Subgroups — C₂³×#14523978 #31719519

Order 17152.

2846 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11761612	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14523978	C₂	1	28	yes	maximal, direct product factor
4	C₂³×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	—	2	7	no	—
6	C₂²×#11761612	C₂²	1	14	yes	—
7	C₂³×C₆₇⋊C₈	—	2	1	no	—
8	C₂³×C₆₇⋊C₈	C₄	1	14	yes	—
9	C₂³×C₆₇⋊C₈	C₄	1	2	yes	—
10	C₂³×C₆₇⋊C₈	C₂²	1	14	yes	—
11	C₂×#14523978	C₂²	1	112	yes	direct product factor
12	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	—	2	7	no	—
14	C₂²×M₁₆×C₆₇	C₂²	1	14	yes	—
15	C₂³×C₈×C₆₇	—	2	1	no	—
16	C₂×#11761612	—	2	42	no	—
17	C₂×#11761612	C₂³	1	28	yes	—
18	C₂²×C₆₇⋊C₈	—	2	14	no	—
19	C₂²×C₆₇⋊C₈	C₂×C₄	1	84	yes	—
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	28	yes	—
21	C₂²×C₆₇⋊C₈	C₂³	1	28	yes	—
22	#14523978	C₂³	1	64	yes	direct product factor
23	C₂⁵×C₆₇	Q₈	1	1	yes	characteristic
24	C₂³×C₄×C₆₇	C₂×C₄	1	7	yes	—
25	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	D₄	1	7	yes	—
27	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	Q₈	1	7	yes	—
29	C₂³×C₄×C₆₇	C₂³	1	7	yes	—
30	C₂×M₁₆×C₆₇	—	2	42	no	—
31	C₂×M₁₆×C₆₇	C₂³	1	28	yes	—
32	C₂²×C₈×C₆₇	—	2	14	no	—
33	#11761612	—	2	28	no	—
34	#11761612	C₂⁴	1	8	yes	—
35	C₂×C₆₇⋊C₈	—	2	28	no	—
36	C₂×C₆₇⋊C₈	C₂²×C₄	1	56	yes	—
37	C₂×C₆₇⋊C₈	C₂²×C₄	1	56	yes	—
38	C₂×C₆₇⋊C₈	C₂⁴	1	8	yes	—
39	C₂⁴×C₆₇	—	2	8	no	—
40	C₂⁴×C₆₇	C₄⋊C₄	1	7	yes	—
41	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂⁴×C₆₇	C₂×Q₈	1	7	yes	—
43	C₂²×C₄×C₆₇	C₄⋊C₄	1	42	yes	—
44	C₂²×C₄×C₆₇	C₄⋊C₄	1	7	yes	—
45	C₂²×C₄×C₆₇	C₄⋊C₄	1	7	yes	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	21	yes	—
47	C₂²×C₄×C₆₇	C₂²×C₄	1	7	yes	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	7	yes	—
49	C₂²×C₄×C₆₇	C₂×D₄	1	21	yes	—
50	C₂²×C₄×C₆₇	C₂×Q₈	1	21	yes	—
51	C₂²×C₄×C₆₇	C₂⁴	1	7	yes	—
52	M₁₆×C₆₇	—	2	28	no	—
53	M₁₆×C₆₇	C₂⁴	1	8	yes	—
54	C₂×C₈×C₆₇	—	2	28	no	—
55	C₂³×C₆₇	—	2	28	no	—
56	C₂³×C₆₇	—	2	28	no	—
57	C₂³×C₆₇	#108	1	8	yes	complemented
58	C₂³×C₆₇	C₂×C₄⋊C₄	1	21	yes	—
59	C₂³×C₆₇	C₂×C₄⋊C₄	1	7	yes	—
60	C₂³×C₆₇	C₂²×Q₈	1	7	yes	—
61	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	42	yes	—
62	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	21	yes	—
63	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	21	yes	—
64	C₂×C₄×C₆₇	C₂³×C₄	1	7	yes	—
65	C₂×C₄×C₆₇	C₂³×C₄	1	7	yes	—
66	C₂×C₄×C₆₇	C₂²×D₄	1	7	yes	—
67	C₂×C₄×C₆₇	C₂²×D₄	1	7	yes	—
68	C₂×C₄×C₆₇	C₂²×Q₈	1	7	yes	—
69	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
70	C₈×C₆₇	—	2	8	no	—
71	C₆₇⋊C₈	—	2	8	no	—
72	C₆₇⋊C₈	C₂³×C₄	1	16	yes	—
73	C₂²×C₆₇	—	2	42	no	—
74	C₂²×C₆₇	—	2	14	no	—
75	C₂²×C₆₇	C₂×#108	1	28	yes	complemented
76	C₂²×C₆₇	C₂²×C₄⋊C₄	1	7	yes	—
77	C₂²×C₆₇	C₂²×C₄⋊C₄	1	7	yes	—
78	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
79	C₄×C₆₇	C₂²×C₄⋊C₄	1	7	yes	—
80	C₄×C₆₇	C₂²×C₄⋊C₄	1	7	yes	—
81	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
82	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
83	C₂³×#108	—	67	1	no	maximal, Sylow
84	C₂×C₆₇	—	2	7	no	—
85	C₂×C₆₇	—	2	1	no	—
86	C₂×C₆₇	C₂²×#108	1	14	yes	complemented
87	C₂×C₆₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
88	C₂³×M₁₆	—	67	1	no	—
89	C₂³×M₁₆	D₆₇	1	1	yes	characteristic
90	C₂⁴×C₈	—	67	1	no	—
91	C₂²×#108	—	67	28	no	—
92	C₆₇	C₂³×#108	1	1	yes	characteristic, complemented, Sylow
93	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
94	C₂²×M₁₆	—	2	7	no	—
95	C₂²×M₁₆	—	67	14	no	—
96	C₂²×M₁₆	—	134	7	no	—
97	C₂²×M₁₆	C₂×D₆₇	1	14	yes	—
98	C₂³×C₈	—	2	1	no	—
99	C₂³×C₈	—	67	2	no	—
100	C₂³×C₈	—	67	14	no	—
101	C₂³×C₈	—	67	14	no	—
102	C₂³×C₈	—	134	1	no	—
103	C₂×#108	—	67	112	no	—
104	C₂⁵	Dic₁₃₄	1	1	yes	characteristic
105	C₂³×C₄	Dic₁₃₄	1	7	yes	—
106	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
107	C₂³×C₄	C₄×D₆₇	1	7	yes	—
108	C₂³×C₄	C₂²⋊D₆₇	1	7	yes	—
109	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
110	C₂³×C₄	C₂²×D₆₇	1	7	yes	—
111	C₂×M₁₆	—	2	42	no	—
112	C₂×M₁₆	—	67	28	no	—
113	C₂×M₁₆	—	134	42	no	—
114	C₂×M₁₆	C₂²×D₆₇	1	28	yes	—
115	C₂²×C₈	—	2	14	no	—
116	C₂²×C₈	—	67	84	no	—
117	C₂²×C₈	—	67	28	no	—
118	C₂²×C₈	—	67	28	no	—
119	C₂²×C₈	—	134	14	no	—
120	#108	—	67	64	no	—
121	C₂⁴	—	2	8	no	—
122	C₂⁴	Dic₆₇⋊C₄	1	7	yes	—
123	C₂⁴	Dic₆₇⋊C₄	1	1	yes	characteristic
124	C₂⁴	C₂×Dic₁₃₄	1	7	yes	—
125	C₂²×C₄	Dic₆₇⋊C₄	1	42	yes	—
126	C₂²×C₄	Dic₆₇⋊C₄	1	7	yes	—
127	C₂²×C₄	Dic₆₇⋊C₄	1	7	yes	—
128	C₂²×C₄	C₂×C₄×D₆₇	1	21	yes	—
129	C₂²×C₄	C₂×C₄×D₆₇	1	7	yes	—
130	C₂²×C₄	C₂×C₂²⋊D₆₇	1	7	yes	—
131	C₂²×C₄	C₂×C₂²⋊D₆₇	1	21	yes	—
132	C₂²×C₄	C₂×Dic₁₃₄	1	21	yes	—
133	C₂²×C₄	C₂³×D₆₇	1	7	yes	—
134	M₁₆	—	2	28	no	—
135	M₁₆	—	67	8	no	—
136	M₁₆	—	134	28	no	—
137	M₁₆	C₂³×D₆₇	1	8	yes	—
138	C₂×C₈	—	2	28	no	—
139	C₂×C₈	—	67	56	no	—
140	C₂×C₈	—	67	56	no	—
141	C₂×C₈	—	67	8	no	—
142	C₂×C₈	—	134	28	no	—
143	C₂³	—	2	28	no	—
144	C₂³	—	2	28	no	—
145	C₂³	#14523978	1	8	yes	direct product factor
146	C₂³	C₂×Dic₆₇⋊C₄	1	21	yes	—
147	C₂³	C₂×Dic₆₇⋊C₄	1	7	yes	—
148	C₂³	C₂²×Dic₁₃₄	1	7	yes	—
149	C₂×C₄	C₂×Dic₆₇⋊C₄	1	42	yes	—
150	C₂×C₄	C₂×Dic₆₇⋊C₄	1	21	yes	—
151	C₂×C₄	C₂×Dic₆₇⋊C₄	1	21	yes	—
152	C₂×C₄	C₂²×C₄×D₆₇	1	7	yes	—
153	C₂×C₄	C₂²×C₄×D₆₇	1	7	yes	—
154	C₂×C₄	C₂²×C₂²⋊D₆₇	1	7	yes	—
155	C₂×C₄	C₂²×C₂²⋊D₆₇	1	7	yes	—
156	C₂×C₄	C₂²×Dic₁₃₄	1	7	yes	—
157	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
158	C₈	—	2	8	no	—
159	C₈	—	67	16	no	—
160	C₈	—	134	8	no	—
161	C₂²	—	2	42	no	—
162	C₂²	—	2	14	no	—
163	C₂²	C₂×#14523978	1	28	yes	direct product factor
164	C₂²	C₂²×Dic₆₇⋊C₄	1	7	yes	—
165	C₂²	C₂²×Dic₆₇⋊C₄	1	7	yes	—
166	C₂²	C₂³×Dic₁₃₄	1	1	yes	characteristic
167	C₄	C₂²×Dic₆₇⋊C₄	1	7	yes	—
168	C₄	C₂²×Dic₆₇⋊C₄	1	7	yes	—
169	C₄	C₂³×C₄×D₆₇	1	1	yes	characteristic
170	C₄	C₂³×C₂²⋊D₆₇	1	1	yes	characteristic
171	C₂	—	2	7	no	—
172	C₂	—	2	1	no	—
173	C₂	C₂²×#14523978	1	14	yes	direct product factor
174	C₂	C₂³×Dic₆₇⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂³×M₁₆×C₆₇	C₂
socle	C₂⁴×C₆₇	C₄⋊C₄
prime radical	C₂⁵×C₆₇	Q₈
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂³×M₁₆	D₆₇
hypocenter	C₆₇	C₂³×#108
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂²×#14523978	C₂	direct (N × H)
11. C₂×#14523978	C₂²	direct (N × H)
22. #14523978	C₂³	direct (N × H)
57. C₂³×C₆₇	#108	semidirect
75. C₂²×C₆₇	C₂×#108	semidirect
86. C₂×C₆₇	C₂²×#108	semidirect
92. C₆₇	C₂³×#108	semidirect
145. C₂³	#14523978	direct (N × H)
163. C₂²	C₂×#14523978	direct (N × H)
173. C₂	C₂²×#14523978	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11761612	2
C₂⁴×C₆₇⋊C₈	2
C₂²×#14523978	2
C₂³×M₁₆×C₆₇	2
C₂³×#108	67

Subgroups — C23×#14523978 #31719519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14523978 #31719519