	yes	—
82	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
83	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
84	Dic₆₇	—	2	16	no	—
85	Dic₆₇	—	4	8	no	—
86	C₂³×#103	—	67	1	no	maximal, Sylow
87	C₂×C₆₇	C₂²×#103	1	28	yes	complemented
88	C₂×C₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₆₇	C₂³×SD₁₆	1	1	yes	characteristic
90	C₂×C₆₇	C₂³×Q₁₆	1	1	yes	characteristic
91	C₂⁴×Q₈	—	67	1	no	—
92	C₂⁴×C₈	D₆₇	1	1	yes	characteristic
93	C₂³×C₄⋊C₄	—	67	1	no	—
94	C₂²×#103	—	67	28	no	—
95	C₆₇	C₂³×#103	1	1	yes	characteristic, complemented, Sylow
96	C₂³×Q₈	—	67	16	no	—
97	C₂³×Q₈	—	67	14	no	—
98	C₂³×Q₈	—	134	7	no	—
99	C₂³×Q₈	—	134	8	no	—
100	C₂⁴×C₄	—	134	1	no	—
101	C₂⁴×C₄	—	134	1	no	—
102	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
103	C₂³×C₈	—	2	8	no	—
104	C₂³×C₈	C₂×D₆₇	1	14	yes	—
105	C₂²×C₄⋊C₄	—	67	14	no	—
106	C₂²×C₄⋊C₄	—	134	7	no	—
107	C₂×#103	—	67	112	no	—
108	C₂⁵	D₂₆₈	1	1	yes	characteristic
109	C₂²×Q₈	—	67	112	no	—
110	C₂²×Q₈	—	67	28	no	—
111	C₂²×Q₈	—	134	112	no	—
112	C₂²×Q₈	—	134	56	no	—
113	C₂²×Q₈	—	134	42	no	—
114	C₂³×C₄	—	268	8	no	—
115	C₂³×C₄	—	134	14	no	—
116	C₂³×C₄	—	134	16	no	—
117	C₂³×C₄	—	134	14	no	—
118	C₂³×C₄	C₄×D₆₇	1	8	yes	—
119	C₂³×C₄	D₂₆₈	1	7	yes	—
120	C₂³×C₄	C₂²⋊D₆₇	1	8	yes	—
121	C₂³×C₄	C₂²×D₆₇	1	7	yes	—
122	C₂²×C₈	—	2	56	no	—
123	C₂²×C₈	C₂²×D₆₇	1	28	yes	—
124	C₂×C₄⋊C₄	—	67	28	no	—
125	C₂×C₄⋊C₄	—	134	42	no	—
126	#103	—	67	64	no	—
127	C₂⁴	(C₂×D₆₇)⋊C₄	1	8	yes	—
128	C₂⁴	#11761617	1	8	yes	—
129	C₂⁴	Dic₂₆₈	1	8	yes	—
130	C₂⁴	C₂×D₂₆₈	1	7	yes	—
131	C₂×Q₈	—	67	112	no	—
132	C₂×Q₈	—	67	8	no	—
133	C₂×Q₈	—	134	336	no	—
134	C₂×Q₈	—	134	56	no	—
135	C₂×Q₈	—	134	28	no	—
136	C₂²×C₄	—	268	56	no	—
137	C₂²×C₄	—	134	28	no	—
138	C₂²×C₄	—	134	112	no	—
139	C₂²×C₄	—	134	28	no	—
140	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	56	yes	—
141	C₂²×C₄	C₂×C₄×D₆₇	1	28	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₆₇	1	28	yes	—
143	C₂²×C₄	C₂×D₂₆₈	1	21	yes	—
144	C₂²×C₄	C₂³×D₆₇	1	7	yes	—
145	C₂×C₈	—	2	56	no	—
146	C₂×C₈	C₂³×D₆₇	1	8	yes	—
147	C₄⋊C₄	—	67	8	no	—
148	C₄⋊C₄	—	134	28	no	—
149	C₂³	#14523968	1	64	yes	direct product factor
150	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	28	yes	—
151	C₂³	C₂×#11761617	1	28	yes	—
152	C₂³	C₂×Dic₂₆₈	1	28	yes	—
153	C₂³	C₂²×D₂₆₈	1	7	yes	—
154	Q₈	—	67	16	no	—
155	Q₈	—	134	112	no	—
156	Q₈	—	134	8	no	—
157	C₂×C₄	—	268	56	no	—
158	C₂×C₄	—	134	8	no	—
159	C₂×C₄	—	134	112	no	—
160	C₂×C₄	—	134	8	no	—
161	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	84	yes	—
162	C₂×C₄	C₂²×C₄×D₆₇	1	14	yes	—
163	C₂×C₄	C₂²×C₂²⋊D₆₇	1	14	yes	—
164	C₂×C₄	C₂²×D₂₆₈	1	7	yes	—
165	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
166	C₈	—	2	8	no	—
167	C₂²	C₂×#14523968	1	112	yes	direct product factor
168	C₂²	C₂²×(C₂×D₆₇)⋊C₄	1	14	yes	—
169	C₂²	C₂²×#11761617	1	14	yes	—
170	C₂²	C₂²×Dic₂₆₈	1	14	yes	—
171	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
172	C₄	—	268	8	no	—
173	C₄	—	134	16	no	—
174	C₄	C₂²×(C₂×D₆₇)⋊C₄	1	14	yes	—
175	C₄	C₂³×C₄×D₆₇	1	1	yes	characteristic
176	C₄	C₂³×C₂²⋊D₆₇	1	1	yes	characteristic
177	C₂	C₂²×#14523968	1	28	yes	direct product factor
178	C₂	C₂³×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
179	C₂	C₂³×#11761617	1	1	yes	characteristic
180	C₂	C₂³×Dic₂₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	D₂₆₈
commutator	C₄×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂⁴×C₈×C₆₇	C₂
socle, prime radical	C₂⁵×C₆₇	D₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂⁴×C₈	D₆₇
hypocenter	C₆₇	C₂³×#103
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
3. C₂²×#14523968	C₂	direct (N × H)
7. C₂³×Dic₁₃₄	C₄	semidirect
13. C₂×#14523968	C₂²	direct (N × H)
20. C₂²×Dic₁₃₄	C₂×C₄	semidirect
28. #14523968	C₂³	direct (N × H)
39. C₂×Dic₁₃₄	C₂²×C₄	semidirect
58. C₂³×C₆₇	#103	semidirect
74. Dic₁₃₄	C₂³×C₄	semidirect
76. C₂²×C₆₇	C₂×#103	semidirect
87. C₂×C₆₇	C₂²×#103	semidirect
95. C₆₇	C₂³×#103	semidirect
149. C₂³	#14523968	direct (N × H)
167. C₂²	C₂×#14523968	direct (N × H)
177. C₂	C₂²×#14523968	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₁₃₄	2
C₂³×C₄⋊Dic₆₇	2
C₂²×#14523968	2
C₂⁴×C₈×C₆₇	2
C₂³×#103	67

Subgroups — C23×#14523968 #31719344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14523968 #31719344